Finite Group Databasebeta

← #10813705

Subgroups — #10813705

Order 768.

812 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83812C211yesmaximal, characteristic
2#83810C211yesmaximal, characteristic, complemented
3#79260C211yesmaximal, characteristic, complemented
4#78990C211yesmaximal, characteristic
5C2×#4824C211yesmaximal, characteristic
6C3×#1861C211yesmaximal, characteristic, complemented
7#70725C211yesmaximal, characteristic, complemented
8#70530C211yesmaximal, characteristic, complemented
9#70518C211yesmaximal, characteristic, complemented
10#70396C211yesmaximal, characteristic, complemented
11#70375C211yesmaximal, characteristic, complemented
12#69032C211yesmaximal, characteristic, complemented
13#69015C211yesmaximal, characteristic, complemented
14#68773C211yesmaximal, characteristic, complemented
15#68741C211yesmaximal, characteristic, complemented
16#22694—31nomaximal, Sylow
17C3×#568C2211yescharacteristic
18C2×#950C2211yescharacteristic
19#5819C2211yescharacteristic
20C2×#948C2211yescharacteristic
21#5745C2211yescharacteristic
22#5745C2211yescharacteristic
23C2×#907C2211yescharacteristic
24C3×#430C2211yescharacteristic
25C3×#423C2211yescharacteristic
26C3×#421C2211yescharacteristic
27C2×C3×#104C2211yescharacteristic
28#5236C2211yescharacteristic
29#5145—22no—
30#5144—22no—
31#5139C2211yescharacteristic
32#5125C2211yescharacteristic
33#5124—22no—
34#5124C2211yescharacteristic
35#5124C2211yescharacteristic
36C2×#822—22no—
37C2×#822C2211yescharacteristic
38C2×#822C2211yescharacteristic
39C2×#822C2211yescharacteristic
40C2×#822C2211yescharacteristic
41C2×#822C2211yescharacteristic
42C2×#822C2211yescharacteristic
43#5122C2211yescharacteristic
44#5005C2211yescharacteristic
45#4998C2211yescharacteristic
46#4997C2211yescharacteristic
47C2×#805C2211yescharacteristic
48#4924C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50#4893—22no—
51D3×#108—22no—
52#4824—22no—
53#4824C412yescomplemented
54#4824C412yescomplemented
55#4824C412yescomplemented
56#4824C412yescomplemented
57#4823—22no—
58C3×#344C2211yescharacteristic
59C3×#339C2211yescharacteristic
60#4559C2211yescharacteristic
61#4556C2211yescharacteristic
62#4533C2211yescharacteristic
63#4523C2211yescharacteristic
64#2957—31no—
65#2957—31no—
66C2×#443—31no—
67#2916—31no—
68#2916—31no—
69#1953—31no—
70#1953—31no—
71#1861—31no—
72#1861—31no—
73#1861—31no—
74#1861D311yescharacteristic, complemented
75#1848—31no—
76#1814—31no—
77#1814—31no—
78#1772—31no—
79C2×C3×#41C2311yescharacteristic
80C2×#235C2311yescharacteristic
81C3×#131—22no—
82C3×#131—41no—
83C2×C3×M16—22no—
84C2×C3×M16C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C3×#117C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90#950C2×C412yes—
91#950C2×C412yes—
92#950D412yes—
93#950D412yes—
94#948—21no—
95#948—21no—
96#948—22no—
97#948—41no—
98#948C2×C412yes—
99#948C2×C412yes—
100C2×#216Q811yescharacteristic
101C2×#215Q811yescharacteristic
102C2×#214Q811yescharacteristic
103C2×#213Q811yescharacteristic
104#926C2311yescharacteristic
105C2×#208—21no—
106C2×#208—22no—
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C2×#208C2311yescharacteristic
110C2×#208C2311yescharacteristic
111C22×C3⋊C8—21no—
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#907—21no—
115#907—21no—
116#907—22no—
117#907—41no—
118#907C2×C412yes—
119#907C2×C412yes—
120D3×M16—22no—
121D3×M16—22no—
122#901—22no—
123#901—41no—
124C2×ΓL(1,𝔽25)—21no—
125C2×D3×C8—21no—
126C3×#108—22no—
127C3×C4⋊C8—41no—
128C3×#104—22no—
129C3×#104C2×C412yes—
130C3×#104C2×C412yes—
131#837—22no—
132#825—22no—
133#822—22no—
134#822—22no—
135#822—21no—
136#822—21no—
137#822—21no—
138#822—21no—
139#822—21no—
140#822—21no—
141#822—21no—
142#822—21no—
143#822C2×C412yes—
144#822C2×C412yes—
145#822D412yes—
146#822D412yes—
147#819—22no—
148#815—21no—
149C8×Dic3—21no—
150#808—21no—
151#807—21no—
152#805—22no—
153#805C2×C412yes—
154#805C2×C412yes—
155C4⋊(C3⋊C8)—41no—
156#803—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2×C411yescharacteristic
159C4×C3⋊C8C2×C411yescharacteristic
160C4×C3⋊C8C2×C411yescharacteristic
161C4×C3⋊C8C2×C411yescharacteristic
162C2×#135—31no—
163#568—31no—
164#568—31no—
165#568—31no—
166#568C2×D311yescharacteristic
167C2×#131—31no—
168C2×#131—31no—
169#444—62no—
170#444—62no—
171#443—32no—
172#443—32no—
173#443—32no—
174#443—32no—
175#443—62no—
176#443—62no—
177#430—31no—
178#430—31no—
179#430—31no—
180#430—62no—
181#430—62no—
182#430C2×D311yescharacteristic
183C2×#108—31no—
184C2×#108—31no—
185C2×#108—31no—
186C2×#108—31no—
187C2×#108—31no—
188C2×#108—31no—
189C2×#108—62no—
190C2×#108—62no—
191#427—31no—
192#423—31no—
193#423C2×D311yescharacteristic
194#421—31no—
195#421—31no—
196#421—31no—
197#421C2×D311yescharacteristic
198C2×#104—31no—
199C2×#104C2×D311yescharacteristic
200#405—31no—
201#405—31no—
202C2×C4×C8—31no—
203#344—31no—
204#344—31no—
205#344—31no—
206#344C2×D311yescharacteristic
207#339—31no—
208#339C2×D311yescharacteristic
209C3×#41—21no—
210C3×#41—21no—
211C3×#41C22×C412yes—
212C3×#41C2×D412yes—
213C2×C3×Q8C2×Q811yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4C2×Q811yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4C2411yescharacteristic
219C2×#67—21no—
220#235—21no—
221#235—21no—
222#235C22×C412yes—
223#235C2×D412yes—
224C2×D12C2×Q811yescharacteristic
225C2×C4×D3—21no—
226C2×Dic6C2×Q811yescharacteristic
227C3×M16—21no—
228C3×M16—21no—
229C3×M16—21no—
230C3×M16—21no—
231C3×M16—21no—
232C3×M16—21no—
233C3×M16—42no—
234C3×M16C22×C412yes—
235C3×M16C2×D412yes—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—42no—
241C3×C4⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42C22×C411yescharacteristic
246C3×C42C22×C411yescharacteristic
247#217—21no—
248#216C4⋊C412yes—
249#216C4⋊C412yes—
250#215C4⋊C412yes—
251#215C4⋊C412yes—
252#214C4⋊C412yes—
253#214C4⋊C412yes—
254#213C4⋊C412yes—
255#213C4⋊C412yes—
256C4⋊Dic3—21no—
257C4×Dic3—21no—
258#208—21no—
259#208—21no—
260#208—22no—
261#208—22no—
262#208—21no—
263#208—21no—
264#208—21no—
265#208—21no—
266#208—42no—
267#208C22×C412yes—
268#208C2×D412yes—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—22no—
272C2×C3⋊C8—22no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—42no—
276C2×C3⋊C8C22×C411yescharacteristic
277C2×C3⋊C8C22×C411yescharacteristic
278C2×C3⋊C8C22×C411yescharacteristic
279C2×C3⋊C8C22×C411yescharacteristic
280C2×C3⋊C8C2×D411yescharacteristic
281C2×C3⋊C8C2×D411yescharacteristic
282C2×C3⋊C8C2×D411yescharacteristic
283C2×C3⋊C8C2×D411yescharacteristic
284C2×C3⋊C8C2×Q811yescharacteristic
285C2×C3⋊C8C2×Q811yescharacteristic
286C2×C3⋊C8#4111yescharacteristic
287C2×C3⋊C8#4111yescharacteristic
288ΓL(1,𝔽25)—22no—
289ΓL(1,𝔽25)—22no—
290ΓL(1,𝔽25)—42no—
291ΓL(1,𝔽25)—42no—
292D3×C8—22no—
293D3×C8—22no—
294D3×C8—42no—
295D3×C8—42no—
296C2×#41—31no—
297C2×#41C22×D311yescharacteristic
298#135—32no—
299#135—32no—
300#135—32no—
301#135—32no—
302C2×Q16—31no—
303C2×SD16—31no—
304C2×SD16—31no—
305C2×D8—31no—
306#131—22no—
307#131—32no—
308#131—32no—
309#131—32no—
310#131—32no—
311#131—41no—
312#131—62no—
313#131—61no—
314#131—61no—
315#131—61no—
316#131—61no—
317#131—62no—
318#131—62no—
319#131—121no—
320#131—121no—
321#131—121no—
322C2×M16—22no—
323C2×M16—31no—
324C2×M16—31no—
325C2×M16—31no—
326C2×M16—31no—
327C2×M16—61no—
328C2×M16—61no—
329C2×M16—62no—
330C2×M16—62no—
331C2×M16—62no—
332C2×M16C22×D311yescharacteristic
333C2×M16C22×D311yescharacteristic
334C2×M16C22×D311yescharacteristic
335C2×M16C22×D311yescharacteristic
336C22×C8—31no—
337C22×C8—31no—
338C22×C8—61no—
339C22×C8—61no—
340#117—31no—
341#117C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343#108—22no—
344#108—32no—
345#108—32no—
346#108—32no—
347#108—32no—
348#108—62no—
349#108—61no—
350#108—61no—
351#108—61no—
352#108—61no—
353#108—61no—
354#108—61no—
355#108—61no—
356#108—61no—
357#108—62no—
358#108—62no—
359#107—61no—
360#106—61no—
361C4⋊C8—41no—
362C4⋊C8—121no—
363#104—22no—
364#104—32no—
365#104—32no—
366#104—62no—
367#104—62no—
368#104—62no—
369#104C4×D312yes—
370#104C4×D312yes—
371#97—61no—
372#97—61no—
373C4×C8—31no—
374C4×C8—31no—
375C4×C8—31no—
376C4×C8—31no—
377C4×C8—61no—
378C4×C8—61no—
379C23×C3—21no—
380C23×C32−1+411yescharacteristic
381C22×D3—21no—
382C3×Q8—21no—
383C3×Q8C2×C4⋊C412yes—
384C3×D4—22no—
385C3×D4—22no—
386C3×D4—21no—
387C3×D4C2×C4⋊C412yes—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—42no—
394C2×C3×C4—41no—
395C2×C3×C4C23×C411yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×Q811yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401#67—22no—
402#67—22no—
403C2×Dic3—21no—
404C2×Dic3—41no—
405D12—21no—
406D12C2×C4⋊C412yes—
407C4×D3—22no—
408C4×D3—22no—
409Dic6—21no—
410Dic6C2×C4⋊C412yes—
411C3×C8—22no—
412C3×C8—22no—
413C3×C8—41no—
414C3×C8—41no—
415C3⋊C8—22no—
416C3⋊C8—22no—
417C3⋊C8—41no—
418C3⋊C8—41no—
419C3⋊C8C2×C4⋊C411yescharacteristic
420C3⋊C8C2×C4⋊C411yescharacteristic
421C3⋊C8C2×C4⋊C411yescharacteristic
422C3⋊C8C2×C4⋊C411yescharacteristic
423C3⋊C8C4×D411yescharacteristic
424C3⋊C8C4×D411yescharacteristic
425C3⋊C8C4×D411yescharacteristic
426C3⋊C8C4×D411yescharacteristic
427#41—21no—
428#41—21no—
429#41—32no—
430#41—32no—
431#41—61no—
432#41—61no—
433#41C2×C4×D312yes—
434#41D3×D412yes—
435C2×Q8—31no—
436C2×Q8D3×Q811yescharacteristic
437C2×D4—21no—
438C2×D4—31no—
439C2×D4—61no—
440C2×D4D3×Q811yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—61no—
444C22×C4C23×D311yescharacteristic
445Q16—32no—
446Q16—32no—
447SD16—32no—
448SD16—32no—
449SD16—32no—
450SD16—32no—
451D8—32no—
452D8—32no—
453M16—21no—
454M16—21no—
455M16—21no—
456M16—21no—
457M16—21no—
458M16—21no—
459M16—32no—
460M16—32no—
461M16—42no—
462M16—62no—
463M16—62no—
464M16—62no—
465M16—62no—
466M16—61no—
467M16—61no—
468M16—61no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—122no—
473M16—122no—
474M16—122no—
475M16C2×C4×D312yes—
476M16D3×D412yes—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—31no—
482C2×C8—31no—
483C2×C8—31no—
484C2×C8—31no—
485C2×C8—31no—
486C2×C8—31no—
487C2×C8—31no—
488C2×C8—31no—
489C2×C8—31no—
490C2×C8—31no—
491C2×C8—31no—
492C2×C8—31no—
493C2×C8—42no—
494C2×C8—62no—
495C2×C8—62no—
496C2×C8—62no—
497C2×C8—62no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—122no—
503C2×C8—122no—
504C2×C8—122no—
505C4⋊C4—21no—
506C4⋊C4—61no—
507C22⋊C4—21no—
508C22⋊C4—61no—
509C42—21no—
510C42—21no—
511C42—61no—
512C42C2×C4×D311yescharacteristic
513C42C2×C4×D311yescharacteristic
514C22×C3—22no—
515C22×C3—41no—
516C22×C3—81no—
517C22×C3#52011yescharacteristic
518C22×C3#54711yescharacteristic
519C22×C3D4×Q811yescharacteristic
520C2×D3—22no—
521C2×D3—81no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—41no—
525C3×C4—81no—
526C3×C4C22×C4⋊C411yescharacteristic
527C3×C4C2×C4×D411yescharacteristic
528C3×C4D4211yescharacteristic
529C3×C4#55411yescharacteristic
530Dic3—22no—
531Dic3—81no—
532C23—21no—
533C23—61no—
534C23#101011yescharacteristic
535Q8—21no—
536Q8—32no—
537Q8—61no—
538Q8D3×C4⋊C412yes—
539D4—22no—
540D4—22no—
541D4—21no—
542D4—32no—
543D4—62no—
544D4—62no—
545D4—61no—
546D4D3×C4⋊C412yes—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—61no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—121no—
558C2×C4C22×C4×D311yescharacteristic
559C2×C4C2×D3×D411yescharacteristic
560C2×C4C2×D3×D411yescharacteristic
561C2×C4#100411yescharacteristic
562C2×C4C2×D3×Q811yescharacteristic
563C2×C4D3×#4111yescharacteristic
564C8—22no—
565C8—22no—
566C8—31no—
567C8—31no—
568C8—31no—
569C8—31no—
570C8—31no—
571C8—31no—
572C8—31no—
573C8—31no—
574C8—41no—
575C8—41no—
576C8—62no—
577C8—62no—
578C8—121no—
579C8—121no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—42no—
584C2×C3D4×C4⋊C411yescharacteristic
585D3—42no—
586C22—22no—
587C22—41no—
588C22—62no—
589C22—81no—
590C22—241no—
591C22#554611yescharacteristic
592C22#563211yescharacteristic
593C22#565511yescharacteristic
594C4—22no—
595C4—22no—
596C4—41no—
597C4—62no—
598C4—81no—
599C4—241no—
600C4C2×D3×C4⋊C411yescharacteristic
601C4C4×D3×D411yescharacteristic
602C4#559011yescharacteristic
603C4#563611yescharacteristic
604C3#2269411yescharacteristic, complemented, Sylow
605C2—21no—
606C2—21no—
607C2—21no—
608C2—42no—
609C2—122no—
610C2#7439311yescharacteristic
subgroupG/H
centerC4#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1861C2
socleC2×C3D4×C4⋊C4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1861D3
hypocenterC3#22694
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1861

Complements

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

NH ≅ G/Ntype
2. #83810C2semidirect
3. #79260C2semidirect
6. C3×#1861C2semidirect
7. #70725C2semidirect
8. #70530C2semidirect
9. #70518C2semidirect
10. #70396C2semidirect
11. #70375C2semidirect
12. #69032C2semidirect
13. #69015C2semidirect
14. #68773C2semidirect
15. #68741C2semidirect
53. #4824C4semidirect
54. #4824C4semidirect
55. #4824C4semidirect
56. #4824C4semidirect
74. #1861D3semidirect
604. C3#22694semidirect

Maximal subgroups

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

subgroupindex
#838122
#838102
#792602
#789902
C2×#48242
C3×#18612
#707252
#705302
#705182
#703962
#703752
#690322
#690152
#687732
#687412
#226943