Finite Group Databasebeta

← #10778243

Subgroups — #10778243

Order 768.

592 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81596C211yesmaximal, characteristic, complemented
2#81059C211yesmaximal, characteristic
3#80949C211yesmaximal, characteristic
4#79522C211yesmaximal, characteristic
5C3×#1703C211yesmaximal, characteristic, complemented
6#71241C211yesmaximal, characteristic, complemented
7#71136C211yesmaximal, characteristic, complemented
8#71105C211yesmaximal, characteristic, complemented
9#70875C211yesmaximal, characteristic, complemented
10#68549C211yesmaximal, characteristic, complemented
11#68528C211yesmaximal, characteristic, complemented
12#68508C211yesmaximal, characteristic, complemented
13#68486C211yesmaximal, characteristic, complemented
14#68465C211yesmaximal, characteristic, complemented
15#68424C211yesmaximal, characteristic, complemented
16#19185—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5728C2211yescharacteristic
19C4×#237C2211yescharacteristic
20C3×#487C2211yescharacteristic
21C3×#476C2211yescharacteristic
22C3×#474C2211yescharacteristic
23C3×#436C2211yescharacteristic
24#5128—21no—
25Dic3×M16—21no—
26#5094C2211yescharacteristic
27#5092C2211yescharacteristic
28#5091C2211yescharacteristic
29#5080C2211yescharacteristic
30#5019C2211yescharacteristic
31#5012C2211yescharacteristic
32#5007C2211yescharacteristic
33#4888—21no—
34#4873—21no—
35#4860—21no—
36#4857—21no—
37#4851C2211yescharacteristic
38#4849C2211yescharacteristic
39#4835C2211yescharacteristic
40#4833C2211yescharacteristic
41D3×C4⋊C8C2211yescharacteristic
42#4830C2211yescharacteristic
43#4816—21no—
44#4816—21no—
45#4789—21no—
46#4789—21no—
47#4783—21no—
48#4772—21no—
49#4763—21no—
50#4758—21no—
51#4729—21no—
52#4719—21no—
53C4×ΓL(1,𝔽25)C2211yescharacteristic
54#4681C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56C3×#334C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4546C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4482C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3185—31no—
69#3107—31no—
70#3076—31no—
71#2971—31no—
72#1703D311yescharacteristic, complemented
73#1699—31no—
74#1691—31no—
75#1689—31no—
76#1684—31no—
77#1680—31no—
78#1672—31no—
79#1670—31no—
80#1565—31no—
81#1554—31no—
82#1552—31no—
83C2×#237D411yescharacteristic
84C2×C3×M16—21no—
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4×D4C2311yescharacteristic
88#946—21no—
89#946—21no—
90Q8×Dic3—21no—
91Q8×Dic3—21no—
92C2×#216—21no—
93C2×#216—21no—
94D4×Dic3D411yescharacteristic
95C2×#214—21no—
96C4×#67—21no—
97C2×C4×Dic3—21no—
98C2×#208—21no—
99C2×Dic12—21no—
100C2×ΓL(1,𝔽25)D411yescharacteristic
101C2×ΓL(1,𝔽25)D411yescharacteristic
102C2×D3×C8—21no—
103#897—21no—
104#897—21no—
105#892—21no—
106#892—21no—
107#892D411yescharacteristic
108#890—21no—
109#890—21no—
110#887—21no—
111#887—21no—
112#887D411yescharacteristic
113#67⋊C4—21no—
114#877—21no—
115C42×D3C2311yescharacteristic
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118C4×Dic6C2311yescharacteristic
119C3×#107—21no—
120C3×#106—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×#103—21no—
125C3×#103—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×C22⋊C8—21no—
129C3×C4×C8C2311yescharacteristic
130#832—21no—
131#830—21no—
132(C2×D3)⋊C8—21no—
133(C2×D3)⋊C8—21no—
134#817—21no—
135#816—21no—
136#816—21no—
137#816—21no—
138#816—21no—
139#815—21no—
140#815D411yescharacteristic
141#815D411yescharacteristic
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145C8×Dic3—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148#810—21no—
149#810—21no—
150#810—21no—
151#808—21no—
152#808—21no—
153#808—21no—
154#807—21no—
155#807—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161#533—31no—
162C4×#41—31no—
163#498—31no—
164#494—31no—
165#487—31no—
166#487C2×D311yescharacteristic
167#485—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#476C2×D311yescharacteristic
172#475—31no—
173#475—31no—
174#474—31no—
175#474C2×D311yescharacteristic
176#470—61no—
177#470—61no—
178#462—31no—
179#461—31no—
180#452—61no—
181C8×Q8—31no—
182C8×Q8—31no—
183#441—61no—
184#441—61no—
185C4×Q16—61no—
186C4×SD16—61no—
187#436—31no—
188#436—61no—
189#436—61no—
190#436C2×D311yescharacteristic
191#432—31no—
192#432—61no—
193#431—31no—
194#427—61no—
195C2×C4⋊C8—31no—
196#416—61no—
197C4×M16—31no—
198C4×M16—61no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334C2×D311yescharacteristic
203#333—31no—
204#333C2×D311yescharacteristic
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#332C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C2×#67—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218#237—22no—
219#237—41no—
220C2×C4×D3C2×D411yescharacteristic
221C2×C4×D3C2×D411yescharacteristic
222C2×C4×D3#4111yescharacteristic
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C2×D411yescharacteristic
226C3×M16—42no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8C2×D411yescharacteristic
231C2×C3×C8C2×D411yescharacteristic
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4C2×D411yescharacteristic
237C3×C22⋊C4—21no—
238C3×C42C2411yescharacteristic
239#217—21no—
240#216—42no—
241#216—42no—
242#214—42no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—41no—
245(C2×D3)⋊C4—41no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—41no—
251C4⋊Dic3—41no—
252C4⋊Dic3C2×D411yescharacteristic
253Dic3⋊C4—21no—
254Dic3⋊C4—21no—
255Dic3⋊C4—41no—
256Dic3⋊C4—41no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—41no—
261C4×Dic3—41no—
262C4×Dic3C2×D411yescharacteristic
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3#4111yescharacteristic
265#208—42no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271Dic12—42no—
272ΓL(1,𝔽25)—22no—
273ΓL(1,𝔽25)—41no—
274D3×C8—42no—
275C2×#41—31no—
276C2×Q16—61no—
277C2×Q16—61no—
278C2×Q16—61no—
279C2×SD16—61no—
280C2×M16—21no—
281C2×M16—31no—
282C2×M16—31no—
283C2×M16—61no—
284C22×C8—61no—
285C4⋊Q8—31no—
286C4⋊Q8—31no—
287C4⋊Q8C22×D311yescharacteristic
288C4⋊Q8C22×D311yescharacteristic
289#125—61no—
290#125—61no—
291C22⋊Q8—61no—
292C22⋊Q8—61no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8—61no—
298C4×Q8—61no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×D4—31no—
302C4×D4—61no—
303C4×D4—61no—
304C4×D4C22×D311yescharacteristic
305#117—31no—
306#117—61no—
307#117—61no—
308#117—61no—
309C2×C42—31no—
310C2×C42—61no—
311#107—21no—
312#107—61no—
313#107—61no—
314#106—21no—
315#106—61no—
316#106—61no—
317#106—61no—
318#106—61no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—21no—
329#103—21no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—21no—
338#102—21no—
339#102—61no—
340C22⋊C8—21no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—31no—
345#97—31no—
346#97—61no—
347C4×C8—31no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8—61no—
351C4×C8C22×D311yescharacteristic
352C23×C3—21no—
353C22×D3#12111yescharacteristic
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—81no—
357C3×D4—21no—
358C3×D4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367#67—42no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—42no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—81no—
374C2×Dic3#12111yescharacteristic
375C2×Dic3#12111yescharacteristic
376C2×Dic3#12111yescharacteristic
377C4×D3—21no—
378C4×D3—21no—
379C4×D3—21no—
380C4×D3—21no—
381C4×D3—41no—
382Dic6—21no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386Dic6—41no—
387Dic6#13712yes—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C3⋊C8—21no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8#13512yes—
398#41—62no—
399#41—121no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—31no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×D4—61no—
406C2×D4D3×D411yescharacteristic
407C22×C4—21no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—61no—
412C22×C4—61no—
413Q16—122no—
414Q16—122no—
415Q16—122no—
416SD16—122no—
417M16—42no—
418M16—62no—
419M16—121no—
420M16—122no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—122no—
430C2×C8D3×D411yescharacteristic
431C2×C8D3×D411yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—31no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—121no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C4⋊C4—121no—
447C4⋊C4D3×D411yescharacteristic
448C22⋊C4—21no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—121no—
452C22⋊C4—121no—
453C42—31no—
454C42—31no—
455C42—31no—
456C42—61no—
457C42—61no—
458C42—61no—
459C42—121no—
460C42—121no—
461C42C23×D311yescharacteristic
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3C2×#12111yescharacteristic
466C2×D3—21no—
467C2×D3—21no—
468C2×D3—21no—
469C3×C4—21no—
470C3×C4—41no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4C2×#13511yescharacteristic
474C3×C4C2×#13711yescharacteristic
475C3×C4#57511yescharacteristic
476C3×C4#57511yescharacteristic
477Dic3—21no—
478Dic3—22no—
479Dic3—21no—
480Dic3—21no—
481Dic3—41no—
482Dic3—81no—
483Dic3—81no—
484C23—21no—
485C23—31no—
486Q8—32no—
487Q8—41no—
488Q8—41no—
489Q8—61no—
490Q8—81no—
491Q8—121no—
492Q8—121no—
493Q8—121no—
494Q8—121no—
495D4—21no—
496D4—21no—
497D4—122no—
498C2×C4—21no—
499C2×C4—31no—
500C2×C4—31no—
501C2×C4—31no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—121no—
513C2×C4—122no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—241no—
517C2×C4C2×D3×D411yescharacteristic
518C2×C4C2×D3×D411yescharacteristic
519C2×C4D3×#4111yescharacteristic
520C8—21no—
521C8—21no—
522C8—32no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—61no—
527C8—121no—
528C8—121no—
529C8—121no—
530C2×C3—81no—
531C2×C3#303511yescharacteristic
532C2×C3#305611yescharacteristic
533C2×C3#306911yescharacteristic
534D3—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—61no—
539C22—61no—
540C22—61no—
541C22D3×#12111yescharacteristic
542C4—21no—
543C4—41no—
544C4—61no—
545C4—62no—
546C4—61no—
547C4—61no—
548C4—81no—
549C4—81no—
550C4—121no—
551C4—241no—
552C4—241no—
553C4D3×#13511yescharacteristic
554C4#577311yescharacteristic
555C4D3×#13711yescharacteristic
556C4#577711yescharacteristic
557C3#1918511yescharacteristic, complemented, Sylow
558C2—81no—
559C2—121no—
560C2#8053311yescharacteristic
561C2#8090411yescharacteristic
562C2#8093111yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1703C2
socleC22×C3C2×#121
prime radicalC2×#237D4
hypercenter#1703D3
hypocenterC3#19185
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1703

Complements

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

NH ≅ G/Ntype
1. #81596C2semidirect
5. C3×#1703C2semidirect
6. #71241C2semidirect
7. #71136C2semidirect
8. #71105C2semidirect
9. #70875C2semidirect
10. #68549C2semidirect
11. #68528C2semidirect
12. #68508C2semidirect
13. #68486C2semidirect
14. #68465C2semidirect
15. #68424C2semidirect
72. #1703D3semidirect
557. C3#19185semidirect

Maximal subgroups

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

subgroupindex
#815962
#810592
#809492
#795222
C3×#17032
#712412
#711362
#711052
#708752
#685492
#685282
#685082
#684862
#684652
#684242
#191853