Finite Group Databasebeta

← #10782800

Subgroups — #10782800

Order 768.

580 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#81608C211yesmaximal, characteristic, complemented
2#80976C211yesmaximal, characteristic
3#80435C211yesmaximal, characteristic
4#79410C211yesmaximal, characteristic
5C3×#1679C211yesmaximal, characteristic, complemented
6#71242C211yesmaximal, characteristic, complemented
7#71106C211yesmaximal, characteristic, complemented
8#71041C211yesmaximal, characteristic, complemented
9#70852C211yesmaximal, characteristic, complemented
10#68552C211yesmaximal, characteristic, complemented
11#68534C211yesmaximal, characteristic, complemented
12#68468C211yesmaximal, characteristic, complemented
13#68467C211yesmaximal, characteristic, complemented
14#68425C211yesmaximal, characteristic, complemented
15D3×#332C211yesmaximal, characteristic, complemented
16#19590—31nomaximal, Sylow
17#5728C2211yescharacteristic
18D3×C4⋊Q8C2211yescharacteristic
19#5557C2211yescharacteristic
20C3×#487C2211yescharacteristic
21C3×#474C2211yescharacteristic
22C3×#461C2211yescharacteristic
23C3×C8×D4C2211yescharacteristic
24#5170—21no—
25#5167—21no—
26C8×#67—21no—
27#5102—21no—
28#5094C2211yescharacteristic
29#5090C2211yescharacteristic
30#5079C2211yescharacteristic
31#5078C2211yescharacteristic
32#5019C2211yescharacteristic
33#5019C2211yescharacteristic
34#5012C2211yescharacteristic
35#4887—21no—
36#4877—21no—
37#4849C2211yescharacteristic
38#4849C2211yescharacteristic
39#4835C2211yescharacteristic
40D3×C4⋊C8C2211yescharacteristic
41D3×C4⋊C8C2211yescharacteristic
42#4830C2211yescharacteristic
43#4811—21no—
44#4811—21no—
45#4809—21no—
46#4797—21no—
47#4793—21no—
48#4793—21no—
49#4778—21no—
50#4767—21no—
51#4728—21no—
52#4720—21no—
53#4685C2211yescharacteristic
54#4682C2211yescharacteristic
55#4679C2211yescharacteristic
56C3×#334C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#332C2211yescharacteristic
59#4547C2211yescharacteristic
60#4547C2211yescharacteristic
61#4486C2211yescharacteristic
62#4486C2211yescharacteristic
63#4482C2211yescharacteristic
64#4482C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4458C2211yescharacteristic
68#3248—31no—
69#3230—31no—
70#3218—31no—
71#2994—31no—
72#1700—31no—
73#1699—31no—
74#1694—31no—
75#1693—31no—
76#1680—31no—
77#1679D311yescharacteristic, complemented
78#1674—31no—
79#1673—31no—
80#1573—31no—
81#1552—31no—
82C2×#332—31no—
83C2×D3×Q8—21no—
84C2×C3×SD16—21no—
85C22×C3×C8—21no—
86C3×C4⋊Q8C2311yescharacteristic
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4×D4C2311yescharacteristic
89#946—21no—
90Q8×Dic3—21no—
91#944—21no—
92C2×#216—21no—
93#938D411yescharacteristic
94#936D411yescharacteristic
95C2×#214—21no—
96C2×#214—21no—
97C4×#67—21no—
98#926—21no—
99C2×#208—21no—
100C2×Dic12—21no—
101C2×Dic12—21no—
102C2×D3×C8—21no—
103C2×D3×C8—21no—
104#897—21no—
105#897D411yescharacteristic
106#892—21no—
107D3×C4⋊C4—21no—
108D3×C4⋊C4—21no—
109#890—21no—
110#890D411yescharacteristic
111#888—21no—
112#888—21no—
113#887—21no—
114#883—21no—
115#879—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#835—21no—
132#835—21no—
133#830—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8D411yescharacteristic
136(C2×D3)⋊C8D411yescharacteristic
137#818—21no—
138#818—21no—
139#816—21no—
140#816—21no—
141#815—21no—
142#815—21no—
143Dic3⋊C8—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146Dic3⋊C8—21no—
147Dic3⋊C8D411yescharacteristic
148Dic3⋊C8D411yescharacteristic
149C8×Dic3—21no—
150#810—21no—
151#810—21no—
152#810—21no—
153#808—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—
161C2×C4⋊Q8—31no—
162#529—31no—
163#500—31no—
164#500—31no—
165#497—61no—
166#491—61no—
167#487—31no—
168#487C2×D311yescharacteristic
169#484—61no—
170#484—61no—
171#479—61no—
172#477—61no—
173#477—61no—
174#475—31no—
175#474C2×D311yescharacteristic
176#467—61no—
177#465—61no—
178#464—61no—
179#462—31no—
180#462—31no—
181#462—31no—
182#461—31no—
183#461—31no—
184#461C2×D311yescharacteristic
185#451—61no—
186#448—61no—
187#446—31no—
188C8×Q8—31no—
189#436—31no—
190#435—61no—
191C8×D4—61no—
192C8×D4C2×D311yescharacteristic
193#433—61no—
194#432—31no—
195#431—31no—
196#424—61no—
197C2×C4⋊C8—31no—
198C2×C4⋊C8—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334C2×D311yescharacteristic
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—
216D3×Q8—81no—
217D3×Q8—81no—
218C2×C4×D3—41no—
219C2×C4×D3C2×D411yescharacteristic
220C2×C4×D3C2×D411yescharacteristic
221C2×C4×D3#4111yescharacteristic
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—41no—
225C2×Dic6C2×D411yescharacteristic
226C3×SD16—81no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—81no—
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#217—21no—
242#217—21no—
243#216—81no—
244#214—81no—
245#214—81no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—41no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—41no—
255C4⋊Dic3C2×D411yescharacteristic
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258Dic3⋊C4—41no—
259Dic3⋊C4—41no—
260Dic3⋊C4—41no—
261C4×Dic3—21no—
262C4×Dic3—41no—
263C4×Dic3C2×D411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3#4111yescharacteristic
266#208—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272Dic12—81no—
273Dic12—81no—
274D3×C8—81no—
275D3×C8—81no—
276C22×Q8—61no—
277C2×Q16—61no—
278C2×Q16—61no—
279C2×Q16—61no—
280C2×SD16—21no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×M16—61no—
284C22×C8—21no—
285C22×C8—61no—
286C22×C8—61no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8—61no—
290C4⋊Q8—61no—
291C4⋊Q8—61no—
292C4⋊Q8C22×D311yescharacteristic
293C4⋊Q8C22×D311yescharacteristic
294#126—61no—
295#125—31no—
296#125—61no—
297#124—31no—
298#123—61no—
299C22⋊Q8—31no—
300C22⋊Q8—61no—
301C22⋊Q8—61no—
302#121—31no—
303C4×Q8—31no—
304C4×Q8—61no—
305C4×Q8—61no—
306C4×D4—61no—
307C4×D4C22×D311yescharacteristic
308#117—61no—
309#117—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C42—31no—
313#107—21no—
314#107—61no—
315#107—61no—
316#107—61no—
317#107—61no—
318#106—61no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8—61no—
327C4⋊C8—61no—
328C4⋊C8C22×D311yescharacteristic
329C4⋊C8C22×D311yescharacteristic
330C4⋊C8C22×D311yescharacteristic
331#103—21no—
332#103—21no—
333#103—21no—
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#102—21no—
343C22⋊C8—21no—
344C22⋊C8—31no—
345C22⋊C8—31no—
346C22⋊C8—61no—
347C22⋊C8—61no—
348#97—61no—
349#97—61no—
350C4×C8—31no—
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—41no—
373C2×Dic3—41no—
374C2×Dic3#12111yescharacteristic
375C2×Dic3#12111yescharacteristic
376C2×Dic3#12111yescharacteristic
377C4×D3—22no—
378C4×D3—21no—
379C4×D3—21no—
380C4×D3—21no—
381C4×D3—21no—
382C4×D3—82no—
383Dic6—21no—
384Dic6—21no—
385Dic6—41no—
386Dic6—41no—
387Dic6—41no—
388Dic6—41no—
389Dic6—82no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—41no—
395C3⋊C8—21no—
396C3⋊C8—21no—
397C3⋊C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—31no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—121no—
406C2×Q8—241no—
407C2×Q8—241no—
408C2×D4—61no—
409C2×D4D3×D411yescharacteristic
410C22×C4—21no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—121no—
415Q16—241no—
416Q16—241no—
417Q16—241no—
418SD16—81no—
419SD16—241no—
420SD16—241no—
421M16—241no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—81no—
431C2×C8—241no—
432C2×C8—241no—
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—61no—
447C4⋊C4—61no—
448C4⋊C4—121no—
449C4⋊C4—121no—
450C4⋊C4—121no—
451C4⋊C4—121no—
452C4⋊C4D3×D411yescharacteristic
453C22⋊C4—21no—
454C22⋊C4—61no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—61no—
458C22⋊C4—121no—
459C42—31no—
460C42—31no—
461C42—31no—
462C42—61no—
463C42—121no—
464C42C23×D311yescharacteristic
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3C2×#12111yescharacteristic
469C2×D3—21no—
470C2×D3—21no—
471C2×D3#47312yes—
472C3×C4—21no—
473C3×C4—41no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4#57611yescharacteristic
477C3×C4#57711yescharacteristic
478C3×C4#57711yescharacteristic
479C3×C4#57811yescharacteristic
480Dic3—21no—
481Dic3—21no—
482Dic3—21no—
483Dic3—41no—
484Dic3—41no—
485Dic3—81no—
486Dic3—81no—
487C23—21no—
488C23—31no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—61no—
494Q8—61no—
495Q8—121no—
496Q8—121no—
497Q8—121no—
498Q8—121no—
499Q8—242no—
500D4—21no—
501D4—21no—
502D4—122no—
503C2×C4—21no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—31no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—62no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—242no—
521C2×C4C2×D3×D411yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4D3×#4111yescharacteristic
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—61no—
530C8—61no—
531C8—121no—
532C8—121no—
533C8—121no—
534C2×C3—81no—
535C2×C3#304611yescharacteristic
536C2×C3#306211yescharacteristic
537C2×C3C2×#47311yescharacteristic
538D3—22no—
539C22—32no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—61no—
544C22—61no—
545C22D3×#12111yescharacteristic
546C4—21no—
547C4—41no—
548C4—61no—
549C4—61no—
550C4—61no—
551C4—81no—
552C4—81no—
553C4—121no—
554C4—121no—
555C4—241no—
556C4—241no—
557C4#576211yescharacteristic
558C4#576911yescharacteristic
559C4#577111yescharacteristic
560C4#577511yescharacteristic
561C3#1959011yescharacteristic, complemented, Sylow
562C2—62no—
563C2—81no—
564C2#8053911yescharacteristic
565C2#8066111yescharacteristic
566C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1679C2
socleC22×C3C2×#121
prime radical#938D4
hypercenter#1679D3
hypocenterC3#19590
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1679

Complements

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

NH ≅ G/Ntype
1. #81608C2semidirect
5. C3×#1679C2semidirect
6. #71242C2semidirect
7. #71106C2semidirect
8. #71041C2semidirect
9. #70852C2semidirect
10. #68552C2semidirect
11. #68534C2semidirect
12. #68468C2semidirect
13. #68467C2semidirect
14. #68425C2semidirect
15. D3×#332C2semidirect
77. #1679D3semidirect
561. C3#19590semidirect

Maximal subgroups

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

subgroupindex
#816082
#809762
#804352
#794102
C3×#16792
#712422
#711062
#710412
#708522
#685522
#685342
#684682
#684672
#684252
D3×#3322
#195903