Finite Group Databasebeta

← #11660814

Subgroups — #11660814

Order 768.

1760 conjugacy classes of subgroups.

The subgroup lattice has 610 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
4#79921C211yesmaximal, characteristic, complemented
5#79920C211yesmaximal, characteristic, complemented
6#79905C211yesmaximal, characteristic, complemented
7C8×#239C211yesmaximal, characteristic, complemented
8#79431C213yesmaximal, complemented
9#79430C213yesmaximal, complemented
10#79403C213yesmaximal, complemented
11#79401C213yesmaximal, complemented
12#79295C213yesmaximal, complemented
13#79283C213yesmaximal, complemented
14#79017C213yesmaximal, complemented
15#79000C213yesmaximal, complemented
16#60301—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5834C411yescharacteristic
19#5831C411yescharacteristic
20#5576C411yescharacteristic
21C4×#239C2211yescharacteristic
22C4×D3×Q8C2211yescharacteristic
23#5569C411yescharacteristic
24#5559C413yes—
25#5558C413yes—
26#5557C2213yes—
27#5548C2213yes—
28#5533C413yes—
29#5530C413yes—
30#5479C2213yes—
31C4×#235C2213yes—
32C3×C8×Q8C2211yescharacteristic
33C3×C8×Q8C2211yescharacteristic
34C3×C8×D4C2213yes—
35C3×C8×D4C2213yes—
36C3×#431C2213yes—
37C2×C3×C4⋊C8C2213yes—
38#5111C2216yes—
39#5110C2216yes—
40C8×#67C2216yes—
41#5101C2216yes—
42Q8×C3⋊C8C2211yescharacteristic
43Q8×C3⋊C8C2211yescharacteristic
44#5012C2213yes—
45#5012C2213yes—
46#4925C2213yes—
47#4924C2213yes—
48#4840C2213yes—
49#4840C2216yes—
50#4840C2211yescharacteristic
51#4839C2213yes—
52#4839C2213yes—
53#4838C2213yes—
54#4835C2213yes—
55#4835C2213yes—
56#4834C2213yes—
57#4833C2213yes—
58#4833C2213yes—
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#2993—33no—
78#2991—31no—
79#2988D311yescharacteristic, complemented
80#2984—31no—
81#2982—31no—
82#2981—33no—
83#2979—33no—
84#2971—31no—
85#2969—33no—
86#2966—33no—
87C8×#41—31no—
88#2925—33no—
89#2922—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—
117C2×ΓL(1,𝔽25)C2313yes—
118C2×ΓL(1,𝔽25)C2313yes—
119C2×D3×C8C2313yes—
120C2×D3×C8C2313yes—
121#897C2×C413yes—
122#895C2×C413yes—
123#893C2×C413yes—
124#892C2×C413yes—
125D3×C4⋊C4C2×C413yes—
126#890C2×C413yes—
127#888C2×C413yes—
128#887C2×C413yes—
129#885C2×C416yes—
130#884C2×C416yes—
131#883C2×C416yes—
132#879C2×C416yes—
133#876C2×C416yes—
134#875C2×C413yes—
135C4×D12C2×C413yes—
136C4×D12C2×C413yes—
137C4×D12C2313yes—
138#872C2311yescharacteristic
139#872C2313yes—
140C42×D3C2313yes—
141C42×D3C2313yes—
142#870C2×C413yes—
143C4×Dic6C2×C413yes—
144C4×Dic6C2×C413yes—
145C4×Dic6C2313yes—
146C3×C4⋊C8C2313yes—
147C3×C4⋊C8C2313yes—
148C3×C4⋊C8C2313yes—
149C3×C4⋊C8C2311yescharacteristic
150C3×C22⋊C8C2313yes—
151C3×C22⋊C8C2313yes—
152C3×C4×C8C2313yes—
153C3×C4×C8C2313yes—
154#830C2313yes—
155#830C2313yes—
156(C2×D3)⋊C8C2311yescharacteristic
157(C2×D3)⋊C8C2311yescharacteristic
158(C2×D3)⋊C8C2313yes—
159(C2×D3)⋊C8C2313yes—
160(C2×D3)⋊C8C2316yes—
161(C2×D3)⋊C8C2316yes—
162#815C2316yes—
163#815C2311yescharacteristic
164#815C2313yes—
165Dic3⋊C8C2313yes—
166Dic3⋊C8C2313yes—
167Dic3⋊C8C2313yes—
168Dic3⋊C8C2313yes—
169C8×Dic3C2316yes—
170C8×Dic3C2311yescharacteristic
171C8×Dic3C2313yes—
172C4⋊(C3⋊C8)C2313yes—
173C4⋊(C3⋊C8)C2313yes—
174C4⋊(C3⋊C8)C2313yes—
175C4⋊(C3⋊C8)C2311yescharacteristic
176C4×C3⋊C8C2313yes—
177C4×C3⋊C8C2313yes—
178#559—33no—
179#557—31no—
180#556—33no—
181#553—33no—
182#553—33no—
183#550—31no—
184#548—31no—
185#548—31no—
186#532—33no—
187#529—33no—
188#529—33no—
189C4×#41—31no—
190C4×#41—33no—
191C4×#41C2×D311yescharacteristic
192C2×C4×Q8—31no—
193#446—33no—
194#446—33no—
195C8×Q8—31no—
196C8×Q8—31no—
197C8×Q8—33no—
198C8×Q8—33no—
199C8×Q8C2×D311yescharacteristic
200C8×Q8C2×D311yescharacteristic
201#436—33no—
202#436—33no—
203#436—33no—
204#436—36no—
205#436—33no—
206#435—36no—
207#435—36no—
208C8×D4—36no—
209C8×D4—33no—
210C8×D4—33no—
211C8×D4C2×D313yes—
212C8×D4C2×D313yes—
213#433—31no—
214#433—33no—
215#433—36no—
216#433—33no—
217#433—33no—
218#432—33no—
219#432—33no—
220#431—33no—
221#431—33no—
222#431—33no—
223#431—33no—
224#431—33no—
225#431C2×D313yes—
226#424—33no—
227#424—33no—
228#423—33no—
229C2×C4⋊C8C2×D313yes—
230#408—36no—
231#408—33no—
232#408—33no—
233#405—36no—
234C4×M16—33no—
235C2×C4×C8—33no—
236C3×#41—24no—
237C2×C3×Q8C22×C411yescharacteristic
238C2×C3×D4C22×C413yes—
239C22×C3×C4C22×C413yes—
240C22×C3×C4C22×C413yes—
241C22×C3×C4C2413yes—
242C2×#67C22×C416yes—
243C2×#67C22×C413yes—
244C2×#67C22×C413yes—
245#239—24no—
246D3×Q8—24no—
247#235—43no—
248#235—43no—
249C2×D12C22×C413yes—
250C2×D12C22×C413yes—
251C2×C4×D3C22×C413yes—
252C2×C4×D3C22×C413yes—
253C2×C4×D3C22×C413yes—
254C2×C4×D3C22×C413yes—
255C2×C4×D3C2411yescharacteristic
256C2×C4×D3C2413yes—
257C2×Dic6C22×C413yes—
258C2×Dic6C22×C413yes—
259C2×C3×C8—46no—
260C2×C3×C8C2413yes—
261C2×C3×C8C2411yescharacteristic
262C2×C3×C8C2411yescharacteristic
263C2×C3×C8C2413yes—
264C3×C4⋊C4C22×C413yes—
265C3×C4⋊C4C22×C413yes—
266C3×C22⋊C4C22×C416yes—
267C3×C42C22×C413yes—
268C3×C42C22×C413yes—
269C3×C42C2413yes—
270C3×C42C2411yescharacteristic
271#217C22×C413yes—
272#217C22×C416yes—
273#217C22×C413yes—
274(C2×D3)⋊C4C22×C411yescharacteristic
275(C2×D3)⋊C4C22×C411yescharacteristic
276(C2×D3)⋊C4C22×C413yes—
277(C2×D3)⋊C4C22×C413yes—
278(C2×D3)⋊C4C22×C416yes—
279(C2×D3)⋊C4C22×C416yes—
280C4⋊Dic3C22×C413yes—
281C4⋊Dic3C22×C413yes—
282C4⋊Dic3C22×C413yes—
283C4⋊Dic3C22×C413yes—
284Dic3⋊C4C22×C411yescharacteristic
285Dic3⋊C4C22×C411yescharacteristic
286Dic3⋊C4C22×C413yes—
287Dic3⋊C4C22×C413yes—
288Dic3⋊C4C22×C416yes—
289Dic3⋊C4C22×C416yes—
290C4×Dic3C22×C413yes—
291C4×Dic3C22×C413yes—
292C4×Dic3C22×C413yes—
293C4×Dic3C22×C413yes—
294C4×Dic3C2411yescharacteristic
295C4×Dic3C2413yes—
296#208—46no—
297C2×C3⋊C8C2411yescharacteristic
298C2×C3⋊C8C2411yescharacteristic
299C2×C3⋊C8C2413yes—
300C2×C3⋊C8C2413yes—
301ΓL(1,𝔽25)—43no—
302ΓL(1,𝔽25)—43no—
303D3×C8—43no—
304D3×C8—43no—
305C2×#41—31no—
306C2×#41—33no—
307C2×#41C4×D311yescharacteristic
308C22×Q8—31no—
309C2×M16—36no—
310C2×M16—33no—
311C2×M16—33no—
312C22×C8—33no—
313C22×C8—33no—
314C22×C8C22×D316yes—
315C4⋊Q8—33no—
316#127—33no—
317#126—36no—
318#126—36no—
319#125—33no—
320#125—33no—
321#124—33no—
322#124—36no—
323#124—33no—
324#123—36no—
325#123—36no—
326C22⋊Q8—33no—
327C22⋊Q8—33no—
328#121—33no—
329#121—33no—
330#121—33no—
331#121—33no—
332#121—36no—
333C4×Q8—31no—
334C4×Q8—31no—
335C4×Q8—33no—
336C4×Q8—33no—
337C4×Q8—33no—
338C4×Q8—33no—
339C4×Q8C4×D311yescharacteristic
340C4×Q8C22×D311yescharacteristic
341C4×D4—33no—
342C4×D4—33no—
343C4×D4—33no—
344C4×D4—33no—
345C4×D4—36no—
346C4×D4—33no—
347C4×D4—33no—
348C4×D4—33no—
349C4×D4—33no—
350C4×D4C4×D313yes—
351C4×D4C22×D313yes—
352#117—33no—
353#117—33no—
354#117—31no—
355#117—33no—
356#117C4×D313yes—
357C2×C4⋊C4—33no—
358C2×C42—33no—
359C2×C42—33no—
360C2×C42C22×D313yes—
361C4⋊C8—33no—
362C4⋊C8—33no—
363C4⋊C8—33no—
364C4⋊C8—33no—
365C4⋊C8—33no—
366C4⋊C8—33no—
367C4⋊C8—33no—
368C4⋊C8—31no—
369C4⋊C8C22×D313yes—
370C4⋊C8C22×D313yes—
371C4⋊C8C22×D313yes—
372C4⋊C8C22×D311yescharacteristic
373C22⋊C8—33no—
374C22⋊C8—33no—
375C22⋊C8—31no—
376C22⋊C8—33no—
377C22⋊C8—31no—
378C22⋊C8—33no—
379C22⋊C8—36no—
380C22⋊C8—36no—
381C22⋊C8C22×D313yes—
382C22⋊C8C22×D313yes—
383#97—31no—
384#97—33no—
385#97—36no—
386C4×C8—31no—
387C4×C8—33no—
388C4×C8—36no—
389C4×C8—33no—
390C4×C8—33no—
391C4×C8C22×D313yes—
392C4×C8C22×D313yes—
393C23×C3C23×C413yes—
394C22×D3C23×C413yes—
395C22×D3C23×C411yescharacteristic
396C3×Q8#13114yes—
397C3×D4—26no—
398C2×C3×C4—26no—
399C2×C3×C4—23no—
400C2×C3×C4—23no—
401C2×C3×C4—43no—
402C2×C3×C4C23×C413yes—
403C2×C3×C4C23×C411yescharacteristic
404C2×C3×C4C23×C411yescharacteristic
405C2×C3×C4C23×C413yes—
406C2×C3×C4C23×C413yes—
407C2×C3×C4C2511yescharacteristic
408#67—43no—
409#67—43no—
410#67—46no—
411C2×Dic3C23×C411yescharacteristic
412C2×Dic3C23×C411yescharacteristic
413C2×Dic3C23×C411yescharacteristic
414C2×Dic3C23×C413yes—
415C2×Dic3C23×C413yes—
416C2×Dic3C23×C413yes—
417D12—26no—
418D12—43no—
419C4×D3—26no—
420C4×D3—26no—
421C4×D3—23no—
422C4×D3—23no—
423C4×D3—21no—
424C4×D3—21no—
425C4×D3—43no—
426C4×D3—43no—
427Dic6—26no—
428Dic6—43no—
429C3×C8—21no—
430C3×C8—23no—
431C3×C8—21no—
432C3×C8—23no—
433C3⋊C8—21no—
434C3⋊C8—23no—
435C3⋊C8—21no—
436C3⋊C8—23no—
437#41—24no—
438#41—64no—
439#41—123no—
440#41—123no—
441C2×Q8—33no—
442C2×Q8—33no—
443C2×Q8—64no—
444C2×Q8C2×C4×D311yescharacteristic
445C2×D4—33no—
446C2×D4—33no—
447C2×D4—36no—
448C2×D4—33no—
449C2×D4—33no—
450C2×D4C2×C4×D313yes—
451C22×C4—33no—
452C22×C4—33no—
453C22×C4—31no—
454C22×C4—33no—
455C22×C4—33no—
456C22×C4—33no—
457C22×C4C2×C4×D313yes—
458C22×C4C2×C4×D313yes—
459C22×C4C23×D313yes—
460M16—123no—
461M16—123no—
462M16—126no—
463C2×C8—31no—
464C2×C8—31no—
465C2×C8—33no—
466C2×C8—33no—
467C2×C8—46no—
468C2×C8—123no—
469C2×C8—123no—
470C2×C8C23×D313yes—
471C2×C8C23×D311yescharacteristic
472C2×C8C23×D311yescharacteristic
473C2×C8C23×D313yes—
474C4⋊C4—33no—
475C4⋊C4—36no—
476C4⋊C4—31no—
477C4⋊C4—31no—
478C4⋊C4—33no—
479C4⋊C4—33no—
480C4⋊C4—33no—
481C4⋊C4—33no—
482C4⋊C4—33no—
483C4⋊C4—36no—
484C4⋊C4C2×C4×D313yes—
485C4⋊C4C2×C4×D313yes—
486C22⋊C4—33no—
487C22⋊C4—33no—
488C22⋊C4—36no—
489C22⋊C4—31no—
490C22⋊C4—31no—
491C22⋊C4—33no—
492C22⋊C4—36no—
493C22⋊C4—33no—
494C22⋊C4—36no—
495C22⋊C4C2×C4×D316yes—
496C42—33no—
497C42—33no—
498C42—33no—
499C42—33no—
500C42—31no—
501C42—33no—
502C42C2×C4×D313yes—
503C42C2×C4×D313yes—
504C42C23×D313yes—
505C42C23×D311yescharacteristic
506C22×C3—23no—
507C22×C3—23no—
508C22×C3—23no—
509C22×C3C24×C411yescharacteristic
510C2×D3—21no—
511C2×D3—23no—
512C2×D3—21no—
513C2×D3—23no—
514C2×D3—21no—
515C2×D3—23no—
516C3×C4—23no—
517C3×C4—21no—
518C3×C4—23no—
519C3×C4—21no—
520C3×C4C2×#13116yes—
521C3×C4#58511yescharacteristic
522C3×C4#58511yescharacteristic
523Dic3—21no—
524Dic3—23no—
525Dic3—23no—
526Dic3—21no—
527Dic3—21no—
528Dic3—23no—
529C23—33no—
530C23—31no—
531C23C22×C4×D313yes—
532Q8—66no—
533Q8—123no—
534Q8#90114yes—
535D4—26no—
536D4—66no—
537D4—123no—
538D4—123no—
539D4—123no—
540D4—126no—
541C2×C4—23no—
542C2×C4—23no—
543C2×C4—26no—
544C2×C4—31no—
545C2×C4—31no—
546C2×C4—31no—
547C2×C4—33no—
548C2×C4—33no—
549C2×C4—33no—
550C2×C4—43no—
551C2×C4—63no—
552C2×C4—61no—
553C2×C4—63no—
554C2×C4—61no—
555C2×C4—66no—
556C2×C4—66no—
557C2×C4—123no—
558C2×C4—123no—
559C2×C4C22×C4×D313yes—
560C2×C4C22×C4×D311yescharacteristic
561C2×C4C22×C4×D311yescharacteristic
562C2×C4C22×C4×D313yes—
563C2×C4C22×C4×D313yes—
564C2×C4C24×D311yescharacteristic
565C8—23no—
566C8—23no—
567C8—21no—
568C8—21no—
569C8—61no—
570C8—61no—
571C8—63no—
572C8—63no—
573C2×C3—43no—
574C2×C3#342811yescharacteristic
575C2×C3C22×#13111yescharacteristic
576C2×C3#357311yescharacteristic
577D3—41no—
578D3—43no—
579C22—23no—
580C22—23no—
581C22—23no—
582C22—61no—
583C22—63no—
584C22—61no—
585C22—63no—
586C22—61no—
587C22—63no—
588C22C23×C4×D311yescharacteristic
589C4—21no—
590C4—23no—
591C4—21no—
592C4—23no—
593C4—61no—
594C4—63no—
595C4—61no—
596C4—63no—
597C4—61no—
598C4—63no—
599C4C2×#90116yes—
600C4#596611yescharacteristic
601C4#596811yescharacteristic
602C3#6030111yescharacteristic, complemented, Sylow
603C2—43no—
604C2—121no—
605C2—123no—
606C2#8588511yescharacteristic
607C2C22×#90111yescharacteristic
608C2#8716611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC3×#2988C2
prime radical#85842C2
cosocleC2×C3×C4C25
hypercenter#2988D3
hypocenterC3#60301
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. #79921C2semidirect
5. #79920C2semidirect
6. #79905C2semidirect
7. C8×#239C2semidirect
8. #79431C2semidirect
9. #79430C2semidirect
10. #79403C2semidirect
11. #79401C2semidirect
12. #79295C2semidirect
13. #79283C2semidirect
14. #79017C2semidirect
15. #79000C2semidirect
79. #2988D3semidirect
602. C3#60301semidirect

Maximal subgroups

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

subgroupindex
#858422
C3×#29882
#825232
#799212
#799202
#799052
C8×#2392
#794312
#794302
#794032
#794012
#792952
#792832
#790172
#790002
#603013