Finite Group Databasebeta

← #26219521

Subgroups — #26219521

Order 12032.

780 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252189C211yesmaximal, characteristic
2#20252035C211yesmaximal, characteristic, complemented
3#20251814C211yesmaximal, characteristic, complemented
4#20244039C211yesmaximal, characteristic, complemented
5#20244033C211yesmaximal, characteristic, complemented
6#20243988C211yesmaximal, characteristic, complemented
7#20243986C211yesmaximal, characteristic, complemented
8#20243717C211yesmaximal, characteristic
9#20243663C211yesmaximal, characteristic, complemented
10#20243661C211yesmaximal, characteristic, complemented
11#20243293C211yesmaximal, characteristic, complemented
12#104×Dic47C211yesmaximal, characteristic, complemented
13#104×Dic47C211yesmaximal, characteristic, complemented
14#20243135C211yesmaximal, characteristic, complemented
15C47×#1876C211yesmaximal, characteristic
16C2×#13094031C2211yescharacteristic
17#41×Dic47C2211yescharacteristic
18#41×Dic47C2211yescharacteristic
19#15689235C411yescharacteristic, complemented
20#15689235C411yescharacteristic, complemented
21#15689230C411yescharacteristic, complemented
22#15689230C411yescharacteristic, complemented
23#15689224C411yescharacteristic, complemented
24#15689224C411yescharacteristic, complemented
25#15689224C411yescharacteristic, complemented
26#15689224C411yescharacteristic, complemented
27#15689181—21no—
28#15689181—21no—
29#15689177—21no—
30#15689177—21no—
31C2×#13093959C2211yescharacteristic
32C2×#13093959C2211yescharacteristic
33#15689161C2211yescharacteristic
34#15689151C2211yescharacteristic
35#15689149C2211yescharacteristic
36C2×#13093951C2211yescharacteristic
37C2×#13093951C2211yescharacteristic
38#15689119C2211yescharacteristic
39#15689119C2211yescharacteristic
40#15689090C2211yescharacteristic
41#15689081—21no—
42#15689081C2211yescharacteristic
43#15689078—21no—
44#15689078C2211yescharacteristic
45M16×Dic47—21no—
46M16×Dic47C2211yescharacteristic
47M16×Dic47C2211yescharacteristic
48C2×#13093934—21no—
49C2×#13093934C2211yescharacteristic
50C2×#13093934C2211yescharacteristic
51C42×Dic47C2211yescharacteristic
52#15688888C2211yescharacteristic
53#15688884C2211yescharacteristic
54#15688884C2211yescharacteristic
55#15688883C2211yescharacteristic
56#15688883C2211yescharacteristic
57#15688878C2211yescharacteristic
58#15688877C2211yescharacteristic
59#15688877C2211yescharacteristic
60C2×#135×C47C2211yescharacteristic
61C2×#131×C47C2211yescharacteristic
62C2×#104×C47C2211yescharacteristic
63C2×#104×C47C2211yescharacteristic
64C47×#408C2211yescharacteristic
65C47×#341C2211yescharacteristic
66C47×#336C2211yescharacteristic
67#13094031—22no—
68#13094031—22no—
69C2×#10625329—21no—
70C2×#10625329C2311yescharacteristic
71C2×#10625329C2311yescharacteristic
72C22×C47⋊C8—21no—
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75Q8×Dic47C2×C411yescharacteristic
76Q8×Dic47C2×C411yescharacteristic
77Q8×Dic47C2×C411yescharacteristic
78Q8×Dic47C2×C411yescharacteristic
79D4×Dic47—21no—
80D4×Dic47—21no—
81D4×Dic47—21no—
82D4×Dic47—21no—
83D4×Dic47C2×C411yescharacteristic
84D4×Dic47C2×C411yescharacteristic
85D4×Dic47C2×C411yescharacteristic
86D4×Dic47C2×C411yescharacteristic
87#13093983—21no—
88#13093983—21no—
89#13093983C2311yescharacteristic
90#13093983C2311yescharacteristic
91C2×C4×Dic47—21no—
92C2×C4×Dic47—21no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47—21no—
95C2×C4×Dic47C2311yescharacteristic
96C2×C4×Dic47C2311yescharacteristic
97#13093959—22no—
98#13093959—22no—
99#13093958C2×C411yescharacteristic
100#13093958C2×C411yescharacteristic
101#13093956C2×C411yescharacteristic
102#13093956C2×C411yescharacteristic
103Dic47⋊C8—21no—
104Dic47⋊C8—21no—
105Dic47⋊C8—21no—
106Dic47⋊C8—21no—
107#13093951—22no—
108#13093951—22no—
109#13093948C2×C411yescharacteristic
110#13093948C2×C411yescharacteristic
111#13093948C2×C411yescharacteristic
112#13093948C2×C411yescharacteristic
113#13093945C2×C411yescharacteristic
114#13093945C2×C411yescharacteristic
115#13093945C2×C411yescharacteristic
116#13093945C2×C411yescharacteristic
117C22⋊(C47⋊C8)—21no—
118C22⋊(C47⋊C8)—21no—
119#13093934—21no—
120#13093934—21no—
121#13093934—21no—
122#13093934—21no—
123#13093934C2×C411yescharacteristic
124#13093934C2×C411yescharacteristic
125#13093934C2×C411yescharacteristic
126#13093934C2×C411yescharacteristic
127#13093934D411yescharacteristic
128#13093934D411yescharacteristic
129#13093934D411yescharacteristic
130#13093934D411yescharacteristic
131C8×Dic47—21no—
132C8×Dic47—21no—
133C8×Dic47—21no—
134C8×Dic47—21no—
135C2×#41×C47C2311yescharacteristic
136C2×#41×C47C2311yescharacteristic
137#135×C47—22no—
138#135×C47—22no—
139C2×Q16×C47C2×C411yescharacteristic
140C2×SD16×C47C2×C411yescharacteristic
141C2×SD16×C47C2×C411yescharacteristic
142C2×D8×C47C2×C411yescharacteristic
143#131×C47—22no—
144#131×C47—22no—
145C2×M16×C47—21no—
146C2×M16×C47C2311yescharacteristic
147C2×M16×C47C2311yescharacteristic
148C22×C8×C47—21no—
149C22×C8×C47C2311yescharacteristic
150C22×C8×C47C2311yescharacteristic
151C2×C42×C47C2311yescharacteristic
152#104×C47—22no—
153#104×C47—22no—
154C22⋊C8×C47—21no—
155C22⋊C8×C47—21no—
156#41×C47—21no—
157#41×C47—21no—
158#41×C47—21no—
159#41×C47—21no—
160#41×C47—21no—
161#41×C47—21no—
162#41×C47#4112yes—
163#41×C47#4112yes—
164C2×Q8×C47C22×C411yescharacteristic
165C2×Q8×C47C22×C411yescharacteristic
166C2×D4×C47—21no—
167C2×D4×C47—21no—
168C2×D4×C47C22×C411yescharacteristic
169C2×D4×C47C22×C411yescharacteristic
170C22×C4×C47—21no—
171C22×C4×C47—21no—
172C22×C4×C47—21no—
173C22×C4×C47C2411yescharacteristic
174C22×Dic47—21no—
175C22×Dic47—21no—
176C22×Dic47—21no—
177C22×Dic47—21no—
178C22×Dic47—21no—
179C22×Dic47#4111yescharacteristic
180C22×Dic47#4111yescharacteristic
181Q16×C47—22no—
182SD16×C47—22no—
183SD16×C47—22no—
184D8×C47—22no—
185M16×C47—24no—
186M16×C47—21no—
187M16×C47—21no—
188M16×C47—21no—
189M16×C47#4112yes—
190C2×C8×C47—24no—
191C2×C8×C47—21no—
192C2×C8×C47—21no—
193C2×C8×C47—21no—
194C2×C8×C47—21no—
195C2×C8×C47—21no—
196C2×C8×C47—21no—
197C2×C8×C47C22×C411yescharacteristic
198C2×C8×C47C22×C411yescharacteristic
199C2×C8×C47C2×D411yescharacteristic
200C2×C8×C47C2×D411yescharacteristic
201C42×C47—21no—
202C42×C47—21no—
203C22⋊Dic47—21no—
204C22⋊Dic47—21no—
205C22⋊Dic47—21no—
206C22⋊Dic47—21no—
207C22⋊Dic47—21no—
208C22⋊Dic47—21no—
209C4⋊Dic47—21no—
210C4⋊Dic47—21no—
211C4⋊Dic47—21no—
212C4⋊Dic47—21no—
213C4⋊Dic47C22×C411yescharacteristic
214C4⋊Dic47C22×C411yescharacteristic
215C4⋊Dic47C22×C411yescharacteristic
216C4⋊Dic47C22×C411yescharacteristic
217C4×Dic47—21no—
218C4×Dic47—21no—
219C4×Dic47—21no—
220C4×Dic47—21no—
221C4×Dic47—21no—
222C4×Dic47—21no—
223C4×Dic47—21no—
224C4×Dic47—21no—
225C4×Dic47—42no—
226C4×Dic47—42no—
227C4×Dic47C22×C411yescharacteristic
228C4×Dic47C22×C411yescharacteristic
229C4×Dic47C2×D411yescharacteristic
230C4×Dic47C2×D411yescharacteristic
231#10625329—24no—
232#10625329—21no—
233#10625329—21no—
234#10625329—21no—
235#10625329#4112yes—
236C2×C47⋊C8—24no—
237C2×C47⋊C8—21no—
238C2×C47⋊C8—21no—
239C2×C47⋊C8—21no—
240C2×C47⋊C8—21no—
241C2×C47⋊C8—21no—
242C2×C47⋊C8—21no—
243C2×C47⋊C8C22×C411yescharacteristic
244C2×C47⋊C8C22×C411yescharacteristic
245C2×C47⋊C8C2×D411yescharacteristic
246C2×C47⋊C8C2×D411yescharacteristic
247C23×C47—21no—
248C23×C47—21no—
249C23×C47C2×#4111yescharacteristic
250Q8×C47—21no—
251Q8×C47—21no—
252Q8×C47—21no—
253Q8×C47#11712yes—
254D4×C47—22no—
255D4×C47—22no—
256D4×C47—21no—
257D4×C47—21no—
258D4×C47—21no—
259D4×C47—41no—
260D4×C47—41no—
261D4×C47#11712yes—
262C2×C4×C47—21no—
263C2×C4×C47—22no—
264C2×C4×C47—22no—
265C2×C4×C47—22no—
266C2×C4×C47—21no—
267C2×C4×C47—41no—
268C2×C4×C47—41no—
269C2×C4×C47—41no—
270C2×C4×C47—41no—
271C2×C4×C47C23×C411yescharacteristic
272C2×C4×C47C22×D411yescharacteristic
273C2×C4×C47C2×#4111yescharacteristic
274C2×C4×C47C2×#4111yescharacteristic
275C2×C4×C47C2×#4111yescharacteristic
276C2×C4×C47C2×#4111yescharacteristic
277C2×Dic47—22no—
278C2×Dic47—21no—
279C2×Dic47—21no—
280C2×Dic47—21no—
281C2×Dic47—21no—
282C2×Dic47—41no—
283C2×Dic47—41no—
284C2×Dic47—41no—
285C2×Dic47—41no—
286C2×Dic47—81no—
287C2×Dic47—81no—
288C2×Dic47—81no—
289C2×Dic47—81no—
290C2×Dic47C4×D411yescharacteristic
291C2×Dic47C4×D411yescharacteristic
292C2×Dic47C4×D411yescharacteristic
293C2×Dic47C4×D411yescharacteristic
294C8×C47—21no—
295C8×C47—21no—
296C8×C47—21no—
297C8×C47—22no—
298C8×C47—21no—
299C8×C47—41no—
300C47⋊C8—21no—
301C47⋊C8—21no—
302C47⋊C8—21no—
303C47⋊C8—22no—
304C47⋊C8—21no—
305C47⋊C8—41no—
306#21139—471nomaximal, Sylow
307C22×C47—21no—
308C22×C47—21no—
309C22×C47—22no—
310C22×C47—41no—
311C22×C47—41no—
312C22×C47—41no—
313C22×C47—41no—
314C22×C47—41no—
315C22×C47C2×C4×D411yescharacteristic
316C22×C47#52911yescharacteristic
317C22×C47#52911yescharacteristic
318C4×C47—22no—
319C4×C47—42no—
320C4×C47—41no—
321C4×C47C2×#11711yescharacteristic
322C4×C47C4×#4111yescharacteristic
323C4×C47#52511yescharacteristic
324C4×C47#52511yescharacteristic
325Dic47—21no—
326Dic47—21no—
327Dic47—21no—
328Dic47—21no—
329Dic47—42no—
330Dic47—41no—
331Dic47—41no—
332Dic47—41no—
333Dic47—41no—
334#2943—471no—
335#2923—471no—
336#2875—471no—
337#1929—471no—
338#1929—471no—
339#1922—471no—
340#1922—471no—
341#1876—471no—
342#1876D4711yescharacteristic
343#1866—471no—
344#1866—471no—
345#1778—471no—
346C4×#104—471no—
347C4×#104—471no—
348#1745—471no—
349C2×C47—21no—
350C2×C47—21no—
351C2×C47—42no—
352C2×C47—81no—
353C2×C47#229011yescharacteristic
354C2×C47#295611yescharacteristic
355C2×C47#295611yescharacteristic
356C2×#135C2×D4711yescharacteristic
357C2×#131—471no—
358C2×#131C2×D4711yescharacteristic
359C4×#41—471no—
360C4×#41—471no—
361#442—471no—
362#442—471no—
363#441—471no—
364#441—471no—
365#440—471no—
366#440—471no—
367#440—471no—
368#440—471no—
369#433—941no—
370#433—941no—
371#432—941no—
372#432—941no—
373C2×#108—471no—
374C2×#108—471no—
375#427—471no—
376#424—471no—
377#424—471no—
378C2×#104—471no—
379C2×#104—471no—
380C2×#104C2×D4711yescharacteristic
381C2×#104C2×D4711yescharacteristic
382#416—471no—
383#416—471no—
384#408—471no—
385#408C2×D4711yescharacteristic
386#405—471no—
387#405—471no—
388#405—941no—
389#405—941no—
390C4×M16—471no—
391C4×M16—471no—
392C4×M16—941no—
393C2×#97—471no—
394C2×#97—471no—
395C2×#97—941no—
396C43—471no—
397#341—471no—
398#341C2×D4711yescharacteristic
399#339—471no—
400#339—471no—
401#339—471no—
402#339—471no—
403#336—471no—
404#336—471no—
405#336—471no—
406#336C2×D4711yescharacteristic
407C47#2113911yescharacteristic, complemented, Sylow
408C2×#41C22×D4711yescharacteristic
409C2×#41C22×D4711yescharacteristic
410#135—22no—
411#135—22no—
412C2×Q16C4×D4711yescharacteristic
413C2×SD16C4×D4711yescharacteristic
414C2×SD16C4×D4711yescharacteristic
415C2×D8C4×D4711yescharacteristic
416#131—22no—
417#131—22no—
418#131—942no—
419#131—942no—
420C2×M16—21no—
421C2×M16—471no—
422C2×M16—471no—
423C2×M16—941no—
424C2×M16C22×D4711yescharacteristic
425C2×M16C22×D4711yescharacteristic
426C22×C8—21no—
427C22×C8—471no—
428C22×C8—471no—
429C22×C8—941no—
430C22×C8C22×D4711yescharacteristic
431C22×C8C22×D4711yescharacteristic
432C4×Q8—471no—
433C4×Q8—471no—
434C4×Q8—471no—
435C4×Q8—471no—
436C4×D4—471no—
437C4×D4—471no—
438C4×D4—471no—
439C4×D4—471no—
440C4×D4—941no—
441C4×D4—941no—
442C4×D4—941no—
443C4×D4—941no—
444#117—471no—
445#117—471no—
446#117—941no—
447#117—941no—
448C2×C42—471no—
449C2×C42—471no—
450C2×C42—941no—
451C2×C42—941no—
452C2×C42—941no—
453C2×C42—941no—
454C2×C42C22×D4711yescharacteristic
455#108—942no—
456#108—942no—
457#107—471no—
458#107—471no—
459#106—471no—
460#106—471no—
461C4⋊C8—941no—
462C4⋊C8—941no—
463C4⋊C8—941no—
464C4⋊C8—941no—
465#104—22no—
466#104—22no—
467#104—942no—
468#104—942no—
469#103—471no—
470#103—471no—
471#103—471no—
472#103—471no—
473#102—471no—
474#102—471no—
475#102—471no—
476#102—471no—
477C22⋊C8—21no—
478C22⋊C8—21no—
479C22⋊C8—941no—
480C22⋊C8—941no—
481#97—471no—
482#97—471no—
483#97—471no—
484#97—471no—
485#97—471no—
486#97—471no—
487#97—471no—
488#97—471no—
489#97—941no—
490#97—941no—
491#97—941no—
492#97—941no—
493C4×C8—941no—
494C4×C8—941no—
495C4×C8—941no—
496C4×C8—941no—
497#41—21no—
498#41—21no—
499#41—21no—
500#41—21no—
501#41—21no—
502#41—21no—
503#41#1062535212yes—
504#41#1062535212yes—
505C2×Q8C2×C4×D4711yescharacteristic
506C2×Q8C2×C4×D4711yescharacteristic
507C2×D4—21no—
508C2×D4—21no—
509C2×D4C2×C4×D4711yescharacteristic
510C2×D4C2×C4×D4711yescharacteristic
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—471no—
515C22×C4—471no—
516C22×C4—941no—
517C22×C4—941no—
518C22×C4—941no—
519C22×C4—941no—
520C22×C4—941no—
521C22×C4C23×D4711yescharacteristic
522Q16—22no—
523SD16—22no—
524SD16—22no—
525D8—22no—
526M16—24no—
527M16—21no—
528M16—21no—
529M16—21no—
530M16—472no—
531M16—944no—
532M16—941no—
533M16—941no—
534M16—941no—
535M16#1062535212yes—
536C2×C8—24no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—471no—
544C2×C8—471no—
545C2×C8—471no—
546C2×C8—471no—
547C2×C8—944no—
548C2×C8—941no—
549C2×C8—941no—
550C2×C8—941no—
551C2×C8—941no—
552C2×C8—941no—
553C2×C8—941no—
554C2×C8C2×C4×D4711yescharacteristic
555C2×C8C2×C4×D4711yescharacteristic
556C2×C8D4×D4711yescharacteristic
557C2×C8D4×D4711yescharacteristic
558C4⋊C4—471no—
559C4⋊C4—471no—
560C4⋊C4—471no—
561C4⋊C4—471no—
562C4⋊C4—941no—
563C4⋊C4—941no—
564C4⋊C4—941no—
565C4⋊C4—941no—
566C22⋊C4—941no—
567C22⋊C4—941no—
568C22⋊C4—941no—
569C22⋊C4—941no—
570C22⋊C4—941no—
571C22⋊C4—941no—
572C42—21no—
573C42—21no—
574C42—471no—
575C42—471no—
576C42—471no—
577C42—471no—
578C42—941no—
579C42—941no—
580C42—941no—
581C42—941no—
582C42—941no—
583C42—941no—
584C42—941no—
585C42—941no—
586C42—1882no—
587C42—1882no—
588C23—21no—
589C23—21no—
590C23#41×D4711yescharacteristic
591Q8—21no—
592Q8—21no—
593Q8—21no—
594Q8#1309398012yes—
595D4—22no—
596D4—22no—
597D4—21no—
598D4—21no—
599D4—21no—
600D4—41no—
601D4—41no—
602D4#1309398012yes—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—471no—
613C2×C4—471no—
614C2×C4—471no—
615C2×C4—471no—
616C2×C4—942no—
617C2×C4—941no—
618C2×C4—941no—
619C2×C4—941no—
620C2×C4—941no—
621C2×C4—3761no—
622C2×C4—3761no—
623C2×C4—3761no—
624C2×C4—3761no—
625C2×C4—1881no—
626C2×C4—1881no—
627C2×C4—1881no—
628C2×C4—1881no—
629C2×C4C22×C4×D4711yescharacteristic
630C2×C4C2×D4×D4711yescharacteristic
631C2×C4#41×D4711yescharacteristic
632C2×C4C2×#1062535211yescharacteristic
633C2×C4C2×#1062535211yescharacteristic
634C2×C4C2×#1062535211yescharacteristic
635C8—21no—
636C8—21no—
637C8—21no—
638C8—22no—
639C8—21no—
640C8—41no—
641C8—941no—
642C8—941no—
643C8—941no—
644C8—941no—
645C8—942no—
646C8—1881no—
647C22—21no—
648C22—21no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22C4×D4×D4711yescharacteristic
656C22#1568965011yescharacteristic
657C22#1568965411yescharacteristic
658C4—22no—
659C4—42no—
660C4—41no—
661C4—941no—
662C4—941no—
663C4—941no—
664C4—941no—
665C4—1882no—
666C4—1881no—
667C4—1881no—
668C4—1881no—
669C4—1881no—
670C4C2×#1309398011yescharacteristic
671C4C4×#1062535211yescharacteristic
672C4#1568962911yescharacteristic
673C4#1568963311yescharacteristic
674C2—21no—
675C2—21no—
676C2—42no—
677C2—81no—
678C2#2024582811yescharacteristic
679C2#2025229311yescharacteristic
680C2#2025229611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1876C2
socleC22×C47C2×C4×D4
prime radicalC2×#135×C47C22
cosocleC22×C4×C47C24
hypercenter#1876D47
hypocenterC47#21139
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1876

Complements

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

NH ≅ G/Ntype
2. #20252035C2semidirect
3. #20251814C2semidirect
4. #20244039C2semidirect
5. #20244033C2semidirect
6. #20243988C2semidirect
7. #20243986C2semidirect
9. #20243663C2semidirect
10. #20243661C2semidirect
11. #20243293C2semidirect
12. #104×Dic47C2semidirect
13. #104×Dic47C2semidirect
14. #20243135C2semidirect
19. #15689235C4semidirect
20. #15689235C4semidirect
21. #15689230C4semidirect
22. #15689230C4semidirect
23. #15689224C4semidirect
24. #15689224C4semidirect
25. #15689224C4semidirect
26. #15689224C4semidirect
407. C47#21139semidirect

Maximal subgroups

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

subgroupindex
#202521892
#202520352
#202518142
#202440392
#202440332
#202439882
#202439862
#202437172
#202436632
#202436612
#202432932
#104×Dic472
#202431352
C47×#18762
#2113947