Finite Group Databasebeta

← #10778014

Subgroups — #10778014

Order 768.

696 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81565C211yesmaximal, characteristic, complemented
2D3×#476C211yesmaximal, characteristic, complemented
3#80312C211yesmaximal, characteristic
4D4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
5C3×#1677C211yesmaximal, characteristic, complemented
6#71202C211yesmaximal, characteristic, complemented
7#71135C211yesmaximal, characteristic, complemented
8#71006C211yesmaximal, characteristic, complemented
9#70855C211yesmaximal, characteristic, complemented
10#68545C211yesmaximal, characteristic, complemented
11#68528C211yesmaximal, characteristic, complemented
12#68503C211yesmaximal, characteristic, complemented
13#68483C211yesmaximal, characteristic, complemented
14#68400C211yesmaximal, characteristic, complemented
15#68384C211yesmaximal, characteristic, complemented
16#19168—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)—21no—
18D3×C4⋊Q8C2211yescharacteristic
19#5716C2211yescharacteristic
20C4×D3×D4C2211yescharacteristic
21C3×#486C2211yescharacteristic, complemented
22C3×#476C2211yescharacteristic
23C3×#459C2211yescharacteristic
24C3×C8×D4C2211yescharacteristic
25#5156—21no—
26Dic3×D8—21no—
27#5111—21no—
28#5093C2211yescharacteristic, complemented
29#5084C2211yescharacteristic
30#5072C2211yescharacteristic
31#5070C2211yescharacteristic, complemented
32D4×C3⋊C8C2211yescharacteristic
33#5008C2211yescharacteristic
34#5007C2211yescharacteristic
35#4870—21no—
36D3×#106—21no—
37#4846C2211yescharacteristic
38#4842C2211yescharacteristic
39#4838C2211yescharacteristic
40#4837C2211yescharacteristic
41#4833C2211yescharacteristic
42D3×C4⋊C8C2211yescharacteristic
43#4774—21no—
44#4771—21no—
45#4771—21no—
46D3×#102—21no—
47#4765—21no—
48#4765—21no—
49#4761—21no—
50#4756—21no—
51#4730—21no—
52#4726—21no—
53#4688C2211yescharacteristic
54C4×ΓL(1,𝔽25)C2211yescharacteristic
55#4681C2211yescharacteristic, complemented
56C3×#334C2211yescharacteristic, complemented
57C3×#333C2211yescharacteristic, complemented
58C3×#331C2211yescharacteristic, complemented
59#4546C2211yescharacteristic, complemented
60#4544C2211yescharacteristic, complemented
61#4486C2211yescharacteristic, complemented
62#4482C2211yescharacteristic, complemented
63#4481C2211yescharacteristic, complemented
64#4481C2211yescharacteristic, complemented
65#4480C2211yescharacteristic, complemented
66#4479C2211yescharacteristic, complemented
67#4458C2211yescharacteristic, complemented
68#3141—31no—
69#3100—31no—
70C2×#476—31no—
71D4×M16—31no—
72#1703—31no—
73#1703—31no—
74#1697—31no—
75#1697—31no—
76#1683—31no—
77#1683—31no—
78#1677—31no—
79#1677D311yescharacteristic, complemented
80#1565—31no—
81#1548—31no—
82#1547—31no—
83C2×D3×Q8—21no—
84C2×#237—21no—
85C2×D3×D4D411yescharacteristic
86C22×C4×D3—21no—
87C2×C3×D8—21no—
88C22×C3×C8—21no—
89C3×C4⋊Q8C2311yescharacteristic
90C3×#127C2311yescharacteristic
91C3×C4×D4C2311yescharacteristic
92#944—21no—
93#938—21no—
94#938—21no—
95#936—21no—
96D4×Dic3—21no—
97D4×Dic3—21no—
98D4×Dic3D411yescharacteristic
99C2×#213—21no—
100C4×#67—21no—
101C22×C3⋊C8—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×ΓL(1,𝔽25)—21no—
104C2×ΓL(1,𝔽25)—21no—
105C2×ΓL(1,𝔽25)—44no—
106C2×ΓL(1,𝔽25)D411yescharacteristic
107C2×ΓL(1,𝔽25)D411yescharacteristic
108C2×D3×C8—21no—
109#893D411yescharacteristic
110D3×C4⋊C4—21no—
111D3×C4⋊C4—21no—
112D3×C4⋊C4D411yescharacteristic
113#888—21no—
114#888—21no—
115#67⋊C4—21no—
116D3×C22⋊C4—21no—
117C4×D12C2311yescharacteristic
118C42×D3C2311yescharacteristic
119#869C2311yescharacteristic
120#869C2311yescharacteristic
121C3×#106—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×#102—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×#102—21no—
129C3×C22⋊C8—21no—
130C3×C4×C8C2311yescharacteristic
131#832—21no—
132#832—21no—
133#832—21no—
134#832—21no—
135#832—21no—
136#830—21no—
137#821—21no—
138#821—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141(C2×D3)⋊C8—21no—
142#818—21no—
143#817—21no—
144#816—21no—
145#816—21no—
146#815D411yescharacteristic
147#815D411yescharacteristic
148Dic3⋊C8—21no—
149Dic3⋊C8—21no—
150C8×Dic3—21no—
151#810—21no—
152#810—21no—
153#809—21no—
154#808—21no—
155#808—21no—
156#807—21no—
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C22×M16—61no—
162#532—31no—
163C2×C4⋊Q8—31no—
164C2×C4×D4—31no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486C2×D311yescharacteristic, complemented
171#480—61no—
172#476—31no—
173#476—31no—
174#476—31no—
175#476—61no—
176#476—61no—
177#476C2×D311yescharacteristic
178#474—31no—
179#474—31no—
180#459—31no—
181#459C2×D311yescharacteristic
182#447—61no—
183C4×D8—61no—
184C4×D8—61no—
185#436—31no—
186#436—31no—
187#435—61no—
188#435—61no—
189C8×D4—31no—
190C8×D4C2×D311yescharacteristic
191#431—31no—
192C2×#106—61no—
193#423—31no—
194C2×C4⋊C8—31no—
195#417—61no—
196#417—61no—
197C2×#102—61no—
198#407—61no—
199C4×M16—31no—
200#335—31no—
201#335—31no—
202#334—31no—
203#334C2×D311yescharacteristic, complemented
204#333—31no—
205#333—31no—
206#333—31no—
207#333C2×D311yescharacteristic, complemented
208#332—31no—
209#332—31no—
210#331—31no—
211#331C2×D311yescharacteristic, complemented
212C23×D3—21no—
213C2×C3×Q8—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C2×#67—21no—
220C2×#67—41no—
221C22×Dic3—21no—
222C22×Dic3—41no—
223D3×Q8—82no—
224#237—42no—
225#237—42no—
226D3×D4—22no—
227D3×D4—41no—
228C2×D12C2×D411yescharacteristic
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×C4×D3—41no—
235C2×C4×D3C2×D411yescharacteristic
236C2×C4×D3C2×D411yescharacteristic
237C2×C4×D3#4111yescharacteristic
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—41no—
241C3×D8—42no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—42no—
246C2×C3×C8C2×D411yescharacteristic
247C2×C3×C8C2×D411yescharacteristic
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4C2×D411yescharacteristic
251C3×C22⋊C4—21no—
252C3×C42C2411yescharacteristic
253#217—21no—
254#217—41no—
255#217—41no—
256#213—42no—
257(C2×D3)⋊C4—21no—
258(C2×D3)⋊C4—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3C2×D411yescharacteristic
264Dic3⋊C4—21no—
265Dic3⋊C4—41no—
266Dic3⋊C4—41no—
267C4×Dic3C2×D411yescharacteristic
268C4×Dic3C2×D411yescharacteristic
269C4×Dic3#4111yescharacteristic
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—42no—
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276ΓL(1,𝔽25)—22no—
277ΓL(1,𝔽25)—22no—
278ΓL(1,𝔽25)—42no—
279ΓL(1,𝔽25)—42no—
280ΓL(1,𝔽25)—42no—
281D3×C8—42no—
282C2×#41—61no—
283C22×Q8—61no—
284C22×D4—31no—
285C23×C4—61no—
286C2×D8—21no—
287C2×D8—61no—
288C2×M16—31no—
289C2×M16—31no—
290C2×M16—61no—
291C2×M16—61no—
292C2×M16—61no—
293C2×M16—124no—
294C22×C8—21no—
295C22×C8—61no—
296C22×C8—61no—
297C4⋊Q8—31no—
298C4⋊Q8—31no—
299C4⋊Q8—61no—
300C4⋊Q8—61no—
301C4⋊Q8—61no—
302C4⋊Q8C22×D311yescharacteristic
303#127C22×D311yescharacteristic
304#124—61no—
305#121—61no—
306#121—61no—
307C4×D4—31no—
308C4×D4—31no—
309C4×D4—31no—
310C4×D4—61no—
311C4×D4—61no—
312C4×D4—61no—
313C4×D4—61no—
314C4×D4C22×D311yescharacteristic
315C2×C4⋊C4—31no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C42—31no—
320#107—61no—
321#107—61no—
322#106—21no—
323#106—61no—
324#106—61no—
325#106—61no—
326C4⋊C8—31no—
327C4⋊C8—31no—
328C4⋊C8—31no—
329C4⋊C8—61no—
330C4⋊C8—61no—
331C4⋊C8C22×D311yescharacteristic
332C4⋊C8C22×D311yescharacteristic
333C4⋊C8C22×D311yescharacteristic
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#102—21no—
339#102—21no—
340#102—21no—
341#102—21no—
342#102—61no—
343#102—61no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348#102—61no—
349#102—61no—
350C22⋊C8—21no—
351C22⋊C8—61no—
352C22⋊C8—61no—
353C22⋊C8—61no—
354C22⋊C8—61no—
355#97—31no—
356#97—31no—
357C4×C8—31no—
358C4×C8—61no—
359C4×C8C22×D311yescharacteristic
360C23×C3—21no—
361C23×C3—41no—
362C22×D3—21no—
363C22×D3—42no—
364C22×D3—42no—
365C22×D3—81no—
366C22×D3#12111yescharacteristic
367C3×Q8—81no—
368C3×D4—21no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—81no—
374C3×D4#13712yes—
375C2×C3×C4—21no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382#67—42no—
383#67—82no—
384C2×Dic3—21no—
385C2×Dic3—42no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—82no—
389C2×Dic3#12111yescharacteristic
390C2×Dic3#12111yescharacteristic
391C2×Dic3#12111yescharacteristic
392D12—21no—
393D12—21no—
394C4×D3—21no—
395C4×D3—21no—
396C4×D3—41no—
397C4×D3—42no—
398C4×D3—41no—
399C4×D3—41no—
400C4×D3—42no—
401C4×D3—82no—
402C4×D3#13512yes—
403C4×D3AGL(1,ℤ/8)12yes—
404Dic6—41no—
405Dic6—41no—
406Dic6—82no—
407Dic6—81no—
408C3×C8—21no—
409C3×C8—21no—
410C3×C8—41no—
411C3×C8—41no—
412C3×C8—41no—
413C3⋊C8—21no—
414C3⋊C8—21no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—61no—
419#41—122no—
420#41—122no—
421C2×Q8—21no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—121no—
425C2×Q8—242no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—31no—
430C2×D4—61no—
431C2×D4—62no—
432C2×D4—121no—
433C2×D4—121no—
434C2×D4D3×D411yescharacteristic
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—121no—
446C22×C4—121no—
447D8—42no—
448D8—122no—
449M16—62no—
450M16—62no—
451M16—122no—
452M16—122no—
453M16—122no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—31no—
458C2×C8—31no—
459C2×C8—42no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—122no—
464C2×C8—122no—
465C2×C8D3×D411yescharacteristic
466C2×C8D3×D411yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—31no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—121no—
476C4⋊C4—121no—
477C4⋊C4D3×D411yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—61no—
480C22⋊C4—61no—
481C22⋊C4—61no—
482C22⋊C4—121no—
483C22⋊C4—121no—
484C42—31no—
485C42—31no—
486C42—31no—
487C42C23×D311yescharacteristic
488C22×C3—22no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3C2×#12111yescharacteristic
495C2×D3—21no—
496C2×D3—21no—
497C2×D3—41no—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—84no—
501C2×D3#46312yes—
502C3×C4—21no—
503C3×C4—41no—
504C3×C4—81no—
505C3×C4C2×#13511yescharacteristic
506C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
507C3×C4C2×#13711yescharacteristic
508C3×C4#57511yescharacteristic
509Dic3—21no—
510Dic3—21no—
511Dic3—41no—
512Dic3—81no—
513Dic3—81no—
514Dic3#46312yes—
515C23—21no—
516C23—31no—
517C23—41no—
518C23—61no—
519C23—122no—
520C23—122no—
521C23—241no—
522Q8—81no—
523Q8—121no—
524Q8—121no—
525Q8—241no—
526Q8—242no—
527D4—21no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—61no—
533D4—61no—
534D4—81no—
535D4—122no—
536D4—242no—
537D4#91512yes—
538C2×C4—21no—
539C2×C4—31no—
540C2×C4—32no—
541C2×C4—32no—
542C2×C4—31no—
543C2×C4—31no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—121no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—242no—
559C2×C4—242no—
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C8—21no—
564C8—21no—
565C8—41no—
566C8—41no—
567C8—41no—
568C8—61no—
569C8—61no—
570C8—121no—
571C8—121no—
572C8—121no—
573C2×C3—42no—
574C2×C3—161no—
575C2×C3C2×#46311yescharacteristic
576C2×C3#305611yescharacteristic
577C2×C3#306811yescharacteristic
578D3—22no—
579D3—81no—
580C22—22no—
581C22—32no—
582C22—41no—
583C22—41no—
584C22—61no—
585C22—61no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—121no—
590C22—121no—
591C22—121no—
592C22—244no—
593C22D3×#12111yescharacteristic
594C4—21no—
595C4—32no—
596C4—41no—
597C4—61no—
598C4—61no—
599C4—81no—
600C4—121no—
601C4—241no—
602C4—241no—
603C4C2×#91511yescharacteristic
604C4D3×#13511yescharacteristic
605C4#576811yescharacteristic
606C4D3×AGL(1,ℤ/8)11yescharacteristic
607C3#1916811yescharacteristic, complemented, Sylow
608C2—42no—
609C2—62no—
610C2—161no—
611C2—241no—
612C2D3×#46311yescharacteristic
613C2#8065811yescharacteristic
614C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1677C2
socleC22×C3C2×#121
prime radical#80312C2
hypercenter#1677D3
hypocenterC3#19168
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1677

Complements

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

NH ≅ G/Ntype
1. #81565C2semidirect
2. D3×#476C2semidirect
4. D4×ΓL(1,𝔽25)C2semidirect
5. C3×#1677C2semidirect
6. #71202C2semidirect
7. #71135C2semidirect
8. #71006C2semidirect
9. #70855C2semidirect
10. #68545C2semidirect
11. #68528C2semidirect
12. #68503C2semidirect
13. #68483C2semidirect
14. #68400C2semidirect
15. #68384C2semidirect
21. C3×#486C22semidirect
28. #5093C22semidirect
31. #5070C22semidirect
55. #4681C22semidirect
56. C3×#334C22semidirect
57. C3×#333C22semidirect
58. C3×#331C22semidirect
59. #4546C22semidirect
60. #4544C22semidirect
61. #4486C22semidirect
62. #4482C22semidirect
63. #4481C22semidirect
64. #4481C22semidirect
65. #4480C22semidirect
66. #4479C22semidirect
67. #4458C22semidirect
79. #1677D3semidirect
170. #486C2×D3semidirect
203. #334C2×D3semidirect
207. #333C2×D3semidirect
211. #331C2×D3semidirect
607. C3#19168semidirect

Maximal subgroups

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

subgroupindex
#815652
D3×#4762
#803122
D4×ΓL(1,𝔽25)2
C3×#16772
#712022
#711352
#710062
#708552
#685452
#685282
#685032
#684832
#684002
#683842
#191683