Finite Group Databasebeta

← #10778221

Subgroups — #10778221

Order 768.

668 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2971C211yesmaximal, characteristic, complemented
2#82529C211yesmaximal, characteristic, complemented
3#82525C211yesmaximal, characteristic, complemented
4#78535C211yesmaximal, characteristic, complemented
5#70943C211yesmaximal, characteristic, complemented
6#70941C211yesmaximal, characteristic, complemented
7#70940C211yesmaximal, characteristic, complemented
8#70939C211yesmaximal, characteristic, complemented
9#70872C211yesmaximal, characteristic, complemented
10#70870C211yesmaximal, characteristic, complemented
11#70869C211yesmaximal, characteristic, complemented
12#70868C211yesmaximal, characteristic, complemented
13#70817C211yesmaximal, characteristic, complemented
14#70816C211yesmaximal, characteristic, complemented
15#69850C211yesmaximal, characteristic, complemented
16#19184—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18C2×#950—21no—
19#5477C2211yescharacteristic
20#5477C2211yescharacteristic
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#5238—21no—
32#5135—21no—
33#5135—21no—
34#5031C2211yescharacteristic
35#5028—21no—
36#5027C2211yescharacteristic
37#5027C2211yescharacteristic
38C4×#215—21no—
39#5021C2211yescharacteristic
40#5019C2211yescharacteristic
41#5016—21no—
42#5015C2211yescharacteristic
43#5015C2211yescharacteristic
44C4×#213—21no—
45#5008C2211yescharacteristic
46#5003—21no—
47#5002—21no—
48#4925C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4711—21no—
51#4711—21no—
52#4695C2211yescharacteristic
53#4694C2211yescharacteristic
54#4694C2211yescharacteristic
55#4680C2211yescharacteristic
56#4486C2211yescharacteristic
57#4486C2211yescharacteristic
58#4485C2211yescharacteristic
59#4485C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3184—31no—
69#3106—31no—
70#3035—31no—
71#2971D311yescharacteristic, complemented
72#1701—31no—
73#1697—31no—
74#1691—31no—
75#1687—31no—
76#1682—31no—
77#1678—31no—
78#1672—31no—
79#1668—31no—
80#1565—31no—
81#1553—31no—
82#1551—31no—
83C2×C3×#41D411yescharacteristic
84C2×#235—21no—
85C2×#235—21no—
86C2×C3×M16—21no—
87C2×C3×M16—21no—
88C2×C3×M16D411yescharacteristic
89C2×C3×M16D411yescharacteristic
90C3×C4×Q8D411yescharacteristic
91C3×C4×Q8C2311yescharacteristic
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C3×C4×D4D411yescharacteristic
95C3×C4×D4C2311yescharacteristic
96C3×#117—21no—
97C3×#117D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100#950—44no—
101C2×#216—21no—
102C2×#215—21no—
103C2×#215—21no—
104C2×#214—21no—
105C2×#213—21no—
106C2×#213—21no—
107#930—21no—
108#930—21no—
109#930—21no—
110#930—21no—
111C22×C3⋊C8—21no—
112C2×D24—21no—
113C2×D24—21no—
114#875—21no—
115#875—21no—
116#874C2311yescharacteristic
117#874C2311yescharacteristic
118C4×D12—21no—
119C4×D12—21no—
120C4×D12—21no—
121C4×D12—21no—
122#869C2311yescharacteristic
123#869C2311yescharacteristic
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#821—21no—
144#821—21no—
145#817—21no—
146#817—21no—
147#810—21no—
148#810—21no—
149#809—21no—
150#809—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#532—31no—
163#532—31no—
164C4×#41C2×D311yescharacteristic
165#498—31no—
166#493—31no—
167#486—31no—
168#486—31no—
169#482—61no—
170#477—31no—
171#476—31no—
172#469—61no—
173#469—61no—
174#464—61no—
175#463—61no—
176#462—31no—
177#461—31no—
178#460—31no—
179#460—31no—
180#459—31no—
181#459—31no—
182#449—61no—
183C8×Q8C2×D311yescharacteristic
184C8×Q8C2×D311yescharacteristic
185#442—61no—
186#442—61no—
187C4×SD16—61no—
188C4×D8—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#333—31no—
205#333—31no—
206#332—31no—
207#332—31no—
208#331—31no—
209#331—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—
224C2×#67—41no—
225#235—42no—
226#235—42no—
227#235—82no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×C4×D3—41no—
235C2×C4×D3—41no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C3×M16—22no—
239C3×M16—41no—
240C3×M16—42no—
241C3×M16—42no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×D411yescharacteristic
245C2×C3×C8C2×D411yescharacteristic
246C2×C3×C8C2×D411yescharacteristic
247C2×C3×C8C2×D411yescharacteristic
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4C2×D411yescharacteristic
251C3×C4⋊C4C2×D411yescharacteristic
252C3×C22⋊C4—21no—
253C3×C22⋊C4—21no—
254C3×C22⋊C4—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42C2×D411yescharacteristic
259C3×C42C2×D411yescharacteristic
260C3×C42#4111yescharacteristic
261C3×C42C2411yescharacteristic
262#216—42no—
263#215—42no—
264#215—42no—
265#214—42no—
266#213—42no—
267#213—42no—
268(C2×D3)⋊C4—41no—
269(C2×D3)⋊C4—41no—
270(C2×D3)⋊C4—41no—
271(C2×D3)⋊C4—41no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—42no—
281D24—42no—
282D24—42no—
283C2×#41—61no—
284C2×#41—61no—
285C2×#41D1211yescharacteristic
286#135—124no—
287C2×Q16—61no—
288C2×SD16—61no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×D8—61no—
292C2×D8—61no—
293C2×D8—61no—
294C2×D8—61no—
295C2×M16—21no—
296C2×M16—21no—
297C2×M16#6711yescharacteristic
298C2×M16#6711yescharacteristic
299C22×C8—61no—
300C4⋊Q8—31no—
301C4⋊Q8—31no—
302#127—31no—
303#127—31no—
304#124—61no—
305#124—61no—
306#121—61no—
307#121—61no—
308#121—61no—
309#121—61no—
310C4×Q8D1211yescharacteristic
311C4×Q8C22×D311yescharacteristic
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4—61no—
317C4×D4—61no—
318C4×D4D1211yescharacteristic
319C4×D4C22×D311yescharacteristic
320#117—21no—
321#117D1211yescharacteristic
322C2×C42—21no—
323C2×C42C22×D311yescharacteristic
324#107—61no—
325#106—61no—
326#106—61no—
327#106—61no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8—31no—
333C4⋊C8—31no—
334C4⋊C8—61no—
335C4⋊C8C22×D311yescharacteristic
336C4⋊C8C22×D311yescharacteristic
337#103—61no—
338#103—61no—
339#103—61no—
340#102—61no—
341#102—61no—
342#102—61no—
343#102—61no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348#102—61no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8—61no—
353#97—21no—
354#97#6711yescharacteristic
355#97#6711yescharacteristic
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—61no—
359C4×C8C22×D311yescharacteristic
360C4×C8C22×D311yescharacteristic
361C23×C3—21no—
362C23×C3#12111yescharacteristic
363C22×D3—41no—
364C22×D3—41no—
365C3×Q8—21no—
366C3×Q8AGL(1,ℤ/8)12yes—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—42no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—41no—
378C2×C3×C4—42no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—81no—
382C2×C3×C4#12111yescharacteristic
383C2×C3×C4#12111yescharacteristic
384C2×C3×C4#12111yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388#67—82no—
389#67—82no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392D12—41no—
393D12—41no—
394D12—41no—
395D12—41no—
396D12—41no—
397D12—41no—
398D12—41no—
399D12—41no—
400D12—81no—
401D12—81no—
402C4×D3—82no—
403C4×D3—82no—
404Dic6—41no—
405Dic6—41no—
406Dic6—81no—
407C3×C8—21no—
408C3×C8—21no—
409C3×C8—21no—
410C3×C8—41no—
411C3×C8—41no—
412C3×C8#13512yescomplemented
413C3⋊C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417#41—22no—
418#41—41no—
419#41—122no—
420#41—122no—
421#41—242no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8C2×D1211yescharacteristic
425C2×D4—21no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—121no—
433C2×D4—121no—
434C2×D4C2×D1211yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—121no—
439C22×C4—121no—
440C22×C4C2×D1211yescharacteristic
441C22×C4#23511yescharacteristic
442C22×C4C2×#6711yescharacteristic
443Q16—122no—
444SD16—122no—
445SD16—122no—
446SD16—122no—
447D8—122no—
448D8—122no—
449D8—122no—
450D8—122no—
451M16—22no—
452M16—41no—
453M16—42no—
454M16—42no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—122no—
462C2×C8C2×#6711yescharacteristic
463C2×C8C2×#6711yescharacteristic
464C2×C8C2×#6711yescharacteristic
465C2×C8C2×#6711yescharacteristic
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4C2×D1211yescharacteristic
473C4⋊C4C2×D1211yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—121no—
478C22⋊C4—121no—
479C22⋊C4—121no—
480C22⋊C4—121no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42C2×D1211yescharacteristic
485C42#23511yescharacteristic
486C42C2×#6711yescharacteristic
487C42C23×D311yescharacteristic
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3C2×#12111yescharacteristic
495C2×D3—81no—
496C2×D3—81no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—22no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4C2×#13511yescharacteristic
509C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
510C3×C4#57511yescharacteristic
511C3×C4#57511yescharacteristic
512Dic3—81no—
513Dic3—81no—
514C23—21no—
515C23—121no—
516C23—121no—
517C23#93011yescharacteristic
518Q8—21no—
519Q8—121no—
520Q8—121no—
521Q8—241no—
522Q8#90812yes—
523D4—21no—
524D4—21no—
525D4—42no—
526D4—121no—
527D4—121no—
528D4—121no—
529D4—121no—
530D4—121no—
531D4—121no—
532D4—121no—
533D4—121no—
534D4—241no—
535D4—241no—
536D4—242no—
537D4—242no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—81no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—242no—
553C2×C4—242no—
554C2×C4#93011yescharacteristic
555C2×C4#93011yescharacteristic
556C2×C4#93011yescharacteristic
557C2×C4C22×D1211yescharacteristic
558C2×C4C2×#23511yescharacteristic
559C2×C4C22×#6711yescharacteristic
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—121no—
566C8—121no—
567C8—121no—
568C8—121no—
569C8#95012yescomplemented
570C2×C3—41no—
571C2×C3—81no—
572C2×C3#303511yescharacteristic
573C2×C3#305511yescharacteristic
574C2×C3#306811yescharacteristic
575D3—161no—
576D3—161no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—241no—
584C22—241no—
585C22—241no—
586C22—241no—
587C22—241no—
588C22—241no—
589C22C2×#93011yescharacteristic
590C4—21no—
591C4—22no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—241no—
598C4—241no—
599C4C2×#90811yescharacteristic
600C4#574811yescharacteristic
601C4C2×#95011yescharacteristic
602C4#582211yescharacteristic
603C3#1918411yescharacteristic, complemented, Sylow
604C2—41no—
605C2—81no—
606C2—481no—
607C2—481no—
608C2#8252511yescharacteristic
609C2#8377211yescharacteristic
610C2#8381411yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2971C2
socleC22×C3C2×#121
hypercenter#2971D3
hypocenterC3#19184
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. #82529C2semidirect
3. #82525C2semidirect
4. #78535C2semidirect
5. #70943C2semidirect
6. #70941C2semidirect
7. #70940C2semidirect
8. #70939C2semidirect
9. #70872C2semidirect
10. #70870C2semidirect
11. #70869C2semidirect
12. #70868C2semidirect
13. #70817C2semidirect
14. #70816C2semidirect
15. #69850C2semidirect
71. #2971D3semidirect
412. C3×C8#135semidirect
569. C8#950semidirect
603. C3#19184semidirect

Maximal subgroups

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

subgroupindex
C3×#29712
#825292
#825252
#785352
#709432
#709412
#709402
#709392
#708722
#708702
#708692
#708682
#708172
#708162
#698502
#191843