Finite Group Databasebeta

← #10777943

Subgroups — #10777943

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81593C211yesmaximal, characteristic, complemented
2#80997C211yesmaximal, characteristic
3#80457C211yesmaximal, characteristic
4#79938C211yesmaximal, characteristic
5C3×#1700C211yesmaximal, characteristic, complemented
6#71234C211yesmaximal, characteristic, complemented
7#71121C211yesmaximal, characteristic, complemented
8#71057C211yesmaximal, characteristic, complemented
9#70966C211yesmaximal, characteristic, complemented
10#68585C211yesmaximal, characteristic, complemented
11#68567C211yesmaximal, characteristic, complemented
12#68451C211yesmaximal, characteristic, complemented
13#68449C211yesmaximal, characteristic, complemented
14#68419C211yesmaximal, characteristic, complemented
15#68419C211yesmaximal, characteristic, complemented
16#19161—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5728C2211yescharacteristic
19C4×#239C2211yescharacteristic
20C3×#487C2211yescharacteristic
21C3×#475C2211yescharacteristic
22C3×#462C2211yescharacteristic
23C3×#446C2211yescharacteristic
24#5187—21no—
25Dic3×Q16—21no—
26#5092C2211yescharacteristic
27#5087C2211yescharacteristic
28#5083C2211yescharacteristic
29#5078C2211yescharacteristic
30#5031C2211yescharacteristic
31#5024C2211yescharacteristic
32#5020C2211yescharacteristic
33#4868—21no—
34#4852—21no—
35#4846C2211yescharacteristic
36#4841C2211yescharacteristic
37#4838C2211yescharacteristic
38#4836—21no—
39#4833—21no—
40#4833C2211yescharacteristic
41#4833C2211yescharacteristic
42#4833C2211yescharacteristic
43#4814—21no—
44#4813—21no—
45#4813—21no—
46#4808—21no—
47#4790—21no—
48#4790—21no—
49#4790—21no—
50#4788—21no—
51#4707—21no—
52#4706—21no—
53#4688C2211yescharacteristic
54C4×D3×C8C2211yescharacteristic
55#4681C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4547C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4482C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4480C2211yescharacteristic
67#4458C2211yescharacteristic
68#3170—31no—
69#3107—31no—
70#3107—31no—
71#2965—31no—
72#1700—31no—
73#1700D311yescharacteristic, complemented
74#1699—31no—
75#1699—31no—
76#1694—31no—
77#1693—31no—
78#1690—31no—
79#1689—31no—
80#1564—31no—
81#1544—31no—
82#1544—31no—
83C2×#239D411yescharacteristic
84C2×C3×Q16—21no—
85C3×C4⋊Q8C2311yescharacteristic
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4×Q8C2311yescharacteristic
88#946—21no—
89#946—21no—
90Q8×Dic3—21no—
91Q8×Dic3—21no—
92Q8×Dic3D411yescharacteristic
93C2×#216—21no—
94C2×#216—21no—
95C2×#216—21no—
96C2×#215—21no—
97C2×#204—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×ΓL(1,𝔽25)—21no—
100C2×D3×C8D411yescharacteristic
101C2×D3×C8D411yescharacteristic
102#897—21no—
103#897—21no—
104#893—21no—
105#893D411yescharacteristic
106#892—21no—
107#892—21no—
108#892—21no—
109#892D411yescharacteristic
110#890—21no—
111#890—21no—
112#887—21no—
113#887—21no—
114C4×D12—21no—
115C4×D12C2311yescharacteristic
116C42×D3—21no—
117C42×D3C2311yescharacteristic
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C3×#106—21no—
121C3×C4⋊C8—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×#103—21no—
126C3×#103—21no—
127C3×#103—21no—
128C3×#103—21no—
129C3×#97—21no—
130C3×C4×C8C2311yescharacteristic
131#835—21no—
132#835—21no—
133#821—21no—
134(C2×D3)⋊C8—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#816—21no—
141#816—21no—
142#816—21no—
143C8×Dic3—21no—
144C8×Dic3—21no—
145C8×Dic3—21no—
146C8×Dic3D411yescharacteristic
147C8×Dic3D411yescharacteristic
148#810—21no—
149#809—21no—
150#808—21no—
151#807—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158#803—21no—
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161#533—31no—
162C4×#41—31no—
163#500—31no—
164#494—31no—
165#487—31no—
166#487C2×D311yescharacteristic
167#485—61no—
168#485—61no—
169#475—31no—
170#475C2×D311yescharacteristic
171#474—31no—
172#474—31no—
173#467—61no—
174#467—61no—
175#462—31no—
176#462C2×D311yescharacteristic
177#461—31no—
178#461—31no—
179#452—61no—
180#452—61no—
181#446—31no—
182#446C2×D311yescharacteristic
183C4×Q16—61no—
184C4×Q16—61no—
185C4×Q16—61no—
186C4×Q16—61no—
187C4×SD16—61no—
188C4×SD16—61no—
189#436—31no—
190#436—31no—
191#435—61no—
192#435—61no—
193#431—31no—
194#431—31no—
195#431—31no—
196#431—61no—
197C4×M16—61no—
198C2×C4×C8—31no—
199#335—31no—
200#335C2×D311yescharacteristic
201#334—31no—
202#334—31no—
203#332—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332C2×D311yescharacteristic
210#332C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D411yescharacteristic
214#239—41no—
215#239—41no—
216C2×D12—21no—
217C2×D12C2×D411yescharacteristic
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3C2×D411yescharacteristic
222C2×C4×D3C2×D411yescharacteristic
223C2×C4×D3#4111yescharacteristic
224C2×Dic6—21no—
225C2×Dic6—21no—
226C3×Q16—42no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8C2×D411yescharacteristic
231C2×C3×C8C2×D411yescharacteristic
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4C2×D411yescharacteristic
238C3×C42—21no—
239C3×C42C2411yescharacteristic
240#216—42no—
241#216—42no—
242#216—42no—
243#215—42no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—41no—
248(C2×D3)⋊C4—41no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—41no—
255C4⋊Dic3—41no—
256C4⋊Dic3C2×D411yescharacteristic
257Dic3⋊C4—41no—
258Dic3⋊C4—41no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—41no—
262C4×Dic3—41no—
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3#4111yescharacteristic
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271#204—42no—
272ΓL(1,𝔽25)—42no—
273ΓL(1,𝔽25)—42no—
274D3×C8—22no—
275D3×C8—41no—
276C2×#41—31no—
277C2×Q16—21no—
278C2×Q16—61no—
279C2×Q16—61no—
280C2×Q16—61no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×M16—61no—
284C2×M16—61no—
285C22×C8—31no—
286C22×C8—31no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8C22×D311yescharacteristic
290C4⋊Q8C22×D311yescharacteristic
291#125—61no—
292#125—61no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297C4×Q8—31no—
298C4×Q8—61no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×Q8—61no—
302C4×Q8C22×D311yescharacteristic
303C4×D4—31no—
304C4×D4—31no—
305C4×D4—61no—
306C4×D4—61no—
307#117—31no—
308#117—61no—
309#117—61no—
310#117—61no—
311C2×C42—31no—
312C2×C42—61no—
313#107—61no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—21no—
318#106—61no—
319C4⋊C8—21no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—61no—
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—21no—
328#103—21no—
329#103—21no—
330#103—21no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—21no—
345#97—61no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—61no—
350C4×C8—61no—
351C4×C8—61no—
352C4×C8C22×D311yescharacteristic
353C22×D3—21no—
354C22×D3#12111yescharacteristic
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3#12111yescharacteristic
372C2×Dic3#12111yescharacteristic
373C2×Dic3#12111yescharacteristic
374D12—21no—
375D12—21no—
376D12—42no—
377C4×D3—21no—
378C4×D3—21no—
379C4×D3—21no—
380C4×D3—21no—
381C4×D3—22no—
382C4×D3—42no—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—42no—
386Dic6—41no—
387Dic6—41no—
388Dic6—41no—
389Dic6—41no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—41no—
395C3⋊C8—21no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8#13512yes—
400#41—121no—
401#41—121no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8D3×D411yescharacteristic
407C2×D4—31no—
408C2×D4—61no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415Q16—42no—
416Q16—122no—
417Q16—122no—
418Q16—122no—
419SD16—122no—
420SD16—122no—
421M16—122no—
422M16—122no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—31no—
427C2×C8—31no—
428C2×C8—62no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—121no—
433C2×C8D3×D411yescharacteristic
434C2×C8D3×D411yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—31no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—121no—
447C4⋊C4—121no—
448C4⋊C4—121no—
449C4⋊C4—121no—
450C4⋊C4D3×D411yescharacteristic
451C22⋊C4—61no—
452C22⋊C4—61no—
453C22⋊C4—61no—
454C22⋊C4—121no—
455C22⋊C4—121no—
456C42—21no—
457C42—31no—
458C42—31no—
459C42—31no—
460C42—61no—
461C42—61no—
462C42—121no—
463C42—121no—
464C42C23×D311yescharacteristic
465C22×C3C2×#12111yescharacteristic
466C2×D3—21no—
467C2×D3—21no—
468C2×D3—21no—
469C2×D3—41no—
470C2×D3—41no—
471C2×D3—41no—
472C3×C4—21no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4C2×#13511yescharacteristic
478C3×C4#57511yescharacteristic
479C3×C4#57511yescharacteristic
480C3×C4#57511yescharacteristic
481Dic3—21no—
482Dic3—21no—
483Dic3—41no—
484Dic3—81no—
485Dic3—81no—
486Dic3#47312yes—
487C23—31no—
488C23—61no—
489Q8—21no—
490Q8—21no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—41no—
495Q8—121no—
496Q8—121no—
497Q8—121no—
498Q8—121no—
499D4—61no—
500D4—61no—
501D4—122no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—31no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—62no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4D3×#4111yescharacteristic
524C8—21no—
525C8—21no—
526C8—32no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—61no—
531C8—121no—
532C8—121no—
533C8—121no—
534C2×C3#303511yescharacteristic
535C2×C3#305411yescharacteristic
536C2×C3C2×#47311yescharacteristic
537D3—41no—
538D3—81no—
539C22—61no—
540C22—61no—
541C22—61no—
542C22—121no—
543C22—121no—
544C22—121no—
545C22D3×#12111yescharacteristic
546C4—21no—
547C4—32no—
548C4—41no—
549C4—41no—
550C4—61no—
551C4—61no—
552C4—81no—
553C4—81no—
554C4—121no—
555C4—241no—
556C4—241no—
557C4D3×#13511yescharacteristic
558C4#577311yescharacteristic
559C4#577711yescharacteristic
560C4#577711yescharacteristic
561C3#1916111yescharacteristic, complemented, Sylow
562C2—121no—
563C2—241no—
564C2#8059711yescharacteristic
565C2#8080411yescharacteristic
566C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1700C2
socleC22×C3C2×#121
prime radicalC2×#239D4
hypercenter#1700D3
hypocenterC3#19161
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1700

Complements

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

NH ≅ G/Ntype
1. #81593C2semidirect
5. C3×#1700C2semidirect
6. #71234C2semidirect
7. #71121C2semidirect
8. #71057C2semidirect
9. #70966C2semidirect
10. #68585C2semidirect
11. #68567C2semidirect
12. #68451C2semidirect
13. #68449C2semidirect
14. #68419C2semidirect
15. #68419C2semidirect
73. #1700D3semidirect
561. C3#19161semidirect

Maximal subgroups

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

subgroupindex
#815932
#809972
#804572
#799382
C3×#17002
#712342
#711212
#710572
#709662
#685852
#685672
#684512
#684492
#684192
#191613