Finite Group Databasebeta

← #11675359

Subgroups — #11675359

Order 768.

1732 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85830C211yesmaximal, characteristic, complemented
2#85829C211yesmaximal, characteristic, complemented
3C3×#2943C211yesmaximal, characteristic
4#83902C212yesmaximal
5#82533C211yesmaximal, characteristic, complemented
6#82509C211yesmaximal, characteristic, complemented
7#79854C214yesmaximal, complemented
8#79817C214yesmaximal, complemented
9#79758C214yesmaximal, complemented
10#79755C214yesmaximal, complemented
11#79198C212yesmaximal, complemented
12#78828C212yesmaximal, complemented
13#78821C212yesmaximal, complemented
14C2×#4703C211yesmaximal, characteristic, complemented
15#78480C211yesmaximal, characteristic, complemented
16#61338—31nomaximal, Sylow
17C2×C3×#135C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19C3×C4×#41C2211yescharacteristic
20#5827C2212yes—
21#5825C2212yes—
22C2×#951C2212yes—
23#5569C2214yes—
24#5567C2214yes—
25#5542C2214yes—
26#5541—24no—
27#5539—24no—
28#5539C2214yes—
29#5519—22no—
30#5519C2212yes—
31#5518—21no—
32#5518—21no—
33#5516C2212yes—
34#5470—21no—
35#5470—22no—
36#5470C2211yescharacteristic
37C2×#870—21no—
38C2×#870C2211yescharacteristic
39C2×#869C2211yescharacteristic
40C3×#442C2212yes—
41C3×#441C2212yes—
42C3×#440C2212yes—
43C3×#440C2212yes—
44C3×#427C2211yescharacteristic
45C3×#416C2211yescharacteristic
46C3×#416C2211yescharacteristic
47C2×C3×#97C2211yescharacteristic
48#5232C2212yes—
49#5182C2214yes—
50#5166C2214yes—
51#5162C2214yes—
52#5153C2214yes—
53C2×#816C2212yes—
54C2×#816C2212yes—
55#5102C2212yes—
56#5101C2212yes—
57#5031C2214yes—
58#5022C2214yes—
59#5019C2214yes—
60#5009C2214yes—
61#5004C2212yes—
62#4995C2212yes—
63#4924C2211yescharacteristic
64#4877C2214yes—
65#4875C2214yes—
66#4857C2214yes—
67#4854C2214yes—
68#4797C2214yes—
69#4792C2214yes—
70#4790C2214yes—
71#4789C2214yes—
72#4768C2214yes—
73#4765C2214yes—
74#4758C2214yes—
75#4757C2214yes—
76#4703C2214yes—
77#4703C2214yes—
78#4701C2214yes—
79#4701C2214yes—
80#3509—31no—
81#3504—31no—
82#3377—34no—
83#3376—34no—
84#3374—34no—
85#3372—34no—
86#3137—32no—
87C2×#491—31no—
88#3082—32no—
89#3080—31no—
90#3046—32no—
91#3039—31no—
92#2961—31no—
93#2947—32no—
94#2943D311yescharacteristic
95C2×C3×#41C2311yescharacteristic
96C2×C3×#41C2311yescharacteristic
97C22×Dic6C2312yes—
98C3×#135—24no—
99C2×C3×Q16C2311yescharacteristic
100C2×C3×SD16C2311yescharacteristic
101C2×C3×SD16C2311yescharacteristic
102C2×C3×D8C2311yescharacteristic
103C22×C3×C8C2311yescharacteristic
104C22×C3×C8C2311yescharacteristic
105C3×C4×Q8C2312yes—
106C3×C4×Q8C2312yes—
107C3×C4×D4—22no—
108C3×C4×D4—22no—
109C3×C4×D4C2312yes—
110C3×C4×D4C2312yes—
111C3×#117—21no—
112C3×#117—21no—
113C3×#117C2311yescharacteristic
114C3×#117C2311yescharacteristic
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118#951—28no—
119Q8×Dic3C2314yes—
120#944C2314yes—
121C2×#216C2314yes—
122#936C2314yes—
123#935—24no—
124D4×Dic3—24no—
125D4×Dic3C2314yes—
126C2×#214C2314yes—
127#926—22no—
128#926—24no—
129#926C2312yes—
130C2×C4⋊Dic3—22no—
131C2×C4⋊Dic3—22no—
132C2×C4⋊Dic3C2312yes—
133#924—24no—
134#924—22no—
135#924—22no—
136#924—24no—
137#924D414yes—
138C2×Dic3⋊C4—22no—
139C2×Dic3⋊C4—24no—
140C2×#208C2312yes—
141#890—24no—
142#890C2314yes—
143#889—24no—
144#889—24no—
145#888—24no—
146#888C2314yes—
147#887C2314yes—
148#887C2314yes—
149#879—24no—
150#879—24no—
151#878—24no—
152#878—24no—
153#878—24no—
154#877—24no—
155#870—21no—
156#870—21no—
157#870—22no—
158#870—24no—
159#870—24no—
160#870D411yescharacteristic
161#870D411yescharacteristic
162#870C2314yes—
163#869—21no—
164#869—21no—
165#869—22no—
166#869D412yes—
167#869C2314yes—
168C4×Dic6—24no—
169C4×Dic6—22no—
170C4×Dic6—22no—
171C4×Dic6—24no—
172C4×Dic6C2314yes—
173C3×#107C2312yes—
174C3×#106C2312yes—
175C3×#103C2312yes—
176C3×#103C2312yes—
177C3×#102C2312yes—
178C3×#102C2312yes—
179C3×#97C2312yes—
180C3×#97C2312yes—
181#835C2314yes—
182#832C2314yes—
183#816C2314yes—
184#816C2314yes—
185#816C2314yes—
186#816C2314yes—
187#815C2314yes—
188Dic3⋊C8C2314yes—
189Dic3⋊C8C2314yes—
190C8×Dic3C2314yes—
191#810C2314yes—
192#810C2314yes—
193#808C2314yes—
194#807C2314yes—
195C4⋊(C3⋊C8)C2314yes—
196C2×#137—32no—
197C2×#135C2×D311yescharacteristic
198#557—34no—
199#557—34no—
200#556—34no—
201#554—34no—
202#554—64no—
203#552—64no—
204#544—32no—
205#544—62no—
206#541—61no—
207#541—61no—
208#536—32no—
209#533—31no—
210#533—61no—
211#533—62no—
212C2×C4⋊Q8—31no—
213C2×#125—31no—
214C2×#125—61no—
215#520—32no—
216#519—32no—
217C4×#41C2×D311yescharacteristic
218C4×#41C2×D311yescharacteristic
219#491—34no—
220#491—34no—
221#491—34no—
222#489—34no—
223#488—34no—
224#487—34no—
225#479—34no—
226#479—34no—
227#479—34no—
228#478—34no—
229#477—34no—
230#475—34no—
231#464—34no—
232#463—34no—
233#462—34no—
234#462—34no—
235#461—34no—
236#461—34no—
237#446—34no—
238#446—34no—
239#442C2×D312yes—
240#441—34no—
241#441C2×D312yes—
242#440—34no—
243#440C2×D312yes—
244#440C2×D312yes—
245C4×Q16—34no—
246C4×SD16—34no—
247#428—32no—
248#427C2×D311yescharacteristic
249#424—32no—
250#423—31no—
251#419—32no—
252#417—32no—
253#416C2×D311yescharacteristic
254#416C2×D311yescharacteristic
255C2×#103—32no—
256C2×#103—32no—
257#405—32no—
258C2×#97C2×D311yescharacteristic
259C3×#41—22no—
260C3×#41—22no—
261C3×#41—22no—
262C3×#41#4114yes—
263C2×C3×Q8C2411yescharacteristic
264C2×C3×Q8C2411yescharacteristic
265C2×C3×D4—21no—
266C2×C3×D4—21no—
267C2×C3×D4C2411yescharacteristic
268C2×C3×D4C2411yescharacteristic
269C22×C3×C4—21no—
270C22×C3×C4—21no—
271C22×C3×C4—22no—
272C22×C3×C4—22no—
273C22×C3×C4C2×D412yes—
274C22×C3×C4C2411yescharacteristic
275C22×Dic3—22no—
276C22×Dic3—22no—
277C22×Dic3—24no—
278C2×Dic6—24no—
279C2×Dic6—24no—
280C2×Dic6—22no—
281C2×Dic6—24no—
282C2×Dic6C2×D414yes—
283C2×Dic6C2414yes—
284C3×Q16—22no—
285C3×SD16—22no—
286C3×SD16—22no—
287C3×D8—22no—
288C2×C3×C8—22no—
289C2×C3×C8—22no—
290C2×C3×C8C2412yes—
291C2×C3×C8C2411yescharacteristic
292C2×C3×C8C2411yescharacteristic
293C3×C4⋊C4—22no—
294C3×C4⋊C4—22no—
295C3×C4⋊C4C2412yes—
296C3×C4⋊C4C2412yes—
297C3×C22⋊C4—21no—
298C3×C22⋊C4—21no—
299C3×C22⋊C4—22no—
300C3×C22⋊C4—22no—
301C3×C42—21no—
302C3×C42—22no—
303C3×C42—22no—
304C3×C42—21no—
305C3×C42C2×D411yescharacteristic
306C3×C42C2×D411yescharacteristic
307C3×C42C2412yes—
308#217—24no—
309#217—24no—
310#217—24no—
311#217C2×D414yes—
312#216—28no—
313#214—28no—
314C4⋊Dic3—24no—
315C4⋊Dic3—24no—
316C4⋊Dic3—22no—
317C4⋊Dic3—24no—
318C4⋊Dic3—24no—
319C4⋊Dic3C2×D412yes—
320C4⋊Dic3C2×D412yes—
321C4⋊Dic3C2×D414yes—
322C4⋊Dic3C2414yes—
323Dic3⋊C4—24no—
324Dic3⋊C4—24no—
325Dic3⋊C4—24no—
326Dic3⋊C4—24no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329Dic3⋊C4—24no—
330Dic3⋊C4—24no—
331Dic3⋊C4C2×D414yes—
332Dic3⋊C4C2×D414yes—
333C4×Dic3—24no—
334C4×Dic3—24no—
335C4×Dic3—24no—
336C4×Dic3C2414yes—
337#208—24no—
338C2×C3⋊C8C2414yes—
339C2×#41C22×D311yescharacteristic
340C2×#41C22×D311yescharacteristic
341C22×Q8—32no—
342#137—68no—
343#135—24no—
344C2×Q16—34no—
345C2×Q16C22×D311yescharacteristic
346C2×SD16—34no—
347C2×SD16C22×D311yescharacteristic
348C2×SD16C22×D311yescharacteristic
349C2×D8C22×D311yescharacteristic
350C2×M16—32no—
351C22×C8C22×D311yescharacteristic
352C22×C8C22×D311yescharacteristic
353C4⋊Q8—34no—
354C4⋊Q8—34no—
355C4⋊Q8—32no—
356C4⋊Q8—34no—
357C4⋊Q8—62no—
358C4⋊Q8—64no—
359C4⋊Q8—61no—
360C4⋊Q8—61no—
361#126—64no—
362#126—64no—
363#125—34no—
364#125—31no—
365#125—31no—
366#125—34no—
367#125—62no—
368#125—64no—
369#125—64no—
370#125—64no—
371#125—64no—
372#125—64no—
373#125—61no—
374#125—61no—
375#124—34no—
376#123—64no—
377C22⋊Q8—34no—
378C22⋊Q8—64no—
379C22⋊Q8—64no—
380C22⋊Q8—64no—
381C22⋊Q8—62no—
382C22⋊Q8—62no—
383C22⋊Q8—64no—
384C22⋊Q8—64no—
385C4×Q8—34no—
386C4×Q8—34no—
387C4×Q8—34no—
388C4×Q8—34no—
389C4×Q8—62no—
390C4×Q8—62no—
391C4×Q8—64no—
392C4×Q8—64no—
393C4×Q8C22×D312yes—
394C4×Q8C22×D312yes—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—34no—
398C4×D4—64no—
399C4×D4C22×D312yes—
400C4×D4C22×D312yes—
401#117—21no—
402#117—21no—
403#117—32no—
404#117—64no—
405#117—62no—
406#117—64no—
407#117C22×D311yescharacteristic
408#117C22×D311yescharacteristic
409C2×C4⋊C4—32no—
410C2×C4⋊C4—62no—
411C2×C4⋊C4—64no—
412C2×C4⋊C4—62no—
413C2×C4⋊C4—62no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42C22×D311yescharacteristic
417#107—34no—
418#107C22×D312yes—
419#106—34no—
420#106C22×D312yes—
421C4⋊C8—34no—
422C4⋊C8—34no—
423C4⋊C8—34no—
424#103—34no—
425#103—34no—
426#103—34no—
427#103—34no—
428#103—34no—
429#103—34no—
430#103—34no—
431#103C22×D312yes—
432#103C22×D312yes—
433#102—34no—
434#102C22×D312yes—
435#102C22×D312yes—
436#97—34no—
437#97C22×D312yes—
438#97C22×D312yes—
439C4×C8—34no—
440C23×C3—21no—
441C23×C3—21no—
442C23×C3C22×D411yescharacteristic
443C3×Q8—21no—
444C3×Q8—21no—
445C3×Q8—21no—
446C3×Q8C2×#4112yes—
447C3×D4—24no—
448C3×D4—21no—
449C3×D4—21no—
450C3×D4—21no—
451C3×D4—42no—
452C3×D4C2×#4112yes—
453C2×C3×C4—21no—
454C2×C3×C4—24no—
455C2×C3×C4—22no—
456C2×C3×C4—21no—
457C2×C3×C4—22no—
458C2×C3×C4—42no—
459C2×C3×C4—44no—
460C2×C3×C4—42no—
461C2×C3×C4—82no—
462C2×C3×C4C22×D412yes—
463C2×C3×C4C22×D412yes—
464C2×C3×C4C22×D411yescharacteristic
465C2×C3×C4C2×#4112yes—
466C2×C3×C42−1+412yes—
467C2×C3×C4C2511yescharacteristic
468C2×Dic3—24no—
469C2×Dic3—24no—
470C2×Dic3—44no—
471C2×Dic3—84no—
472C2×Dic3—82no—
473C2×Dic3C22×D414yes—
474C2×Dic3C22×D414yes—
475Dic6—24no—
476Dic6—24no—
477Dic6—44no—
478Dic6—44no—
479Dic6—84no—
480C3×C8—21no—
481C3×C8—21no—
482C3×C8—22no—
483C3⋊C8—24no—
484#41—22no—
485#41—22no—
486#41—22no—
487#41#23514yes—
488C2×Q8—34no—
489C2×Q8—34no—
490C2×Q8—64no—
491C2×Q8—62no—
492C2×Q8—64no—
493C2×Q8—64no—
494C2×Q8C23×D311yescharacteristic
495C2×Q8C23×D311yescharacteristic
496C2×D4—21no—
497C2×D4—21no—
498C2×D4C23×D311yescharacteristic
499C2×D4C23×D311yescharacteristic
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—64no—
505C22×C4—62no—
506C22×C4—62no—
507C22×C4D3×D412yes—
508C22×C4C23×D311yescharacteristic
509Q16—22no—
510Q16—68no—
511SD16—22no—
512SD16—22no—
513SD16—68no—
514D8—22no—
515M16—64no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—34no—
519C2×C8C23×D312yes—
520C2×C8C23×D311yescharacteristic
521C2×C8C23×D311yescharacteristic
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—32no—
525C4⋊C4—34no—
526C4⋊C4—32no—
527C4⋊C4—34no—
528C4⋊C4—34no—
529C4⋊C4—34no—
530C4⋊C4—64no—
531C4⋊C4—64no—
532C4⋊C4—64no—
533C4⋊C4—64no—
534C4⋊C4—64no—
535C4⋊C4—64no—
536C4⋊C4—64no—
537C4⋊C4—64no—
538C4⋊C4—62no—
539C4⋊C4—64no—
540C4⋊C4—64no—
541C4⋊C4—64no—
542C4⋊C4—64no—
543C4⋊C4C23×D312yes—
544C4⋊C4C23×D312yes—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—34no—
550C22⋊C4—64no—
551C22⋊C4—64no—
552C22⋊C4—64no—
553C42—21no—
554C42—21no—
555C42—22no—
556C42—22no—
557C42—34no—
558C42—64no—
559C42—64no—
560C42—64no—
561C42D3×D411yescharacteristic
562C42D3×D411yescharacteristic
563C42C23×D312yes—
564C22×C3—22no—
565C22×C3—22no—
566C22×C3—42no—
567C22×C3—42no—
568C22×C3—41no—
569C22×C3#54812yes—
570C22×C3C23×D411yescharacteristic
571C3×C4—22no—
572C3×C4—22no—
573C3×C4—22no—
574C3×C4—42no—
575C3×C4—42no—
576C3×C4—41no—
577C3×C4#54812yes—
578C3×C4C22×#4111yescharacteristic
579C3×C4C2×2−1+411yescharacteristic
580Dic3—24no—
581Dic3—24no—
582Dic3—44no—
583Dic3—44no—
584C23—21no—
585C23—21no—
586C23C2×D3×D411yescharacteristic
587Q8—21no—
588Q8—21no—
589Q8—21no—
590Q8—64no—
591Q8—64no—
592Q8—124no—
593Q8—124no—
594Q8—244no—
595Q8C2×#23512yes—
596D4—24no—
597D4—21no—
598D4—21no—
599D4—21no—
600D4—42no—
601D4C2×#23512yes—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—34no—
608C2×C4—34no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—82no—
615C2×C4—124no—
616C2×C4—244no—
617C2×C4—242no—
618C2×C4C2×#23512yes—
619C2×C4C2×D3×D412yes—
620C2×C4C2×D3×D412yes—
621C2×C4C2×D3×D411yescharacteristic
622C2×C4#101012yes—
623C2×C4C24×D311yescharacteristic
624C8—21no—
625C8—21no—
626C8—22no—
627C8—64no—
628C2×C3—22no—
629C2×C3—42no—
630C2×C3—81no—
631C2×C3C2×#54811yescharacteristic
632C2×C3#358612yes—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22#555212yes—
639C22C22×D3×D411yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—64no—
647C4—64no—
648C4—124no—
649C4—124no—
650C4#555212yes—
651C4C22×#23511yescharacteristic
652C4C2×#101011yescharacteristic
653C3#6133811yescharacteristic, complemented, Sylow
654C2—22no—
655C2—42no—
656C2—81no—
657C2C2×#555211yescharacteristic
658C2#8720112yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2943C2
socleC22×C3C23×D4
prime radicalC2×C3×#135C22
hypercenter#2943D3
hypocenterC3#61338
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2943

Complements

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

NH ≅ G/Ntype
1. #85830C2semidirect
2. #85829C2semidirect
5. #82533C2semidirect
6. #82509C2semidirect
7. #79854C2semidirect
8. #79817C2semidirect
9. #79758C2semidirect
10. #79755C2semidirect
11. #79198C2semidirect
12. #78828C2semidirect
13. #78821C2semidirect
14. C2×#4703C2semidirect
15. #78480C2semidirect
653. C3#61338semidirect

Maximal subgroups

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

subgroupindex
#858302
#858292
C3×#29432
#839022
#825332
#825092
#798542
#798172
#797582
#797552
#791982
#788282
#788212
C2×#47032
#784802
#613383