Finite Group Databasebeta

← #26219532

Subgroups — #26219532

Order 12032.

764 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×Dic47C211yesmaximal, 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#20243289C211yesmaximal, characteristic, complemented
12#20243200C211yesmaximal, characteristic, complemented
13#20243200C211yesmaximal, characteristic, complemented
14#20243138C211yesmaximal, characteristic, complemented
15C47×#1876C211yesmaximal, characteristic
16C2×#13094031C2211yescharacteristic
17#41×Dic47C2211yescharacteristic
18#41×Dic47C2211yescharacteristic
19Q16×Dic47C411yescharacteristic, complemented
20Q16×Dic47C411yescharacteristic, complemented
21SD16×Dic47C411yescharacteristic, complemented
22SD16×Dic47C411yescharacteristic, complemented
23SD16×Dic47C411yescharacteristic, complemented
24SD16×Dic47C411yescharacteristic, complemented
25D8×Dic47C411yescharacteristic, complemented
26D8×Dic47C411yescharacteristic, 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×C8×Dic47C2211yescharacteristic
51#15688969C2211yescharacteristic
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×Dic47C2311yescharacteristic
94C2×C4×Dic47C2311yescharacteristic
95#13093959—22no—
96#13093959—22no—
97#13093958C2×C411yescharacteristic
98#13093958C2×C411yescharacteristic
99#13093956C2×C411yescharacteristic
100#13093956C2×C411yescharacteristic
101Dic47⋊C8—21no—
102Dic47⋊C8—21no—
103Dic47⋊C8—21no—
104Dic47⋊C8—21no—
105#13093951—22no—
106#13093951—22no—
107#13093948C2×C411yescharacteristic
108#13093948C2×C411yescharacteristic
109#13093948C2×C411yescharacteristic
110#13093948C2×C411yescharacteristic
111#13093945C2×C411yescharacteristic
112#13093945C2×C411yescharacteristic
113#13093945C2×C411yescharacteristic
114#13093945C2×C411yescharacteristic
115C22⋊(C47⋊C8)—21no—
116C22⋊(C47⋊C8)—21no—
117#13093934—21no—
118#13093934—21no—
119#13093934—21no—
120#13093934—21no—
121#13093934D411yescharacteristic
122#13093934D411yescharacteristic
123#13093934D411yescharacteristic
124#13093934D411yescharacteristic
125C8×Dic47—21no—
126C8×Dic47—21no—
127C8×Dic47—21no—
128C8×Dic47—21no—
129C8×Dic47C2×C411yescharacteristic
130C8×Dic47C2×C411yescharacteristic
131C8×Dic47C2×C411yescharacteristic
132C8×Dic47C2×C411yescharacteristic
133C47⋊#95—21no—
134C47⋊#95—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×Dic47C22×C411yescharacteristic
226C4×Dic47C22×C411yescharacteristic
227C4×Dic47C2×D411yescharacteristic
228C4×Dic47C2×D411yescharacteristic
229#10625329—24no—
230#10625329—21no—
231#10625329—21no—
232#10625329—21no—
233#10625329#4112yes—
234C2×C47⋊C8—24no—
235C2×C47⋊C8—21no—
236C2×C47⋊C8—21no—
237C2×C47⋊C8—21no—
238C2×C47⋊C8—21no—
239C2×C47⋊C8—21no—
240C2×C47⋊C8—21no—
241C2×C47⋊C8C22×C411yescharacteristic
242C2×C47⋊C8C22×C411yescharacteristic
243C2×C47⋊C8C2×D411yescharacteristic
244C2×C47⋊C8C2×D411yescharacteristic
245C23×C47—21no—
246C23×C47—21no—
247C23×C47C2×#4111yescharacteristic
248Q8×C47—21no—
249Q8×C47—21no—
250Q8×C47—21no—
251Q8×C47#11712yes—
252D4×C47—22no—
253D4×C47—22no—
254D4×C47—21no—
255D4×C47—21no—
256D4×C47—21no—
257D4×C47—41no—
258D4×C47—41no—
259D4×C47#11712yes—
260C2×C4×C47—21no—
261C2×C4×C47—22no—
262C2×C4×C47—22no—
263C2×C4×C47—21no—
264C2×C4×C47—41no—
265C2×C4×C47—41no—
266C2×C4×C47—41no—
267C2×C4×C47—41no—
268C2×C4×C47—41no—
269C2×C4×C47C23×C411yescharacteristic
270C2×C4×C47C22×D411yescharacteristic
271C2×C4×C47C2×#4111yescharacteristic
272C2×C4×C47C2×#4111yescharacteristic
273C2×C4×C47C2×#4111yescharacteristic
274C2×C4×C47C2×#4111yescharacteristic
275C2×Dic47—21no—
276C2×Dic47—21no—
277C2×Dic47—21no—
278C2×Dic47—21no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47—41no—
284C2×Dic47—81no—
285C2×Dic47—81no—
286C2×Dic47—81no—
287C2×Dic47—81no—
288C2×Dic47C4×D411yescharacteristic
289C2×Dic47C4×D411yescharacteristic
290C2×Dic47C4×D411yescharacteristic
291C2×Dic47C4×D411yescharacteristic
292C8×C47—21no—
293C8×C47—21no—
294C8×C47—21no—
295C8×C47—22no—
296C8×C47—21no—
297C8×C47—41no—
298C47⋊C8—21no—
299C47⋊C8—21no—
300C47⋊C8—21no—
301C47⋊C8—22no—
302C47⋊C8—21no—
303C47⋊C8—41no—
304#21140—471nomaximal, Sylow
305C22×C47—21no—
306C22×C47—21no—
307C22×C47—22no—
308C22×C47—41no—
309C22×C47—41no—
310C22×C47—41no—
311C22×C47—41no—
312C22×C47—41no—
313C22×C47C2×C4×D411yescharacteristic
314C22×C47#52911yescharacteristic
315C22×C47#52911yescharacteristic
316C4×C47—22no—
317C4×C47—41no—
318C4×C47—81no—
319C4×C47C2×#11711yescharacteristic
320C4×C47C4×#4111yescharacteristic
321C4×C47#52511yescharacteristic
322C4×C47#52511yescharacteristic
323Dic47—21no—
324Dic47—21no—
325Dic47—21no—
326Dic47—21no—
327Dic47—41no—
328Dic47—41no—
329Dic47—41no—
330Dic47—41no—
331Dic47—81no—
332C4×#135—471no—
333#2923—471no—
334#2875—471no—
335#1929—471no—
336#1929—471no—
337#1922—471no—
338#1922—471no—
339#1876—471no—
340#1876D4711yescharacteristic
341#1866—471no—
342#1866—471no—
343C4×#108—471no—
344#1759—471no—
345#1759—471no—
346#1746—471no—
347C2×C47—21no—
348C2×C47—21no—
349C2×C47—42no—
350C2×C47—81no—
351C2×C47#229011yescharacteristic
352C2×C47#295711yescharacteristic
353C2×C47#295711yescharacteristic
354C2×#135C2×D4711yescharacteristic
355C2×#131—471no—
356C2×#131C2×D4711yescharacteristic
357C4×#41—471no—
358C4×#41—471no—
359C4×Q16—471no—
360C4×Q16—471no—
361C4×SD16—471no—
362C4×SD16—471no—
363C4×SD16—471no—
364C4×SD16—471no—
365C4×D8—471no—
366C4×D8—471no—
367#433—941no—
368#433—941no—
369#432—941no—
370#432—941no—
371C2×#108—471no—
372C2×#108—471no—
373#427—471no—
374#424—471no—
375#424—471no—
376C2×#104—471no—
377C2×#104—471no—
378C2×#104C2×D4711yescharacteristic
379C2×#104C2×D4711yescharacteristic
380#416—471no—
381#416—471no—
382#408—471no—
383#408C2×D4711yescharacteristic
384#405—471no—
385#405—471no—
386#405—941no—
387#405—941no—
388C4×M16—471no—
389C4×M16—471no—
390C4×M16—941no—
391C2×#97—471no—
392C2×#97—941no—
393C2×C4×C8—471no—
394#376—471no—
395#341—471no—
396#341C2×D4711yescharacteristic
397#339—471no—
398#339—471no—
399#339—471no—
400#339—471no—
401#336—471no—
402#336—471no—
403#336—471no—
404#336C2×D4711yescharacteristic
405C47#2114011yescharacteristic, complemented, Sylow
406C2×#41C22×D4711yescharacteristic
407C2×#41C22×D4711yescharacteristic
408#135—22no—
409#135—22no—
410C2×Q16C4×D4711yescharacteristic
411C2×SD16C4×D4711yescharacteristic
412C2×SD16C4×D4711yescharacteristic
413C2×D8C4×D4711yescharacteristic
414#131—22no—
415#131—22no—
416#131—942no—
417#131—942no—
418C2×M16—21no—
419C2×M16—471no—
420C2×M16—471no—
421C2×M16—941no—
422C2×M16C22×D4711yescharacteristic
423C2×M16C22×D4711yescharacteristic
424C22×C8—21no—
425C22×C8—471no—
426C22×C8—471no—
427C22×C8—941no—
428C22×C8C22×D4711yescharacteristic
429C22×C8C22×D4711yescharacteristic
430C4×Q8—471no—
431C4×Q8—471no—
432C4×Q8—471no—
433C4×Q8—471no—
434C4×D4—471no—
435C4×D4—471no—
436C4×D4—471no—
437C4×D4—471no—
438C4×D4—941no—
439C4×D4—941no—
440C4×D4—941no—
441C4×D4—941no—
442#117—471no—
443#117—471no—
444#117—941no—
445#117—941no—
446C2×C42—471no—
447C2×C42—471no—
448C2×C42—941no—
449C2×C42—941no—
450C2×C42C22×D4711yescharacteristic
451#108—942no—
452#108—942no—
453#107—471no—
454#107—471no—
455#106—471no—
456#106—471no—
457C4⋊C8—941no—
458C4⋊C8—941no—
459C4⋊C8—941no—
460C4⋊C8—941no—
461#104—22no—
462#104—22no—
463#104—942no—
464#104—942no—
465#103—471no—
466#103—471no—
467#103—471no—
468#103—471no—
469#102—471no—
470#102—471no—
471#102—471no—
472#102—471no—
473C22⋊C8—21no—
474C22⋊C8—21no—
475C22⋊C8—941no—
476C22⋊C8—941no—
477#97—471no—
478#97—471no—
479#97—471no—
480#97—471no—
481#97—941no—
482#97—941no—
483#97—941no—
484#97—941no—
485C4×C8—471no—
486C4×C8—471no—
487C4×C8—471no—
488C4×C8—471no—
489C4×C8—941no—
490C4×C8—941no—
491C4×C8—941no—
492C4×C8—941no—
493#95—941no—
494#95—941no—
495#41—21no—
496#41—21no—
497#41—21no—
498#41—21no—
499#41—21no—
500#41—21no—
501#41#1062535212yes—
502#41#1062535212yes—
503C2×Q8C2×C4×D4711yescharacteristic
504C2×Q8C2×C4×D4711yescharacteristic
505C2×D4—21no—
506C2×D4—21no—
507C2×D4C2×C4×D4711yescharacteristic
508C2×D4C2×C4×D4711yescharacteristic
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—471no—
513C22×C4—471no—
514C22×C4—941no—
515C22×C4—941no—
516C22×C4—941no—
517C22×C4—941no—
518C22×C4—941no—
519C22×C4C23×D4711yescharacteristic
520Q16—22no—
521SD16—22no—
522SD16—22no—
523D8—22no—
524M16—24no—
525M16—21no—
526M16—21no—
527M16—21no—
528M16—472no—
529M16—944no—
530M16—941no—
531M16—941no—
532M16—941no—
533M16#1062535212yes—
534C2×C8—24no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—471no—
542C2×C8—471no—
543C2×C8—471no—
544C2×C8—471no—
545C2×C8—944no—
546C2×C8—941no—
547C2×C8—941no—
548C2×C8—941no—
549C2×C8—941no—
550C2×C8—941no—
551C2×C8—941no—
552C2×C8C2×C4×D4711yescharacteristic
553C2×C8C2×C4×D4711yescharacteristic
554C2×C8D4×D4711yescharacteristic
555C2×C8D4×D4711yescharacteristic
556C4⋊C4—471no—
557C4⋊C4—471no—
558C4⋊C4—471no—
559C4⋊C4—471no—
560C4⋊C4—941no—
561C4⋊C4—941no—
562C4⋊C4—941no—
563C4⋊C4—941no—
564C22⋊C4—941no—
565C22⋊C4—941no—
566C22⋊C4—941no—
567C22⋊C4—941no—
568C22⋊C4—941no—
569C22⋊C4—941no—
570C42—21no—
571C42—21no—
572C42—471no—
573C42—471no—
574C42—471no—
575C42—471no—
576C42—941no—
577C42—941no—
578C42—941no—
579C42—941no—
580C42—941no—
581C42—941no—
582C42—941no—
583C42—941no—
584C23—21no—
585C23—21no—
586C23#41×D4711yescharacteristic
587Q8—21no—
588Q8—21no—
589Q8—21no—
590Q8#1309398012yes—
591D4—22no—
592D4—22no—
593D4—21no—
594D4—21no—
595D4—21no—
596D4—41no—
597D4—41no—
598D4#1309398012yes—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—471no—
609C2×C4—471no—
610C2×C4—471no—
611C2×C4—471no—
612C2×C4—941no—
613C2×C4—941no—
614C2×C4—941no—
615C2×C4—941no—
616C2×C4—3761no—
617C2×C4—3761no—
618C2×C4—3761no—
619C2×C4—3761no—
620C2×C4—1881no—
621C2×C4—1881no—
622C2×C4—1881no—
623C2×C4—1881no—
624C2×C4—1881no—
625C2×C4C22×C4×D4711yescharacteristic
626C2×C4C2×D4×D4711yescharacteristic
627C2×C4#41×D4711yescharacteristic
628C2×C4C2×#1062535211yescharacteristic
629C2×C4C2×#1062535211yescharacteristic
630C2×C4C2×#1062535211yescharacteristic
631C8—21no—
632C8—21no—
633C8—21no—
634C8—22no—
635C8—21no—
636C8—41no—
637C8—941no—
638C8—941no—
639C8—941no—
640C8—941no—
641C8—942no—
642C8—1881no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C4×D4×D4711yescharacteristic
652C22#1568965011yescharacteristic
653C22#1568965411yescharacteristic
654C4—22no—
655C4—41no—
656C4—81no—
657C4—941no—
658C4—941no—
659C4—941no—
660C4—941no—
661C4—3761no—
662C4—1881no—
663C4—1881no—
664C4—1881no—
665C4—1881no—
666C4C2×#1309398011yescharacteristic
667C4C4×#1062535211yescharacteristic
668C4#1568962911yescharacteristic
669C4#1568963311yescharacteristic
670C2—21no—
671C2—21no—
672C2—42no—
673C2—81no—
674C2#2024582811yescharacteristic
675C2#2025230011yescharacteristic
676C2#2025230211yescharacteristic
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#21140
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. #20243289C2semidirect
12. #20243200C2semidirect
13. #20243200C2semidirect
14. #20243138C2semidirect
19. Q16×Dic47C4semidirect
20. Q16×Dic47C4semidirect
21. SD16×Dic47C4semidirect
22. SD16×Dic47C4semidirect
23. SD16×Dic47C4semidirect
24. SD16×Dic47C4semidirect
25. D8×Dic47C4semidirect
26. D8×Dic47C4semidirect
405. C47#21140semidirect

Maximal subgroups

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

subgroupindex
#135×Dic472
#202520352
#202518142
#202440392
#202440332
#202439882
#202439862
#202437172
#202436632
#202436612
#202432892
#202432002
#202431382
C47×#18762
#2114047