Finite Group Databasebeta

← #10778191

Subgroups — #10778191

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2971C211yesmaximal, characteristic, complemented
2#82534C211yesmaximal, characteristic, complemented
3#82525C211yesmaximal, characteristic
4#78533C211yesmaximal, characteristic, complemented
5#70946C211yesmaximal, characteristic, complemented
6#70944C211yesmaximal, characteristic, complemented
7#70941C211yesmaximal, characteristic, complemented
8#70938C211yesmaximal, characteristic, complemented
9#70875C211yesmaximal, characteristic, complemented
10#70873C211yesmaximal, characteristic, complemented
11#70870C211yesmaximal, characteristic, complemented
12#70867C211yesmaximal, characteristic, complemented
13#70820C211yesmaximal, characteristic, complemented
14#70817C211yesmaximal, characteristic, complemented
15#69850C211yesmaximal, characteristic, complemented
16#19182—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18C2×#950—21no—
19#5477C2211yescharacteristic
20#5470C2211yescharacteristic
21C3×C8×Q8C2211yescharacteristic
22C3×C8×Q8C2211yescharacteristic
23C3×#436—21no—
24C3×#436—21no—
25C3×#436C2211yescharacteristic
26C3×#436C2211yescharacteristic
27C3×#432—21no—
28C3×#432C2211yescharacteristic
29C3×C4×M16—21no—
30C3×C4×M16C2211yescharacteristic
31#5239—21no—
32#5134—21no—
33#5134—21no—
34#5031C2211yescharacteristic
35#5031C2211yescharacteristic
36C4×#216—21no—
37#5028—21no—
38#5027C2211yescharacteristic
39#5020C2211yescharacteristic
40#5019C2211yescharacteristic
41#5019C2211yescharacteristic
42C4×#214—21no—
43#5016—21no—
44#5015C2211yescharacteristic
45#5007C2211yescharacteristic
46#5002—21no—
47#4996—21no—
48#4925C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4710—21no—
51#4710—21no—
52#4694C2211yescharacteristic
53#4693C2211yescharacteristic
54#4682C2211yescharacteristic
55#4681C2211yescharacteristic
56#4486C2211yescharacteristic
57#4486C2211yescharacteristic
58#4486C2211yescharacteristic
59#4486C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3182—31no—
69#3107—31no—
70#3035—31no—
71#2971D311yescharacteristic, complemented
72#1701—31no—
73#1699—31no—
74#1693—31no—
75#1685—31no—
76#1682—31no—
77#1680—31no—
78#1674—31no—
79#1666—31no—
80#1565—31no—
81#1553—31no—
82#1552—31no—
83C2×C3×#41D411yescharacteristic
84C2×#235—21no—
85C2×C3×M16—21no—
86C2×C3×M16—21no—
87C2×C3×M16D411yescharacteristic
88C2×C3×M16D411yescharacteristic
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
99#950—44no—
100C2×#216—21no—
101C2×#216—21no—
102C2×#215—21no—
103C2×#214—21no—
104C2×#214—21no—
105C2×#213—21no—
106#930—21no—
107#930—21no—
108#926—21no—
109#924—21no—
110#924—21no—
111C22×C3⋊C8—21no—
112C2×#204—21no—
113C2×#204—21no—
114#875—21no—
115#874C2311yescharacteristic
116C4×D12—21no—
117C4×D12—21no—
118#870—21no—
119#869C2311yescharacteristic
120#869C2311yescharacteristic
121#869C2311yescharacteristic
122C4×Dic6—21no—
123C4×Dic6—21no—
124C3×C4⋊C8—21no—
125C3×C4⋊C8—21no—
126C3×C4⋊C8C2311yescharacteristic
127C3×C4⋊C8C2311yescharacteristic
128C3×C22⋊C8—21no—
129C3×C22⋊C8—21no—
130C3×C22⋊C8—21no—
131C3×#97—21no—
132C3×#97D411yescharacteristic
133C3×#97D411yescharacteristic
134C3×C4×C8—21no—
135C3×C4×C8—21no—
136C3×C4×C8C2311yescharacteristic
137C3×C4×C8C2311yescharacteristic
138#835—21no—
139#832—21no—
140#830—21no—
141#821—21no—
142#821—21no—
143#818—21no—
144#818—21no—
145#816—21no—
146#816—21no—
147#810—21no—
148#810—21no—
149#810—21no—
150#810—21no—
151#809—21no—
152#809—21no—
153#808—21no—
154#807—21no—
155C4⋊(C3⋊C8)—21no—
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4⋊(C3⋊C8)C2311yescharacteristic
160C4×C3⋊C8—21no—
161C2×#135—61no—
162#533—31no—
163#532—31no—
164C4×#41C2×D311yescharacteristic
165#500—31no—
166#492—31no—
167#487—31no—
168#486—31no—
169#485—61no—
170#475—31no—
171#474—31no—
172#468—61no—
173#468—61no—
174#464—61no—
175#463—61no—
176#462—31no—
177#462—31no—
178#461—31no—
179#461—31no—
180#460—31no—
181#459—31no—
182#452—61no—
183C8×Q8C2×D311yescharacteristic
184C8×Q8C2×D311yescharacteristic
185#440—61no—
186#440—61no—
187C4×Q16—61no—
188C4×SD16—61no—
189#436—21no—
190#436—21no—
191#436C2×D311yescharacteristic
192#436C2×D311yescharacteristic
193#432—21no—
194#432C2×D311yescharacteristic
195#431—31no—
196C2×C4⋊C8—31no—
197#416—61no—
198C4×M16—21no—
199C4×M16C2×D311yescharacteristic
200#335—31no—
201#335—31no—
202#334—31no—
203#334—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#331—31no—
211#331—31no—
212C3×#41—22no—
213C3×#41—41no—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×D4—21no—
216C2×C3×D4C2×D411yescharacteristic
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4#4111yescharacteristic
223C2×#67—41no—
224#235—42no—
225#235—42no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×C4×D3—41no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C3×M16—22no—
234C3×M16—41no—
235C3×M16—42no—
236C3×M16—42no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8C2×D411yescharacteristic
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42#4111yescharacteristic
256C3×C42C2411yescharacteristic
257#217—41no—
258#216—42no—
259#216—42no—
260#215—42no—
261#214—42no—
262#214—42no—
263#213—42no—
264(C2×D3)⋊C4—41no—
265(C2×D3)⋊C4—41no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272Dic3⋊C4—41no—
273Dic3⋊C4—41no—
274C4×Dic3—41no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—42no—
280#204—42no—
281#204—42no—
282C2×#41—61no—
283C2×#41D1211yescharacteristic
284#135—124no—
285C2×Q16—61no—
286C2×Q16—61no—
287C2×SD16—61no—
288C2×SD16—61no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×D8—61no—
293C2×M16—21no—
294C2×M16—21no—
295C2×M16#6711yescharacteristic
296C2×M16#6711yescharacteristic
297C22×C8—61no—
298C4⋊Q8—31no—
299C4⋊Q8—31no—
300C4⋊Q8—31no—
301#127—31no—
302#125—61no—
303#124—61no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306#121—61no—
307#121—61no—
308C4×Q8—61no—
309C4×Q8—61no—
310C4×Q8D1211yescharacteristic
311C4×Q8C22×D311yescharacteristic
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4D1211yescharacteristic
317C4×D4C22×D311yescharacteristic
318#117—21no—
319#117—61no—
320#117D1211yescharacteristic
321C2×C42—21no—
322C2×C42C22×D311yescharacteristic
323#107—61no—
324#107—61no—
325#107—61no—
326#106—61no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8—31no—
333C4⋊C8—61no—
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#103—61no—
343#102—61no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348C22⋊C8—21no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—61no—
352#97—21no—
353#97#6711yescharacteristic
354#97#6711yescharacteristic
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—61no—
358C4×C8C22×D311yescharacteristic
359C4×C8C22×D311yescharacteristic
360C23×C3—21no—
361C23×C3#12111yescharacteristic
362C22×D3—41no—
363C3×Q8—21no—
364C3×Q8AGL(1,ℤ/8)12yes—
365C3×D4—21no—
366C3×D4—21no—
367C3×D4—42no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—81no—
380C2×C3×C4#12111yescharacteristic
381C2×C3×C4#12111yescharacteristic
382C2×C3×C4#12111yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386#67—82no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390D12—41no—
391D12—41no—
392D12—41no—
393D12—41no—
394D12—81no—
395C4×D3—82no—
396Dic6—41no—
397Dic6—41no—
398Dic6—41no—
399Dic6—41no—
400Dic6—41no—
401Dic6—41no—
402C3×C8—21no—
403C3×C8—21no—
404C3×C8—21no—
405C3×C8—41no—
406C3×C8—41no—
407C3×C8#13512yescomplemented
408C3⋊C8—41no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412#41—22no—
413#41—41no—
414#41—122no—
415#41—122no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8C2×D1211yescharacteristic
420C2×D4—21no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—121no—
425C2×D4C2×D1211yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—121no—
430C22×C4C2×D1211yescharacteristic
431C22×C4#23511yescharacteristic
432C22×C4C2×#6711yescharacteristic
433Q16—122no—
434Q16—122no—
435SD16—122no—
436SD16—122no—
437SD16—122no—
438SD16—122no—
439SD16—122no—
440D8—122no—
441M16—22no—
442M16—41no—
443M16—42no—
444M16—42no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—122no—
452C2×C8C2×#6711yescharacteristic
453C2×C8C2×#6711yescharacteristic
454C2×C8C2×#6711yescharacteristic
455C2×C8C2×#6711yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—121no—
465C4⋊C4—121no—
466C4⋊C4C2×D1211yescharacteristic
467C4⋊C4C2×D1211yescharacteristic
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—121no—
472C22⋊C4—121no—
473C22⋊C4—121no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—121no—
478C42C2×D1211yescharacteristic
479C42#23511yescharacteristic
480C42C2×#6711yescharacteristic
481C42C23×D311yescharacteristic
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3C2×#12111yescharacteristic
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4C2×#13511yescharacteristic
500C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
501C3×C4#57511yescharacteristic
502C3×C4#57511yescharacteristic
503Dic3—81no—
504Dic3—81no—
505Dic3—81no—
506C23—21no—
507C23—121no—
508C23#93011yescharacteristic
509Q8—21no—
510Q8—121no—
511Q8—121no—
512Q8—121no—
513Q8—121no—
514Q8—121no—
515Q8—121no—
516Q8#90812yes—
517D4—21no—
518D4—21no—
519D4—42no—
520D4—121no—
521D4—121no—
522D4—121no—
523D4—121no—
524D4—241no—
525D4—242no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—81no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—242no—
542C2×C4#93011yescharacteristic
543C2×C4#93011yescharacteristic
544C2×C4#93011yescharacteristic
545C2×C4C22×D1211yescharacteristic
546C2×C4C2×#23511yescharacteristic
547C2×C4C22×#6711yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8—41no—
552C8—41no—
553C8—121no—
554C8—121no—
555C8—121no—
556C8—121no—
557C8#95012yescomplemented
558C2×C3—41no—
559C2×C3—81no—
560C2×C3#303511yescharacteristic
561C2×C3#305511yescharacteristic
562C2×C3#306911yescharacteristic
563D3—161no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—241no—
571C22—241no—
572C22—241no—
573C22C2×#93011yescharacteristic
574C4—21no—
575C4—22no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—41no—
580C4—41no—
581C4—241no—
582C4—241no—
583C4—241no—
584C4C2×#90811yescharacteristic
585C4#574811yescharacteristic
586C4C2×#95011yescharacteristic
587C4#582211yescharacteristic
588C3#1918211yescharacteristic, complemented, Sylow
589C2—41no—
590C2—81no—
591C2—481no—
592C2#8252511yescharacteristic
593C2#8377211yescharacteristic
594C2#8381611yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2971C2
socleC22×C3C2×#121
prime radical#82525C2
hypercenter#2971D3
hypocenterC3#19182
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2971

Complements

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

NH ≅ G/Ntype
1. C3×#2971C2semidirect
2. #82534C2semidirect
4. #78533C2semidirect
5. #70946C2semidirect
6. #70944C2semidirect
7. #70941C2semidirect
8. #70938C2semidirect
9. #70875C2semidirect
10. #70873C2semidirect
11. #70870C2semidirect
12. #70867C2semidirect
13. #70820C2semidirect
14. #70817C2semidirect
15. #69850C2semidirect
71. #2971D3semidirect
407. C3×C8#135semidirect
557. C8#950semidirect
588. C3#19182semidirect

Maximal subgroups

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

subgroupindex
C3×#29712
#825342
#825252
#785332
#709462
#709442
#709412
#709382
#708752
#708732
#708702
#708672
#708202
#708172
#698502
#191823