Finite Group Databasebeta

← #10780310

Subgroups — #10780310

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83365C211yesmaximal, characteristic
2#83337C211yesmaximal, characteristic, complemented
3#79323C211yesmaximal, characteristic
4#79280C211yesmaximal, characteristic
5C3×#1554C211yesmaximal, characteristic, complemented
6#70795C211yesmaximal, characteristic, complemented
7#70780C211yesmaximal, characteristic, complemented
8#68515C211yesmaximal, characteristic, complemented
9#68511C211yesmaximal, characteristic, complemented
10#68503C211yesmaximal, characteristic, complemented
11#68499C211yesmaximal, characteristic, complemented
12#68477C211yesmaximal, characteristic, complemented
13#68469C211yesmaximal, characteristic, complemented
14#68453C211yesmaximal, characteristic, complemented
15#68445C211yesmaximal, characteristic, complemented
16#19381—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18#5479C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#432C2211yescharacteristic
21C3×#431C2211yescharacteristic
22#5093C2211yescharacteristic
23#5093C2211yescharacteristic
24#5093C2211yescharacteristic
25#5093C2211yescharacteristic
26#5091C2211yescharacteristic
27#5090C2211yescharacteristic
28#5084C2211yescharacteristic
29#5083C2211yescharacteristic
30#5069—21no—
31#5069—21no—
32#5052—21no—
33#5049—21no—
34#5028—21no—
35#5022—21no—
36#5004—21no—
37#5004—21no—
38#4924C2211yescharacteristic
39C4×#208C2211yescharacteristic
40#4851C2211yescharacteristic
41#4851C2211yescharacteristic
42#4846C2211yescharacteristic
43#4846C2211yescharacteristic
44#4838C2211yescharacteristic
45#4834C2211yescharacteristic
46#4831C2211yescharacteristic
47#4830C2211yescharacteristic
48#4735—21no—
49#4729—21no—
50#4728—21no—
51#4722—21no—
52#4714—21no—
53#4702—21no—
54#4690—21no—
55#4689—21no—
56C3×#333C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#332C2211yescharacteristic
59C3×#332C2211yescharacteristic
60#4482C2211yescharacteristic
61#4482C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3252—31no—
69#3239—31no—
70#3139—31no—
71#2969—31no—
72#1704—31no—
73#1703—31no—
74#1700—31no—
75#1699—31no—
76#1684—31no—
77#1683—31no—
78#1680—31no—
79#1679—31no—
80#1569—31no—
81#1554D311yescharacteristic, complemented
82#1549—31no—
83C22×Dic6—21no—
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#125—21no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8—21no—
89C3×C4×Q8—21no—
90C3×C4×Q8—21no—
91C3×#117—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#216—21no—
94C2×#215—21no—
95#930D411yescharacteristic
96#929—21no—
97C4×#67—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100#924D411yescharacteristic
101C2×#208—21no—
102C2×#208D411yescharacteristic
103C2×#208D411yescharacteristic
104C2×ΓL(1,𝔽25)—21no—
105C2×D3×C8—21no—
106#876—21no—
107#875D411yescharacteristic
108C4×D12C2311yescharacteristic
109#872—21no—
110#870D411yescharacteristic
111#869—21no—
112#869—21no—
113#869—21no—
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C4×Dic6C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C22⋊C8—21no—
122C3×C22⋊C8—21no—
123C3×#97—21no—
124C3×C4×C8—21no—
125#835—21no—
126#835—21no—
127#821—21no—
128#821—21no—
129(C2×D3)⋊C8—21no—
130(C2×D3)⋊C8—21no—
131#818—21no—
132#818—21no—
133#817—21no—
134#817—21no—
135#816—21no—
136#816—21no—
137#815—21no—
138Dic3⋊C8—21no—
139Dic3⋊C8—21no—
140C8×Dic3—21no—
141#810—21no—
142#810—21no—
143#810—21no—
144#810—21no—
145#810—21no—
146#810—21no—
147#810—21no—
148#810—21no—
149#810—21no—
150#809—21no—
151#808—21no—
152#807—21no—
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156#803D411yescharacteristic
157#803D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533C2×D311yescharacteristic
161C2×C4⋊Q8—31no—
162#529—31no—
163#501—61no—
164#491—61no—
165#491—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#487—31no—
170#487—31no—
171#484—61no—
172#481—61no—
173#479—61no—
174#477—31no—
175#476—31no—
176#475—31no—
177#475—31no—
178#474—31no—
179#474—31no—
180#470—61no—
181#468—61no—
182#464—61no—
183#462—31no—
184#461—31no—
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432C2×D311yescharacteristic
193#431C2×D311yescharacteristic
194#423—31no—
195#419—61no—
196#419—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333C2×D311yescharacteristic
204#333C2×D311yescharacteristic
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332C2×D311yescharacteristic
210#332C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4#4111yescharacteristic
216C2×#67—21no—
217C22×Dic3—41no—
218C2×D12C2×D411yescharacteristic
219C2×C4×D3—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—42no—
223C2×Dic6—42no—
224C2×Dic6—41no—
225C2×Dic6C2×D411yescharacteristic
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4—41no—
236C3×C22⋊C4—41no—
237C3×C42—41no—
238C3×C42C2×D411yescharacteristic
239C3×C42C2×D411yescharacteristic
240C3×C42#4111yescharacteristic
241C3×C42C2411yescharacteristic
242#217—21no—
243#216—81no—
244#215—81no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—41no—
254C4⋊Dic3—41no—
255C4⋊Dic3C2×D411yescharacteristic
256C4⋊Dic3C2×D411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261C4×Dic3—21no—
262#208—41no—
263#208—41no—
264#208—81no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8C2×D411yescharacteristic
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271ΓL(1,𝔽25)—42no—
272D3×C8—42no—
273C22×Q8—61no—
274C2×Q16—61no—
275C2×SD16—61no—
276C2×M16—31no—
277C2×M16—31no—
278C2×M16—61no—
279C2×M16—61no—
280C22×C8—61no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283C4⋊Q8—61no—
284C4⋊Q8—61no—
285C4⋊Q8—61no—
286C4⋊Q8C22×D311yescharacteristic
287C4⋊Q8C22×D311yescharacteristic
288#126—61no—
289#125—21no—
290#125—31no—
291#124—31no—
292#123—61no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—31no—
296#121—31no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—31no—
300C4×D4—31no—
301C4×D4—61no—
302#117—21no—
303#117—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C42C22×D311yescharacteristic
307#107—61no—
308#107—61no—
309#107—61no—
310#106—61no—
311#106—61no—
312#106—61no—
313C4⋊C8—31no—
314C4⋊C8—31no—
315C4⋊C8—61no—
316C4⋊C8—61no—
317C4⋊C8—61no—
318C4⋊C8C22×D311yescharacteristic
319C4⋊C8C22×D311yescharacteristic
320C4⋊C8C22×D311yescharacteristic
321C4⋊C8C22×D311yescharacteristic
322#103—61no—
323#103—61no—
324#103—61no—
325#103—61no—
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#102—61no—
336#102—61no—
337#102—61no—
338C22⋊C8—21no—
339C22⋊C8—21no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342#97—21no—
343#97—31no—
344#97—31no—
345#97—61no—
346C4×C8—21no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—61no—
350C23×C3#12111yescharacteristic
351C22×D3—21no—
352C3×Q8—41no—
353C3×Q8—41no—
354C3×Q8—81no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4#12111yescharacteristic
363C2×C3×C4#12111yescharacteristic
364C2×C3×C4#12111yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368#67—81no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—82no—
375D12—21no—
376D12—21no—
377C4×D3—41no—
378C4×D3—41no—
379Dic6—21no—
380Dic6—21no—
381Dic6—41no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—82no—
386C3×C8—41no—
387C3×C8—41no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—21no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—31no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—122no—
402C2×Q8—122no—
403C2×Q8—121no—
404C2×D4—31no—
405C2×D4—61no—
406C22×C4—61no—
407C22×C4—121no—
408C22×C4#23511yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411Q16—241no—
412SD16—241no—
413M16—122no—
414M16—121no—
415M16—121no—
416M16—241no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—31no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—122no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—31no—
433C4⋊C4—31no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C22⋊C4—41no—
447C22⋊C4—61no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—61no—
452C42—41no—
453C42—61no—
454C42#23511yescharacteristic
455C42D3×D411yescharacteristic
456C42D3×D411yescharacteristic
457C42C23×D311yescharacteristic
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3C2×#12111yescharacteristic
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4#57511yescharacteristic
472C3×C4#57511yescharacteristic
473C3×C4#57711yescharacteristic
474C3×C4#57811yescharacteristic
475Dic3—41no—
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—61no—
481C23#88411yescharacteristic
482Q8—41no—
483Q8—41no—
484Q8—61no—
485Q8—61no—
486Q8—81no—
487Q8—121no—
488Q8—121no—
489Q8—121no—
490Q8—121no—
491Q8—242no—
492D4—61no—
493D4—61no—
494D4—241no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—242no—
510C2×C4#88411yescharacteristic
511C2×C4#88411yescharacteristic
512C2×C4#88411yescharacteristic
513C2×C4C2×#23511yescharacteristic
514C2×C4C2×D3×D411yescharacteristic
515C2×C4C2×D3×D411yescharacteristic
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—61no—
521C8—61no—
522C8—61no—
523C8—61no—
524C8—121no—
525C8—121no—
526C2×C3—41no—
527C2×C3#304011yescharacteristic
528C2×C3#306311yescharacteristic
529C2×C3#306911yescharacteristic
530D3—81no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—121no—
535C22—121no—
536C22—121no—
537C22C2×#88411yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—81no—
543C4—81no—
544C4—121no—
545C4—121no—
546C4—121no—
547C4—241no—
548C4—241no—
549C4#577011yescharacteristic
550C4#577111yescharacteristic
551C4#577711yescharacteristic
552C4#577711yescharacteristic
553C3#1938111yescharacteristic, complemented, Sylow
554C2—41no—
555C2—241no—
556C2#7872111yescharacteristic
557C2#7884111yescharacteristic
558C2#7897511yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1554C2
socleC22×C3C2×#121
prime radical#930D4
hypercenter#1554D3
hypocenterC3#19381
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1554

Complements

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

NH ≅ G/Ntype
2. #83337C2semidirect
5. C3×#1554C2semidirect
6. #70795C2semidirect
7. #70780C2semidirect
8. #68515C2semidirect
9. #68511C2semidirect
10. #68503C2semidirect
11. #68499C2semidirect
12. #68477C2semidirect
13. #68469C2semidirect
14. #68453C2semidirect
15. #68445C2semidirect
81. #1554D3semidirect
553. C3#19381semidirect

Maximal subgroups

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

subgroupindex
#833652
#833372
#793232
#792802
C3×#15542
#707952
#707802
#685152
#685112
#685032
#684992
#684772
#684692
#684532
#684452
#193813