Finite Group Databasebeta

← #10785124

Subgroups — #10785124

Order 768.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79972C211yesmaximal, characteristic, complemented
2#79688C211yesmaximal, characteristic
3#79623C211yesmaximal, characteristic, complemented
4#79442C211yesmaximal, characteristic
5C3×#1582C211yesmaximal, characteristic, complemented
6#70966C211yesmaximal, characteristic, complemented
7#70855C211yesmaximal, characteristic, complemented
8#68547C211yesmaximal, characteristic, complemented
9#68539C211yesmaximal, characteristic, complemented
10#68270C211yesmaximal, characteristic, complemented
11#68256C211yesmaximal, characteristic, complemented
12#68205C211yesmaximal, characteristic, complemented
13#68184C211yesmaximal, characteristic
14#68096C211yesmaximal, characteristic, complemented
15#68076C211yesmaximal, characteristic, complemented
16#19794—31nomaximal, Sylow
17#5579C2211yescharacteristic
18#5570C2211yescharacteristic
19#5551C2211yescharacteristic
20#5542C2211yescharacteristic
21C3×#446C2211yescharacteristic
22C3×C4×SD16C2211yescharacteristic
23C3×C8×D4C2211yescharacteristic
24#5171—21no—
25#5166—21no—
26#5116—21no—
27#5113—21no—
28#5031C2211yescharacteristic
29#5020C2211yescharacteristic
30#5008C2211yescharacteristic
31#5007C2211yescharacteristic
32D3×#106—21no—
33#4857—21no—
34#4850C2211yescharacteristic
35#4849C2211yescharacteristic
36#4846C2211yescharacteristic
37#4841—21no—
38#4841C2211yescharacteristic
39#4829—21no—
40#4805—21no—
41#4790—21no—
42#4777—21no—
43#4768—21no—
44#4733—21no—
45#4723—21no—
46#4710—21no—
47#4703—21no—
48C4×#204C2211yescharacteristic
49C8×D12C2211yescharacteristic
50#4682C2211yescharacteristic
51#4681C2211yescharacteristic
52#4681C2211yescharacteristic
53#4681C2211yescharacteristic
54#4681C2211yescharacteristic
55#4680C2211yescharacteristic
56C8×Dic6C2211yescharacteristic
57C3×#334C2211yescharacteristic
58C3×Q8⋊C8C2211yescharacteristic
59C3×D4⋊C8C2211yescharacteristic
60C3×#322C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4481C2211yescharacteristic
65#4459C2211yescharacteristic
66#4458C2211yescharacteristic
67#4456C2211yescharacteristic
68#3372—31no—
69#3364—31no—
70#3335—31no—
71#3302—31no—
72#1718—31no—
73#1700—31no—
74#1683—31no—
75#1679—31no—
76#1672—31no—
77#1648—31no—
78#1647—31no—
79#1644—31no—
80#1638—31no—
81#1582D311yescharacteristic, complemented
82SD16⋊C8—31no—
83C2×#235—21no—
84C2×C3×SD16D411yescharacteristic
85C22×C3×C8—21no—
86C3×C4×Q8C2311yescharacteristic
87C3×C4×D4C2311yescharacteristic
88#946—21no—
89Q8×Dic3—21no—
90C2×#216—21no—
91#938—21no—
92#936—21no—
93#926—21no—
94#924—21no—
95#924—21no—
96C2×Dic12—21no—
97C2×#204—21no—
98C2×#204D411yescharacteristic
99C2×D3×C8—21no—
100#896—41no—
101D3×C4⋊C4—21no—
102D3×C4⋊C4—21no—
103#890—21no—
104#890—41no—
105#887—21no—
106#883—41no—
107#879—41no—
108C4×D12—21no—
109C4×D12C2311yescharacteristic
110#870—21no—
111#869—21no—
112#869C2311yescharacteristic
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6—21no—
117C4×Dic6C2311yescharacteristic
118C3×#106D411yescharacteristic
119C3×C4⋊C8—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×#103D411yescharacteristic
123C3×#102D411yescharacteristic
124C3×C22⋊C8—21no—
125C3×#97—21no—
126C3×C4×C8C2311yescharacteristic
127C3×C4×C8C2311yescharacteristic
128C3×C4×C8C2311yescharacteristic
129#835—21no—
130#832—21no—
131#832—21no—
132#821—21no—
133#821D411yescharacteristic
134(C2×D3)⋊C8—21no—
135#818—21no—
136#818—21no—
137#818—41no—
138#817—21no—
139#817—21no—
140#817D411yescharacteristic
141#816—21no—
142#816—21no—
143#816—21no—
144#816—21no—
145#816—21no—
146#816—21no—
147#816—21no—
148#816—21no—
149#816—21no—
150#816—41no—
151#816D411yescharacteristic
152Dic3⋊C8—21no—
153C8×Dic3—21no—
154#810—21no—
155#808—21no—
156#808—21no—
157#807—21no—
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160#557—31no—
161#556—31no—
162#554—31no—
163#547—31no—
164#500—31no—
165#498—31no—
166#491—61no—
167#488—61no—
168#487—31no—
169#487—31no—
170#487—31no—
171#487—31no—
172#485—61no—
173#483—61no—
174#481—61no—
175#479—61no—
176#479—61no—
177#477—31no—
178#476—31no—
179#475—31no—
180#474—31no—
181#474—31no—
182#467—61no—
183#465—61no—
184#464—61no—
185#462—31no—
186#462—31no—
187#461—31no—
188#461—61no—
189#452—61no—
190#446C2×D311yescharacteristic
191C8×Q8—31no—
192#440—61no—
193C4×Q16—61no—
194C4×SD16—31no—
195C4×SD16C2×D311yescharacteristic
196C8×D4—31no—
197C8×D4C2×D311yescharacteristic
198C2×#106—61no—
199#416—61no—
200#335—31no—
201#334C2×D311yescharacteristic
202#333—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206Q8⋊C8—31no—
207Q8⋊C8C2×D311yescharacteristic
208D4⋊C8—31no—
209D4⋊C8C2×D311yescharacteristic
210#322C2×D311yescharacteristic
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×D4C2×D411yescharacteristic
213C22×C3×C4—21no—
214C2×#67—41no—
215#235—42no—
216#235—81no—
217C2×D12C2×D411yescharacteristic
218C2×C4×D3—21no—
219C2×C4×D3—41no—
220C2×C4×D3—41no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C2×D411yescharacteristic
226C3×SD16—41no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—81no—
231C2×C3×C8C2×D411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C3×C4⋊C4—21no—
236C3×C4⋊C4C2×D411yescharacteristic
237C3×C4⋊C4C2×D411yescharacteristic
238C3×C22⋊C4—21no—
239C3×C42—21no—
240C3×C42C2411yescharacteristic
241#217—41no—
242#217—41no—
243#217—41no—
244#216—42no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—41no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—41no—
256C4⋊Dic3—41no—
257C4⋊Dic3C2×D411yescharacteristic
258C4⋊Dic3C2×D411yescharacteristic
259Dic3⋊C4—21no—
260Dic3⋊C4—41no—
261Dic3⋊C4—41no—
262Dic3⋊C4—41no—
263Dic3⋊C4—41no—
264Dic3⋊C4—41no—
265C4×Dic3—21no—
266C4×Dic3—41no—
267C4×Dic3—41no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270Dic12—81no—
271#204—22no—
272#204—42no—
273D3×C8—42no—
274C2×#41—61no—
275C2×Q16—61no—
276C2×Q16—61no—
277C2×SD16—31no—
278C2×SD16—61no—
279C2×SD16D1211yescharacteristic
280C22×C8—21no—
281C22×C8—61no—
282C4⋊Q8—31no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8—31no—
286C4⋊Q8—61no—
287#126—121no—
288#125—61no—
289#125—61no—
290#125—121no—
291#124—61no—
292#123—121no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295C22⋊Q8—61no—
296C22⋊Q8—121no—
297#121—61no—
298C4×Q8—31no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×Q8—61no—
302C4×Q8C22×D311yescharacteristic
303C4×D4—31no—
304C4×D4—61no—
305C4×D4C22×D311yescharacteristic
306#117—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309#107—61no—
310#107—61no—
311#107—61no—
312#107—121no—
313#106—31no—
314#106—61no—
315#106—61no—
316#106—61no—
317#106—61no—
318#106D1211yescharacteristic
319C4⋊C8—21no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8C22×D311yescharacteristic
324C4⋊C8C22×D311yescharacteristic
325#103—31no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#103—121no—
338#103D1211yescharacteristic
339#102—31no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102D1211yescharacteristic
344C22⋊C8—21no—
345C22⋊C8—61no—
346#97—21no—
347C4×C8—61no—
348C4×C8C22×D311yescharacteristic
349C4×C8C22×D311yescharacteristic
350C4×C8C22×D311yescharacteristic
351C23×C3—21no—
352C22×D3—21no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×D4—21no—
356C3×D4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365#67—82no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373D12—21no—
374D12#13512yes—
375C4×D3—41no—
376C4×D3—41no—
377C4×D3—82no—
378C4×D3—161no—
379Dic6—21no—
380Dic6—21no—
381Dic6—41no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386Dic6—41no—
387Dic6—81no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397#41—122no—
398#41—241no—
399C2×Q8—31no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8C2×D1211yescharacteristic
405C2×D4—31no—
406C2×D4—121no—
407C2×D4C2×D1211yescharacteristic
408C22×C4—21no—
409C22×C4—61no—
410C22×C4—121no—
411C22×C4—121no—
412Q16—122no—
413Q16—241no—
414SD16—41no—
415SD16—62no—
416SD16—122no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—81no—
423C2×C8—122no—
424C2×C8C2×D1211yescharacteristic
425C2×C8C2×D1211yescharacteristic
426C2×C8D3×D411yescharacteristic
427C2×C8D3×D411yescharacteristic
428C4⋊C4—21no—
429C4⋊C4—31no—
430C4⋊C4—31no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—121no—
441C4⋊C4—121no—
442C4⋊C4—121no—
443C4⋊C4—121no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C4⋊C4—121no—
447C4⋊C4C2×D1211yescharacteristic
448C4⋊C4C2×D1211yescharacteristic
449C22⋊C4—21no—
450C22⋊C4—61no—
451C22⋊C4—121no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C22⋊C4—121no—
455C42—21no—
456C42—61no—
457C42—121no—
458C42—121no—
459C42C23×D311yescharacteristic
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3D4211yescharacteristic
464C2×D3—22no—
465C2×D3—41no—
466C2×D3—41no—
467C3×C4—21no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4C2×#13511yescharacteristic
472C3×C4#57511yescharacteristic
473C3×C4#57711yescharacteristic
474C3×C4#57811yescharacteristic
475Dic3—41no—
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480Dic3—81no—
481Dic3—81no—
482C23—21no—
483C23—61no—
484Q8—21no—
485Q8—21no—
486Q8—61no—
487Q8—61no—
488Q8—121no—
489Q8—121no—
490Q8—121no—
491Q8—121no—
492Q8—121no—
493Q8—121no—
494Q8—241no—
495D4—21no—
496D4—21no—
497D4—32no—
498D4—61no—
499D4—242no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—242no—
515C2×C4—481no—
516C2×C4C22×D1211yescharacteristic
517C2×C4C2×D3×D411yescharacteristic
518C2×C4#100911yescharacteristic
519C8—21no—
520C8—21no—
521C8—21no—
522C8—21no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—121no—
527C8—121no—
528C2×C3—81no—
529C2×C3#326911yescharacteristic
530C2×C3#328211yescharacteristic
531C2×C3#328911yescharacteristic
532D3—42no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—62no—
537C22—121no—
538C22—121no—
539C22D4×D1211yescharacteristic
540C4—21no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—121no—
545C4—121no—
546C4—121no—
547C4—241no—
548C4—241no—
549C4—241no—
550C4—241no—
551C4#574811yescharacteristic
552C4#575311yescharacteristic
553C4#576211yescharacteristic
554C4D3×#13511yescharacteristic
555C3#1979411yescharacteristic, complemented, Sylow
556C2—81no—
557C2—122no—
558C2#7949811yescharacteristic
559C2#7967411yescharacteristic
560C2#7990111yescharacteristic
subgroupG/H
centerC22D4×D12
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1582C2
socleC22×C3D42
prime radical#5551C22
hypercenter#1582D3
hypocenterC3#19794
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1582

Complements

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

NH ≅ G/Ntype
1. #79972C2semidirect
3. #79623C2semidirect
5. C3×#1582C2semidirect
6. #70966C2semidirect
7. #70855C2semidirect
8. #68547C2semidirect
9. #68539C2semidirect
10. #68270C2semidirect
11. #68256C2semidirect
12. #68205C2semidirect
14. #68096C2semidirect
15. #68076C2semidirect
81. #1582D3semidirect
555. C3#19794semidirect

Maximal subgroups

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

subgroupindex
#799722
#796882
#796232
#794422
C3×#15822
#709662
#708552
#685472
#685392
#682702
#682562
#682052
#681842
#680962
#680762
#197943