Finite Group Databasebeta

← #10777940

Subgroups — #10777940

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
1C3×#3107C211yesmaximal, characteristic
2#83344C211yesmaximal, characteristic, complemented
3#82534C211yesmaximal, characteristic
4#82517C211yesmaximal, characteristic
5#71829C211yesmaximal, characteristic, complemented
6#71122C211yesmaximal, characteristic, complemented
7#71121C211yesmaximal, characteristic, complemented
8#71106C211yesmaximal, characteristic, complemented
9#71105C211yesmaximal, characteristic, complemented
10#71058C211yesmaximal, characteristic, complemented
11#71057C211yesmaximal, characteristic, complemented
12#71042C211yesmaximal, characteristic, complemented
13#71041C211yesmaximal, characteristic, complemented
14#70777C211yesmaximal, characteristic, complemented
15#70745C211yesmaximal, characteristic, complemented
16#19161—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#485—21no—
21C3×#475C2211yescharacteristic
22C3×#474C2211yescharacteristic
23C3×#462C2211yescharacteristic
24C3×#461C2211yescharacteristic
25C3×#452—21no—
26C3×C4×Q16—21no—
27C3×C4×SD16—21no—
28C3×#431C2211yescharacteristic
29#5239—21no—
30#5093C2211yescharacteristic
31#5092C2211yescharacteristic
32#5079C2211yescharacteristic
33#5078C2211yescharacteristic
34#5051—21no—
35#5051—21no—
36#5031C2211yescharacteristic
37C4×#216—21no—
38C4×#216—21no—
39C4×#216—21no—
40#5024C2211yescharacteristic
41#5024C2211yescharacteristic
42#5020C2211yescharacteristic
43#5019C2211yescharacteristic
44C4×#214—21no—
45#5012C2211yescharacteristic
46#5012C2211yescharacteristic
47#5011—21no—
48#5011—21no—
49#5007C2211yescharacteristic
50#4996—21no—
51#4925—21no—
52#4925C2211yescharacteristic
53#4925C2211yescharacteristic
54C4×#208—21no—
55C2×C4×C3⋊C8C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4547C2211yescharacteristic
59#4547C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3170—31no—
69#3107—31no—
70#3107D311yescharacteristic
71#2965—31no—
72#1700—31no—
73#1700—31no—
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×C3×#41D411yescharacteristic
84C2×C3×Q16—21no—
85C2×C3×SD16—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×#125—21no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8D411yescharacteristic
94C3×C4×Q8C2311yescharacteristic
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4D411yescharacteristic
98C3×C4×D4C2311yescharacteristic
99C3×#117—21no—
100C3×#117—21no—
101C3×#117D411yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C2×#216—21no—
105C2×#216—21no—
106C2×#216—21no—
107C2×#214—21no—
108#926—21no—
109#924—21no—
110#924—21no—
111C2×#208—21no—
112C2×#208—21no—
113C22×C3⋊C8D411yescharacteristic
114C22×C3⋊C8D411yescharacteristic
115#870—21no—
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118C4×Dic6—21no—
119C4×Dic6—21no—
120C3×#107—21no—
121C3×#106—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×#103—21no—
125C3×#103—21no—
126C3×#103—21no—
127C3×#102—21no—
128C3×C22⋊C8—21no—
129C3×C4×C8—21no—
130#835—21no—
131#835—21no—
132#835—21no—
133#832—21no—
134#830—21no—
135#830—21no—
136#830—21no—
137#830—21no—
138#810—21no—
139#810—21no—
140#810—21no—
141#810—21no—
142#808—21no—
143#807—21no—
144#807—21no—
145#807—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152#803—21no—
153#803—21no—
154C4×C3⋊C8—21no—
155C4×C3⋊C8—21no—
156C4×C3⋊C8D411yescharacteristic
157C4×C3⋊C8D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161#533C2×D311yescharacteristic
162C4×#41C2×D311yescharacteristic
163#500—31no—
164#494—31no—
165#487—31no—
166#487—31no—
167#485—21no—
168#485—61no—
169#475—31no—
170#475C2×D311yescharacteristic
171#474—31no—
172#474C2×D311yescharacteristic
173#467—61no—
174#467—61no—
175#462—31no—
176#462C2×D311yescharacteristic
177#461—31no—
178#461C2×D311yescharacteristic
179#452—21no—
180#452—61no—
181#446—31no—
182#446—31no—
183C4×Q16—21no—
184C4×Q16—61no—
185C4×Q16—61no—
186C4×Q16—61no—
187C4×SD16—21no—
188C4×SD16—61no—
189#436—31no—
190#436—31no—
191#435—61no—
192#435—61no—
193#431—31no—
194#431—31no—
195#431—61no—
196#431C2×D311yescharacteristic
197C4×M16—61no—
198C2×C4×C8—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#332—31no—
208#332—31no—
209#332—31no—
210#332—31no—
211C3×#41—41no—
212C3×#41—41no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4#4111yescharacteristic
224C2×Dic6—21no—
225C2×Dic6—21no—
226C3×Q16—42no—
227C3×SD16—42no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—41no—
238C3×C4⋊C4C2×D411yescharacteristic
239C3×C4⋊C4C2×D411yescharacteristic
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—41no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—41no—
248C3×C42C2×D411yescharacteristic
249C3×C42C2×D411yescharacteristic
250C3×C42#4111yescharacteristic
251C3×C42C2411yescharacteristic
252#217—41no—
253#216—42no—
254#216—42no—
255#216—42no—
256#214—42no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261Dic3⋊C4—41no—
262Dic3⋊C4—41no—
263C4×Dic3—41no—
264#208—42no—
265#208—42no—
266C2×C3⋊C8—22no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—41no—
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×#41#6711yescharacteristic
277C2×Q16—21no—
278C2×Q16—61no—
279C2×Q16—61no—
280C2×Q16—61no—
281C2×SD16—21no—
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—21no—
292#125—61no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×Q8#6711yescharacteristic
302C4×Q8C22×D311yescharacteristic
303C4×D4—21no—
304C4×D4—21no—
305C4×D4#6711yescharacteristic
306C4×D4C22×D311yescharacteristic
307#117—21no—
308#117—21no—
309#117—61no—
310#117#6711yescharacteristic
311C2×C42—21no—
312C2×C42C22×D311yescharacteristic
313#107—21no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—21no—
318#106—61no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—61no—
324C4⋊C8—61no—
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—21no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8—61no—
353C23×C3—21no—
354C23×C3#12111yescharacteristic
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—42no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—42no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—42no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4#12111yescharacteristic
379C2×C3×C4#12111yescharacteristic
380C2×C3×C4#12111yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×Dic3—41no—
385C2×Dic3—41no—
386Dic6—41no—
387Dic6—41no—
388Dic6—41no—
389Dic6—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8#13512yes—
400#41—41no—
401#41—41no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8C2×#6711yescharacteristic
407C2×D4—21no—
408C2×D4C2×#6711yescharacteristic
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4D3×D411yescharacteristic
413C22×C4#23711yescharacteristic
414C22×C4C2×#6711yescharacteristic
415Q16—42no—
416Q16—122no—
417Q16—122no—
418Q16—122no—
419SD16—42no—
420SD16—122no—
421M16—122no—
422M16—122no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—31no—
428C2×C8—31no—
429C2×C8—62no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—121no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—121no—
448C4⋊C4—121no—
449C4⋊C4C2×#6711yescharacteristic
450C4⋊C4C2×#6711yescharacteristic
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—41no—
455C22⋊C4—121no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—41no—
460C42—121no—
461C42D3×D411yescharacteristic
462C42#23711yescharacteristic
463C42C2×#6711yescharacteristic
464C42C23×D311yescharacteristic
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3C2×#12111yescharacteristic
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4#47312yes—
481C3×C4C2×#13511yescharacteristic
482C3×C4#57511yescharacteristic
483C3×C4#57511yescharacteristic
484C3×C4#57511yescharacteristic
485Dic3—81no—
486Dic3—81no—
487C23—21no—
488C23#93911yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—41no—
495Q8—121no—
496Q8—121no—
497Q8—121no—
498Q8—121no—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4#93911yescharacteristic
519C2×C4#93911yescharacteristic
520C2×C4#93911yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×#23711yescharacteristic
523C2×C4C22×#6711yescharacteristic
524C8—32no—
525C8—41no—
526C8—41no—
527C8—61no—
528C8—61no—
529C8—61no—
530C8—121no—
531C8—121no—
532C8—121no—
533C8—121no—
534C2×C3—41no—
535C2×C3—81no—
536C2×C3#303511yescharacteristic
537C2×C3#305411yescharacteristic
538C2×C3C2×#47311yescharacteristic
539C22—21no—
540C22—21no—
541C22—21no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×#93911yescharacteristic
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—81no—
553C4—81no—
554C4—241no—
555C4—241no—
556C4#520412yes—
557C4D3×#13511yescharacteristic
558C4#576811yescharacteristic
559C4#582211yescharacteristic
560C4#582211yescharacteristic
561C3#1916111yescharacteristic, complemented, Sylow
562C2—41no—
563C2—81no—
564C2#8292311yescharacteristic
565C2#8390011yescharacteristic
566C2C2×#520411yescharacteristic
subgroupG/H
centerC2×C4#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3107C2
socleC22×C3C2×#121
prime radicalC2×C3×#41D4
hypercenter#3107D3
hypocenterC3#19161
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #83344C2semidirect
5. #71829C2semidirect
6. #71122C2semidirect
7. #71121C2semidirect
8. #71106C2semidirect
9. #71105C2semidirect
10. #71058C2semidirect
11. #71057C2semidirect
12. #71042C2semidirect
13. #71041C2semidirect
14. #70777C2semidirect
15. #70745C2semidirect
561. C3#19161semidirect

Maximal subgroups

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

subgroupindex
C3×#31072
#833442
#825342
#825172
#718292
#711222
#711212
#711062
#711052
#710582
#710572
#710422
#710412
#707772
#707452
#191613