Finite Group Databasebeta

← #10786196

Subgroups — #10786196

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#81902C211yesmaximal, characteristic, complemented
2#81087C211yesmaximal, characteristic
3#80421C211yesmaximal, characteristic
4#79419C211yesmaximal, characteristic, complemented
5C3×#1671C211yesmaximal, characteristic, complemented
6#71341C211yesmaximal, characteristic, complemented
7#71147C211yesmaximal, characteristic, complemented
8#71049C211yesmaximal, characteristic, complemented
9#70841C211yesmaximal, characteristic
10#68542C211yesmaximal, characteristic, complemented
11#68517C211yesmaximal, characteristic, complemented
12#68511C211yesmaximal, characteristic, complemented
13#68494C211yesmaximal, characteristic, complemented
14#68465C211yesmaximal, characteristic, complemented
15#68437C211yesmaximal, characteristic, complemented
16#19866—31nomaximal, Sylow
17#5732C2211yescharacteristic
18#5730C2211yescharacteristic
19#5542C2211yescharacteristic
20#5542C2211yescharacteristic
21C3×#498C2211yescharacteristic
22C3×#476C2211yescharacteristic
23C3×#461C2211yescharacteristic
24C3×C8×D4C2211yescharacteristic
25#5168—21no—
26#5166—21no—
27#5116—21no—
28#5106—21no—
29#5094C2211yescharacteristic
30#5093C2211yescharacteristic
31#5092C2211yescharacteristic
32#5091C2211yescharacteristic
33#5079C2211yescharacteristic
34#5078C2211yescharacteristic
35C4×#213C2211yescharacteristic
36#4868—21no—
37#4867—21no—
38#4862—21no—
39#4861—21no—
40#4857—21no—
41#4852—21no—
42#4851C2211yescharacteristic
43#4851C2211yescharacteristic
44#4841C2211yescharacteristic
45#4841C2211yescharacteristic
46#4830C2211yescharacteristic
47#4830C2211yescharacteristic
48#4798—21no—
49#4790—21no—
50#4787—21no—
51#4768—21no—
52#4723—21no—
53#4723—21no—
54C4×Dic12C2211yescharacteristic
55#4688C2211yescharacteristic
56#4682C2211yescharacteristic
57C3×#334C2211yescharacteristic
58C3×#333C2211yescharacteristic
59C3×#332C2211yescharacteristic
60D4⋊(C3⋊C8)C2211yescharacteristic
61#4484C2211yescharacteristic
62#4484C2211yescharacteristic
63#4483C2211yescharacteristic
64#4482C2211yescharacteristic
65#4482C2211yescharacteristic
66#4479C2211yescharacteristic
67#4455C2211yescharacteristic
68#3393—31no—
69#3360—31no—
70#3356—31no—
71#3342—31no—
72#1723—31no—
73#1693—31no—
74#1684—31no—
75#1671D311yescharacteristic, complemented
76#1670—31no—
77#1650—31no—
78#1650—31no—
79#1643—31no—
80#1638—31no—
81#1581—31no—
82#1579—31no—
83C2×C3×SD16—21no—
84C22×C3×C8—21no—
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4×D4C2311yescharacteristic
88#947—21no—
89#946—41no—
90Q8×Dic3—21no—
91C2×#216—21no—
92C2×#216—21no—
93#936—21no—
94#936—21no—
95C2×#213D411yescharacteristic
96#926—21no—
97#926—21no—
98#924—21no—
99#924—21no—
100C2×Dic12D411yescharacteristic
101C2×#204—21no—
102C2×#204—21no—
103C2×ΓL(1,𝔽25)—21no—
104#898—41no—
105#897—21no—
106#897—21no—
107#893—21no—
108#892—21no—
109D3×C4⋊C4—21no—
110#890—21no—
111#887—21no—
112#887—21no—
113#879—41no—
114#879—41no—
115C4×D12C2311yescharacteristic
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118C4×Dic6C2311yescharacteristic
119C4×Dic6C2311yescharacteristic
120C3×#106—21no—
121C3×#106—21no—
122C3×#106—21no—
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×C4⋊C8C2311yescharacteristic
126C3×#103—21no—
127C3×#102—21no—
128C3×C22⋊C8—21no—
129C3×C4×C8C2311yescharacteristic
130#835—41no—
131#832D411yescharacteristic
132(C2×D3)⋊C8—21no—
133#818—21no—
134#818—21no—
135#818D411yescharacteristic
136#817—21no—
137#817—21no—
138#816—21no—
139#816—21no—
140#816—21no—
141#816—21no—
142#816D411yescharacteristic
143#816D411yescharacteristic
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148#810—21no—
149#810—21no—
150#810—21no—
151#810—21no—
152#809D411yescharacteristic
153#808—21no—
154#807—21no—
155#807—41no—
156#807D411yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#556—31no—
161#556—31no—
162#556—31no—
163#554—31no—
164#500—31no—
165#500—31no—
166#498C2×D311yescharacteristic
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#476C2×D311yescharacteristic
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#461C2×D311yescharacteristic
190C8×Q8—31no—
191C8×Q8—31no—
192C4×Q16—31no—
193C4×Q16—61no—
194C4×SD16—61no—
195C4×D8—31no—
196#436—31no—
197C8×D4C2×D311yescharacteristic
198#428—61no—
199#427—61no—
200#335—31no—
201#334C2×D311yescharacteristic
202#333C2×D311yescharacteristic
203#332—31no—
204#332—31no—
205#332C2×D311yescharacteristic
206Q8⋊C8—31no—
207Q8⋊C8—31no—
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C2×D12C2×D411yescharacteristic
216C2×C4×D3—21no—
217C2×C4×D3—41no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6C2×D411yescharacteristic
221C2×Dic6C2×D411yescharacteristic
222C3×SD16—81no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—42no—
227C2×C3×C8C2×D411yescharacteristic
228C2×C3×C8C2×D411yescharacteristic
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4C2×D411yescharacteristic
234C3×C22⋊C4—21no—
235C3×C42C2411yescharacteristic
236#217—41no—
237#217—41no—
238#217—41no—
239#217—41no—
240#216—42no—
241#216—81no—
242#213—41no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—41no—
245(C2×D3)⋊C4—41no—
246(C2×D3)⋊C4—41no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—41no—
252C4⋊Dic3—41no—
253C4⋊Dic3C2×D411yescharacteristic
254C4⋊Dic3C2×D411yescharacteristic
255C4⋊Dic3C2×D411yescharacteristic
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—41no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—41no—
264C4×Dic3—41no—
265C4×Dic3—41no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270Dic12—22no—
271#204—42no—
272#204—81no—
273ΓL(1,𝔽25)—42no—
274C2×Q16—31no—
275C2×Q16—61no—
276C2×Q16—61no—
277C2×SD16—21no—
278C2×SD16—61no—
279C2×SD16—61no—
280C2×D8—31no—
281C2×M16—61no—
282C22×C8—21no—
283C4⋊Q8—31no—
284C4⋊Q8—31no—
285C4⋊Q8C22×D311yescharacteristic
286C4⋊Q8C22×D311yescharacteristic
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—31no—
300C4×Q8—31no—
301C4×Q8—61no—
302C4×Q8—61no—
303C4×Q8—61no—
304C4×D4—31no—
305C4×D4—61no—
306C4×D4C22×D311yescharacteristic
307#117—61no—
308#117—61no—
309#117—61no—
310C2×C4⋊C4—61no—
311#107—31no—
312#107—31no—
313#107—61no—
314#107—61no—
315#107—61no—
316#107—121no—
317#106—21no—
318#106—21no—
319#106—21no—
320#106—61no—
321#106—61no—
322#106—61no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—21no—
330#103—31no—
331#103—31no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—121no—
341#102—21no—
342#102—31no—
343#102—31no—
344C22⋊C8—21no—
345C22⋊C8—61no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8C22×D311yescharacteristic
351C23×C3—21no—
352C22×D3—21no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×Q8—81no—
356C3×D4—21no—
357C3×D4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4H(5,𝔽2)11yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373D12—21no—
374D12—21no—
375C4×D3—41no—
376C4×D3—41no—
377C4×D3—161no—
378Dic6—21no—
379Dic6—21no—
380Dic6—21no—
381Dic6—21no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386C3×C8—21no—
387C3×C8—41no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8#13512yes—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—31no—
398C2×Q8—31no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×D4—31no—
402C2×D4D3×D411yescharacteristic
403C22×C4—21no—
404C22×C4—61no—
405C22×C4—121no—
406Q16—62no—
407Q16—122no—
408Q16—241no—
409SD16—81no—
410SD16—122no—
411SD16—241no—
412D8—121no—
413M16—122no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—31no—
418C2×C8—31no—
419C2×C8—42no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8D3×D411yescharacteristic
423C2×C8D3×D411yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—31no—
429C4⋊C4—31no—
430C4⋊C4—31no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—121no—
438C4⋊C4—121no—
439C4⋊C4—121no—
440C4⋊C4—121no—
441C4⋊C4—121no—
442C4⋊C4D3×D411yescharacteristic
443C22⋊C4—21no—
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—61no—
453C42—61no—
454C42—121no—
455C42—121no—
456C42—121no—
457C42C23×D311yescharacteristic
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3D4211yescharacteristic
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—41no—
465C3×C4—21no—
466C3×C4—41no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4C2×#13511yescharacteristic
470C3×C4#57511yescharacteristic
471C3×C4#57711yescharacteristic
472C3×C4#57811yescharacteristic
473Dic3—41no—
474Dic3—41no—
475Dic3—41no—
476Dic3—41no—
477Dic3—41no—
478Dic3—81no—
479Dic3—81no—
480C23—21no—
481C23—61no—
482Q8—41no—
483Q8—41no—
484Q8—61no—
485Q8—61no—
486Q8—61no—
487Q8—61no—
488Q8—81no—
489Q8—121no—
490Q8—121no—
491Q8—121no—
492Q8—121no—
493D4—21no—
494D4—21no—
495D4—61no—
496D4—61no—
497C2×C4—21no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—481no—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4#100911yescharacteristic
515C8—21no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—61no—
520C8—61no—
521C8—121no—
522C8—121no—
523C8#91612yes—
524C2×C3—81no—
525C2×C3#327111yescharacteristic
526C2×C3#328311yescharacteristic
527C2×C3#328911yescharacteristic
528D3—81no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—121no—
533C22—121no—
534C22—121no—
535C22#560911yescharacteristic
536C4—21no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—121no—
541C4—121no—
542C4—121no—
543C4—121no—
544C4—121no—
545C4—241no—
546C4—241no—
547C4C2×#91611yescharacteristic
548C4#576211yescharacteristic
549C4#576811yescharacteristic
550C4#577911yescharacteristic
551C3#1986611yescharacteristic, complemented, Sylow
552C2—81no—
553C2—241no—
554C2#8044911yescharacteristic
555C2#8068811yescharacteristic
556C2#8093311yescharacteristic
subgroupG/H
centerC22#5609
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1671C2
socleC22×C3D42
prime radicalC2×#213D4
hypercenter#1671D3
hypocenterC3#19866
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1671

Complements

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

NH ≅ G/Ntype
1. #81902C2semidirect
4. #79419C2semidirect
5. C3×#1671C2semidirect
6. #71341C2semidirect
7. #71147C2semidirect
8. #71049C2semidirect
10. #68542C2semidirect
11. #68517C2semidirect
12. #68511C2semidirect
13. #68494C2semidirect
14. #68465C2semidirect
15. #68437C2semidirect
75. #1671D3semidirect
551. C3#19866semidirect

Maximal subgroups

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

subgroupindex
#819022
#810872
#804212
#794192
C3×#16712
#713412
#711472
#710492
#708412
#685422
#685172
#685112
#684942
#684652
#684372
#198663