Finite Group Databasebeta

← #10777918

Subgroups — #10777918

Order 768.

620 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2965C211yesmaximal, characteristic, complemented
2#82534C211yesmaximal, characteristic, complemented
3#82529C211yesmaximal, characteristic
4#78468C211yesmaximal, characteristic, complemented
5#70967C211yesmaximal, characteristic, complemented
6#70963C211yesmaximal, characteristic, complemented
7#70961C211yesmaximal, characteristic, complemented
8#70958C211yesmaximal, characteristic, complemented
9#70876C211yesmaximal, characteristic, complemented
10#70872C211yesmaximal, characteristic, complemented
11#70870C211yesmaximal, characteristic, complemented
12#70867C211yesmaximal, characteristic, complemented
13#70777C211yesmaximal, characteristic, complemented
14#70773C211yesmaximal, characteristic, complemented
15#69729C211yesmaximal, characteristic, complemented
16#19159—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5477C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#446C2211yescharacteristic
21C3×#446C2211yescharacteristic
22C3×#436C2211yescharacteristic
23C3×#436C2211yescharacteristic
24C3×#435—21no—
25C3×#435—21no—
26C3×#431—21no—
27C3×#431C2211yescharacteristic
28C3×C4×M16—21no—
29C2×C3×C4×C8C2211yescharacteristic
30#5239—21no—
31#5238—21no—
32#5114—21no—
33#5114—21no—
34#5031C2211yescharacteristic
35C4×#216—21no—
36#5027C2211yescharacteristic
37C4×#215—21no—
38#5021C2211yescharacteristic
39#5020C2211yescharacteristic
40#5019C2211yescharacteristic
41C4×#214—21no—
42#5015C2211yescharacteristic
43C4×#213—21no—
44#5008C2211yescharacteristic
45#5007C2211yescharacteristic
46#5003—21no—
47#4996—21no—
48#4925C2211yescharacteristic
49#4925C2211yescharacteristic
50#4694C2211yescharacteristic
51#4693C2211yescharacteristic
52C4×#204—21no—
53C4×#204—21no—
54#4681C2211yescharacteristic
55#4680C2211yescharacteristic
56#4486C2211yescharacteristic
57#4486C2211yescharacteristic
58#4485C2211yescharacteristic
59#4485C2211yescharacteristic
60#4482C2211yescharacteristic
61#4482C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3167—31no—
69#3107—31no—
70#3106—31no—
71#2965D311yescharacteristic, complemented
72#1704—31no—
73#1703—31no—
74#1702—31no—
75#1701—31no—
76#1692—31no—
77#1691—31no—
78#1686—31no—
79#1685—31no—
80#1564—31no—
81#1544—31no—
82#1543—31no—
83C2×C3×#41D411yescharacteristic
84C2×#235—21no—
85C2×C3×M16—21no—
86C2×C3×M16—21no—
87C22×C3×C8D411yescharacteristic
88C22×C3×C8D411yescharacteristic
89C3×C4×Q8D411yescharacteristic
90C3×C4×Q8C2311yescharacteristic
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C3×C4×D4D411yescharacteristic
94C3×C4×D4C2311yescharacteristic
95C3×#117—21no—
96C3×#117D411yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#216—21no—
100C2×#215—21no—
101C2×#214—21no—
102C2×#213—21no—
103#930—21no—
104#930—21no—
105#926—21no—
106#924—21no—
107#924—21no—
108C2×#204—21no—
109C2×#204—21no—
110#875—21no—
111#874C2311yescharacteristic
112C4×D12—21no—
113C4×D12—21no—
114#870—21no—
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118C4×Dic6—21no—
119C4×Dic6—21no—
120C3×C4⋊C8—21no—
121C3×C4⋊C8—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×C22⋊C8—21no—
125C3×C22⋊C8—21no—
126C3×C22⋊C8—21no—
127C3×#97—21no—
128C3×#97—21no—
129C3×C4×C8—21no—
130C3×C4×C8D411yescharacteristic
131C3×C4×C8D411yescharacteristic
132C3×C4×C8C2311yescharacteristic
133C3×C4×C8C2311yescharacteristic
134#835—21no—
135#835—21no—
136#832—21no—
137#832—21no—
138#830—21no—
139#830—21no—
140#821—21no—
141#821—21no—
142#817—21no—
143#817—21no—
144#816—21no—
145#816—21no—
146#810—21no—
147#810—21no—
148#809—21no—
149#809—21no—
150#808—21no—
151#808—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4×C3⋊C8—21no—
159C4×C3⋊C8—21no—
160#533—31no—
161#532—31no—
162C4×#41C2×D311yescharacteristic
163#498—31no—
164#492—31no—
165#487—31no—
166#486—31no—
167#485—61no—
168#482—61no—
169#477—31no—
170#476—31no—
171#475—31no—
172#474—31no—
173#465—61no—
174#465—61no—
175#462—31no—
176#461—31no—
177#460—31no—
178#459—31no—
179#452—61no—
180#449—61no—
181#446C2×D311yescharacteristic
182#446C2×D311yescharacteristic
183C4×Q16—61no—
184C4×SD16—61no—
185C4×SD16—61no—
186C4×SD16—61no—
187C4×SD16—61no—
188C4×D8—61no—
189#436C2×D311yescharacteristic
190#436C2×D311yescharacteristic
191#435—21no—
192#435—21no—
193#431—21no—
194#431—31no—
195#431—31no—
196#431C2×D311yescharacteristic
197C4×M16—21no—
198C2×C4×C8C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C3×#41—41no—
212C3×#41—41no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4C2×D411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4111yescharacteristic
222C2×#67—41no—
223#235—82no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×C4×D3—41no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C3×M16—42no—
232C3×M16—42no—
233C2×C3×C8—22no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—41no—
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42#4111yescharacteristic
254C3×C42C2411yescharacteristic
255#217—41no—
256#216—42no—
257#215—42no—
258#214—42no—
259#213—42no—
260(C2×D3)⋊C4—41no—
261(C2×D3)⋊C4—41no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268Dic3⋊C4—41no—
269Dic3⋊C4—41no—
270C4×Dic3—41no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275#204—42no—
276#204—42no—
277C2×#41—61no—
278C2×#41D1211yescharacteristic
279C2×Q16—61no—
280C2×SD16—61no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×SD16—61no—
284C2×D8—61no—
285C2×M16—21no—
286C2×M16—21no—
287C22×C8#6711yescharacteristic
288C22×C8#6711yescharacteristic
289C4⋊Q8—31no—
290C4⋊Q8—31no—
291C4⋊Q8—31no—
292#127—31no—
293#125—61no—
294#124—61no—
295C22⋊Q8—61no—
296C22⋊Q8—61no—
297#121—61no—
298#121—61no—
299C4×Q8—61no—
300C4×Q8—61no—
301C4×Q8D1211yescharacteristic
302C4×Q8C22×D311yescharacteristic
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—61no—
306C4×D4—61no—
307C4×D4D1211yescharacteristic
308C4×D4C22×D311yescharacteristic
309#117—21no—
310#117—61no—
311#117D1211yescharacteristic
312C2×C42—21no—
313C2×C42C22×D311yescharacteristic
314#107—61no—
315#107—61no—
316#106—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—31no—
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—61no—
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—21no—
346#97—21no—
347C4×C8—21no—
348C4×C8—61no—
349C4×C8—61no—
350C4×C8#6711yescharacteristic
351C4×C8#6711yescharacteristic
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C23×C3—21no—
355C23×C3#12111yescharacteristic
356C22×D3—41no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—42no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4#12111yescharacteristic
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4#12111yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380#67—82no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384D12—41no—
385D12—41no—
386D12—41no—
387D12—41no—
388D12—81no—
389C4×D3—82no—
390Dic6—41no—
391Dic6—41no—
392Dic6—41no—
393Dic6—41no—
394Dic6—81no—
395C3×C8—21no—
396C3×C8—21no—
397C3×C8—21no—
398C3×C8—41no—
399C3×C8—41no—
400C3×C8#13512yes—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405#41—41no—
406#41—41no—
407#41—242no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8C2×D1211yescharacteristic
412C2×D4—21no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—121no—
417C2×D4C2×D1211yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—121no—
422C22×C4C2×D1211yescharacteristic
423C22×C4#23511yescharacteristic
424C22×C4C2×#6711yescharacteristic
425Q16—122no—
426SD16—122no—
427SD16—122no—
428SD16—122no—
429SD16—122no—
430D8—122no—
431M16—42no—
432M16—42no—
433C2×C8—22no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—41no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8C2×#6711yescharacteristic
442C2×C8C2×#6711yescharacteristic
443C2×C8C2×#6711yescharacteristic
444C2×C8C2×#6711yescharacteristic
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—121no—
454C4⋊C4—121no—
455C4⋊C4C2×D1211yescharacteristic
456C4⋊C4C2×D1211yescharacteristic
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—121no—
461C22⋊C4—121no—
462C22⋊C4—121no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—121no—
467C42C2×D1211yescharacteristic
468C42#23511yescharacteristic
469C42C2×#6711yescharacteristic
470C42C23×D311yescharacteristic
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3C2×#12111yescharacteristic
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—81no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4#47112yes—
488C3×C4C2×#13511yescharacteristic
489C3×C4#57511yescharacteristic
490C3×C4#57511yescharacteristic
491C3×C4#57511yescharacteristic
492Dic3—81no—
493Dic3—81no—
494Dic3—81no—
495C23—21no—
496C23—121no—
497C23#93011yescharacteristic
498Q8—21no—
499Q8—21no—
500Q8—121no—
501Q8—121no—
502Q8—121no—
503Q8—121no—
504Q8—241no—
505D4—21no—
506D4—21no—
507D4—42no—
508D4—121no—
509D4—121no—
510D4—121no—
511D4—121no—
512D4—241no—
513D4—242no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—242no—
530C2×C4#93011yescharacteristic
531C2×C4#93011yescharacteristic
532C2×C4#93011yescharacteristic
533C2×C4C22×D1211yescharacteristic
534C2×C4C2×#23511yescharacteristic
535C2×C4C22×#6711yescharacteristic
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—121no—
542C8—121no—
543C8—121no—
544C8—121no—
545C8#95012yes—
546C2×C3—41no—
547C2×C3—81no—
548C2×C3#303511yescharacteristic
549C2×C3#305411yescharacteristic
550C2×C3C2×#47111yescharacteristic
551D3—161no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—241no—
559C22—241no—
560C22—241no—
561C22C2×#93011yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—241no—
569C4—241no—
570C4—241no—
571C4#514112yes—
572C4#574811yescharacteristic
573C4#574811yescharacteristic
574C4C2×#95011yescharacteristic
575C4#582211yescharacteristic
576C3#1915911yescharacteristic, complemented, Sylow
577C2—41no—
578C2—81no—
579C2—481no—
580C2#8252511yescharacteristic
581C2#8377111yescharacteristic
582C2C2×#514111yescharacteristic
subgroupG/H
centerC2×C4#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2965C2
socleC22×C3C2×#121
prime radical#82529C2
hypercenter#2965D3
hypocenterC3#19159
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. C3×#2965C2semidirect
2. #82534C2semidirect
4. #78468C2semidirect
5. #70967C2semidirect
6. #70963C2semidirect
7. #70961C2semidirect
8. #70958C2semidirect
9. #70876C2semidirect
10. #70872C2semidirect
11. #70870C2semidirect
12. #70867C2semidirect
13. #70777C2semidirect
14. #70773C2semidirect
15. #69729C2semidirect
71. #2965D3semidirect
576. C3#19159semidirect

Maximal subgroups

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

subgroupindex
C3×#29652
#825342
#825292
#784682
#709672
#709632
#709612
#709582
#708762
#708722
#708702
#708672
#707772
#707732
#697292
#191593