Finite Group Databasebeta

← #10777941

Subgroups — #10777941

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#80949C211yesmaximal, characteristic
3#80409C211yesmaximal, characteristic
4#79517C211yesmaximal, characteristic
5C3×#1699C211yesmaximal, characteristic, complemented
6#71242C211yesmaximal, characteristic, complemented
7#71105C211yesmaximal, characteristic, complemented
8#71041C211yesmaximal, characteristic, complemented
9#70875C211yesmaximal, characteristic, complemented
10#68595C211yesmaximal, characteristic, complemented
11#68567C211yesmaximal, characteristic, complemented
12#68475C211yesmaximal, characteristic, complemented
13#68473C211yesmaximal, characteristic, complemented
14#68419C211yesmaximal, characteristic, complemented
15#68419C211yesmaximal, characteristic, complemented
16#19161—31nomaximal, Sylow
17#5728C2211yescharacteristic
18#5728C2211yescharacteristic
19C4×#237C2211yescharacteristic
20C3×#487C2211yescharacteristic
21C3×#474C2211yescharacteristic
22C3×#461C2211yescharacteristic
23C3×#436C2211yescharacteristic
24#5173—21no—
25Dic3×SD16—21no—
26#5127—21no—
27Dic3×M16—21no—
28#5094C2211yescharacteristic
29#5092C2211yescharacteristic
30#5090C2211yescharacteristic
31#5078C2211yescharacteristic
32#5019C2211yescharacteristic
33#5012C2211yescharacteristic
34#5007C2211yescharacteristic
35#4888—21no—
36#4873—21no—
37#4851C2211yescharacteristic
38#4849C2211yescharacteristic
39#4833C2211yescharacteristic
40#4833C2211yescharacteristic
41#4833C2211yescharacteristic
42#4831C2211yescharacteristic
43#4815—21no—
44#4813—21no—
45#4813—21no—
46#4790—21no—
47#4790—21no—
48#4790—21no—
49#4783—21no—
50#4758—21no—
51#4728—21no—
52#4718—21no—
53C4×D3×C8C2211yescharacteristic
54#4681C2211yescharacteristic
55#4679C2211yescharacteristic
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#1700—31no—
74#1699—31no—
75#1699D311yescharacteristic, complemented
76#1694—31no—
77#1693—31no—
78#1690—31no—
79#1689—31no—
80#1564—31no—
81#1544—31no—
82#1544—31no—
83C2×#237D411yescharacteristic
84C2×C3×SD16—21no—
85C2×C3×M16—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4×D4C2311yescharacteristic
89#946—21no—
90#946—21no—
91Q8×Dic3—21no—
92Q8×Dic3—21no—
93C2×#216—21no—
94C2×#216—21no—
95C2×#216—21no—
96D4×Dic3D411yescharacteristic
97C2×#214—21no—
98C4×#67—21no—
99C2×C4×Dic3—21no—
100C2×#208—21no—
101C2×Dic12—21no—
102C2×D3×C8D411yescharacteristic
103C2×D3×C8D411yescharacteristic
104#897—21no—
105#897—21no—
106#892—21no—
107#892—21no—
108#892D411yescharacteristic
109#890—21no—
110#890—21no—
111#887—21no—
112#887—21no—
113#887D411yescharacteristic
114#67⋊C4—21no—
115#877—21no—
116C42×D3C2311yescharacteristic
117#869C2311yescharacteristic
118#869C2311yescharacteristic
119C4×Dic6C2311yescharacteristic
120C3×#107—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×#103—21no—
125C3×#103—21no—
126C3×#103—21no—
127C3×#102—21no—
128C3×C22⋊C8—21no—
129C3×C4×C8C2311yescharacteristic
130#835—21no—
131#832—21no—
132#830—21no—
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136#817—21no—
137#816—21no—
138#816—21no—
139#816—21no—
140#816—21no—
141#815—21no—
142#815—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145C8×Dic3—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148C8×Dic3D411yescharacteristic
149C8×Dic3D411yescharacteristic
150#810—21no—
151#810—21no—
152#808—21no—
153#807—21no—
154#807—21no—
155#807—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
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#475—31no—
171#474—31no—
172#474C2×D311yescharacteristic
173#467—61no—
174#467—61no—
175#462—31no—
176#462—31no—
177#461—31no—
178#461C2×D311yescharacteristic
179#452—61no—
180#452—61no—
181#446—31no—
182#446—31no—
183C4×Q16—61no—
184C4×Q16—61no—
185C4×Q16—61no—
186C4×Q16—61no—
187C4×SD16—61no—
188C4×SD16—61no—
189#436—31no—
190#436C2×D311yescharacteristic
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×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C2×#67—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218#237—41no—
219#237—41no—
220C2×C4×D3C2×D411yescharacteristic
221C2×C4×D3C2×D411yescharacteristic
222C2×C4×D3#4111yescharacteristic
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C2×D411yescharacteristic
226C3×SD16—42no—
227C3×M16—42no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8C2×D411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4C2×D411yescharacteristic
238C3×C22⋊C4—21no—
239C3×C42C2411yescharacteristic
240#217—21no—
241#216—42no—
242#216—42no—
243#216—42no—
244#214—42no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—41no—
247(C2×D3)⋊C4—41no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—41no—
253C4⋊Dic3—41no—
254C4⋊Dic3C2×D411yescharacteristic
255Dic3⋊C4—21no—
256Dic3⋊C4—21no—
257Dic3⋊C4—41no—
258Dic3⋊C4—41no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—41no—
263C4×Dic3—41no—
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3#4111yescharacteristic
267#208—42no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273Dic12—42no—
274D3×C8—22no—
275D3×C8—41no—
276C2×#41—31no—
277C2×Q16—61no—
278C2×Q16—61no—
279C2×Q16—61no—
280C2×Q16—61no—
281C2×SD16—21no—
282C2×SD16—61no—
283C2×M16—21no—
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—31no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×Q8—61no—
302C4×Q8—61no—
303C4×D4—31no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4C22×D311yescharacteristic
307#117—31no—
308#117—61no—
309#117—61no—
310#117—61no—
311C2×C42—31no—
312C2×C42—61no—
313#107—21no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—61no—
318#106—61no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—21no—
328#103—21no—
329#103—21no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#102—21no—
338#102—61no—
339C22⋊C8—21no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—61no—
345#97—61no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—61no—
350C4×C8—61no—
351C4×C8—61no—
352C4×C8C22×D311yescharacteristic
353C23×C3—21no—
354C22×D3#12111yescharacteristic
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×D4—21no—
360C3×D4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369#67—42no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3#12111yescharacteristic
377C2×Dic3#12111yescharacteristic
378C2×Dic3#12111yescharacteristic
379C4×D3—21no—
380C4×D3—21no—
381C4×D3—21no—
382C4×D3—21no—
383C4×D3—22no—
384Dic6—21no—
385Dic6—21no—
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—31no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×D4—61no—
408C2×D4D3×D411yescharacteristic
409C22×C4—21no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—61no—
414C22×C4—61no—
415Q16—122no—
416Q16—122no—
417Q16—122no—
418Q16—122no—
419SD16—42no—
420SD16—122no—
421M16—42no—
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—31no—
440C4⋊C4—61no—
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—21no—
452C22⋊C4—61no—
453C22⋊C4—61no—
454C22⋊C4—121no—
455C22⋊C4—121no—
456C42—31no—
457C42—31no—
458C42—31no—
459C42—61no—
460C42—61no—
461C42—61no—
462C42—121no—
463C42—121no—
464C42C23×D311yescharacteristic
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3C2×#12111yescharacteristic
469C2×D3—21no—
470C2×D3—21no—
471C2×D3—21no—
472C3×C4—21no—
473C3×C4—41no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4C2×#13511yescharacteristic
477C3×C4#57511yescharacteristic
478C3×C4#57511yescharacteristic
479C3×C4#57511yescharacteristic
480Dic3—21no—
481Dic3—21no—
482Dic3—41no—
483Dic3—41no—
484Dic3—81no—
485Dic3—81no—
486Dic3#47312yes—
487C23—21no—
488C23—31no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—61no—
494Q8—61no—
495Q8—121no—
496Q8—121no—
497Q8—121no—
498Q8—121no—
499D4—21no—
500D4—21no—
501D4—122no—
502C2×C4—21no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—62no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—122no—
518C2×C4—122no—
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—81no—
535C2×C3#303511yescharacteristic
536C2×C3#305411yescharacteristic
537C2×C3C2×#47311yescharacteristic
538D3—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—61no—
543C22—61no—
544C22—61no—
545C22D3×#12111yescharacteristic
546C4—21no—
547C4—32no—
548C4—41no—
549C4—61no—
550C4—61no—
551C4—81no—
552C4—81no—
553C4—121no—
554C4—121no—
555C4—241no—
556C4—241no—
557C4D3×#13511yescharacteristic
558C4#577311yescharacteristic
559C4#577311yescharacteristic
560C4#577711yescharacteristic
561C3#1916111yescharacteristic, complemented, Sylow
562C2—81no—
563C2—121no—
564C2#8053311yescharacteristic
565C2#8080411yescharacteristic
566C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1699C2
socleC22×C3C2×#121
prime radicalC2×#237D4
hypercenter#1699D3
hypocenterC3#19161
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1699

Complements

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

NH ≅ G/Ntype
1. #81593C2semidirect
5. C3×#1699C2semidirect
6. #71242C2semidirect
7. #71105C2semidirect
8. #71041C2semidirect
9. #70875C2semidirect
10. #68595C2semidirect
11. #68567C2semidirect
12. #68475C2semidirect
13. #68473C2semidirect
14. #68419C2semidirect
15. #68419C2semidirect
75. #1699D3semidirect
561. C3#19161semidirect

Maximal subgroups

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

subgroupindex
#815932
#809492
#804092
#795172
C3×#16992
#712422
#711052
#710412
#708752
#685952
#685672
#684752
#684732
#684192
#191613