Finite Group Databasebeta

← #10778147

Subgroups — #10778147

Order 768.

616 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81936C211yesmaximal, characteristic, complemented
2#81059C211yesmaximal, characteristic, complemented
3#81045C211yesmaximal, characteristic
4#79404C211yesmaximal, characteristic, complemented
5C3×#1675C211yesmaximal, characteristic, complemented
6#71380C211yesmaximal, characteristic, complemented
7#71135C211yesmaximal, characteristic, complemented
8#71134C211yesmaximal, characteristic, complemented
9#70846C211yesmaximal, characteristic, complemented
10#68592C211yesmaximal, characteristic, complemented
11#68568C211yesmaximal, characteristic, complemented
12#68515C211yesmaximal, characteristic, complemented
13#68514C211yesmaximal, characteristic, complemented
14#68486C211yesmaximal, characteristic, complemented
15#68485C211yesmaximal, characteristic, complemented
16#19179—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5725C2211yescharacteristic
19C4×#237C2211yescharacteristic
20C3×#500C2211yescharacteristic
21C3×#476C2211yescharacteristic
22C3×#476C2211yescharacteristic
23C3×C8×D4C2211yescharacteristic
24#5111—21no—
25C2×#815—21no—
26#5093C2211yescharacteristic
27#5091C2211yescharacteristic
28#5091C2211yescharacteristic
29#5088C2211yescharacteristic
30#5015C2211yescharacteristic
31D4×C3⋊C8C2211yescharacteristic
32#5007C2211yescharacteristic
33#4889—21no—
34#4889—21no—
35#4886—21no—
36#4886—21no—
37#4866—21no—
38#4860—21no—
39#4857—21no—
40#4852—21no—
41#4851C2211yescharacteristic
42#4848C2211yescharacteristic
43#4835C2211yescharacteristic
44#4833C2211yescharacteristic
45D3×C4⋊C8C2211yescharacteristic
46#4831C2211yescharacteristic
47#4781—21no—
48#4772—21no—
49#4763—21no—
50#4756—21no—
51#4730—21no—
52#4719—21no—
53#4693C2211yescharacteristic
54C4×ΓL(1,𝔽25)C2211yescharacteristic
55#4679C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#333C2211yescharacteristic
59#4546C2211yescharacteristic
60#4546C2211yescharacteristic
61#4486C2211yescharacteristic
62#4485C2211yescharacteristic
63#4485C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4457C2211yescharacteristic
68#3182—31no—
69#3106—31no—
70#3076—31no—
71#2970—31no—
72#1704—31no—
73#1698—31no—
74#1696—31no—
75#1686—31no—
76#1683—31no—
77#1677—31no—
78#1675D311yescharacteristic, complemented
79#1665—31no—
80#1565—31no—
81#1554—31no—
82#1551—31no—
83C2×#239—21no—
84C2×#237D411yescharacteristic
85C22×C3×C8—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4×D4C2311yescharacteristic
89#947—21no—
90#946—21no—
91Q8×Dic3—21no—
92D4×Dic3D411yescharacteristic
93C2×#213—21no—
94C4×#67—21no—
95C2×C4×Dic3—21no—
96C22×C3⋊C8—21no—
97C2×#204—21no—
98C2×#204—21no—
99C2×#204—21no—
100C2×ΓL(1,𝔽25)D411yescharacteristic
101C2×ΓL(1,𝔽25)D411yescharacteristic
102C2×D3×C8—21no—
103#897—21no—
104#895—21no—
105#895—21no—
106#893—21no—
107#893—21no—
108#892—21no—
109#892D411yescharacteristic
110#890—21no—
111#887—21no—
112#887D411yescharacteristic
113#67⋊C4—21no—
114#877—21no—
115#874C2311yescharacteristic
116C42×D3C2311yescharacteristic
117#869C2311yescharacteristic
118C4×Dic6C2311yescharacteristic
119C3×#107—21no—
120C3×#107—21no—
121C3×#106—21no—
122C3×#106—21no—
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×C4⋊C8C2311yescharacteristic
126C3×#102—21no—
127C3×#102—21no—
128C3×C22⋊C8—21no—
129C3×C4×C8C2311yescharacteristic
130#832—21no—
131#830—21no—
132#821—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135#817—21no—
136#816—21no—
137#815—21no—
138#815—21no—
139#815—21no—
140#815D411yescharacteristic
141#815D411yescharacteristic
142Dic3⋊C8—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145C8×Dic3—21no—
146#810—21no—
147#810—21no—
148#810—21no—
149#810—21no—
150#809—21no—
151#809—21no—
152#809—21no—
153#807—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#532—31no—
162C4×#41—31no—
163#500C2×D311yescharacteristic
164#492—31no—
165#487—31no—
166#486—31no—
167#482—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#477—31no—
172#476C2×D311yescharacteristic
173#476C2×D311yescharacteristic
174#475—31no—
175#474—31no—
176#468—61no—
177#468—61no—
178#460—31no—
179#459—31no—
180#449—61no—
181#446—31no—
182#446—31no—
183#440—61no—
184#440—61no—
185C4×SD16—61no—
186C4×D8—61no—
187#435—61no—
188#435—61no—
189C8×D4—31no—
190C8×D4C2×D311yescharacteristic
191#432—31no—
192#432—61no—
193#431—31no—
194#427—61no—
195C2×C4⋊C8—31no—
196#416—61no—
197C4×M16—31no—
198C2×#97—61no—
199#335—31no—
200#335C2×D311yescharacteristic
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333C2×D311yescharacteristic
206#333C2×D311yescharacteristic
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
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#239—82no—
219#237—22no—
220#237—41no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—41no—
225C2×C4×D3—41no—
226C2×C4×D3C2×D411yescharacteristic
227C2×C4×D3C2×D411yescharacteristic
228C2×C4×D3#4111yescharacteristic
229C2×Dic6—21no—
230C2×Dic6C2×D411yescharacteristic
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—42no—
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4C2×D411yescharacteristic
242C3×C22⋊C4—21no—
243C3×C42C2411yescharacteristic
244#217—21no—
245#213—42no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—41no—
248(C2×D3)⋊C4—41no—
249(C2×D3)⋊C4—41no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—41no—
253C4⋊Dic3C2×D411yescharacteristic
254Dic3⋊C4—21no—
255Dic3⋊C4—21no—
256Dic3⋊C4—41no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—41no—
261C4×Dic3C2×D411yescharacteristic
262C4×Dic3C2×D411yescharacteristic
263C4×Dic3#4111yescharacteristic
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—42no—
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270#204—42no—
271#204—42no—
272#204—42no—
273ΓL(1,𝔽25)—22no—
274ΓL(1,𝔽25)—41no—
275D3×C8—42no—
276C2×#41—31no—
277C2×#41—61no—
278C2×SD16—61no—
279C2×SD16—61no—
280C2×SD16—61no—
281C2×D8—61no—
282C2×M16—31no—
283C2×M16—31no—
284C22×C8—21no—
285C22×C8—61no—
286C22×C8—61no—
287C4⋊Q8—31no—
288C4⋊Q8C22×D311yescharacteristic
289C4⋊Q8C22×D311yescharacteristic
290#127—31no—
291#125—61no—
292#124—61no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295#121—61no—
296#121—61no—
297C4×Q8—31no—
298C4×Q8—31no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×D4—31no—
302C4×D4—61no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4C22×D311yescharacteristic
307#117—31no—
308#117—61no—
309#117—61no—
310C2×C42—31no—
311C2×C42—61no—
312#107—21no—
313#107—21no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—21no—
318#106—21no—
319#106—61no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—21no—
335#102—21no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341C22⋊C8—21no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—31no—
346#97—31no—
347#97—61no—
348#97—61no—
349#97—61no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8C22×D311yescharacteristic
353C23×C3—21no—
354C22×D3—41no—
355C22×D3#12111yescharacteristic
356C3×Q8—81no—
357C3×Q8—81no—
358C3×D4—21no—
359C3×D4#13712yes—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368#67—42no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—42no—
372C2×Dic3—41no—
373C2×Dic3—81no—
374C2×Dic3#12111yescharacteristic
375C2×Dic3#12111yescharacteristic
376C2×Dic3#12111yescharacteristic
377D12—41no—
378D12—41no—
379D12—41no—
380D12—41no—
381D12—81no—
382D12—82no—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—21no—
386C4×D3—21no—
387C4×D3—41no—
388C4×D3—82no—
389Dic6—21no—
390Dic6—21no—
391Dic6—41no—
392Dic6—41no—
393C3×C8—21no—
394C3×C8—41no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8#13512yes—
398C3⋊C8—21no—
399C3⋊C8—21no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403#41—62no—
404#41—121no—
405#41—242no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—31no—
409C2×Q8—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—121no—
415C2×D4D3×D411yescharacteristic
416C22×C4—21no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—121no—
423SD16—122no—
424SD16—122no—
425SD16—122no—
426D8—122no—
427M16—62no—
428M16—121no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—42no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—122no—
439C2×C8—122no—
440C2×C8D3×D411yescharacteristic
441C2×C8D3×D411yescharacteristic
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—31no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—121no—
452C4⋊C4—121no—
453C4⋊C4D3×D411yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C22⋊C4—121no—
460C42—31no—
461C42—31no—
462C42—31no—
463C42—61no—
464C42—61no—
465C42—61no—
466C42—121no—
467C42C23×D311yescharacteristic
468C22×C3—22no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3C2×#12111yescharacteristic
472C2×D3—21no—
473C2×D3—21no—
474C2×D3—21no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—21no—
479C3×C4—41no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4C2×#13511yescharacteristic
483C3×C4C2×#13711yescharacteristic
484C3×C4#57511yescharacteristic
485C3×C4#57511yescharacteristic
486Dic3—21no—
487Dic3—21no—
488Dic3—21no—
489Dic3—41no—
490Dic3—41no—
491Dic3—81no—
492C23—21no—
493C23—31no—
494C23—121no—
495Q8—61no—
496Q8—61no—
497Q8—81no—
498Q8—81no—
499Q8—121no—
500Q8—121no—
501D4—21no—
502D4—122no—
503D4—121no—
504D4—121no—
505D4—121no—
506D4—121no—
507D4—241no—
508D4—242no—
509D4#91812yes—
510C2×C4—21no—
511C2×C4—31no—
512C2×C4—31no—
513C2×C4—31no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—121no—
527C2×C4—242no—
528C2×C4—241no—
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×D3×D411yescharacteristic
531C2×C4D3×#4111yescharacteristic
532C8—21no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—61no—
537C8—61no—
538C8—121no—
539C8—121no—
540C8—121no—
541C8#91612yes—
542C2×C3—42no—
543C2×C3#303511yescharacteristic
544C2×C3#305611yescharacteristic
545C2×C3#306811yescharacteristic
546D3—41no—
547D3—161no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—61no—
552C22—61no—
553C22—61no—
554C22—241no—
555C22—241no—
556C22—241no—
557C22D3×#12111yescharacteristic
558C4—21no—
559C4—41no—
560C4—61no—
561C4—61no—
562C4—61no—
563C4—81no—
564C4—81no—
565C4—121no—
566C4—121no—
567C4—241no—
568C4C2×#91611yescharacteristic
569C4C2×#91811yescharacteristic
570C4#576811yescharacteristic
571C4#577711yescharacteristic
572C3#1917911yescharacteristic, complemented, Sylow
573C2—42no—
574C2—121no—
575C2—481no—
576C2#8042711yescharacteristic
577C2#8075911yescharacteristic
578C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1675C2
socleC22×C3C2×#121
prime radical#81045C2
hypercenter#1675D3
hypocenterC3#19179
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1675

Complements

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

NH ≅ G/Ntype
1. #81936C2semidirect
2. #81059C2semidirect
4. #79404C2semidirect
5. C3×#1675C2semidirect
6. #71380C2semidirect
7. #71135C2semidirect
8. #71134C2semidirect
9. #70846C2semidirect
10. #68592C2semidirect
11. #68568C2semidirect
12. #68515C2semidirect
13. #68514C2semidirect
14. #68486C2semidirect
15. #68485C2semidirect
78. #1675D3semidirect
572. C3#19179semidirect

Maximal subgroups

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

subgroupindex
#819362
#810592
#810452
#794042
C3×#16752
#713802
#711352
#711342
#708462
#685922
#685682
#685152
#685142
#684862
#684852
#191793