Finite Group Databasebeta

← #10836663

Subgroups — #10836663

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79240C211yesmaximal, characteristic
2#78946C211yesmaximal, characteristic
3C2×#4793C211yesmaximal, characteristic, complemented
4#78552C211yesmaximal, characteristic
5C3×#1930C211yesmaximal, characteristic, complemented
6#70669C211yesmaximal, characteristic, complemented
7#70338C211yesmaximal, characteristic, complemented
8#70264C211yesmaximal, characteristic
9#70126C211yesmaximal, characteristic
10#70097C211yesmaximal, characteristic, complemented
11#69894C211yesmaximal, characteristic, complemented
12#69010C211yesmaximal, characteristic
13#68974C211yesmaximal, characteristic, complemented
14#68921C211yesmaximal, characteristic
15#68920C211yesmaximal, characteristic, complemented
16#24722—31nomaximal, Sylow
17C2×Q8×Dic3C2211yescharacteristic
18C2×#942C2211yescharacteristic
19#5522C2211yescharacteristic
20C2×#888C2211yescharacteristic
21C3×#428C2211yescharacteristic
22C3×#419C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#101C2211yescharacteristic
25C3×#405C2211yescharacteristic
26C2×#836C2211yescharacteristic
27C2×#835C2211yescharacteristic
28#5133C2211yescharacteristic
29#5133C2211yescharacteristic
30#5129C2211yescharacteristic
31C2×#822C2211yescharacteristic
32#5118C2211yescharacteristic
33#5118C2211yescharacteristic
34Dic3×M16C2211yescharacteristic
35#5110C2211yescharacteristic
36C2×#819C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4997C2211yescharacteristic
40#4995C2211yescharacteristic
41#4995C2211yescharacteristic
42#4966C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4890—21no—
45#4875—21no—
46#4870—21no—
47#4854—21no—
48#4817—21no—
49#4793—21no—
50#4793—21no—
51#4793—21no—
52#4793—21no—
53#4790—21no—
54#4789—21no—
55#4788—21no—
56#4709—21no—
57#4705—21no—
58#4690—21no—
59#4685—21no—
60C3×#343C2211yescharacteristic
61C3×#341C2211yescharacteristic
62#4558C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4530C2211yescharacteristic
68#3079—31no—
69C2×#477—31no—
70C4×#137—31no—
71#2925—31no—
72#2078—31no—
73#2078—31no—
74#2052—31no—
75#2052—31no—
76#1997—31no—
77#1952—31no—
78#1952—31no—
79#1950—31no—
80#1930—31no—
81#1930D311yescharacteristic, complemented
82#1801—31no—
83C22×C3×Q8C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C22×Dic6—21no—
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×#117C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92Q8×Dic3—21no—
93Q8×Dic3—21no—
94Q8×Dic3—21no—
95Q8×Dic3—21no—
96C2×#216—21no—
97#942—22no—
98#942—22no—
99C2×#215—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#893D411yescharacteristic
111#893D411yescharacteristic
112#888—21no—
113#888—21no—
114#888—21no—
115#888—21no—
116#887D411yescharacteristic
117#887D411yescharacteristic
118#67⋊C4—21no—
119#67⋊C4—21no—
120#872—21no—
121C42×D3—21no—
122C3×#107—21no—
123C3×#106—21no—
124C3×#103—21no—
125C3×#103—21no—
126C3×#103—21no—
127C3×#103—21no—
128C3×#101—22no—
129C3×#97—21no—
130C3×C4×C8—21no—
131#836—22no—
132#835—21no—
133#835—21no—
134#822—22no—
135#821—21no—
136#821—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141#819—22no—
142#818—21no—
143#818—21no—
144#816—21no—
145#816—21no—
146#815—21no—
147Dic3⋊C8—21no—
148Dic3⋊C8—21no—
149Dic3⋊C8—21no—
150Dic3⋊C8—21no—
151Dic3⋊C8D411yescharacteristic
152Dic3⋊C8D411yescharacteristic
153Dic3⋊C8D411yescharacteristic
154Dic3⋊C8D411yescharacteristic
155C8×Dic3—21no—
156#810—21no—
157#809—21no—
158#808—21no—
159#808—21no—
160#807—21no—
161#807—21no—
162#807—21no—
163#807—21no—
164#803—21no—
165C4×C3⋊C8—21no—
166C2×#137—31no—
167C2×C4⋊Q8—31no—
168C4×#41—31no—
169C2×C4×Q8—31no—
170#477—61no—
171#477—61no—
172#477—61no—
173#477—61no—
174#477—61no—
175#476—61no—
176#475—61no—
177#474—61no—
178#441—61no—
179#440—61no—
180C4×Q16—61no—
181C4×SD16—61no—
182#433—61no—
183#433—61no—
184#432—61no—
185#431—61no—
186C2×#108—31no—
187C2×#108—31no—
188#428—31no—
189#428—31no—
190#428C2×D311yescharacteristic
191C2×#107—31no—
192C2×#107—31no—
193#423—31no—
194#423—31no—
195C2×C4⋊C8—31no—
196#419C2×D311yescharacteristic
197#417—31no—
198#417—31no—
199#417—31no—
200C2×#103—31no—
201C2×#103C2×D311yescharacteristic
202C2×#101—31no—
203C2×#101C2×D311yescharacteristic
204#408—31no—
205#408—31no—
206#405—31no—
207#405C2×D311yescharacteristic
208C4×M16—31no—
209#343—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#343—31no—
214#343C2×D311yescharacteristic
215#341—31no—
216#341C2×D311yescharacteristic
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8#4112yes—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4C2411yescharacteristic
226C2×#67—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3#4111yescharacteristic
230C22×Dic3#4111yescharacteristic
231#235—21no—
232#235—21no—
233#235—21no—
234#235—21no—
235C2×D12C2×D411yescharacteristic
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×Dic6—41no—
240C2×Dic6—41no—
241C2×Dic6—82no—
242C2×Dic6C2×D411yescharacteristic
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8C2×D411yescharacteristic
255C2×C3×C8C2×D411yescharacteristic
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C22⋊C4—21no—
261C3×C42—21no—
262#216—24no—
263#215—24no—
264(C2×D3)⋊C4—21no—
265(C2×D3)⋊C4—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—41no—
271C4⋊Dic3—41no—
272Dic3⋊C4—21no—
273Dic3⋊C4—21no—
274Dic3⋊C4—41no—
275Dic3⋊C4—41no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—41no—
279C4×Dic3—41no—
280C4×Dic3C2×D411yescharacteristic
281C4×Dic3C2×D411yescharacteristic
282C4×Dic3C2×D411yescharacteristic
283C4×Dic3C2×D411yescharacteristic
284#208—21no—
285#208—21no—
286#208—21no—
287#208—21no—
288#208—21no—
289#208#4112yes—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8C2×D411yescharacteristic
296C2×C3⋊C8C2×D411yescharacteristic
297C2×#41—31no—
298C22×Q8—61no—
299C22×Q8C22×D311yescharacteristic
300#137—62no—
301#137—62no—
302C2×Q16—61no—
303C2×SD16—61no—
304C2×M16—31no—
305C2×M16—31no—
306C2×M16—31no—
307C2×M16C22×D311yescharacteristic
308C2×M16C22×D311yescharacteristic
309C2×M16C22×D311yescharacteristic
310C22×C8—31no—
311C22×C8C22×D311yescharacteristic
312C4⋊Q8—61no—
313C4⋊Q8—61no—
314C4⋊Q8—61no—
315C4⋊Q8—61no—
316C4×Q8—31no—
317C4×Q8—31no—
318C4×Q8—61no—
319C4×Q8—61no—
320C4×Q8—61no—
321C4×Q8—61no—
322C4×D4—31no—
323C4×D4—31no—
324C4×D4—61no—
325C4×D4—61no—
326#117—61no—
327#117C22×D311yescharacteristic
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C42—31no—
334C2×C42—61no—
335C2×C42—61no—
336#108—62no—
337#108—62no—
338#107—21no—
339#107—61no—
340#107—61no—
341#107—61no—
342#107—61no—
343#107—61no—
344#107—61no—
345#106—21no—
346#106—61no—
347#106—61no—
348C4⋊C8—31no—
349C4⋊C8—31no—
350C4⋊C8—31no—
351C4⋊C8—31no—
352C4⋊C8—61no—
353C4⋊C8—61no—
354C4⋊C8—61no—
355C4⋊C8—61no—
356#103—21no—
357#103—21no—
358#103—21no—
359#103—21no—
360#103—61no—
361#103—61no—
362#103—61no—
363#103—61no—
364#103—61no—
365#102—61no—
366#102—61no—
367#102—61no—
368#101—22no—
369#101—62no—
370C22⋊C8—61no—
371C22⋊C8—61no—
372C22⋊C8—61no—
373C22⋊C8—61no—
374#97—21no—
375#97—61no—
376#97—61no—
377C4×C8—21no—
378C4×C8—61no—
379C4×C8—61no—
380C23×C3C2×#4111yescharacteristic
381C22×D3—21no—
382C3×Q8—22no—
383C3×Q8—22no—
384C3×Q8—41no—
385C3×Q8—41no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—81no—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C42−1+411yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399#67—41no—
400#67—41no—
401C2×Dic3—22no—
402C2×Dic3—21no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—81no—
407C2×Dic3—81no—
408C2×Dic3#12111yescharacteristic
409C2×Dic3#12111yescharacteristic
410C2×Dic3#12111yescharacteristic
411C2×Dic3#12111yescharacteristic
412D12—21no—
413D12—21no—
414C4×D3—41no—
415C4×D3—41no—
416C4×D3—42no—
417C4×D3—81no—
418Dic6—21no—
419Dic6—21no—
420Dic6—82no—
421Dic6—82no—
422C3×C8—21no—
423C3×C8—21no—
424C3×C8—41no—
425C3×C8—41no—
426C3×C8—41no—
427C3⋊C8—22no—
428C3⋊C8—21no—
429C3⋊C8—21no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432#41—61no—
433#41—61no—
434#41—61no—
435#41—61no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—31no—
442C2×Q8—121no—
443C2×Q8—121no—
444C2×Q8—242no—
445C2×Q8#23512yes—
446C2×D4—31no—
447C2×D4—61no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4C23×D311yescharacteristic
458Q16—64no—
459SD16—64no—
460M16—21no—
461M16—21no—
462M16—21no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—32no—
467M16—61no—
468M16—61no—
469M16—61no—
470M16—61no—
471M16—61no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—31no—
478C2×C8—31no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8D3×D411yescharacteristic
485C2×C8D3×D411yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—121no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C4⋊C4—121no—
498C4⋊C4D3×D411yescharacteristic
499C4⋊C4D3×D411yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—61no—
502C22⋊C4—61no—
503C42—21no—
504C42—31no—
505C42—31no—
506C42—31no—
507C42—31no—
508C42—61no—
509C42—61no—
510C42—121no—
511C42—121no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3C2×#12111yescharacteristic
515C22×C3#55211yescharacteristic
516C22×C3#55611yescharacteristic
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—41no—
520C3×C4—42no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—81no—
524C3×C4#52511yescharacteristic
525C3×C4#53611yescharacteristic
526C3×C4#54711yescharacteristic
527C3×C4#54811yescharacteristic
528Dic3—21no—
529Dic3—21no—
530Dic3—21no—
531Dic3—41no—
532Dic3—81no—
533Dic3—81no—
534Dic3#47312yes—
535C23—61no—
536C23D3×#4111yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—61no—
542Q8—61no—
543Q8—242no—
544Q8—242no—
545D4—61no—
546D4—61no—
547D4—121no—
548D4—121no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—81no—
562C2×C4—122no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—241no—
569C2×C4—241no—
570C2×C4—241no—
571C2×C4C2×#23511yescharacteristic
572C2×C4C2×D3×D411yescharacteristic
573C2×C4C2×D3×D411yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4#101011yescharacteristic
576C2×C4#101011yescharacteristic
577C8—21no—
578C8—21no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—61no—
583C8—61no—
584C8—62no—
585C8—121no—
586C8—121no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3#241511yescharacteristic
590C2×C3C2×#47311yescharacteristic
591C2×C3#306811yescharacteristic
592D3—81no—
593C22—21no—
594C22—21no—
595C22—121no—
596C22—121no—
597C22—121no—
598C22#553111yescharacteristic
599C22D3×#12111yescharacteristic
600C22#570211yescharacteristic
601C4—32no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—61no—
606C4—61no—
607C4—61no—
608C4—81no—
609C4—121no—
610C4—241no—
611C4—241no—
612C4#554711yescharacteristic
613C4#555211yescharacteristic
614C4#565511yescharacteristic
615C4#567511yescharacteristic
616C3#2472211yescharacteristic, complemented, Sylow
617C2—21no—
618C2—21no—
619C2—241no—
620C2#7549611yescharacteristic
621C2#8093011yescharacteristic
622C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1930C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenter#1930D3
hypocenterC3#24722
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1930

Complements

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

NH ≅ G/Ntype
3. C2×#4793C2semidirect
5. C3×#1930C2semidirect
6. #70669C2semidirect
7. #70338C2semidirect
10. #70097C2semidirect
11. #69894C2semidirect
13. #68974C2semidirect
15. #68920C2semidirect
81. #1930D3semidirect
616. C3#24722semidirect

Maximal subgroups

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

subgroupindex
#792402
#789462
C2×#47932
#785522
C3×#19302
#706692
#703382
#702642
#701262
#700972
#698942
#690102
#689742
#689212
#689202
#247223