Finite Group Databasebeta

← #11660461

Subgroups — #11660461

Order 768.

1862 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85842C211yesmaximal, characteristic
2C3×#2988C211yesmaximal, characteristic, complemented
3#82523C211yesmaximal, characteristic, complemented
4Q8×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
5D3×C8×Q8C211yesmaximal, characteristic, complemented
6#79907C211yesmaximal, characteristic, complemented
7#79906C211yesmaximal, characteristic, complemented
8#79425C213yesmaximal, complemented
9#79422C213yesmaximal, complemented
10#79410C213yesmaximal, complemented
11#79408C213yesmaximal, complemented
12#79295C213yesmaximal, complemented
13#79283C213yesmaximal, complemented
14#79017C213yesmaximal, complemented
15#79012C213yesmaximal, complemented
16#60274—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5834C411yescharacteristic
19#5831C411yescharacteristic
20C2×#901C2216yes—
21#5576C411yescharacteristic
22C4×#239C2211yescharacteristic
23C4×D3×Q8C2211yescharacteristic
24#5569C411yescharacteristic
25#5559C413yes—
26#5558C413yes—
27#5557C2213yes—
28#5548C2213yes—
29#5533C413yes—
30#5530C413yes—
31#5479C2213yes—
32C4×#235C2213yes—
33C3×C8×Q8C2211yescharacteristic
34C3×C8×Q8C2211yescharacteristic
35C3×C8×D4C2213yes—
36C3×C8×D4C2213yes—
37C3×#431C2213yes—
38C2×C3×C4⋊C8C2213yes—
39#5111C2216yes—
40C8×#67C2216yes—
41#5102C2216yes—
42Q8×C3⋊C8C2211yescharacteristic
43Q8×C3⋊C8C2211yescharacteristic
44#5012C2213yes—
45#5012C2213yes—
46#4925C2213yes—
47#4924C2213yes—
48#4840C2213yes—
49#4840C2213yes—
50#4839C2213yes—
51#4839C2216yes—
52#4839C2211yescharacteristic
53#4838C2213yes—
54#4837C2213yes—
55#4837C2213yes—
56#4834C2213yes—
57D3×C4⋊C8C2213yes—
58D3×C4⋊C8C2213yes—
59#4831C2213yes—
60#4830C2213yes—
61#4729C2216yes—
62#4728C2216yes—
63#4727C2213yes—
64#4727C2213yes—
65#4720C2213yes—
66#4720C2213yes—
67#4688C2213yes—
68C4×ΓL(1,𝔽25)C2213yes—
69C8×D12C2213yes—
70#4685C2213yes—
71#4685C2213yes—
72C4×D3×C8C2213yes—
73#4679C2213yes—
74C8×Dic6C2213yes—
75#3480—31no—
76#2994—33no—
77#2994—33no—
78#2993—33no—
79#2993—33no—
80#2991—31no—
81#2991—31no—
82#2988—31no—
83#2988D311yescharacteristic, complemented
84Q8×M16—31no—
85C2×C8×Q8—31no—
86#2935—33no—
87#2931—33no—
88#2922—33no—
89#2906—33no—
90C2×C3×#41C2×C411yescharacteristic
91C2×#239C2×C411yescharacteristic
92C2×D3×Q8C2×C411yescharacteristic
93C2×#235C2313yes—
94C22×C3×C8C2316yes—
95C3×C4×Q8C2×C411yescharacteristic
96C3×C4×Q8C2311yescharacteristic
97C3×C4×D4C2×C413yes—
98C3×C4×D4C2313yes—
99C3×#117C2×C413yes—
100C2×C3×C42C2313yes—
101Q8×Dic3C2×C411yescharacteristic
102Q8×Dic3C2×C411yescharacteristic
103#940C2×C413yes—
104#938C2×C413yes—
105#938C2×C413yes—
106#936C2×C413yes—
107#930C2×C413yes—
108#929C2×C416yes—
109C4×#67C2×C413yes—
110C4×#67C2×C413yes—
111C4×#67C2×C413yes—
112C4×#67C2×C413yes—
113C4×#67C2316yes—
114#926C2313yes—
115#924C2×C413yes—
116C2×#208C2316yes—
117#901—412no—
118C2×ΓL(1,𝔽25)C2311yescharacteristic
119C2×ΓL(1,𝔽25)C2313yes—
120C2×ΓL(1,𝔽25)C2316yes—
121C2×D3×C8C2311yescharacteristic
122C2×D3×C8C2313yes—
123C2×D3×C8C2316yes—
124#897C2×C413yes—
125#895C2×C413yes—
126#893C2×C413yes—
127#892C2×C413yes—
128D3×C4⋊C4C2×C413yes—
129#890C2×C413yes—
130#888C2×C413yes—
131#887C2×C413yes—
132#885C2×C416yes—
133#884C2×C416yes—
134#883C2×C416yes—
135#879C2×C416yes—
136#876C2×C416yes—
137#875C2×C413yes—
138C4×D12C2×C413yes—
139C4×D12C2×C413yes—
140C4×D12C2313yes—
141#872C2311yescharacteristic
142#872C2313yes—
143C42×D3C2313yes—
144C42×D3C2313yes—
145#870C2×C413yes—
146C4×Dic6C2×C413yes—
147C4×Dic6C2×C413yes—
148C4×Dic6C2313yes—
149C3×C4⋊C8C2313yes—
150C3×C4⋊C8C2313yes—
151C3×C4⋊C8C2313yes—
152C3×C4⋊C8C2311yescharacteristic
153C3×C22⋊C8C2313yes—
154C3×C22⋊C8C2313yes—
155C3×C4×C8C2313yes—
156C3×C4×C8C2313yes—
157#830C2313yes—
158#830C2313yes—
159(C2×D3)⋊C8C2313yes—
160(C2×D3)⋊C8C2313yes—
161(C2×D3)⋊C8C2313yes—
162(C2×D3)⋊C8C2313yes—
163#815C2313yes—
164#815C2313yes—
165Dic3⋊C8C2316yes—
166Dic3⋊C8C2316yes—
167Dic3⋊C8C2311yescharacteristic
168Dic3⋊C8C2311yescharacteristic
169Dic3⋊C8C2313yes—
170Dic3⋊C8C2313yes—
171C8×Dic3C2313yes—
172C8×Dic3C2313yes—
173C4⋊(C3⋊C8)C2313yes—
174C4⋊(C3⋊C8)C2313yes—
175C4⋊(C3⋊C8)C2313yes—
176C4⋊(C3⋊C8)C2311yescharacteristic
177C4×C3⋊C8C2313yes—
178C4×C3⋊C8C2313yes—
179C2×#131—36no—
180#559—33no—
181#557—31no—
182#556—33no—
183#553—33no—
184#553—33no—
185#550—31no—
186#548—31no—
187#548—31no—
188#532—33no—
189#529—33no—
190#529—33no—
191C4×#41—31no—
192C4×#41—33no—
193C4×#41C2×D311yescharacteristic
194C2×C4×Q8—31no—
195#446—33no—
196#446—33no—
197C8×Q8—31no—
198C8×Q8—31no—
199C8×Q8—33no—
200C8×Q8—33no—
201C8×Q8C2×D311yescharacteristic
202C8×Q8C2×D311yescharacteristic
203#436—33no—
204#436—33no—
205#436—33no—
206#436—36no—
207#436—33no—
208#435—36no—
209#435—36no—
210C8×D4—36no—
211C8×D4—33no—
212C8×D4—33no—
213C8×D4C2×D313yes—
214C8×D4C2×D313yes—
215#433—33no—
216#433—33no—
217#433—33no—
218#433—33no—
219#431—33no—
220#431—33no—
221#431—33no—
222#431C2×D313yes—
223#424—36no—
224#424—33no—
225#424—36no—
226#424—31no—
227#423—33no—
228#423—33no—
229#423—33no—
230C2×C4⋊C8—33no—
231C2×C4⋊C8—33no—
232C2×C4⋊C8C2×D313yes—
233#408—33no—
234#408—33no—
235C4×M16—33no—
236C2×C4×C8—33no—
237C3×#41—24no—
238C2×C3×Q8C22×C411yescharacteristic
239C2×C3×D4C22×C413yes—
240C22×C3×C4C22×C413yes—
241C22×C3×C4C22×C413yes—
242C22×C3×C4C2413yes—
243C2×#67C22×C413yes—
244C2×#67C22×C413yes—
245C2×#67C22×C416yes—
246#239—24no—
247D3×Q8—24no—
248#235—26no—
249#235—26no—
250C2×D12C22×C413yes—
251C2×D12C22×C413yes—
252C2×C4×D3C22×C413yes—
253C2×C4×D3C22×C413yes—
254C2×C4×D3C22×C413yes—
255C2×C4×D3C22×C413yes—
256C2×C4×D3C2413yes—
257C2×C4×D3C2411yescharacteristic
258C2×Dic6C22×C413yes—
259C2×Dic6C22×C413yes—
260C2×C3×C8—46no—
261C2×C3×C8C2411yescharacteristic
262C2×C3×C8C2413yes—
263C2×C3×C8C2413yes—
264C2×C3×C8C2411yescharacteristic
265C3×C4⋊C4C22×C413yes—
266C3×C4⋊C4C22×C413yes—
267C3×C22⋊C4C22×C416yes—
268C3×C42C22×C413yes—
269C3×C42C22×C413yes—
270C3×C42C2413yes—
271C3×C42C2411yescharacteristic
272#217C22×C413yes—
273#217C22×C416yes—
274#217C22×C413yes—
275(C2×D3)⋊C4C22×C411yescharacteristic
276(C2×D3)⋊C4C22×C411yescharacteristic
277(C2×D3)⋊C4C22×C413yes—
278(C2×D3)⋊C4C22×C413yes—
279(C2×D3)⋊C4C22×C416yes—
280(C2×D3)⋊C4C22×C416yes—
281C4⋊Dic3C22×C413yes—
282C4⋊Dic3C22×C413yes—
283C4⋊Dic3C22×C413yes—
284C4⋊Dic3C22×C413yes—
285Dic3⋊C4C22×C411yescharacteristic
286Dic3⋊C4C22×C411yescharacteristic
287Dic3⋊C4C22×C413yes—
288Dic3⋊C4C22×C413yes—
289Dic3⋊C4C22×C416yes—
290Dic3⋊C4C22×C416yes—
291C4×Dic3C22×C413yes—
292C4×Dic3C22×C413yes—
293C4×Dic3C22×C413yes—
294C4×Dic3C22×C413yes—
295C4×Dic3C2413yes—
296C4×Dic3C2411yescharacteristic
297#208—46no—
298C2×C3⋊C8C2411yescharacteristic
299C2×C3⋊C8C2411yescharacteristic
300C2×C3⋊C8C2413yes—
301C2×C3⋊C8C2413yes—
302ΓL(1,𝔽25)—26no—
303ΓL(1,𝔽25)—22no—
304ΓL(1,𝔽25)—46no—
305D3×C8—26no—
306D3×C8—22no—
307D3×C8—46no—
308C2×#41—31no—
309C2×#41—33no—
310C2×#41C4×D311yescharacteristic
311C22×Q8—31no—
312#131—1212no—
313C2×M16—36no—
314C2×M16—33no—
315C2×M16—31no—
316C2×M16—36no—
317C22×C8—33no—
318C22×C8—31no—
319C22×C8—36no—
320C22×C8C22×D316yes—
321C4⋊Q8—33no—
322#127—33no—
323#126—36no—
324#126—36no—
325#125—33no—
326#125—33no—
327#124—33no—
328#124—36no—
329#124—33no—
330#123—36no—
331#123—36no—
332C22⋊Q8—33no—
333C22⋊Q8—33no—
334#121—33no—
335#121—33no—
336#121—33no—
337#121—33no—
338#121—36no—
339C4×Q8—31no—
340C4×Q8—31no—
341C4×Q8—33no—
342C4×Q8—33no—
343C4×Q8—33no—
344C4×Q8—33no—
345C4×Q8C4×D311yescharacteristic
346C4×Q8C22×D311yescharacteristic
347C4×D4—33no—
348C4×D4—33no—
349C4×D4—33no—
350C4×D4—33no—
351C4×D4—36no—
352C4×D4—33no—
353C4×D4—33no—
354C4×D4—33no—
355C4×D4—33no—
356C4×D4C4×D313yes—
357C4×D4C22×D313yes—
358#117—33no—
359#117—33no—
360#117—31no—
361#117—33no—
362#117C4×D313yes—
363C2×C4⋊C4—33no—
364C2×C42—33no—
365C2×C42—33no—
366C2×C42C22×D313yes—
367C4⋊C8—33no—
368C4⋊C8—36no—
369C4⋊C8—33no—
370C4⋊C8—36no—
371C4⋊C8—31no—
372C4⋊C8—31no—
373C4⋊C8—33no—
374C4⋊C8—33no—
375C4⋊C8—33no—
376C4⋊C8—31no—
377C4⋊C8C22×D313yes—
378C4⋊C8C22×D313yes—
379C4⋊C8C22×D313yes—
380C4⋊C8C22×D311yescharacteristic
381C22⋊C8—33no—
382C22⋊C8—33no—
383C22⋊C8—33no—
384C22⋊C8—33no—
385C22⋊C8—33no—
386C22⋊C8—33no—
387C22⋊C8C22×D313yes—
388C22⋊C8C22×D313yes—
389#97—33no—
390#97—33no—
391C4×C8—33no—
392C4×C8—33no—
393C4×C8—33no—
394C4×C8—33no—
395C4×C8C22×D313yes—
396C4×C8C22×D313yes—
397C23×C3C23×C413yes—
398C22×D3C23×C413yes—
399C22×D3C23×C411yescharacteristic
400C3×Q8#13114yescomplemented
401C3×D4—26no—
402C2×C3×C4—26no—
403C2×C3×C4—23no—
404C2×C3×C4—23no—
405C2×C3×C4—43no—
406C2×C3×C4C23×C413yes—
407C2×C3×C4C23×C411yescharacteristic
408C2×C3×C4C23×C411yescharacteristic
409C2×C3×C4C23×C413yes—
410C2×C3×C4C23×C413yes—
411C2×C3×C4C2511yescharacteristic
412#67—26no—
413#67—26no—
414#67—46no—
415C2×Dic3C23×C411yescharacteristic
416C2×Dic3C23×C411yescharacteristic
417C2×Dic3C23×C413yes—
418C2×Dic3C23×C413yes—
419C2×Dic3C23×C413yes—
420C2×Dic3C23×C411yescharacteristic
421D12—26no—
422D12—26no—
423C4×D3—26no—
424C4×D3—26no—
425C4×D3—23no—
426C4×D3—23no—
427C4×D3—43no—
428C4×D3—43no—
429C4×D3H(5,𝔽2)12yes—
430C4×D32−1+412yes—
431Dic6—26no—
432Dic6—26no—
433C3×C8—23no—
434C3×C8—23no—
435C3×C8—21no—
436C3×C8—21no—
437C3⋊C8—23no—
438C3⋊C8—23no—
439C3⋊C8—21no—
440C3⋊C8—21no—
441#41—24no—
442#41—66no—
443#41—66no—
444#41—64no—
445C2×Q8—33no—
446C2×Q8—33no—
447C2×Q8—64no—
448C2×Q8C2×C4×D311yescharacteristic
449C2×D4—33no—
450C2×D4—33no—
451C2×D4—36no—
452C2×D4—33no—
453C2×D4—33no—
454C2×D4C2×C4×D313yes—
455C22×C4—31no—
456C22×C4—33no—
457C22×C4—33no—
458C22×C4—33no—
459C22×C4—33no—
460C22×C4—33no—
461C22×C4C2×C4×D313yes—
462C22×C4C2×C4×D313yes—
463C22×C4C23×D313yes—
464M16—66no—
465M16—62no—
466M16—126no—
467M16—126no—
468C2×C8—31no—
469C2×C8—31no—
470C2×C8—33no—
471C2×C8—33no—
472C2×C8—46no—
473C2×C8—66no—
474C2×C8—62no—
475C2×C8—126no—
476C2×C8C23×D311yescharacteristic
477C2×C8C23×D313yes—
478C2×C8C23×D311yescharacteristic
479C2×C8C23×D313yes—
480C4⋊C4—33no—
481C4⋊C4—36no—
482C4⋊C4—31no—
483C4⋊C4—31no—
484C4⋊C4—33no—
485C4⋊C4—33no—
486C4⋊C4—33no—
487C4⋊C4—33no—
488C4⋊C4—33no—
489C4⋊C4—36no—
490C4⋊C4C2×C4×D313yes—
491C4⋊C4C2×C4×D313yes—
492C22⋊C4—33no—
493C22⋊C4—33no—
494C22⋊C4—36no—
495C22⋊C4—31no—
496C22⋊C4—31no—
497C22⋊C4—33no—
498C22⋊C4—36no—
499C22⋊C4—33no—
500C22⋊C4—36no—
501C22⋊C4C2×C4×D316yes—
502C42—33no—
503C42—33no—
504C42—33no—
505C42—33no—
506C42—33no—
507C42—31no—
508C42C2×C4×D313yes—
509C42C2×C4×D313yes—
510C42C23×D313yes—
511C42C23×D311yescharacteristic
512C22×C3—23no—
513C22×C3—23no—
514C22×C3—23no—
515C22×C3C24×C411yescharacteristic
516C2×D3—21no—
517C2×D3—21no—
518C2×D3—23no—
519C2×D3—23no—
520C2×D3—23no—
521C2×D3#52012yes—
522C3×C4—23no—
523C3×C4—21no—
524C3×C4—23no—
525C3×C4—21no—
526C3×C4C2×#13116yes—
527C3×C4C2×H(5,𝔽2)11yescharacteristic
528C3×C4C2×2−1+411yescharacteristic
529Dic3—23no—
530Dic3—23no—
531Dic3—21no—
532Dic3—21no—
533Dic3—23no—
534Dic3#52012yes—
535C23—31no—
536C23—33no—
537C23C22×C4×D313yes—
538Q8—66no—
539Q8—66no—
540Q8#90114yescomplemented
541D4—26no—
542D4—66no—
543D4—66no—
544D4—66no—
545D4—66no—
546D4—126no—
547C2×C4—23no—
548C2×C4—23no—
549C2×C4—26no—
550C2×C4—31no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—33no—
556C2×C4—33no—
557C2×C4—33no—
558C2×C4—43no—
559C2×C4—63no—
560C2×C4—63no—
561C2×C4—66no—
562C2×C4—66no—
563C2×C4—123no—
564C2×C4—123no—
565C2×C4C22×C4×D313yes—
566C2×C4C22×C4×D311yescharacteristic
567C2×C4C22×C4×D311yescharacteristic
568C2×C4C22×C4×D313yes—
569C2×C4C22×C4×D313yes—
570C2×C4C24×D311yescharacteristic
571C8—23no—
572C8—23no—
573C8—21no—
574C8—21no—
575C8—63no—
576C8—63no—
577C8—61no—
578C8—61no—
579C2×C3—43no—
580C2×C3C2×#52011yescharacteristic
581C2×C3C22×#13111yescharacteristic
582C2×C3#357311yescharacteristic
583D3—22no—
584D3—43no—
585C22—23no—
586C22—23no—
587C22—23no—
588C22—32no—
589C22—61no—
590C22—61no—
591C22—63no—
592C22—63no—
593C22—63no—
594C22C23×C4×D311yescharacteristic
595C4—21no—
596C4—21no—
597C4—23no—
598C4—23no—
599C4—32no—
600C4—61no—
601C4—63no—
602C4—61no—
603C4—63no—
604C4—63no—
605C4C2×#90116yes—
606C4D3×H(5,𝔽2)11yescharacteristic
607C4D3×2−1+411yescharacteristic
608C3#6027411yescharacteristic, complemented, Sylow
609C2—43no—
610C2—62no—
611C2—123no—
612C2D3×#52011yescharacteristic
613C2C22×#90111yescharacteristic
614C2#8716611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC3×#2988C2
prime radical#85842C2
cosocleC2×C3×C4C25
hypercenter#2988D3
hypocenterC3#60274
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #2988

Complements

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

NH ≅ G/Ntype
2. C3×#2988C2semidirect
3. #82523C2semidirect
4. Q8×ΓL(1,𝔽25)C2semidirect
5. D3×C8×Q8C2semidirect
6. #79907C2semidirect
7. #79906C2semidirect
8. #79425C2semidirect
9. #79422C2semidirect
10. #79410C2semidirect
11. #79408C2semidirect
12. #79295C2semidirect
13. #79283C2semidirect
14. #79017C2semidirect
15. #79012C2semidirect
83. #2988D3semidirect
400. C3×Q8#131semidirect
540. Q8#901semidirect
608. C3#60274semidirect

Maximal subgroups

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

subgroupindex
#858422
C3×#29882
#825232
Q8×ΓL(1,𝔽25)2
D3×C8×Q82
#799072
#799062
#794252
#794222
#794102
#794082
#792952
#792832
#790172
#790122
#602743