Finite Group Databasebeta

← #10780332

Subgroups — #10780332

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83312C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic, complemented
3#79358C211yesmaximal, characteristic, complemented
4#79280C211yesmaximal, characteristic, complemented
5C3×#1553C211yesmaximal, characteristic, complemented
6#70795C211yesmaximal, characteristic, complemented
7#70778C211yesmaximal, characteristic, complemented
8#68476C211yesmaximal, characteristic, complemented
9#68468C211yesmaximal, characteristic, complemented
10#68452C211yesmaximal, characteristic, complemented
11#68444C211yesmaximal, characteristic, complemented
12#68415C211yesmaximal, characteristic, complemented
13#68411C211yesmaximal, characteristic, complemented
14#68401C211yesmaximal, characteristic, complemented
15#68397C211yesmaximal, characteristic, complemented
16#19383—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18#5479C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#432C2211yescharacteristic
21C3×#431C2211yescharacteristic
22#5232—21no—
23#5232—21no—
24#5094C2211yescharacteristic
25#5087C2211yescharacteristic
26#5079C2211yescharacteristic
27#5079C2211yescharacteristic
28#5075C2211yescharacteristic
29#5072C2211yescharacteristic
30#5070C2211yescharacteristic
31#5070C2211yescharacteristic
32#5055—21no—
33#5055—21no—
34#5042—21no—
35#5039—21no—
36#5016—21no—
37#5009—21no—
38#4924C2211yescharacteristic
39C4×#208C2211yescharacteristic
40#4849C2211yescharacteristic
41#4849C2211yescharacteristic
42#4841C2211yescharacteristic
43#4841C2211yescharacteristic
44#4838C2211yescharacteristic
45#4834C2211yescharacteristic
46#4831C2211yescharacteristic
47#4830C2211yescharacteristic
48#4739—21no—
49#4738—21no—
50#4729—21no—
51#4728—21no—
52#4715—21no—
53#4703—21no—
54#4690—21no—
55#4689—21no—
56C3×#332C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#331C2211yescharacteristic
59C3×#331C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4486C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3251—31no—
69#3218—31no—
70#3141—31no—
71#2969—31no—
72#1702—31no—
73#1701—31no—
74#1700—31no—
75#1699—31no—
76#1682—31no—
77#1681—31no—
78#1680—31no—
79#1679—31no—
80#1569—31no—
81#1553D311yescharacteristic, complemented
82#1546—31no—
83C2×C3×#41—21no—
84C22×Dic6—21no—
85C3×C4⋊Q8C2311yescharacteristic
86C3×#127C2311yescharacteristic
87C3×#124—21no—
88C3×#121—21no—
89C3×#121—21no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#214—21no—
94C2×#213—21no—
95#930D411yescharacteristic
96#929—21no—
97C4×#67—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100#924D411yescharacteristic
101C2×#208—21no—
102C2×#208D411yescharacteristic
103C2×#208D411yescharacteristic
104C2×Dic12—21no—
105C2×Dic12—21no—
106C2×#204—21no—
107C2×#204—21no—
108C2×ΓL(1,𝔽25)—21no—
109C2×D3×C8—21no—
110#876—21no—
111#875D411yescharacteristic
112C4×D12C2311yescharacteristic
113#872—21no—
114#870D411yescharacteristic
115#869—21no—
116#869—21no—
117#869—21no—
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C4×Dic6C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×C22⋊C8—21no—
126C3×C22⋊C8—21no—
127C3×#97—21no—
128C3×C4×C8—21no—
129#835—21no—
130#835—21no—
131#835—21no—
132#835—21no—
133#832—21no—
134#832—21no—
135#832—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141#816—21no—
142#816—21no—
143#816—21no—
144#816—21no—
145#815—21no—
146Dic3⋊C8—21no—
147Dic3⋊C8—21no—
148C8×Dic3—21no—
149#810—21no—
150#809—21no—
151#808—21no—
152#807—21no—
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156#803D411yescharacteristic
157#803D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532C2×D311yescharacteristic
161C2×C4⋊Q8—31no—
162#529—31no—
163#497—61no—
164#491—61no—
165#489—61no—
166#489—61no—
167#487—31no—
168#487—31no—
169#486—31no—
170#486—31no—
171#478—61no—
172#475—31no—
173#474—31no—
174#469—61no—
175#468—61no—
176#463—61no—
177#462—31no—
178#462—31no—
179#461—31no—
180#461—31no—
181#460—31no—
182#459—31no—
183#451—61no—
184#448—61no—
185#446—31no—
186C8×Q8—31no—
187#436—31no—
188#436—61no—
189#435—61no—
190C8×D4—31no—
191#433—61no—
192#432C2×D311yescharacteristic
193#431C2×D311yescharacteristic
194#423—31no—
195#417—61no—
196#417—61no—
197#405—61no—
198C4×M16—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332C2×D311yescharacteristic
208#332C2×D311yescharacteristic
209#331C2×D311yescharacteristic
210#331C2×D311yescharacteristic
211C3×#41—42no—
212C3×#41—42no—
213C2×C3×Q8—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—41no—
218C22×C3×C4—41no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4111yescharacteristic
222C2×#67—21no—
223C22×Dic3—41no—
224C2×D12C2×D411yescharacteristic
225C2×C4×D3—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—41no—
229C2×Dic6—81no—
230C2×Dic6—81no—
231C2×Dic6C2×D411yescharacteristic
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C22⋊C4—41no—
239C3×C22⋊C4—41no—
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42#4111yescharacteristic
243C3×C42C2411yescharacteristic
244#217—21no—
245#214—81no—
246#213—81no—
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—41no—
256C4⋊Dic3—41no—
257C4⋊Dic3C2×D411yescharacteristic
258C4⋊Dic3C2×D411yescharacteristic
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263C4×Dic3—21no—
264#208—41no—
265#208—41no—
266#208—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273Dic12—81no—
274Dic12—81no—
275#204—81no—
276#204—81no—
277ΓL(1,𝔽25)—42no—
278D3×C8—42no—
279C2×#41—21no—
280C22×Q8—61no—
281C2×Q16—61no—
282C2×Q16—61no—
283C2×SD16—61no—
284C2×SD16—61no—
285C2×SD16—61no—
286C2×D8—61no—
287C2×M16—31no—
288C2×M16—31no—
289C2×M16—61no—
290C2×M16—61no—
291C22×C8—61no—
292C4⋊Q8—31no—
293C4⋊Q8—31no—
294C4⋊Q8—61no—
295C4⋊Q8—61no—
296C4⋊Q8—61no—
297C4⋊Q8C22×D311yescharacteristic
298#127C22×D311yescharacteristic
299#126—61no—
300#125—31no—
301#124—21no—
302#124—31no—
303#123—61no—
304C22⋊Q8—31no—
305#121—21no—
306#121—21no—
307#121—31no—
308C4×Q8—31no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—31no—
312C4×D4—61no—
313#117—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C42C22×D311yescharacteristic
317#107—61no—
318#106—61no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102—61no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—61no—
347C22⋊C8—61no—
348#97—21no—
349#97—31no—
350#97—31no—
351#97—61no—
352C4×C8—21no—
353C4×C8—31no—
354C4×C8—31no—
355C4×C8—61no—
356C23×C3—41no—
357C23×C3#12111yescharacteristic
358C22×D3—21no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—81no—
366C3×D4—82no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—82no—
374C2×C3×C4#12111yescharacteristic
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4#12111yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380#67—81no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—82no—
387D12—21no—
388D12—21no—
389C4×D3—41no—
390C4×D3—41no—
391Dic6—21no—
392Dic6—21no—
393Dic6—41no—
394Dic6—41no—
395Dic6—41no—
396Dic6—41no—
397Dic6—82no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8—41no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C3⋊C8—21no—
405C3⋊C8—21no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408#41—42no—
409#41—42no—
410C2×Q8—21no—
411C2×Q8—31no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—121no—
415C2×Q8—241no—
416C2×Q8—241no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—31no—
421C2×D4—41no—
422C2×D4—61no—
423C22×C4—41no—
424C22×C4—61no—
425C22×C4—121no—
426C22×C4#23511yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4D3×D411yescharacteristic
429Q16—241no—
430Q16—241no—
431SD16—241no—
432SD16—241no—
433SD16—241no—
434D8—241no—
435M16—122no—
436M16—121no—
437M16—121no—
438M16—241no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—31no—
444C2×C8—31no—
445C2×C8—31no—
446C2×C8—31no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—122no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—31no—
453C4⋊C4—31no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—121no—
463C4⋊C4—121no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C42—61no—
472C42#23511yescharacteristic
473C42D3×D411yescharacteristic
474C42D3×D411yescharacteristic
475C42C23×D311yescharacteristic
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3C2×#12111yescharacteristic
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—41no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—81no—
491C3×C4#57511yescharacteristic
492C3×C4#57511yescharacteristic
493C3×C4#57611yescharacteristic
494C3×C4#57711yescharacteristic
495Dic3—41no—
496Dic3—41no—
497Dic3—41no—
498Dic3—81no—
499Dic3—81no—
500C23—41no—
501C23—61no—
502C23#88411yescharacteristic
503Q8—41no—
504Q8—41no—
505Q8—61no—
506Q8—61no—
507Q8—121no—
508Q8—121no—
509Q8—121no—
510Q8—121no—
511Q8—242no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—61no—
517D4—61no—
518D4—81no—
519D4—82no—
520D4—241no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—82no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—242no—
536C2×C4#88411yescharacteristic
537C2×C4#88411yescharacteristic
538C2×C4#88411yescharacteristic
539C2×C4C2×#23511yescharacteristic
540C2×C4C2×D3×D411yescharacteristic
541C2×C4C2×D3×D411yescharacteristic
542C8—41no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—61no—
547C8—61no—
548C8—61no—
549C8—61no—
550C8—121no—
551C8—121no—
552C2×C3—41no—
553C2×C3—161no—
554C2×C3#304011yescharacteristic
555C2×C3#306211yescharacteristic
556C2×C3#306911yescharacteristic
557D3—81no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—121no—
565C22—121no—
566C22—121no—
567C22C2×#88411yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—81no—
573C4—121no—
574C4—121no—
575C4—121no—
576C4—241no—
577C4—241no—
578C4#576911yescharacteristic
579C4#577011yescharacteristic
580C4#577311yescharacteristic
581C4#577311yescharacteristic
582C3#1938311yescharacteristic, complemented, Sylow
583C2—41no—
584C2—161no—
585C2—241no—
586C2#7872111yescharacteristic
587C2#7884011yescharacteristic
588C2#7894311yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1553C2
socleC22×C3C2×#121
prime radical#83312C2
hypercenter#1553D3
hypocenterC3#19383
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1553

Complements

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

NH ≅ G/Ntype
2. #83283C2semidirect
3. #79358C2semidirect
4. #79280C2semidirect
5. C3×#1553C2semidirect
6. #70795C2semidirect
7. #70778C2semidirect
8. #68476C2semidirect
9. #68468C2semidirect
10. #68452C2semidirect
11. #68444C2semidirect
12. #68415C2semidirect
13. #68411C2semidirect
14. #68401C2semidirect
15. #68397C2semidirect
81. #1553D3semidirect
582. C3#19383semidirect

Maximal subgroups

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

subgroupindex
#833122
#832832
#793582
#792802
C3×#15532
#707952
#707782
#684762
#684682
#684522
#684442
#684152
#684112
#684012
#683972
#193833