Finite Group Databasebeta

← #26223682

Subgroups — #26223682

Order 12032.

768 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253299C211yesmaximal, characteristic
2#20252094C211yesmaximal, characteristic
3#20251804C211yesmaximal, characteristic, complemented
4#20244033C211yesmaximal, characteristic, complemented
5#20243988C211yesmaximal, characteristic, complemented
6#20243987C211yesmaximal, characteristic, complemented
7#20243687C211yesmaximal, characteristic
8#20243591C211yesmaximal, characteristic, complemented
9#20243486C211yesmaximal, characteristic, complemented
10#20243271C211yesmaximal, characteristic, complemented
11#20243270C211yesmaximal, characteristic, complemented
12#20243233C211yesmaximal, characteristic, complemented
13#20243228C211yesmaximal, characteristic, complemented
14#20243170C211yesmaximal, characteristic, complemented
15C47×#1929C211yesmaximal, characteristic, complemented
16C2×#13094031C2211yescharacteristic
17C22×#10625324C2211yescharacteristic
18C2×M16×D47C2211yescharacteristic
19#15689630C2211yescharacteristic
20#117×D47C2211yescharacteristic
21#15689463C411yescharacteristic, complemented
22#15689462C411yescharacteristic, complemented
23#15689457C411yescharacteristic, complemented
24#15689452C411yescharacteristic, complemented
25#15689446C411yescharacteristic, complemented
26#15689445C411yescharacteristic, complemented
27#15689440C411yescharacteristic, complemented
28#15689435C411yescharacteristic, complemented
29#15689198—21no—
30#15689198—21no—
31#15689191—21no—
32#15689191—21no—
33C2×#13093959C2211yescharacteristic
34#15689161C2211yescharacteristic
35#15689159C2211yescharacteristic
36#15689149C2211yescharacteristic
37C2×#13093951C2211yescharacteristic
38#15689119C2211yescharacteristic
39#15689086C2211yescharacteristic
40C2×(C2×D47)⋊C8C2211yescharacteristic
41#15689081C2211yescharacteristic
42#15689081C2211yescharacteristic
43#15689080—21no—
44#15689079C2211yescharacteristic
45#15689078C2211yescharacteristic
46#15689077—21no—
47#15689075—21no—
48C4×#10625324—21no—
49C2×#13093935C2211yescharacteristic
50C2×C8×Dic47C2211yescharacteristic
51C4×(C2×D47)⋊C4C2211yescharacteristic
52#15688888C2211yescharacteristic
53#15688887C2211yescharacteristic
54#15688884C2211yescharacteristic
55#15688884C2211yescharacteristic
56#15688883C2211yescharacteristic
57#15688883C2211yescharacteristic
58#15688882C2211yescharacteristic
59#15688877C2211yescharacteristic
60C2×#108×C47C2211yescharacteristic
61C2×#104×C47C2211yescharacteristic
62C47×#416C2211yescharacteristic
63C47×#408C2211yescharacteristic
64C47×#405C2211yescharacteristic
65C47×#339C2211yescharacteristic
66C47×#336C2211yescharacteristic
67C22×C4×D47C2311yescharacteristic
68#13094031—22no—
69#13094031—22no—
70C2×#10625329—21no—
71C2×#10625329C2311yescharacteristic
72C2×#10625329C2311yescharacteristic
73C2×#10625324—21no—
74C2×#10625324—22no—
75C2×#10625324—22no—
76C2×#10625324D411yescharacteristic
77C2×#10625324D411yescharacteristic
78C2×#10625324D411yescharacteristic
79C2×#10625324D411yescharacteristic
80M16×D47—22no—
81M16×D47—22no—
82C22×C47⋊C8—21no—
83C22×C47⋊C8C2311yescharacteristic
84C22×C47⋊C8C2311yescharacteristic
85C2×C8×D47—21no—
86#13094005—21no—
87#13094000C2×C411yescharacteristic
88#13094000C2×C411yescharacteristic
89#13093995C2×C411yescharacteristic
90#13093995C2×C411yescharacteristic
91#13093992—21no—
92C4×C22⋊D47—21no—
93C4×C22⋊D47—21no—
94#13093983C2311yescharacteristic
95#13093983C2311yescharacteristic
96#13093982C2×C411yescharacteristic
97#13093982C2×C411yescharacteristic
98#13093981—21no—
99#13093980—21no—
100C4⋊C4×D47C2×C411yescharacteristic
101C4⋊C4×D47C2×C411yescharacteristic
102C2×(C2×D47)⋊C4—21no—
103C22⋊C4×D47—21no—
104C2×C4×Dic47C2311yescharacteristic
105C42×D47—21no—
106#13093959—22no—
107#13093958C2×C411yescharacteristic
108#13093958C2×C411yescharacteristic
109#13093957C2×C411yescharacteristic
110#13093957C2×C411yescharacteristic
111#13093956C2×C411yescharacteristic
112#13093956C2×C411yescharacteristic
113#13093955C2×C411yescharacteristic
114#13093955C2×C411yescharacteristic
115Dic47⋊C8—21no—
116Dic47⋊C8—21no—
117#13093951—22no—
118#13093948C2×C411yescharacteristic
119#13093948C2×C411yescharacteristic
120#13093945C2×C411yescharacteristic
121#13093945C2×C411yescharacteristic
122(C2×D47)⋊C8—21no—
123(C2×D47)⋊C8—21no—
124(C2×D47)⋊C8C2×C411yescharacteristic
125(C2×D47)⋊C8C2×C411yescharacteristic
126(C2×D47)⋊C8C2×C411yescharacteristic
127(C2×D47)⋊C8C2×C411yescharacteristic
128#13093935—21no—
129#13093935—22no—
130#13093934—21no—
131#13093934—21no—
132#13093934—21no—
133C4×C47⋊C8—21no—
134C8×Dic47—21no—
135C8×Dic47—21no—
136C8×Dic47—21no—
137C8×Dic47—22no—
138C47⋊#95—21no—
139C2×#41×C47C2311yescharacteristic
140C2×M16×C47C2311yescharacteristic
141C2×M16×C47C2311yescharacteristic
142C22×C8×C47C2311yescharacteristic
143C22×C8×C47C2311yescharacteristic
144#117×C47C2311yescharacteristic
145C2×C42×C47C2311yescharacteristic
146#108×C47—22no—
147#104×C47—22no—
148#103×C47C2×C411yescharacteristic
149#103×C47C2×C411yescharacteristic
150#102×C47C2×C411yescharacteristic
151#102×C47C2×C411yescharacteristic
152C22⋊C8×C47—21no—
153C22⋊C8×C47—21no—
154#97×C47—21no—
155C4×C8×C47—21no—
156C23×D47#4111yescharacteristic
157#41×C47—21no—
158#41×C47—21no—
159#41×C47—21no—
160#41×C47—21no—
161C2×Q8×C47C22×C411yescharacteristic
162C2×D4×C47—21no—
163C2×D4×C47C22×C411yescharacteristic
164C22×C4×C47—21no—
165C22×C4×C47—21no—
166C22×C4×C47C2411yescharacteristic
167C2×C22⋊D47—21no—
168C2×C22⋊D47—21no—
169C22×Dic47—21no—
170C22×Dic47#4111yescharacteristic
171C2×C4×D47—21no—
172C2×C4×D47—21no—
173C2×C4×D47—21no—
174C2×C4×D47—21no—
175C2×C4×D47—21no—
176C2×C4×D47—21no—
177C2×C4×D47C22×C411yescharacteristic
178C2×C4×D47C22×C411yescharacteristic
179C2×C4×D47C2×D411yescharacteristic
180C2×C4×D47C2×D411yescharacteristic
181M16×C47—21no—
182M16×C47—21no—
183M16×C47—21no—
184M16×C47#4112yes—
185C2×C8×C47—21no—
186C2×C8×C47—21no—
187C2×C8×C47—21no—
188C2×C8×C47—21no—
189C2×C8×C47—21no—
190C2×C8×C47—21no—
191C2×C8×C47C22×C411yescharacteristic
192C2×C8×C47C22×C411yescharacteristic
193C2×C8×C47C2×D411yescharacteristic
194C2×C8×C47C2×D411yescharacteristic
195C4⋊C4×C47C22×C411yescharacteristic
196C4⋊C4×C47C22×C411yescharacteristic
197C22⋊C4×C47—21no—
198C42×C47—21no—
199C42×C47—21no—
200C42×C47—21no—
201C22⋊Dic47—21no—
202C22⋊Dic47—21no—
203C22⋊Dic47—21no—
204C22⋊Dic47—21no—
205(C2×D47)⋊C4—21no—
206(C2×D47)⋊C4—21no—
207(C2×D47)⋊C4—21no—
208(C2×D47)⋊C4—21no—
209(C2×D47)⋊C4—42no—
210C4⋊Dic47C22×C411yescharacteristic
211C4⋊Dic47C22×C411yescharacteristic
212C4⋊Dic47C22×C411yescharacteristic
213C4⋊Dic47C22×C411yescharacteristic
214Dic47⋊C4—21no—
215Dic47⋊C4—21no—
216Dic47⋊C4—21no—
217Dic47⋊C4—21no—
218C4×Dic47—21no—
219C4×Dic47—21no—
220C4×Dic47—21no—
221C4×Dic47—21no—
222C4×Dic47—21no—
223C4×Dic47—21no—
224#10625329—21no—
225#10625329—21no—
226#10625329—21no—
227#10625329—42no—
228#10625329#4112yes—
229C2×C47⋊C8—21no—
230C2×C47⋊C8—21no—
231C2×C47⋊C8—21no—
232C2×C47⋊C8—21no—
233C2×C47⋊C8—42no—
234C2×C47⋊C8C22×C411yescharacteristic
235C2×C47⋊C8C22×C411yescharacteristic
236C2×C47⋊C8C2×D411yescharacteristic
237C2×C47⋊C8C2×D411yescharacteristic
238C2×C47⋊C8#4112yes—
239C2×C47⋊C8#4112yes—
240#10625324—24no—
241#10625324—22no—
242#10625324—22no—
243#10625324—22no—
244#10625324—22no—
245C8×D47—24no—
246C23×C47—21no—
247C23×C47C2×#4111yescharacteristic
248C22×D47—21no—
249C22×D47—21no—
250C22×D47—41no—
251C22×D47—41no—
252C22×D47C4×D411yescharacteristic
253C22×D47C4×D411yescharacteristic
254Q8×C47—21no—
255Q8×C47—21no—
256D4×C47—21no—
257D4×C47—21no—
258D4×C47—41no—
259D4×C47—41no—
260C2×C4×C47—21no—
261C2×C4×C47—21no—
262C2×C4×C47—21no—
263C2×C4×C47—41no—
264C2×C4×C47—41no—
265C2×C4×C47—41no—
266C2×C4×C47—41no—
267C2×C4×C47—41no—
268C2×C4×C47C23×C411yescharacteristic
269C2×C4×C47C22×D411yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×C4×C47C2×#4111yescharacteristic
272C2×C4×C47C2×#4111yescharacteristic
273C2×C4×C47C2×#4111yescharacteristic
274C22⋊D47—81no—
275C22⋊D47—81no—
276C2×Dic47—21no—
277C2×Dic47—21no—
278C2×Dic47—22no—
279C2×Dic47—21no—
280C2×Dic47—21no—
281C2×Dic47—21no—
282C2×Dic47—21no—
283C2×Dic47—41no—
284C2×Dic47—41no—
285C2×Dic47C4×D411yescharacteristic
286C2×Dic47C4×D411yescharacteristic
287C4×D47—21no—
288C4×D47—21no—
289C4×D47—21no—
290C4×D47—21no—
291C4×D47—21no—
292C4×D47—21no—
293C4×D47—21no—
294C4×D47—21no—
295C4×D47—81no—
296C4×D47—81no—
297C8×C47—21no—
298C8×C47—21no—
299C8×C47—21no—
300C8×C47—21no—
301C8×C47—22no—
302C8×C47—41no—
303C47⋊C8—21no—
304C47⋊C8—21no—
305C47⋊C8—22no—
306C47⋊C8—41no—
307C47⋊C8#12312yes—
308C47⋊C8#12312yes—
309#21594—471nomaximal, Sylow
310C22×C47—21no—
311C22×C47—21no—
312C22×C47—41no—
313C22×C47—41no—
314C22×C47—41no—
315C22×C47C2×C4×D411yescharacteristic
316C22×C47#52911yescharacteristic
317C22×C47#52911yescharacteristic
318C2×D47—21no—
319C2×D47—21no—
320C2×D47—21no—
321C2×D47—21no—
322C2×D47—21no—
323C2×D47—21no—
324C2×D47—81no—
325C2×D47—81no—
326C4×C47—41no—
327C4×C47—41no—
328C4×C47—41no—
329C4×C47—81no—
330C4×C47C4×#4111yescharacteristic
331C4×C47C4×#4111yescharacteristic
332C4×C47C2×#12311yescharacteristic
333C4×C47#52511yescharacteristic
334Dic47—21no—
335Dic47—21no—
336Dic47—42no—
337Dic47—41no—
338Dic47—41no—
339Dic47—41no—
340Dic47—41no—
341#3091—471no—
342#2930—471no—
343#2873—471no—
344#1929—471no—
345#1929D4711yescharacteristic, complemented
346#1922—471no—
347#1922—471no—
348#1871—471no—
349#1855—471no—
350#1826—471no—
351#1773—471no—
352#1772—471no—
353#1765—471no—
354#1764—471no—
355#1753—471no—
356C2×C47—21no—
357C2×C47—21no—
358C2×C47—81no—
359C2×C47C4×#12311yescharacteristic
360C2×C47#295611yescharacteristic
361C2×C47#295711yescharacteristic
362D47—41no—
363D47—41no—
364C2×#131—471no—
365C22×M16—471no—
366C22×M16—471no—
367#525—471no—
368C2×#117—471no—
369#485—471no—
370#485—471no—
371#484—471no—
372#483—471no—
373#482—471no—
374#482—471no—
375#481—471no—
376#480—471no—
377#436—941no—
378#436—941no—
379#435—941no—
380#435—941no—
381C2×#108—471no—
382C2×#108C2×D4711yescharacteristic
383#427—471no—
384#427—471no—
385#424—471no—
386C2×#104—471no—
387C2×#104C2×D4711yescharacteristic
388#416—471no—
389#416C2×D4711yescharacteristic
390#408C2×D4711yescharacteristic
391#407—471no—
392C2×C22⋊C8—471no—
393#405—471no—
394#405—471no—
395#405—471no—
396#405—471no—
397#405—941no—
398#405—941no—
399#405C2×D4711yescharacteristic
400C4×M16—941no—
401C4×M16—941no—
402C2×#97—471no—
403C2×C4×C8—471no—
404C4×C22⋊C4—471no—
405#341—471no—
406#341—471no—
407#339—471no—
408#339—471no—
409#339—471no—
410#339—471no—
411#339—471no—
412#339C2×D4711yescharacteristic
413#336—471no—
414#336C2×D4711yescharacteristic
415C47#2159411yescharacteristic, complemented, Sylow
416C2×#41C22×D4711yescharacteristic
417C23×C4—471no—
418#131—942no—
419#131—942no—
420C2×M16—471no—
421C2×M16—471no—
422C2×M16—471no—
423C2×M16—471no—
424C2×M16—471no—
425C2×M16—471no—
426C2×M16—942no—
427C2×M16—942no—
428C2×M16—941no—
429C2×M16—941no—
430C2×M16—942no—
431C2×M16—942no—
432C2×M16C22×D4711yescharacteristic
433C2×M16C22×D4711yescharacteristic
434C22×C8—471no—
435C22×C8—471no—
436C22×C8—941no—
437C22×C8—941no—
438C22×C8C22×D4711yescharacteristic
439C22×C8C22×D4711yescharacteristic
440#123—941no—
441C22⋊Q8—471no—
442C22⋊Q8—471no—
443#121—471no—
444#121—471no—
445#120—941no—
446C4×D4—941no—
447C4×D4—941no—
448#117—471no—
449#117—471no—
450#117—471no—
451#117—471no—
452#117—941no—
453#117—941no—
454#117C22×D4711yescharacteristic
455C2×C4⋊C4—471no—
456C2×C4⋊C4—471no—
457C2×C22⋊C4—941no—
458C2×C22⋊C4—941no—
459C2×C42—471no—
460C2×C42—941no—
461C2×C42C22×D4711yescharacteristic
462#108—22no—
463#108—942no—
464#107—471no—
465#107—471no—
466#107—471no—
467#107—471no—
468#106—471no—
469#106—471no—
470#106—471no—
471#106—471no—
472C4⋊C8—941no—
473C4⋊C8—941no—
474#104—22no—
475#104—942no—
476#103—471no—
477#103—471no—
478#103C4×D4711yescharacteristic
479#103C4×D4711yescharacteristic
480#102—471no—
481#102—471no—
482#102C4×D4711yescharacteristic
483#102C4×D4711yescharacteristic
484C22⋊C8—21no—
485C22⋊C8—21no—
486C22⋊C8—471no—
487C22⋊C8—471no—
488C22⋊C8—471no—
489C22⋊C8—471no—
490C22⋊C8—941no—
491C22⋊C8—941no—
492#97—21no—
493#97—942no—
494#97—941no—
495#97—941no—
496#97—941no—
497#97—941no—
498C4×C8—21no—
499C4×C8—941no—
500C4×C8—942no—
501C4×C8—941no—
502C4×C8—941no—
503C4×C8—941no—
504#95—941no—
505C24—471no—
506#41—21no—
507#41—21no—
508#41—21no—
509#41—21no—
510C2×Q8C2×C4×D4711yescharacteristic
511C2×D4—21no—
512C2×D4—941no—
513C2×D4—941no—
514C2×D4C2×C4×D4711yescharacteristic
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—471no—
518C22×C4—471no—
519C22×C4—471no—
520C22×C4—471no—
521C22×C4—471no—
522C22×C4—941no—
523C22×C4—941no—
524C22×C4—941no—
525C22×C4—941no—
526C22×C4—941no—
527C22×C4—941no—
528C22×C4—941no—
529C22×C4C23×D4711yescharacteristic
530M16—21no—
531M16—21no—
532M16—21no—
533M16—472no—
534M16—942no—
535M16—942no—
536M16—942no—
537M16—942no—
538M16—944no—
539M16—941no—
540M16—941no—
541M16—941no—
542M16—1882no—
543M16#1062535212yes—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—471no—
551C2×C8—471no—
552C2×C8—472no—
553C2×C8—472no—
554C2×C8—471no—
555C2×C8—471no—
556C2×C8—944no—
557C2×C8—941no—
558C2×C8—941no—
559C2×C8—941no—
560C2×C8—941no—
561C2×C8—1882no—
562C2×C8C2×C4×D4711yescharacteristic
563C2×C8C2×C4×D4711yescharacteristic
564C2×C8D4×D4711yescharacteristic
565C2×C8D4×D4711yescharacteristic
566C4⋊C4—471no—
567C4⋊C4—471no—
568C4⋊C4—471no—
569C4⋊C4—471no—
570C4⋊C4—941no—
571C4⋊C4—941no—
572C4⋊C4—941no—
573C4⋊C4—941no—
574C4⋊C4C2×C4×D4711yescharacteristic
575C4⋊C4C2×C4×D4711yescharacteristic
576C22⋊C4—21no—
577C22⋊C4—941no—
578C22⋊C4—941no—
579C22⋊C4—941no—
580C22⋊C4—941no—
581C22⋊C4—941no—
582C22⋊C4—941no—
583C22⋊C4—941no—
584C22⋊C4—941no—
585C22⋊C4—1882no—
586C42—21no—
587C42—21no—
588C42—21no—
589C42—941no—
590C42—941no—
591C42—941no—
592C42—941no—
593C42—941no—
594C42—941no—
595C23—21no—
596C23—471no—
597C23—471no—
598C23—941no—
599C23—941no—
600C23—1881no—
601C23—1881no—
602C23#41×D4711yescharacteristic
603Q8—21no—
604Q8—21no—
605D4—21no—
606D4—21no—
607D4—41no—
608D4—41no—
609D4—3761no—
610D4—3761no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—471no—
620C2×C4—471no—
621C2×C4—941no—
622C2×C4—941no—
623C2×C4—941no—
624C2×C4—941no—
625C2×C4—941no—
626C2×C4—941no—
627C2×C4—941no—
628C2×C4—941no—
629C2×C4—941no—
630C2×C4—941no—
631C2×C4—942no—
632C2×C4—941no—
633C2×C4—941no—
634C2×C4—941no—
635C2×C4—941no—
636C2×C4—3761no—
637C2×C4—3761no—
638C2×C4—1881no—
639C2×C4—1881no—
640C2×C4C22×C4×D4711yescharacteristic
641C2×C4C2×D4×D4711yescharacteristic
642C2×C4#41×D4711yescharacteristic
643C2×C4#41×D4711yescharacteristic
644C2×C4#41×D4711yescharacteristic
645C2×C4C2×#1062535211yescharacteristic
646C8—21no—
647C8—21no—
648C8—21no—
649C8—21no—
650C8—22no—
651C8—41no—
652C8—472no—
653C8—472no—
654C8—941no—
655C8—941no—
656C8—942no—
657C8—1881no—
658C22—21no—
659C22—21no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—941no—
664C22—941no—
665C22—941no—
666C22—941no—
667C22—941no—
668C22—941no—
669C22—3761no—
670C22—3761no—
671C22C4×D4×D4711yescharacteristic
672C22#1568965211yescharacteristic
673C22#1568965611yescharacteristic
674C4—41no—
675C4—41no—
676C4—41no—
677C4—81no—
678C4—941no—
679C4—941no—
680C4—1882no—
681C4—1881no—
682C4—1881no—
683C4—1881no—
684C4—1881no—
685C4C4×#1062535211yescharacteristic
686C4#1568959111yescharacteristic
687C4#123×D4711yescharacteristic
688C4#1568963311yescharacteristic
689C2—21no—
690C2—21no—
691C2—81no—
692C2—1881no—
693C2—1881no—
694C2#2024591511yescharacteristic
695C2#2025229611yescharacteristic
696C2#2025230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1929C2
socleC22×C47C2×C4×D4
prime radical#15689630C22
cosocleC22×C4×C47C24
hypercenter#1929D47
hypocenterC47#21594
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1929

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20251804C2semidirect
4. #20244033C2semidirect
5. #20243988C2semidirect
6. #20243987C2semidirect
8. #20243591C2semidirect
9. #20243486C2semidirect
10. #20243271C2semidirect
11. #20243270C2semidirect
12. #20243233C2semidirect
13. #20243228C2semidirect
14. #20243170C2semidirect
15. C47×#1929C2semidirect
21. #15689463C4semidirect
22. #15689462C4semidirect
23. #15689457C4semidirect
24. #15689452C4semidirect
25. #15689446C4semidirect
26. #15689445C4semidirect
27. #15689440C4semidirect
28. #15689435C4semidirect
345. #1929D47semidirect
415. C47#21594semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202532992
#202520942
#202518042
#202440332
#202439882
#202439872
#202436872
#202435912
#202434862
#202432712
#202432702
#202432332
#202432282
#202431702
C47×#19292
#2159447