Finite Group Databasebeta

← #10777913

Subgroups — #10777913

Order 768.

620 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81752C211yesmaximal, characteristic, complemented
2#80997C211yesmaximal, characteristic, complemented
3#80361C211yesmaximal, characteristic
4#79938C211yesmaximal, characteristic, complemented
5C3×#1686C211yesmaximal, characteristic, complemented
6#71268C211yesmaximal, characteristic, complemented
7#71123C211yesmaximal, characteristic, complemented
8#71027C211yesmaximal, characteristic, complemented
9#70963C211yesmaximal, characteristic, complemented
10#68583C211yesmaximal, characteristic, complemented
11#68567C211yesmaximal, characteristic, complemented
12#68455C211yesmaximal, characteristic, complemented
13#68419C211yesmaximal, characteristic, complemented
14#68401C211yesmaximal, characteristic, complemented
15#68382C211yesmaximal, characteristic, complemented
16#19159—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5716C2211yescharacteristic
19C4×#239C2211yescharacteristic
20C3×#492C2211yescharacteristic
21C3×#475C2211yescharacteristic
22C3×#460C2211yescharacteristic
23C3×#446C2211yescharacteristic
24#5172—21no—
25#5171—21no—
26#5171—21no—
27Dic3×SD16—21no—
28Dic3×SD16—21no—
29Dic3×SD16—21no—
30#5087C2211yescharacteristic
31#5079C2211yescharacteristic
32#5072C2211yescharacteristic
33#5070C2211yescharacteristic
34#5031C2211yescharacteristic
35#5024C2211yescharacteristic
36#5021C2211yescharacteristic
37#4868—21no—
38#4852—21no—
39#4842C2211yescharacteristic
40#4841C2211yescharacteristic
41#4838C2211yescharacteristic
42#4836—21no—
43#4833—21no—
44#4833C2211yescharacteristic
45#4833C2211yescharacteristic
46#4833C2211yescharacteristic
47#4808—21no—
48#4790—21no—
49#4779—21no—
50#4756—21no—
51#4707—21no—
52#4706—21no—
53#4688C2211yescharacteristic
54C4×D3×C8C2211yescharacteristic
55#4680C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#331C2211yescharacteristic
59#4547C2211yescharacteristic
60#4544C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4480C2211yescharacteristic
67#4458C2211yescharacteristic
68#3167—31no—
69#3107—31no—
70#3106—31no—
71#2965—31no—
72#1704—31no—
73#1703—31no—
74#1702—31no—
75#1701—31no—
76#1692—31no—
77#1691—31no—
78#1686D311yescharacteristic, complemented
79#1685—31no—
80#1564—31no—
81#1544—31no—
82#1543—31no—
83C2×#239D411yescharacteristic
84C2×#237—21no—
85C2×C3×SD16—21no—
86C2×C3×SD16—21no—
87C2×C3×SD16—21no—
88C3×C4⋊Q8C2311yescharacteristic
89C3×#127C2311yescharacteristic
90C3×C4×Q8C2311yescharacteristic
91#946—21no—
92Q8×Dic3—21no—
93Q8×Dic3D411yescharacteristic
94C2×#216—21no—
95#938—21no—
96#938—21no—
97#936—21no—
98D4×Dic3—21no—
99D4×Dic3—21no—
100C2×#213—21no—
101C2×#204—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×ΓL(1,𝔽25)—21no—
104C2×D3×C8D411yescharacteristic
105C2×D3×C8D411yescharacteristic
106#897—21no—
107#893—21no—
108#893D411yescharacteristic
109#892—21no—
110#892—21no—
111#892D411yescharacteristic
112#890—21no—
113#887—21no—
114C4×D12—21no—
115C4×D12C2311yescharacteristic
116C42×D3—21no—
117C42×D3C2311yescharacteristic
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C3×#106—21no—
121C3×C4⋊C8—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×#103—21no—
126C3×#102—21no—
127C3×#97—21no—
128C3×C4×C8C2311yescharacteristic
129#835—21no—
130#835—21no—
131#835—21no—
132#832—21no—
133#832—21no—
134#832—21no—
135#821—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141#817—21no—
142#817—21no—
143#817—21no—
144#816—21no—
145C8×Dic3—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148C8×Dic3D411yescharacteristic
149C8×Dic3D411yescharacteristic
150#810—21no—
151#809—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158#803—21no—
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161#532—31no—
162C4×#41—31no—
163#498—31no—
164#492C2×D311yescharacteristic
165#487—31no—
166#486—31no—
167#485—61no—
168#482—61no—
169#477—31no—
170#476—31no—
171#475C2×D311yescharacteristic
172#474—31no—
173#465—61no—
174#465—61no—
175#462—31no—
176#461—31no—
177#460C2×D311yescharacteristic
178#459—31no—
179#452—61no—
180#449—61no—
181#446—31no—
182#446C2×D311yescharacteristic
183C4×Q16—61no—
184C4×SD16—61no—
185C4×SD16—61no—
186C4×SD16—61no—
187C4×SD16—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#435—61no—
192#435—61no—
193#431—31no—
194#431—31no—
195#431—31no—
196#431—61no—
197C4×M16—61no—
198C2×C4×C8—31no—
199#335—31no—
200#335C2×D311yescharacteristic
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332C2×D311yescharacteristic
209#331—31no—
210#331C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D411yescharacteristic
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×#67—41no—
217C22×Dic3—41no—
218#239—41no—
219#239—41no—
220#237—82no—
221C2×D12—21no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3C2×D411yescharacteristic
227C2×C4×D3C2×D411yescharacteristic
228C2×C4×D3#4111yescharacteristic
229C2×Dic6—21no—
230C2×Dic6—21no—
231C3×SD16—42no—
232C3×SD16—42no—
233C3×SD16—42no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2×D411yescharacteristic
243C3×C42—21no—
244C3×C42C2411yescharacteristic
245#217—41no—
246#217—41no—
247#216—42no—
248#213—42no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—41no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—41no—
259C4⋊Dic3C2×D411yescharacteristic
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—41no—
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3#4111yescharacteristic
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272#204—42no—
273ΓL(1,𝔽25)—42no—
274ΓL(1,𝔽25)—42no—
275D3×C8—22no—
276D3×C8—41no—
277C2×#41—31no—
278C2×#41—61no—
279C2×Q16—61no—
280C2×SD16—21no—
281C2×SD16—21no—
282C2×SD16—21no—
283C2×SD16—61no—
284C2×D8—61no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8—31no—
288C22×C8—31no—
289C4⋊Q8—31no—
290C4⋊Q8—31no—
291C4⋊Q8C22×D311yescharacteristic
292#127C22×D311yescharacteristic
293#125—61no—
294#124—61no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297#121—61no—
298#121—61no—
299C4×Q8—31no—
300C4×Q8—61no—
301C4×Q8—61no—
302C4×Q8C22×D311yescharacteristic
303C4×D4—31no—
304C4×D4—31no—
305C4×D4—61no—
306C4×D4—61no—
307C4×D4—61no—
308C4×D4—61no—
309#117—31no—
310#117—61no—
311#117—61no—
312C2×C42—31no—
313C2×C42—61no—
314#107—61no—
315#107—61no—
316#106—21no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—21no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—21no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—21no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—21no—
346#97—61no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—61no—
351C4×C8—61no—
352C4×C8—61no—
353C4×C8C22×D311yescharacteristic
354C23×C3—41no—
355C22×D3—21no—
356C22×D3#12111yescharacteristic
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—81no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—41no—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—82no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—82no—
377C2×Dic3#12111yescharacteristic
378C2×Dic3#12111yescharacteristic
379C2×Dic3#12111yescharacteristic
380D12—21no—
381D12—21no—
382D12—42no—
383C4×D3—21no—
384C4×D3—21no—
385C4×D3—21no—
386C4×D3—21no—
387C4×D3—22no—
388C4×D3—42no—
389C4×D3—41no—
390C4×D3—41no—
391C4×D3—42no—
392Dic6—41no—
393Dic6—41no—
394Dic6—81no—
395C3×C8—21no—
396C3×C8—41no—
397C3×C8—41no—
398C3×C8—41no—
399C3×C8#13512yes—
400C3⋊C8—21no—
401C3⋊C8—21no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405#41—121no—
406#41—121no—
407#41—242no—
408C2×Q8—21no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8D3×D411yescharacteristic
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—31no—
416C2×D4—61no—
417C2×D4—121no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—121no—
425Q16—122no—
426SD16—42no—
427SD16—42no—
428SD16—42no—
429SD16—122no—
430D8—122no—
431M16—122no—
432M16—122no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—31no—
437C2×C8—31no—
438C2×C8—62no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—121no—
443C2×C8D3×D411yescharacteristic
444C2×C8D3×D411yescharacteristic
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—31no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—121no—
455C4⋊C4—121no—
456C4⋊C4D3×D411yescharacteristic
457C22⋊C4—61no—
458C22⋊C4—61no—
459C22⋊C4—61no—
460C22⋊C4—121no—
461C22⋊C4—121no—
462C22⋊C4—121no—
463C42—21no—
464C42—31no—
465C42—31no—
466C42—31no—
467C42—61no—
468C42—61no—
469C42—121no—
470C42C23×D311yescharacteristic
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3C2×#12111yescharacteristic
475C2×D3—21no—
476C2×D3—21no—
477C2×D3—21no—
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—41no—
481C3×C4—21no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—81no—
485C3×C4C2×#13511yescharacteristic
486C3×C4#57511yescharacteristic
487C3×C4#57511yescharacteristic
488C3×C4#57511yescharacteristic
489Dic3—21no—
490Dic3—21no—
491Dic3—41no—
492Dic3—81no—
493Dic3—81no—
494Dic3#47112yes—
495C23—31no—
496C23—41no—
497C23—61no—
498Q8—21no—
499Q8—21no—
500Q8—41no—
501Q8—41no—
502Q8—121no—
503Q8—121no—
504Q8—241no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—61no—
510D4—61no—
511D4—81no—
512D4—122no—
513D4—242no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—31no—
517C2×C4—31no—
518C2×C4—31no—
519C2×C4—41no—
520C2×C4—62no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—242no—
533C2×C4C2×D3×D411yescharacteristic
534C2×C4C2×D3×D411yescharacteristic
535C2×C4D3×#4111yescharacteristic
536C8—21no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—61no—
541C8—61no—
542C8—121no—
543C8—121no—
544C8—121no—
545C8#91612yes—
546C2×C3—161no—
547C2×C3#303511yescharacteristic
548C2×C3#305411yescharacteristic
549C2×C3C2×#47111yescharacteristic
550D3—41no—
551D3—81no—
552C22—61no—
553C22—61no—
554C22—61no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—121no—
559C22—121no—
560C22—121no—
561C22D3×#12111yescharacteristic
562C4—21no—
563C4—32no—
564C4—41no—
565C4—41no—
566C4—61no—
567C4—61no—
568C4—81no—
569C4—121no—
570C4—241no—
571C4—241no—
572C4C2×#91611yescharacteristic
573C4#577311yescharacteristic
574C4#577311yescharacteristic
575C4#577711yescharacteristic
576C3#1915911yescharacteristic, complemented, Sylow
577C2—121no—
578C2—161no—
579C2—241no—
580C2#8038211yescharacteristic
581C2#8080411yescharacteristic
582C2D3×#47111yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1686C2
socleC22×C3C2×#121
prime radical#80361C2
hypercenter#1686D3
hypocenterC3#19159
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1686

Complements

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

NH ≅ G/Ntype
1. #81752C2semidirect
2. #80997C2semidirect
4. #79938C2semidirect
5. C3×#1686C2semidirect
6. #71268C2semidirect
7. #71123C2semidirect
8. #71027C2semidirect
9. #70963C2semidirect
10. #68583C2semidirect
11. #68567C2semidirect
12. #68455C2semidirect
13. #68419C2semidirect
14. #68401C2semidirect
15. #68382C2semidirect
78. #1686D3semidirect
576. C3#19159semidirect

Maximal subgroups

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

subgroupindex
#817522
#809972
#803612
#799382
C3×#16862
#712682
#711232
#710272
#709632
#685832
#685672
#684552
#684192
#684012
#683822
#191593