Finite Group Databasebeta

← #10802005

Subgroups — #10802005

Order 768.

914 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82923C211yesmaximal, characteristic
2#82916C211yesmaximal, characteristic, complemented
3C2×#4997C211yesmaximal, characteristic, complemented
4C2×#4824C211yesmaximal, characteristic, complemented
5#78626C211yesmaximal, characteristic, complemented
6C3×#1897C211yesmaximal, characteristic, complemented
7#71591C211yesmaximal, characteristic, complemented
8#70374C211yesmaximal, characteristic, complemented
9#70372C211yesmaximal, characteristic, complemented
10#70369C211yesmaximal, characteristic, complemented
11#69983C211yesmaximal, characteristic, complemented
12#68748C211yesmaximal, characteristic, complemented
13#68746C211yesmaximal, characteristic, complemented
14#68741C211yesmaximal, characteristic, complemented
15#68739C211yesmaximal, characteristic, complemented
16#21548—31nomaximal, Sylow
17C3×#525C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#950C2211yescharacteristic
20C2×#948C2211yescharacteristic
21#5791C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C2×#907C2211yescharacteristic
25C2×C3×#104C2211yescharacteristic
26C2×C3×#104C2211yescharacteristic
27C3×#407C2211yescharacteristic
28#5234C2211yescharacteristic
29#5233C2211yescharacteristic
30#5207C2211yescharacteristic
31#5129C2211yescharacteristic
32C2×#822C2211yescharacteristic
33C2×#822C2211yescharacteristic
34#5122C2211yescharacteristic
35#5119C2211yescharacteristic
36#5052C412yescomplemented
37#5049C412yescomplemented
38#5042C412yescomplemented
39#5039C412yescomplemented
40#5011—22no—
41D4×C3⋊C8—22no—
42#5002C2211yescharacteristic
43#4999C2211yescharacteristic
44#4998C2211yescharacteristic
45#4997—24no—
46#4997C2211yescharacteristic
47C2×#805C2211yescharacteristic
48C2×#805C2211yescharacteristic
49C2×#803C2211yescharacteristic
50C2×#803C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4824—24no—
54#4730—22no—
55#67⋊C8—22no—
56C3×#339C2211yescharacteristic
57C3×#339C2211yescharacteristic
58#4558C2211yescharacteristic
59#4558C2211yescharacteristic
60#4523C2211yescharacteristic
61#4523C2211yescharacteristic
62#3054—31no—
63C2×#443—31no—
64#2929—31no—
65#2929—31no—
66C2×#405—31no—
67#1975—31no—
68#1946—31no—
69#1938—31no—
70#1938—31no—
71#1897—31no—
72#1897D311yescharacteristic, complemented
73#1811—31no—
74#1772—31no—
75#1764—31no—
76#1764—31no—
77C2×C3×#41C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C2×#235C2311yescharacteristic
80C2×C3×M16C2311yescharacteristic
81C2×C3×M16C2311yescharacteristic
82C3×#123—21no—
83C3×C22⋊Q8C2×C412yes—
84C3×#121C2×C412yes—
85C3×#120—21no—
86C3×C4×D4—22no—
87C3×#117—22no—
88C3×#117—22no—
89C3×#117C2311yescharacteristic
90C2×C3×C4⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94#953—21no—
95#950—22no—
96#950—22no—
97#948—22no—
98#948—22no—
99C2×#216C2×C411yescharacteristic
100C2×#215C2×C411yescharacteristic
101C2×#214C2×C411yescharacteristic
102C2×#213C2×C411yescharacteristic
103#930C2×C412yes—
104#929—21no—
105C4×#67—22no—
106#926C2311yescharacteristic
107#924C2×C412yes—
108C2×#208—22no—
109C2×#208—21no—
110C2×#208—22no—
111C2×#208C2×C412yes—
112C2×#208C2×C412yes—
113C2×#208C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C22×C3⋊C8—22no—
116C22×C3⋊C8—22no—
117C22×C3⋊C8—21no—
118C22×C3⋊C8D412yes—
119C22×C3⋊C8D412yes—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#907—22no—
125#907—22no—
126C2×ΓL(1,𝔽25)—21no—
127C2×D3×C8—21no—
128C3×#104—22no—
129C3×#104—22no—
130C3×C22⋊C8—22no—
131#835C2×C412yes—
132#832C2×C412yes—
133#830—22no—
134#830—22no—
135#822—22no—
136#822—22no—
137(C2×D3)⋊C8—22no—
138#815—21no—
139Dic3⋊C8—22no—
140C8×Dic3—21no—
141#810C2×C412yes—
142#809C2×C412yes—
143#808C2×C412yes—
144#807C2×C412yes—
145#805—22no—
146#805—22no—
147C4⋊(C3⋊C8)—22no—
148#803—22no—
149#803—22no—
150#803—21no—
151C4×C3⋊C8—22no—
152C4×C3⋊C8—21no—
153C4×C3⋊C8D414yescomplemented
154C2×#135—31no—
155C2×#131—31no—
156C2×#131—31no—
157C22×M16—31no—
158C23×C8—31no—
159#525—31no—
160#525C2×D311yescharacteristic
161C2×#117C2×D311yescharacteristic
162#470—32no—
163#469—32no—
164#468—32no—
165#468—32no—
166#443—64no—
167#435—62no—
168#435—62no—
169C8×D4—62no—
170C8×D4—62no—
171C2×#108—31no—
172C2×#108—31no—
173#427—31no—
174#424—31no—
175#424—31no—
176C2×#104—31no—
177C2×#104—31no—
178C2×#104C2×D311yescharacteristic
179C2×#104C2×D311yescharacteristic
180#416—31no—
181#416—31no—
182#408—31no—
183#408—31no—
184#407—31no—
185#407C2×D311yescharacteristic
186#405—31no—
187#405—31no—
188#405—64no—
189C2×#97—31no—
190C2×#97—31no—
191C2×C4×C8—31no—
192C2×C4×C8—31no—
193#341—31no—
194#341—31no—
195#339—31no—
196#339—31no—
197#339C2×D311yescharacteristic
198#339C2×D311yescharacteristic
199C24×C3#4111yescharacteristic
200C3×#41—21no—
201C3×#41—21no—
202C3×#41—21no—
203C3×#41—21no—
204C2×C3×Q8C22×C411yescharacteristic
205C2×C3×D4—22no—
206C2×C3×D4—21no—
207C2×C3×D4C22×C411yescharacteristic
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C22×C412yes—
216C22×C3×C4C2×D412yes—
217C22×C3×C4#4111yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C2×#67—22no—
220C2×#67—21no—
221#235—21no—
222#235—21no—
223#235—21no—
224#235—21no—
225C2×D12C22×C411yescharacteristic
226C2×C4×D3—21no—
227C2×Dic6C22×C411yescharacteristic
228C3×M16—21no—
229C3×M16—21no—
230C3×M16—21no—
231C3×M16—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4C22×C412yes—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—22no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246#217—22no—
247#217—21no—
248#216—22no—
249#215—22no—
250#214—22no—
251#213—22no—
252(C2×D3)⋊C4—22no—
253C4⋊Dic3C22×C412yes—
254Dic3⋊C4—22no—
255C4×Dic3—21no—
256#208—24no—
257#208—24no—
258#208—21no—
259#208—21no—
260#208—21no—
261#208—21no—
262#208—42no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—24no—
268C2×C3⋊C8—24no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—42no—
272C2×C3⋊C8C22×C412yes—
273C2×C3⋊C8C22×C411yescharacteristic
274C2×C3⋊C8C22×C411yescharacteristic
275C2×C3⋊C8C2×D412yes—
276C2×C3⋊C8C2×D412yes—
277C2×C3⋊C8C2×D412yes—
278C2×C3⋊C8C2×D411yescharacteristic
279C2×C3⋊C8C2×D411yescharacteristic
280C2×C3⋊C8#4112yes—
281C2×C3⋊C8#4112yes—
282ΓL(1,𝔽25)—42no—
283D3×C8—42no—
284C2×#41—31no—
285C2×#41C22×D311yescharacteristic
286C23×C4C22×D311yescharacteristic
287#135—62no—
288#135—62no—
289C2×Q16—31no—
290C2×SD16—31no—
291C2×SD16—31no—
292C2×D8—31no—
293#131—62no—
294#131—62no—
295#131—62no—
296#131—62no—
297C2×M16—32no—
298C2×M16—32no—
299C2×M16—31no—
300C2×M16—31no—
301C2×M16—61no—
302C2×M16—62no—
303C2×M16—62no—
304C2×M16—61no—
305C2×M16C22×D311yescharacteristic
306C2×M16C22×D311yescharacteristic
307C22×C8—31no—
308C22×C8—31no—
309C22×C8—32no—
310C22×C8—32no—
311C22×C8—31no—
312C22×C8—31no—
313C22×C8—61no—
314C22×C8—62no—
315C22×C8—62no—
316C22×C8—61no—
317#123—21no—
318#123—61no—
319C22⋊Q8—32no—
320C22⋊Q8C4×D312yes—
321#121—32no—
322#121C4×D312yes—
323#120—21no—
324#120—61no—
325C4×D4—22no—
326C4×D4—62no—
327#117—22no—
328#117—22no—
329#117—31no—
330#117C22×D311yescharacteristic
331C2×C4⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335#108—62no—
336#108—62no—
337#107—32no—
338#106—32no—
339C4⋊C8—62no—
340C4⋊C8—62no—
341#104—22no—
342#104—22no—
343#104—62no—
344#104—62no—
345#103—32no—
346#103—32no—
347#102—32no—
348#102—32no—
349C22⋊C8—22no—
350C22⋊C8—62no—
351C22⋊C8—62no—
352C22⋊C8—62no—
353#97—62no—
354#97—61no—
355#97—61no—
356#97—62no—
357C4×C8—34no—
358C4×C8—62no—
359C4×C8—61no—
360C4×C8—61no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—42no—
365C23×C3C4×D412yes—
366C23×C3C2×#4111yescharacteristic
367C22×D3—21no—
368C3×Q8—21no—
369C3×Q8—21no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—44no—
373C3×D4—41no—
374C3×D4—41no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4C4×D412yes—
390C2×C3×C4C23×C411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396#67—44no—
397#67—41no—
398#67—41no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401D12—21no—
402D12—21no—
403C4×D3—41no—
404C4×D3—41no—
405Dic6—21no—
406Dic6—21no—
407C3×C8—41no—
408C3×C8—41no—
409C3⋊C8—22no—
410C3⋊C8—22no—
411C3⋊C8—41no—
412C3⋊C8—41no—
413C3⋊C8C4×D412yes—
414C3⋊C8C4×D412yes—
415C3⋊C8#12112yes—
416C3⋊C8#12112yes—
417C24#23711yescharacteristic
418#41—21no—
419#41—21no—
420#41—21no—
421#41—21no—
422#41—61no—
423#41—61no—
424#41—61no—
425#41—61no—
426C2×Q8—31no—
427C2×Q8C2×C4×D311yescharacteristic
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—31no—
431C2×D4—62no—
432C2×D4—61no—
433C2×D4C2×C4×D311yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—61no—
442C22×C4C2×C4×D312yes—
443C22×C4D3×D412yes—
444C22×C4#23711yescharacteristic
445C22×C4C23×D311yescharacteristic
446Q16—62no—
447SD16—62no—
448SD16—62no—
449D8—62no—
450M16—21no—
451M16—21no—
452M16—21no—
453M16—21no—
454M16—64no—
455M16—64no—
456M16—61no—
457M16—61no—
458M16—61no—
459M16—61no—
460M16—122no—
461M16—122no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—31no—
465C2×C8—31no—
466C2×C8—32no—
467C2×C8—32no—
468C2×C8—31no—
469C2×C8—31no—
470C2×C8—32no—
471C2×C8—32no—
472C2×C8—32no—
473C2×C8—32no—
474C2×C8—64no—
475C2×C8—64no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—122no—
483C2×C8—122no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—32no—
487C4⋊C4—62no—
488C4⋊C4C2×C4×D312yes—
489C22⋊C4—24no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—22no—
493C22⋊C4—62no—
494C22⋊C4—61no—
495C22⋊C4—62no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—61no—
500C42D3×D412yes—
501C42#23712yes—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—41no—
510C22×C3C2×C4×D411yescharacteristic
511C22×C3#52511yescharacteristic
512C22×C3#52911yescharacteristic
513C2×D3—42no—
514C2×D3—41no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4C2×C4×D411yescharacteristic
521C3×C4C4×#4111yescharacteristic
522C3×C4C2×#12111yescharacteristic
523C3×C4#53211yescharacteristic
524Dic3—42no—
525Dic3—41no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—42no—
530C23—61no—
531C23#67⋊C412yes—
532C23C2×#23711yescharacteristic
533Q8—21no—
534Q8—21no—
535Q8—61no—
536Q8—61no—
537D4—21no—
538D4—21no—
539D4—44no—
540D4—41no—
541D4—41no—
542D4—61no—
543D4—61no—
544D4—124no—
545D4—121no—
546D4—121no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—62no—
562C2×C4—61no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4#67⋊C412yes—
566C2×C4C22×C4×D311yescharacteristic
567C2×C4C2×D3×D411yescharacteristic
568C2×C4C2×D3×D411yescharacteristic
569C2×C4C2×#23711yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C8—32no—
573C8—32no—
574C8—32no—
575C8—32no—
576C8—41no—
577C8—41no—
578C8—62no—
579C8—62no—
580C8—121no—
581C8—121no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—42no—
585C2×C3—81no—
586C2×C3C4×#12111yescharacteristic
587C2×C3#295512yes—
588D3—81no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—122no—
598C22—121no—
599C22C2×#67⋊C411yescharacteristic
600C22#558611yescharacteristic
601C22#564911yescharacteristic
602C4—22no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—122no—
608C4—121no—
609C4C4×#23711yescharacteristic
610C4C4×D3×D411yescharacteristic
611C4#559611yescharacteristic
612C4D3×#12111yescharacteristic
613C3#2154811yescharacteristic, complemented, Sylow
614C2—21no—
615C2—21no—
616C2—42no—
617C2—81no—
618C2—241no—
619C2#7466011yescharacteristic
620C2#7898912yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1897C2
socleC22×C3C2×C4×D4
prime radical#82923C2
cosocleC22×C3×C4C24
hypercenter#1897D3
hypocenterC3#21548
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1897

Complements

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

NH ≅ G/Ntype
2. #82916C2semidirect
3. C2×#4997C2semidirect
4. C2×#4824C2semidirect
5. #78626C2semidirect
6. C3×#1897C2semidirect
7. #71591C2semidirect
8. #70374C2semidirect
9. #70372C2semidirect
10. #70369C2semidirect
11. #69983C2semidirect
12. #68748C2semidirect
13. #68746C2semidirect
14. #68741C2semidirect
15. #68739C2semidirect
36. #5052C4semidirect
37. #5049C4semidirect
38. #5042C4semidirect
39. #5039C4semidirect
72. #1897D3semidirect
153. C4×C3⋊C8D4semidirect
613. C3#21548semidirect

Maximal subgroups

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

subgroupindex
#829232
#829162
C2×#49972
C2×#48242
#786262
C3×#18972
#715912
#703742
#703722
#703692
#699832
#687482
#687462
#687412
#687392
#215483