Finite Group Databasebeta

← #10778126

Subgroups — #10778126

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81754C211yesmaximal, characteristic, complemented
2#80997C211yesmaximal, characteristic, complemented
3#80382C211yesmaximal, characteristic
4#79943C211yesmaximal, characteristic, complemented
5C3×#1686C211yesmaximal, characteristic, complemented
6#71267C211yesmaximal, characteristic, complemented
7#71122C211yesmaximal, characteristic, complemented
8#71027C211yesmaximal, characteristic, complemented
9#70964C211yesmaximal, characteristic, complemented
10#68578C211yesmaximal, characteristic, complemented
11#68568C211yesmaximal, characteristic, complemented
12#68444C211yesmaximal, characteristic, complemented
13#68424C211yesmaximal, characteristic, complemented
14#68397C211yesmaximal, characteristic, complemented
15#68386C211yesmaximal, characteristic, complemented
16#19178—31nomaximal, Sylow
17C2×#916—21no—
18#5728C2211yescharacteristic
19#5716C2211yescharacteristic
20C4×#239C2211yescharacteristic
21C3×#492C2211yescharacteristic
22C3×#475C2211yescharacteristic
23C3×#460C2211yescharacteristic
24C3×#446C2211yescharacteristic
25#5174—21no—
26#5174—21no—
27#5166—21no—
28#5164—21no—
29#5164—21no—
30#5087C2211yescharacteristic
31#5079C2211yescharacteristic
32#5072C2211yescharacteristic
33#5070C2211yescharacteristic
34#5031C2211yescharacteristic
35#5031C2211yescharacteristic
36#5024C2211yescharacteristic
37#4868—21no—
38#4852—21no—
39#4841C2211yescharacteristic
40#4841C2211yescharacteristic
41#4836—21no—
42#4835—21no—
43#4835C2211yescharacteristic
44#4834C2211yescharacteristic
45#4833C2211yescharacteristic
46D3×C4⋊C8C2211yescharacteristic
47#4808—21no—
48#4790—21no—
49#4787—21no—
50#4772—21no—
51#4706—21no—
52D3×#97—21no—
53C4×ΓL(1,𝔽25)C2211yescharacteristic
54C8×D12C2211yescharacteristic
55#4682C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#331C2211yescharacteristic
59#4547C2211yescharacteristic
60#4544C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4482C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4457C2211yescharacteristic
68#3182—31no—
69#3107—31no—
70#3035—31no—
71#2970—31no—
72#1702—31no—
73#1700—31no—
74#1694—31no—
75#1686D311yescharacteristic, complemented
76#1681—31no—
77#1679—31no—
78#1673—31no—
79#1665—31no—
80#1565—31no—
81#1553—31no—
82#1552—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—
95C2×#216—21no—
96#938—21no—
97#938—21no—
98#936—21no—
99D4×Dic3—21no—
100D4×Dic3—21no—
101C2×#213—21no—
102#916—44no—
103C2×#204—21no—
104C2×#204—21no—
105C2×ΓL(1,𝔽25)D411yescharacteristic
106C2×ΓL(1,𝔽25)D411yescharacteristic
107C2×D3×C8—21no—
108C2×D3×C8—21no—
109C2×D3×C8—21no—
110#897—21no—
111#893—21no—
112#893D411yescharacteristic
113#892—21no—
114#892—21no—
115#892D411yescharacteristic
116#890—21no—
117#887—21no—
118C4×D12—21no—
119C4×D12C2311yescharacteristic
120C42×D3—21no—
121C42×D3C2311yescharacteristic
122#869C2311yescharacteristic
123#869C2311yescharacteristic
124C3×#106—21no—
125C3×C4⋊C8—21no—
126C3×C4⋊C8C2311yescharacteristic
127C3×C4⋊C8C2311yescharacteristic
128C3×C4⋊C8C2311yescharacteristic
129C3×#103—21no—
130C3×#102—21no—
131C3×#97—21no—
132C3×C4×C8C2311yescharacteristic
133#835—21no—
134#835—21no—
135#832—21no—
136#832—21no—
137#832—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141(C2×D3)⋊C8—21no—
142#818—21no—
143#818—21no—
144#816—21no—
145#816—21no—
146#815—21no—
147#815D411yescharacteristic
148#815D411yescharacteristic
149Dic3⋊C8—21no—
150C8×Dic3—21no—
151#810—21no—
152#810—21no—
153#809—21no—
154#807—21no—
155C4⋊(C3⋊C8)—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159#803—21no—
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—61no—
162#533—31no—
163#532—31no—
164C4×#41—31no—
165#500—31no—
166#492C2×D311yescharacteristic
167#487—31no—
168#486—31no—
169#485—61no—
170#475C2×D311yescharacteristic
171#474—31no—
172#468—61no—
173#468—61no—
174#464—61no—
175#463—61no—
176#462—31no—
177#462—31no—
178#461—31no—
179#461—31no—
180#460C2×D311yescharacteristic
181#459—31no—
182#452—61no—
183#446—31no—
184#446C2×D311yescharacteristic
185#440—61no—
186#440—61no—
187C4×Q16—61no—
188C4×SD16—61no—
189#435—61no—
190#435—61no—
191C8×D4—31no—
192C8×D4—31no—
193#432—31no—
194#432—61no—
195#431—31no—
196C2×C4⋊C8—31no—
197#416—61no—
198C4×M16—31no—
199C2×#97—61no—
200#335—31no—
201#335C2×D311yescharacteristic
202#334—31no—
203#334—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332C2×D311yescharacteristic
210#331—31no—
211#331C2×D311yescharacteristic
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×#67—41no—
218C22×Dic3—41no—
219#239—22no—
220#239—41no—
221#237—42no—
222#237—42no—
223C2×D12—21no—
224C2×D12C2×D411yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3C2×D411yescharacteristic
229C2×C4×D3C2×D411yescharacteristic
230C2×C4×D3#4111yescharacteristic
231C2×Dic6—21no—
232C2×Dic6—21no—
233C3×SD16—42no—
234C3×SD16—42no—
235C3×SD16—42no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C42—21no—
246C3×C42C2411yescharacteristic
247#217—41no—
248#217—41no—
249#216—42no—
250#216—42no—
251#213—42no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—21no—
255(C2×D3)⋊C4—41no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—41no—
262C4⋊Dic3C2×D411yescharacteristic
263Dic3⋊C4—41no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—41no—
267C4×Dic3C2×D411yescharacteristic
268C4×Dic3C2×D411yescharacteristic
269C4×Dic3#4111yescharacteristic
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275#204—42no—
276#204—42no—
277ΓL(1,𝔽25)—22no—
278ΓL(1,𝔽25)—41no—
279D3×C8—42no—
280D3×C8—42no—
281D3×C8—42no—
282C2×#41—31no—
283C2×#41—61no—
284#135—124no—
285C2×Q16—61no—
286C2×Q16—61no—
287C2×SD16—21no—
288C2×SD16—21no—
289C2×SD16—21no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×D8—61no—
293C2×M16—31no—
294C2×M16—31no—
295C22×C8—61no—
296C22×C8—61no—
297C22×C8—61no—
298C4⋊Q8—31no—
299C4⋊Q8—31no—
300C4⋊Q8C22×D311yescharacteristic
301#127C22×D311yescharacteristic
302#125—61no—
303#124—61no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306#121—61no—
307#121—61no—
308C4×Q8—31no—
309C4×Q8—61no—
310C4×Q8—61no—
311C4×Q8C22×D311yescharacteristic
312C4×D4—31no—
313C4×D4—31no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—61no—
317C4×D4—61no—
318#117—31no—
319#117—61no—
320#117—61no—
321C2×C42—31no—
322C2×C42—61no—
323#107—61no—
324#107—61no—
325#107—61no—
326#106—21no—
327C4⋊C8—21no—
328C4⋊C8—31no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—61no—
332C4⋊C8—61no—
333C4⋊C8C22×D311yescharacteristic
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336#103—21no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#103—61no—
343#102—21no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348C22⋊C8—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352#97—21no—
353#97—31no—
354#97—31no—
355#97—61no—
356#97—61no—
357C4×C8—31no—
358C4×C8—61no—
359C4×C8C22×D311yescharacteristic
360C23×C3—41no—
361C22×D3—21no—
362C22×D3#12111yescharacteristic
363C3×Q8—21no—
364C3×Q8—21no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—81no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378#67—82no—
379C2×Dic3—21no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—82no—
383C2×Dic3#12111yescharacteristic
384C2×Dic3#12111yescharacteristic
385C2×Dic3#12111yescharacteristic
386D12—21no—
387D12—42no—
388D12AGL(1,ℤ/8)12yes—
389C4×D3—21no—
390C4×D3—21no—
391C4×D3—21no—
392C4×D3—21no—
393C4×D3—41no—
394C4×D3—42no—
395C4×D3—41no—
396C4×D3—41no—
397C4×D3—81no—
398Dic6—41no—
399Dic6—41no—
400Dic6—41no—
401Dic6—41no—
402C3×C8—21no—
403C3×C8—41no—
404C3×C8—41no—
405C3×C8—41no—
406C3×C8#13512yescomplemented
407C3⋊C8—21no—
408C3⋊C8—21no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412#41—62no—
413#41—121no—
414#41—122no—
415#41—122no—
416C2×Q8—21no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8D3×D411yescharacteristic
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—31no—
424C2×D4—61no—
425C2×D4—121no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—121no—
433Q16—122no—
434Q16—122no—
435SD16—42no—
436SD16—42no—
437SD16—42no—
438SD16—122no—
439SD16—122no—
440D8—122no—
441M16—62no—
442M16—121no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—31no—
447C2×C8—31no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—122no—
452C2×C8—122no—
453C2×C8—122no—
454C2×C8D3×D411yescharacteristic
455C2×C8D3×D411yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—31no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—121no—
466C4⋊C4—121no—
467C4⋊C4D3×D411yescharacteristic
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—121no—
472C22⋊C4—121no—
473C22⋊C4—121no—
474C42—21no—
475C42—31no—
476C42—31no—
477C42—31no—
478C42—61no—
479C42—61no—
480C42—121no—
481C42C23×D311yescharacteristic
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3C2×#12111yescharacteristic
486C2×D3—21no—
487C2×D3—21no—
488C2×D3—22no—
489C2×D3—21no—
490C2×D3—41no—
491C2×D3—41no—
492C3×C4—21no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—81no—
496C3×C4C2×#13511yescharacteristic
497C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
498C3×C4#57511yescharacteristic
499C3×C4#57511yescharacteristic
500Dic3—21no—
501Dic3—21no—
502Dic3—21no—
503Dic3—41no—
504Dic3—81no—
505Dic3—81no—
506C23—31no—
507C23—41no—
508C23—61no—
509Q8—21no—
510Q8—21no—
511Q8—41no—
512Q8—41no—
513Q8—121no—
514Q8—121no—
515Q8—121no—
516Q8—121no—
517D4—32no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—61no—
523D4—81no—
524D4—122no—
525D4—242no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—31no—
531C2×C4—41no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—121no—
538C2×C4—122no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—241no—
544C2×C4—242no—
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C8—21no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—61no—
553C8—61no—
554C8—121no—
555C8—121no—
556C8—121no—
557C8#91612yescomplemented
558C2×C3—161no—
559C2×C3#303511yescharacteristic
560C2×C3#305511yescharacteristic
561C2×C3#306911yescharacteristic
562D3—41no—
563D3—42no—
564C22—61no—
565C22—61no—
566C22—62no—
567C22—61no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—121no—
572C22—121no—
573C22D3×#12111yescharacteristic
574C4—21no—
575C4—41no—
576C4—41no—
577C4—61no—
578C4—61no—
579C4—61no—
580C4—81no—
581C4—121no—
582C4—241no—
583C4—241no—
584C4C2×#91611yescharacteristic
585C4D3×AGL(1,ℤ/8)11yescharacteristic
586C4#577311yescharacteristic
587C4#577711yescharacteristic
588C3#1917811yescharacteristic, complemented, Sylow
589C2—121no—
590C2—122no—
591C2—161no—
592C2#8038211yescharacteristic
593C2#8080511yescharacteristic
594C2#8093111yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1686C2
socleC22×C3C2×#121
prime radical#80382C2
hypercenter#1686D3
hypocenterC3#19178
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. #81754C2semidirect
2. #80997C2semidirect
4. #79943C2semidirect
5. C3×#1686C2semidirect
6. #71267C2semidirect
7. #71122C2semidirect
8. #71027C2semidirect
9. #70964C2semidirect
10. #68578C2semidirect
11. #68568C2semidirect
12. #68444C2semidirect
13. #68424C2semidirect
14. #68397C2semidirect
15. #68386C2semidirect
75. #1686D3semidirect
406. C3×C8#135semidirect
557. C8#916semidirect
588. C3#19178semidirect

Maximal subgroups

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

subgroupindex
#817542
#809972
#803822
#799432
C3×#16862
#712672
#711222
#710272
#709642
#685782
#685682
#684442
#684242
#683972
#683862
#191783