Finite Group Databasebeta

← #10777930

Subgroups — #10777930

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81557C211yesmaximal, characteristic, complemented
2#81045C211yesmaximal, characteristic, complemented
3#80313C211yesmaximal, characteristic, complemented
4#79517C211yesmaximal, characteristic, complemented
5C3×#1697C211yesmaximal, characteristic, complemented
6#71210C211yesmaximal, characteristic, complemented
7#71137C211yesmaximal, characteristic, complemented
8#71009C211yesmaximal, characteristic, complemented
9#70869C211yesmaximal, characteristic, complemented
10#68594C211yesmaximal, characteristic, complemented
11#68567C211yesmaximal, characteristic, complemented
12#68513C211yesmaximal, characteristic, complemented
13#68482C211yesmaximal, characteristic, complemented
14#68413C211yesmaximal, characteristic, complemented
15#68382C211yesmaximal, characteristic, complemented
16#19160—31nomaximal, Sylow
17#5725C2211yescharacteristic
18#5716C2211yescharacteristic
19C4×#237C2211yescharacteristic
20C3×#486C2211yescharacteristic
21C3×#476C2211yescharacteristic
22C3×#459C2211yescharacteristic
23C3×#436C2211yescharacteristic
24#5158—21no—
25Dic3×D8—21no—
26#5127—21no—
27Dic3×M16—21no—
28#5091C2211yescharacteristic
29#5078C2211yescharacteristic
30#5075C2211yescharacteristic
31#5073C2211yescharacteristic
32#5015C2211yescharacteristic
33#5012C2211yescharacteristic
34#5008C2211yescharacteristic
35#4866—21no—
36#4852—21no—
37#4850C2211yescharacteristic
38#4848C2211yescharacteristic
39#4833C2211yescharacteristic
40#4833C2211yescharacteristic
41#4833C2211yescharacteristic
42#4831C2211yescharacteristic
43#4784—21no—
44#4781—21no—
45#4775—21no—
46#4775—21no—
47#4758—21no—
48#4756—21no—
49#4756—21no—
50#4756—21no—
51#4728—21no—
52#4718—21no—
53#4694C2211yescharacteristic
54C4×D3×C8C2211yescharacteristic
55#4679C2211yescharacteristic
56C3×#335C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#331C2211yescharacteristic
59#4546C2211yescharacteristic
60#4544C2211yescharacteristic
61#4485C2211yescharacteristic
62#4485C2211yescharacteristic
63#4481C2211yescharacteristic
64#4481C2211yescharacteristic
65#4480C2211yescharacteristic
66#4480C2211yescharacteristic
67#4458C2211yescharacteristic
68#3169—31no—
69#3106—31no—
70#3106—31no—
71#2965—31no—
72#1698—31no—
73#1698—31no—
74#1697—31no—
75#1697D311yescharacteristic, complemented
76#1696—31no—
77#1695—31no—
78#1688—31no—
79#1687—31no—
80#1564—31no—
81#1543—31no—
82#1543—31no—
83C2×#239—21no—
84C2×#237—21no—
85C2×#237D411yescharacteristic
86C2×C3×D8—21no—
87C2×C3×M16—21no—
88C3×C4⋊Q8C2311yescharacteristic
89C3×#127C2311yescharacteristic
90C3×C4×D4C2311yescharacteristic
91#947—21no—
92#938—21no—
93#938—21no—
94#936—21no—
95D4×Dic3—21no—
96D4×Dic3—21no—
97D4×Dic3D411yescharacteristic
98C2×#214—21no—
99C2×#213—21no—
100C2×#213—21no—
101C2×#213—21no—
102C4×#67—21no—
103C2×C4×Dic3—21no—
104C2×#208—21no—
105C2×#204—21no—
106C2×D3×C8D411yescharacteristic
107C2×D3×C8D411yescharacteristic
108#895—21no—
109#895—21no—
110#893—21no—
111#893—21no—
112#892D411yescharacteristic
113#887D411yescharacteristic
114#67⋊C4—21no—
115#877—21no—
116#874C2311yescharacteristic
117C42×D3C2311yescharacteristic
118#869C2311yescharacteristic
119C4×Dic6C2311yescharacteristic
120C3×#106—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×#102—21no—
125C3×#102—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×C22⋊C8—21no—
129C3×C4×C8C2311yescharacteristic
130#832—21no—
131#832—21no—
132#830—21no—
133#821—21no—
134#821—21no—
135#821—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139#818—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#809—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#532—31no—
161#532—31no—
162C4×#41—31no—
163#500—31no—
164#493—31no—
165#486—31no—
166#486C2×D311yescharacteristic
167#482—61no—
168#482—61no—
169#477—31no—
170#477—31no—
171#476—31no—
172#476C2×D311yescharacteristic
173#466—61no—
174#466—61no—
175#460—31no—
176#460—31no—
177#459—31no—
178#459C2×D311yescharacteristic
179#449—61no—
180#449—61no—
181#446—31no—
182#446—31no—
183C4×SD16—61no—
184C4×SD16—61no—
185C4×D8—61no—
186C4×D8—61no—
187C4×D8—61no—
188C4×D8—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#333—31no—
204#333—31no—
205#333—31no—
206#333C2×D311yescharacteristic
207#331—31no—
208#331—31no—
209#331—31no—
210#331C2×D311yescharacteristic
211C2×C3×Q8—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4C2×D411yescharacteristic
216C22×C3×C4—21no—
217C2×#67—21no—
218C2×#67—41no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—41no—
222#239—82no—
223#237—41no—
224#237—41no—
225#237—82no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—41no—
230C2×C4×D3—41no—
231C2×C4×D3C2×D411yescharacteristic
232C2×C4×D3C2×D411yescharacteristic
233C2×C4×D3#4111yescharacteristic
234C2×Dic6—21no—
235C2×Dic6C2×D411yescharacteristic
236C3×D8—42no—
237C3×M16—42no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8C2×D411yescharacteristic
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C22⋊C4—21no—
247C3×C42C2411yescharacteristic
248#217—21no—
249#217—41no—
250#217—41no—
251#214—42no—
252#213—42no—
253#213—42no—
254#213—42no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—41no—
257(C2×D3)⋊C4—41no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2×D411yescharacteristic
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3C2×D411yescharacteristic
268C4×Dic3#4111yescharacteristic
269#208—42no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275#204—42no—
276D3×C8—22no—
277D3×C8—41no—
278C2×#41—31no—
279C2×#41—61no—
280C2×#41—61no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×D8—21no—
284C2×D8—61no—
285C2×D8—61no—
286C2×D8—61no—
287C2×M16—21no—
288C2×M16—61no—
289C22×C8—31no—
290C22×C8—31no—
291C4⋊Q8—31no—
292C4⋊Q8C22×D311yescharacteristic
293#127—31no—
294#127C22×D311yescharacteristic
295#124—61no—
296#124—61no—
297#121—61no—
298#121—61no—
299#121—61no—
300#121—61no—
301C4×Q8—31no—
302C4×Q8—31no—
303C4×D4—31no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4—61no—
307C4×D4—61no—
308C4×D4—61no—
309C4×D4—61no—
310C4×D4C22×D311yescharacteristic
311#117—31no—
312#117—61no—
313C2×C42—31no—
314C2×C42—61no—
315#107—61no—
316#107—61no—
317#107—61no—
318#107—61no—
319#106—21no—
320#106—61no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—61no—
330#103—61no—
331#102—21no—
332#102—21no—
333#102—21no—
334#102—21no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341C22⋊C8—21no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346#97—61no—
347#97—61no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8—61no—
353C4×C8—61no—
354C4×C8C22×D311yescharacteristic
355C23×C3—21no—
356C23×C3—41no—
357C22×D3—41no—
358C22×D3#12111yescharacteristic
359C3×Q8—81no—
360C3×D4—21no—
361C3×D4—21no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—41no—
366C3×D4—81no—
367C2×C3×C4—21no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374#67—42no—
375#67—82no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—41no—
381C2×Dic3—82no—
382C2×Dic3#12111yescharacteristic
383C2×Dic3#12111yescharacteristic
384C2×Dic3#12111yescharacteristic
385D12—41no—
386D12—41no—
387D12—41no—
388D12—41no—
389D12—81no—
390D12—82no—
391C4×D3—21no—
392C4×D3—21no—
393C4×D3—21no—
394C4×D3—21no—
395C4×D3—22no—
396C4×D3—82no—
397Dic6—21no—
398Dic6—21no—
399Dic6—81no—
400C3×C8—21no—
401C3×C8—21no—
402C3×C8—41no—
403C3×C8—41no—
404C3×C8—41no—
405C3⋊C8—21no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8#13512yes—
410#41—121no—
411#41—121no—
412#41—242no—
413#41—242no—
414C2×Q8—21no—
415C2×Q8—31no—
416C2×Q8—61no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—121no—
425C2×D4—121no—
426C2×D4D3×D411yescharacteristic
427C22×C4—21no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—121no—
434C22×C4—121no—
435SD16—122no—
436SD16—122no—
437D8—42no—
438D8—122no—
439D8—122no—
440D8—122no—
441M16—42no—
442M16—122no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—31no—
447C2×C8—31no—
448C2×C8—62no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—121no—
453C2×C8D3×D411yescharacteristic
454C2×C8D3×D411yescharacteristic
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—31no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4D3×D411yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—61no—
465C22⋊C4—61no—
466C22⋊C4—121no—
467C22⋊C4—121no—
468C22⋊C4—121no—
469C22⋊C4—121no—
470C42—31no—
471C42—31no—
472C42—31no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42C23×D311yescharacteristic
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3C2×#12111yescharacteristic
484C2×D3—21no—
485C2×D3—21no—
486C2×D3—21no—
487C2×D3—81no—
488C2×D3—81no—
489C2×D3—81no—
490C3×C4—21no—
491C3×C4—41no—
492C3×C4—81no—
493C3×C4C2×#13511yescharacteristic
494C3×C4#57511yescharacteristic
495C3×C4#57511yescharacteristic
496C3×C4#57511yescharacteristic
497Dic3—21no—
498Dic3—21no—
499Dic3—41no—
500Dic3—41no—
501Dic3—81no—
502Dic3#47212yes—
503C23—21no—
504C23—31no—
505C23—41no—
506C23—121no—
507Q8—61no—
508Q8—61no—
509Q8—81no—
510Q8—241no—
511D4—21no—
512D4—21no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—81no—
518D4—122no—
519D4—121no—
520D4—121no—
521D4—121no—
522D4—121no—
523D4—241no—
524D4—242no—
525D4—242no—
526C2×C4—21no—
527C2×C4—31no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—121no—
543C2×C4—242no—
544C2×C4—242no—
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C8—21no—
549C8—21no—
550C8—32no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—61no—
555C8—121no—
556C8—121no—
557C8—121no—
558C2×C3—81no—
559C2×C3—161no—
560C2×C3#303511yescharacteristic
561C2×C3#305411yescharacteristic
562C2×C3C2×#47211yescharacteristic
563D3—41no—
564D3—161no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—61no—
569C22—61no—
570C22—61no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—241no—
575C22—241no—
576C22—241no—
577C22D3×#12111yescharacteristic
578C4—21no—
579C4—32no—
580C4—41no—
581C4—61no—
582C4—61no—
583C4—81no—
584C4—121no—
585C4—121no—
586C4—241no—
587C4D3×#13511yescharacteristic
588C4#577311yescharacteristic
589C4#577311yescharacteristic
590C4#577711yescharacteristic
591C3#1916011yescharacteristic, complemented, Sylow
592C2—81no—
593C2—121no—
594C2—161no—
595C2—481no—
596C2#8053311yescharacteristic
597C2#8080411yescharacteristic
598C2D3×#47211yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1697C2
socleC22×C3C2×#121
hypercenter#1697D3
hypocenterC3#19160
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1697

Complements

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

NH ≅ G/Ntype
1. #81557C2semidirect
2. #81045C2semidirect
3. #80313C2semidirect
4. #79517C2semidirect
5. C3×#1697C2semidirect
6. #71210C2semidirect
7. #71137C2semidirect
8. #71009C2semidirect
9. #70869C2semidirect
10. #68594C2semidirect
11. #68567C2semidirect
12. #68513C2semidirect
13. #68482C2semidirect
14. #68413C2semidirect
15. #68382C2semidirect
75. #1697D3semidirect
591. C3#19160semidirect

Maximal subgroups

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

subgroupindex
#815572
#810452
#803132
#795172
C3×#16972
#712102
#711372
#710092
#708692
#685942
#685672
#685132
#684822
#684132
#683822
#191603