Finite Group Databasebeta

← #10780314

Subgroups — #10780314

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81565C211yesmaximal, characteristic, complemented
2#81115C211yesmaximal, characteristic, complemented
3#80402C211yesmaximal, characteristic
4#79942C211yesmaximal, characteristic, complemented
5C3×#1698C211yesmaximal, characteristic, complemented
6#71200C211yesmaximal, characteristic, complemented
7#71151C211yesmaximal, characteristic, complemented
8#71022C211yesmaximal, characteristic, complemented
9#70967C211yesmaximal, characteristic, complemented
10#68581C211yesmaximal, characteristic, complemented
11#68571C211yesmaximal, characteristic, complemented
12#68503C211yesmaximal, characteristic, complemented
13#68483C211yesmaximal, characteristic, complemented
14#68400C211yesmaximal, characteristic, complemented
15#68386C211yesmaximal, characteristic, complemented
16#19382—31nomaximal, Sylow
17D3×C4⋊Q8C2211yescharacteristic
18#5716C2211yescharacteristic
19#5580C2211yescharacteristic
20C3×#486C2211yescharacteristic
21C3×#477C2211yescharacteristic
22C3×#460C2211yescharacteristic
23C3×#446C2211yescharacteristic
24#5174—21no—
25#5165—21no—
26#5093C2211yescharacteristic
27#5084C2211yescharacteristic
28#5072C2211yescharacteristic
29#5070C2211yescharacteristic
30Q8×C3⋊C8C2211yescharacteristic
31#5021C2211yescharacteristic
32#5020C2211yescharacteristic
33#4882—21no—
34#4876—21no—
35#4846C2211yescharacteristic
36#4842C2211yescharacteristic
37#4840—21no—
38#4838—21no—
39#4838C2211yescharacteristic
40#4837C2211yescharacteristic
41#4835C2211yescharacteristic
42#4833C2211yescharacteristic
43#4805—21no—
44#4794—21no—
45#4782—21no—
46#4771—21no—
47#4771—21no—
48#4766—21no—
49#4765—21no—
50#4765—21no—
51#4708—21no—
52#4706—21no—
53C4×ΓL(1,𝔽25)C2211yescharacteristic
54C8×D12C2211yescharacteristic
55#4681C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#331C2211yescharacteristic
59#4546C2211yescharacteristic
60#4544C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4457C2211yescharacteristic
68#3251—31no—
69#3238—31no—
70#3141—31no—
71#2969—31no—
72#1704—31no—
73#1703—31no—
74#1698D311yescharacteristic, complemented
75#1697—31no—
76#1684—31no—
77#1683—31no—
78#1678—31no—
79#1677—31no—
80#1569—31no—
81#1553—31no—
82#1548—31no—
83C2×D3×Q8—21no—
84C2×#237—21no—
85C2×C3×SD16—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×#127C2311yescharacteristic
88C3×C4×Q8C2311yescharacteristic
89#947D411yescharacteristic
90#946D411yescharacteristic
91#944—21no—
92#938—21no—
93#938—21no—
94#936—21no—
95D4×Dic3—21no—
96D4×Dic3—21no—
97C2×#213—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×ΓL(1,𝔽25)—21no—
100C2×ΓL(1,𝔽25)D411yescharacteristic
101C2×ΓL(1,𝔽25)D411yescharacteristic
102C2×D3×C8—21no—
103#898—21no—
104#895D411yescharacteristic
105#894—21no—
106D3×C4⋊C4—21no—
107D3×C4⋊C4—21no—
108#890D411yescharacteristic
109#888—21no—
110#888—21no—
111C4×D12—21no—
112C4×D12C2311yescharacteristic
113#872—21no—
114C42×D3C2311yescharacteristic
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C3×#107—21no—
118C3×C4⋊C8—21no—
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×#103—21no—
123C3×#102—21no—
124C3×#102—21no—
125C3×#102—21no—
126C3×#97—21no—
127C3×C4×C8C2311yescharacteristic
128#835—21no—
129#835—21no—
130#832—21no—
131#832—21no—
132#832—21no—
133#821—21no—
134#821—21no—
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139#818—21no—
140#817—21no—
141#816—21no—
142#816—21no—
143#815—21no—
144#815D411yescharacteristic
145#815D411yescharacteristic
146Dic3⋊C8—21no—
147C8×Dic3—21no—
148#810—21no—
149#810—21no—
150#809—21no—
151#808—21no—
152#808—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8—21no—
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8—31no—
162#529—31no—
163#501—61no—
164#489—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486C2×D311yescharacteristic
171#483—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477C2×D311yescharacteristic
176#476—31no—
177#476—31no—
178#475—31no—
179#474—31no—
180#469—61no—
181#468—61no—
182#463—61no—
183#460C2×D311yescharacteristic
184#459—31no—
185#446C2×D311yescharacteristic
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432—31no—
193#431—31no—
194#423—31no—
195#417—61no—
196#417—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335C2×D311yescharacteristic
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333C2×D311yescharacteristic
207#332—31no—
208#332—31no—
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—
218D3×Q8—82no—
219#237—42no—
220#237—42no—
221C2×D12—21no—
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—41no—
225C2×C4×D3C2×D411yescharacteristic
226C2×C4×D3C2×D411yescharacteristic
227C2×C4×D3#4111yescharacteristic
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—41no—
231C3×SD16—81no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4C2×D411yescharacteristic
241C3×C42—21no—
242C3×C42C2411yescharacteristic
243#217—41no—
244#217—41no—
245#213—81no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3C2×D411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3C2×D411yescharacteristic
262C4×Dic3C2×D411yescharacteristic
263C4×Dic3#4111yescharacteristic
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8C2×D411yescharacteristic
268C2×C3⋊C8C2×D411yescharacteristic
269ΓL(1,𝔽25)—41no—
270ΓL(1,𝔽25)—41no—
271ΓL(1,𝔽25)—42no—
272ΓL(1,𝔽25)—81no—
273D3×C8—42no—
274C2×#41—61no—
275C22×Q8—61no—
276C2×SD16—21no—
277C2×D8—61no—
278C2×M16—31no—
279C2×M16—31no—
280C2×M16—61no—
281C2×M16—61no—
282C22×C8—61no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8—61no—
286C4⋊Q8—61no—
287C4⋊Q8—61no—
288C4⋊Q8C22×D311yescharacteristic
289#127C22×D311yescharacteristic
290#126—61no—
291#125—31no—
292#124—31no—
293#124—61no—
294#123—61no—
295C22⋊Q8—31no—
296#121—31no—
297#121—61no—
298#121—61no—
299C4×Q8C22×D311yescharacteristic
300C4×D4—31no—
301C4×D4—61no—
302C4×D4—61no—
303C4×D4—61no—
304#117—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C42—31no—
308#107—21no—
309#107—61no—
310#107—61no—
311#106—61no—
312#106—61no—
313#106—61no—
314C4⋊C8—21no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—61no—
319C4⋊C8—61no—
320C4⋊C8C22×D311yescharacteristic
321C4⋊C8C22×D311yescharacteristic
322C4⋊C8C22×D311yescharacteristic
323#103—21no—
324#103—61no—
325#103—61no—
326#103—61no—
327#103—61no—
328#103—61no—
329#103—61no—
330#102—21no—
331#102—21no—
332#102—21no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343#97—21no—
344#97—31no—
345#97—31no—
346#97—61no—
347C4×C8—31no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8C22×D311yescharacteristic
351C23×C3—41no—
352C22×D3—21no—
353C22×D3#12111yescharacteristic
354C3×Q8—21no—
355C3×Q8—21no—
356C3×Q8—81no—
357C3×D4—41no—
358C3×D4—41no—
359C3×D4—41no—
360C3×D4—41no—
361C3×D4—81no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368#67—82no—
369C2×Dic3—21no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—82no—
373C2×Dic3#12111yescharacteristic
374C2×Dic3#12111yescharacteristic
375C2×Dic3#12111yescharacteristic
376D12—21no—
377D12—21no—
378D12—81no—
379C4×D3—21no—
380C4×D3—21no—
381C4×D3—21no—
382C4×D3—21no—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—41no—
386C4×D3—82no—
387Dic6—41no—
388Dic6—41no—
389Dic6—82no—
390Dic6—81no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—41no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401#41—122no—
402#41—122no—
403C2×Q8—21no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—121no—
407C2×Q8—242no—
408C2×Q8D3×D411yescharacteristic
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—31no—
413C2×D4—61no—
414C2×D4—121no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—61no—
419C22×C4—121no—
420C22×C4—121no—
421SD16—81no—
422D8—241no—
423M16—121no—
424M16—121no—
425M16—122no—
426M16—241no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—122no—
436C2×C8D3×D411yescharacteristic
437C2×C8D3×D411yescharacteristic
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—31no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—121no—
450C4⋊C4—121no—
451C4⋊C4D3×D411yescharacteristic
452C22⋊C4—61no—
453C22⋊C4—61no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C42—21no—
460C42—31no—
461C42—31no—
462C42—31no—
463C42C23×D311yescharacteristic
464C22×C3—81no—
465C22×C3—81no—
466C22×C3—81no—
467C22×C3C2×#12111yescharacteristic
468C2×D3—21no—
469C2×D3—21no—
470C2×D3—21no—
471C2×D3—41no—
472C2×D3—41no—
473C2×D3—41no—
474C3×C4—21no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—81no—
478C3×C4#57511yescharacteristic
479C3×C4#57511yescharacteristic
480C3×C4#57711yescharacteristic
481C3×C4#57811yescharacteristic
482Dic3—21no—
483Dic3—21no—
484Dic3—21no—
485Dic3—41no—
486Dic3—81no—
487Dic3—81no—
488C23—31no—
489C23—41no—
490C23—61no—
491Q8—21no—
492Q8—21no—
493Q8—81no—
494Q8—121no—
495Q8—121no—
496Q8—241no—
497Q8—242no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—61no—
503D4—61no—
504D4—81no—
505D4—241no—
506D4—242no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—31no—
510C2×C4—31no—
511C2×C4—31no—
512C2×C4—41no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—242no—
524C2×C4—242no—
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×D3×D411yescharacteristic
527C2×C4D3×#4111yescharacteristic
528C8—21no—
529C8—21no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—61no—
534C8—61no—
535C8—121no—
536C8—121no—
537C8—121no—
538C2×C3—161no—
539C2×C3#304011yescharacteristic
540C2×C3#306311yescharacteristic
541C2×C3#306811yescharacteristic
542D3—41no—
543D3—81no—
544C22—61no—
545C22—61no—
546C22—61no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—121no—
551C22—121no—
552C22—121no—
553C22D3×#12111yescharacteristic
554C4—21no—
555C4—41no—
556C4—41no—
557C4—61no—
558C4—61no—
559C4—61no—
560C4—81no—
561C4—121no—
562C4—241no—
563C4—241no—
564C4#576811yescharacteristic
565C4#577511yescharacteristic
566C4#577711yescharacteristic
567C4#577911yescharacteristic
568C3#1938211yescharacteristic, complemented, Sylow
569C2—121no—
570C2—161no—
571C2—241no—
572C2#8059911yescharacteristic
573C2#8081011yescharacteristic
574C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1698C2
socleC22×C3C2×#121
prime radical#80402C2
hypercenter#1698D3
hypocenterC3#19382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1698

Complements

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

NH ≅ G/Ntype
1. #81565C2semidirect
2. #81115C2semidirect
4. #79942C2semidirect
5. C3×#1698C2semidirect
6. #71200C2semidirect
7. #71151C2semidirect
8. #71022C2semidirect
9. #70967C2semidirect
10. #68581C2semidirect
11. #68571C2semidirect
12. #68503C2semidirect
13. #68483C2semidirect
14. #68400C2semidirect
15. #68386C2semidirect
74. #1698D3semidirect
568. C3#19382semidirect

Maximal subgroups

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

subgroupindex
#815652
#811152
#804022
#799422
C3×#16982
#712002
#711512
#710222
#709672
#685812
#685712
#685032
#684832
#684002
#683862
#193823