Finite Group Databasebeta

← #10786201

Subgroups — #10786201

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79926C211yesmaximal, characteristic
2#79912C211yesmaximal, characteristic
3#79745C211yesmaximal, characteristic, complemented
4#79743C211yesmaximal, characteristic, complemented
5C3×#1579C211yesmaximal, characteristic, complemented
6#70947C211yesmaximal, characteristic, complemented
7#70942C211yesmaximal, characteristic, complemented
8#68542C211yesmaximal, characteristic, complemented
9#68537C211yesmaximal, characteristic, complemented
10#68250C211yesmaximal, characteristic, complemented
11#68250C211yesmaximal, characteristic, complemented
12#68249C211yesmaximal, characteristic, complemented
13#68247C211yesmaximal, characteristic, complemented
14#68099C211yesmaximal, characteristic, complemented
15#68078C211yesmaximal, characteristic
16#19866—31nomaximal, Sylow
17#5584C2211yescharacteristic
18#5584C2211yescharacteristic
19#5584C2211yescharacteristic
20#5579C2211yescharacteristic
21C3×C8×Q8C2211yescharacteristic
22C3×C8×Q8C2211yescharacteristic
23C3×C4×Q16C2211yescharacteristic
24#5189—21no—
25#5188—21no—
26#5021C2211yescharacteristic
27#5020C2211yescharacteristic
28#5020C2211yescharacteristic
29#5020C2211yescharacteristic
30#4881—21no—
31#4880—21no—
32#4845—21no—
33#4845—21no—
34#4844—21no—
35#4842C2211yescharacteristic
36#4841C2211yescharacteristic
37#4841C2211yescharacteristic
38#4841C2211yescharacteristic
39#4829—21no—
40#4815—21no—
41#4814—21no—
42#4814—21no—
43#4805—21no—
44#4699C2211yescharacteristic
45C4×Dic12—21no—
46#4697—21no—
47#4696—21no—
48C4×D24C2211yescharacteristic
49C4×#204—21no—
50#4688C2211yescharacteristic
51C8×D12C2211yescharacteristic
52#4682C2211yescharacteristic
53#4682C2211yescharacteristic
54#4681C2211yescharacteristic
55#4681C2211yescharacteristic
56#4680C2211yescharacteristic
57C3×#334C2211yescharacteristic
58C3×Q8⋊C8C2211yescharacteristic
59C3×Q8⋊C8C2211yescharacteristic
60C3×#322C2211yescharacteristic
61#4482C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4481C2211yescharacteristic
65#4459C2211yescharacteristic
66#4459C2211yescharacteristic
67#4458C2211yescharacteristic
68#3393—31no—
69#3360—31no—
70#3356—31no—
71#3342—31no—
72#1723—31no—
73#1693—31no—
74#1684—31no—
75#1671—31no—
76#1670—31no—
77#1650—31no—
78#1650—31no—
79#1643—31no—
80#1638—31no—
81#1581—31no—
82#1579D311yescharacteristic, complemented
83C2×C3×Q16D411yescharacteristic
84C3×C4×Q8C2311yescharacteristic
85C3×C4×Q8C2311yescharacteristic
86#946—21no—
87#946—21no—
88#946—21no—
89#946—21no—
90C2×Dic12—21no—
91C2×Dic12—21no—
92C2×D24D411yescharacteristic
93C2×#204—21no—
94C2×#204—21no—
95C2×#204—21no—
96C2×ΓL(1,𝔽25)—21no—
97C2×D3×C8—21no—
98#898—41no—
99#898—41no—
100#898—41no—
101#896—41no—
102#892—21no—
103#892—21no—
104#892—21no—
105D3×C4⋊C4—21no—
106#875—21no—
107#875—21no—
108#875—21no—
109C4×D12—21no—
110C4×D12C2311yescharacteristic
111C4×D12C2311yescharacteristic
112#870—21no—
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C4×Dic6—21no—
118C4×Dic6—21no—
119C4×Dic6—21no—
120C3×#107D411yescharacteristic
121C3×C4⋊C8—21no—
122C3×C4⋊C8—21no—
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×#103D411yescharacteristic
126C3×#103D411yescharacteristic
127C3×C4×C8—21no—
128C3×C4×C8—21no—
129C3×C4×C8C2311yescharacteristic
130C3×C4×C8C2311yescharacteristic
131C3×C4×C8C2311yescharacteristic
132#835—21no—
133#835—21no—
134#835—21no—
135#835—21no—
136#821—21no—
137#821D411yescharacteristic
138#821D411yescharacteristic
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141#818—21no—
142#818—21no—
143#818—41no—
144#818D411yescharacteristic
145#817—21no—
146#817—21no—
147#817—21no—
148#816—21no—
149#816—21no—
150#816—21no—
151#816—21no—
152#816—21no—
153#816—41no—
154#808—21no—
155#808—21no—
156#808—21no—
157#807—21no—
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160#556—31no—
161#556—31no—
162#556—31no—
163#554—31no—
164#500—31no—
165#500—31no—
166#498—31no—
167#495—61no—
168#494—31no—
169#488—61no—
170#487—31no—
171#487—31no—
172#485—61no—
173#485—61no—
174#485—61no—
175#483—61no—
176#479—61no—
177#477—31no—
178#476—31no—
179#475—31no—
180#475—31no—
181#475—31no—
182#470—61no—
183#467—61no—
184#464—61no—
185#464—61no—
186#463—61no—
187#461—31no—
188#461—31no—
189#461—31no—
190C8×Q8C2×D311yescharacteristic
191C8×Q8C2×D311yescharacteristic
192C4×Q16—61no—
193C4×Q16C2×D311yescharacteristic
194C4×SD16—61no—
195C4×D8—31no—
196#436—31no—
197C8×D4—31no—
198#428—61no—
199#427—61no—
200#335—31no—
201#334C2×D311yescharacteristic
202#333—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206Q8⋊C8C2×D311yescharacteristic
207Q8⋊C8C2×D311yescharacteristic
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322C2×D311yescharacteristic
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×Q8C2×D411yescharacteristic
213C2×D12C2×D411yescharacteristic
214C2×D12C2×D411yescharacteristic
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—41no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C3×Q16—22no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8C2×D411yescharacteristic
227C2×C3×C8C2×D411yescharacteristic
228C2×C3×C8C2×D411yescharacteristic
229C2×C3×C8C2×D411yescharacteristic
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4C2×D411yescharacteristic
233C3×C4⋊C4C2×D411yescharacteristic
234C3×C42—21no—
235C3×C42—21no—
236C3×C42C2411yescharacteristic
237(C2×D3)⋊C4—21no—
238(C2×D3)⋊C4—21no—
239(C2×D3)⋊C4—41no—
240(C2×D3)⋊C4—41no—
241(C2×D3)⋊C4—41no—
242(C2×D3)⋊C4—41no—
243(C2×D3)⋊C4—41no—
244(C2×D3)⋊C4—41no—
245(C2×D3)⋊C4—41no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3C2×D411yescharacteristic
255C4⋊Dic3C2×D411yescharacteristic
256Dic3⋊C4—41no—
257Dic3⋊C4—41no—
258Dic3⋊C4—41no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—41no—
262C4×Dic3—41no—
263C4×Dic3—41no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266Dic12—42no—
267Dic12—81no—
268D24—41no—
269#204—42no—
270#204—81no—
271#204—81no—
272ΓL(1,𝔽25)—42no—
273D3×C8—42no—
274C2×Q16—61no—
275C2×Q16—61no—
276C2×Q16D1211yescharacteristic
277C2×SD16—61no—
278C2×SD16—61no—
279C2×SD16—61no—
280C2×D8—31no—
281C2×M16—61no—
282C22×C8—61no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8—31no—
286C4⋊Q8—31no—
287#126—121no—
288#126—121no—
289#126—121no—
290#125—61no—
291#124—61no—
292#124—61no—
293#124—61no—
294C22⋊Q8—61no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297C22⋊Q8—61no—
298C22⋊Q8—121no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×Q8—61no—
302C4×Q8C22×D311yescharacteristic
303C4×Q8C22×D311yescharacteristic
304C4×D4—31no—
305C4×D4—31no—
306C4×D4—61no—
307#117—61no—
308#117—61no—
309#117—61no—
310C2×C4⋊C4—61no—
311#107—31no—
312#107—61no—
313#107—61no—
314#107—61no—
315#107—121no—
316#107D1211yescharacteristic
317#106—61no—
318#106—61no—
319#106—61no—
320#106—61no—
321#106—61no—
322#106—61no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—31no—
326C4⋊C8—31no—
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
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#103—61no—
338#103—121no—
339#103D1211yescharacteristic
340#103D1211yescharacteristic
341#102—31no—
342#102—31no—
343#102—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8C22×D311yescharacteristic
349C4×C8C22×D311yescharacteristic
350C4×C8C22×D311yescharacteristic
351C22×D3—21no—
352C22×D3—21no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×Q8—21no—
356C3×Q8—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4H(5,𝔽2)11yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370D12—21no—
371D12—21no—
372D12—21no—
373D12—21no—
374C4×D3—41no—
375C4×D3—41no—
376C4×D3—41no—
377C4×D3—41no—
378C4×D3—161no—
379Dic6—41no—
380Dic6—41no—
381Dic6—41no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—81no—
386C3×C8—21no—
387C3×C8—21no—
388C3×C8—21no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8#13512yes—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8C2×D1211yescharacteristic
400C2×Q8C2×D1211yescharacteristic
401C2×D4—31no—
402C2×D4—31no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—121no—
406Q16—22no—
407Q16—122no—
408Q16—241no—
409SD16—122no—
410SD16—241no—
411SD16—241no—
412D8—121no—
413M16—122no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—122no—
420C2×C8C2×D1211yescharacteristic
421C2×C8C2×D1211yescharacteristic
422C2×C8D3×D411yescharacteristic
423C2×C8D3×D411yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—31no—
427C4⋊C4—31no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—121no—
437C4⋊C4—121no—
438C4⋊C4—121no—
439C4⋊C4—121no—
440C4⋊C4—121no—
441C4⋊C4C2×D1211yescharacteristic
442C4⋊C4C2×D1211yescharacteristic
443C22⋊C4—61no—
444C22⋊C4—61no—
445C22⋊C4—121no—
446C22⋊C4—121no—
447C22⋊C4—121no—
448C22⋊C4—121no—
449C22⋊C4—121no—
450C22⋊C4—121no—
451C22⋊C4—121no—
452C42—21no—
453C42—21no—
454C42—121no—
455C42—121no—
456C42—121no—
457C42C23×D311yescharacteristic
458C22×C3D4211yescharacteristic
459C2×D3—41no—
460C2×D3—41no—
461C2×D3—41no—
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C3×C4—21no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4C2×#13511yescharacteristic
471C3×C4#57511yescharacteristic
472C3×C4#57711yescharacteristic
473C3×C4#57811yescharacteristic
474Dic3—41no—
475Dic3—41no—
476Dic3—81no—
477Dic3—81no—
478Dic3—81no—
479Dic3—81no—
480C23—61no—
481C23—61no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—21no—
486Q8—121no—
487Q8—121no—
488Q8—121no—
489Q8—121no—
490Q8—121no—
491Q8—121no—
492Q8—241no—
493D4—61no—
494D4—61no—
495D4—61no—
496D4—61no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—481no—
512C2×C4C22×D1211yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4#100911yescharacteristic
515C8—21no—
516C8—21no—
517C8—21no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—121no—
522C8—121no—
523C8#90412yes—
524C2×C3#327111yescharacteristic
525C2×C3#328311yescharacteristic
526C2×C3#328911yescharacteristic
527D3—81no—
528D3—81no—
529C22—121no—
530C22—121no—
531C22—121no—
532C22—121no—
533C22—121no—
534C22—121no—
535C22D4×D1211yescharacteristic
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—121no—
542C4—121no—
543C4—241no—
544C4—241no—
545C4—241no—
546C4—241no—
547C4C2×#90411yescharacteristic
548C4#575311yescharacteristic
549C4#577711yescharacteristic
550C4#577811yescharacteristic
551C3#1986611yescharacteristic, complemented, Sylow
552C2—241no—
553C2—241no—
554C2#7943611yescharacteristic
555C2#7984011yescharacteristic
556C2#7990111yescharacteristic
subgroupG/H
centerC22D4×D12
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1579C2
socleC22×C3D42
prime radicalC2×D24D4
hypercenter#1579D3
hypocenterC3#19866
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1579

Complements

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

NH ≅ G/Ntype
3. #79745C2semidirect
4. #79743C2semidirect
5. C3×#1579C2semidirect
6. #70947C2semidirect
7. #70942C2semidirect
8. #68542C2semidirect
9. #68537C2semidirect
10. #68250C2semidirect
11. #68250C2semidirect
12. #68249C2semidirect
13. #68247C2semidirect
14. #68099C2semidirect
82. #1579D3semidirect
551. C3#19866semidirect

Maximal subgroups

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

subgroupindex
#799262
#799122
#797452
#797432
C3×#15792
#709472
#709422
#685422
#685372
#682502
#682492
#682472
#680992
#680782
#198663