Finite Group Databasebeta

← #10778215

Subgroups — #10778215

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×#3184C211yesmaximal, characteristic, complemented
2#83299C211yesmaximal, characteristic, complemented
3#83296C211yesmaximal, characteristic, complemented
4#78512C211yesmaximal, characteristic, complemented
5#71808C211yesmaximal, characteristic, complemented
6#71807C211yesmaximal, characteristic, complemented
7#71348C211yesmaximal, characteristic, complemented
8#71345C211yesmaximal, characteristic, complemented
9#71286C211yesmaximal, characteristic, complemented
10#71283C211yesmaximal, characteristic, complemented
11#71209C211yesmaximal, characteristic, complemented
12#71208C211yesmaximal, characteristic, complemented
13#71203C211yesmaximal, characteristic, complemented
14#71202C211yesmaximal, characteristic, complemented
15#69850C211yesmaximal, characteristic, complemented
16#19184—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C3×#532C2211yescharacteristic
19C2×#950—21no—
20C4×#235C2211yescharacteristic
21C3×#498C2211yescharacteristic
22C3×#493C2211yescharacteristic
23C3×#486C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#469—21no—
26C3×#469—21no—
27C3×#442—21no—
28C3×#442—21no—
29C3×C4×M16C2211yescharacteristic
30#5233—21no—
31#5128—21no—
32#5128—21no—
33#5091C2211yescharacteristic
34#5090C2211yescharacteristic
35#5084C2211yescharacteristic
36#5083C2211yescharacteristic
37#5075C2211yescharacteristic
38#5075C2211yescharacteristic
39#5072C2211yescharacteristic
40#5072C2211yescharacteristic
41#5057—21no—
42#5054—21no—
43#5037—21no—
44#5035—21no—
45C4×#214—21no—
46C4×#213—21no—
47#4926C2211yescharacteristic
48#4925C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4707—21no—
51#4705—21no—
52#4688C2211yescharacteristic
53#4688C2211yescharacteristic
54C8×Dic6C2211yescharacteristic
55C8×Dic6C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4546C2211yescharacteristic
59#4546C2211yescharacteristic
60#4544C2211yescharacteristic
61#4544C2211yescharacteristic
62#4544C2211yescharacteristic
63#4544C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3184D311yescharacteristic, complemented
69#3106—31no—
70#3035—31no—
71#2971—31no—
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×#41—21no—
84C2×C3×#41—21no—
85C2×#235D411yescharacteristic
86C2×C3×D8—21no—
87C2×C3×D8—21no—
88C2×C3×M16D411yescharacteristic
89C2×C3×M16D411yescharacteristic
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4⋊Q8C2311yescharacteristic
92C3×#127C2311yescharacteristic
93C3×#127C2311yescharacteristic
94C3×#124—21no—
95C3×#124—21no—
96C3×#121—21no—
97C3×#121—21no—
98C3×#121—21no—
99C3×#121—21no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C2×C3×C42C2311yescharacteristic
105#950—44no—
106C2×#216—21no—
107C2×#215—21no—
108C2×#214—21no—
109C2×#214—21no—
110C2×#213—21no—
111C2×#213—21no—
112C4×#67—21no—
113C4×#67—21no—
114#926D411yescharacteristic
115C22×C3⋊C8—21no—
116C2×ΓL(1,𝔽25)—21no—
117C2×ΓL(1,𝔽25)—21no—
118C4×D12D411yescharacteristic
119C4×D12C2311yescharacteristic
120#872—21no—
121C42×D3—21no—
122C4×Dic6D411yescharacteristic
123C4×Dic6C2311yescharacteristic
124C3×#106—21no—
125C3×#106—21no—
126C3×#102—21no—
127C3×#102—21no—
128C3×#102—21no—
129C3×#102—21no—
130C3×#97D411yescharacteristic
131C3×#97D411yescharacteristic
132C3×C4×C8C2311yescharacteristic
133C3×C4×C8C2311yescharacteristic
134#835—21no—
135#835—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139#832—21no—
140#830—21no—
141#830—21no—
142(C2×D3)⋊C8—21no—
143(C2×D3)⋊C8—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148#810—21no—
149#809—21no—
150#808—21no—
151#807—21no—
152C4⋊(C3⋊C8)—21no—
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159#803—21no—
160C4×C3⋊C8—21no—
161C2×#135—61no—
162#532C2×D311yescharacteristic
163#532C2×D311yescharacteristic
164C4×#41—31no—
165#498C2×D311yescharacteristic
166#493C2×D311yescharacteristic
167#486C2×D311yescharacteristic
168#486C2×D311yescharacteristic
169#482—61no—
170#477—31no—
171#476—31no—
172#469—21no—
173#469—21no—
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×Q8—31no—
184C8×Q8—31no—
185#442—21no—
186#442—21no—
187C4×SD16—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#436—61no—
192#436—61no—
193#432—31no—
194#432—61no—
195#431—31no—
196C2×C4⋊C8—31no—
197#416—61no—
198C4×M16—61no—
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—42no—
213C3×#41—42no—
214C3×#41—82no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—41no—
224C2×C3×D4—41no—
225C22×C3×C4—41no—
226C22×C3×C4—41no—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4#4111yescharacteristic
230C2×#67—21no—
231#235—22no—
232#235—41no—
233C2×D12C2×D411yescharacteristic
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×Dic6C2×D411yescharacteristic
238C3×D8—42no—
239C3×D8—42no—
240C3×M16—22no—
241C3×M16—41no—
242C2×C3×C8C2×D411yescharacteristic
243C2×C3×C8C2×D411yescharacteristic
244C2×C3×C8C2×D411yescharacteristic
245C2×C3×C8C2×D411yescharacteristic
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C22⋊C4—41no—
251C3×C22⋊C4—41no—
252C3×C22⋊C4—41no—
253C3×C22⋊C4—41no—
254C3×C42C2×D411yescharacteristic
255C3×C42C2×D411yescharacteristic
256C3×C42#4111yescharacteristic
257C3×C42C2411yescharacteristic
258#217—21no—
259#216—42no—
260#215—42no—
261#214—42no—
262#214—42no—
263#213—42no—
264#213—42no—
265(C2×D3)⋊C4—21no—
266(C2×D3)⋊C4—21no—
267C4⋊Dic3C2×D411yescharacteristic
268C4⋊Dic3C2×D411yescharacteristic
269Dic3⋊C4—21no—
270Dic3⋊C4—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—42no—
281ΓL(1,𝔽25)—42no—
282ΓL(1,𝔽25)—42no—
283C2×#41—21no—
284C2×#41—21no—
285C2×#41—31no—
286#135—124no—
287C2×Q16—61no—
288C2×SD16—61no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×D8—21no—
292C2×D8—21no—
293C2×D8—61no—
294C2×D8—61no—
295C2×M16—61no—
296C2×M16—61no—
297C2×M16#6711yescharacteristic
298C2×M16#6711yescharacteristic
299C22×C8—61no—
300C4⋊Q8C22×D311yescharacteristic
301C4⋊Q8C22×D311yescharacteristic
302#127C22×D311yescharacteristic
303#127C22×D311yescharacteristic
304#124—21no—
305#124—21no—
306#121—21no—
307#121—21no—
308#121—21no—
309#121—21no—
310C4×Q8—31no—
311C4×Q8—31no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—31no—
317C4×D4—31no—
318C4×D4—61no—
319C4×D4—61no—
320#117—31no—
321#117—61no—
322C2×C42—61no—
323C2×C42C22×D311yescharacteristic
324#107—61no—
325#106—21no—
326#106—21no—
327#106—61no—
328C4⋊C8—31no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8—31no—
333C4⋊C8—31no—
334C4⋊C8—61no—
335C4⋊C8—61no—
336C4⋊C8—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#102—21no—
341#102—21no—
342#102—21no—
343#102—21no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348#102—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352C22⋊C8—61no—
353#97—61no—
354#97#6711yescharacteristic
355#97#6711yescharacteristic
356C4×C8—61no—
357C4×C8—61no—
358C4×C8—61no—
359C4×C8C22×D311yescharacteristic
360C4×C8C22×D311yescharacteristic
361C23×C3—41no—
362C23×C3—41no—
363C23×C3#12111yescharacteristic
364C22×D3—21no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—81no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—81no—
377C3×D4—81no—
378C3×D4—82no—
379C3×D4—82no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—82no—
388C2×C3×C4—82no—
389C2×C3×C4#12111yescharacteristic
390C2×C3×C4#12111yescharacteristic
391C2×C3×C4#12111yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395#67—42no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399D12—21no—
400D12—21no—
401C4×D3—42no—
402C4×D3—41no—
403C4×D3—41no—
404C4×D3—81no—
405Dic6—21no—
406Dic6AGL(1,ℤ/8)12yes—
407C3×C8—21no—
408C3×C8—21no—
409C3×C8—21no—
410C3×C8#13512yescomplemented
411C3⋊C8—41no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417#41—42no—
418#41—42no—
419#41—62no—
420#41—82no—
421#41—121no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—31no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—31no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—61no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4D3×D411yescharacteristic
441C22×C4#23711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443Q16—122no—
444SD16—122no—
445SD16—122no—
446SD16—122no—
447D8—42no—
448D8—42no—
449D8—122no—
450D8—122no—
451M16—22no—
452M16—41no—
453M16—122no—
454M16—122no—
455C2×C8—61no—
456C2×C8—61no—
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—21no—
469C4⋊C4—21no—
470C4⋊C4—31no—
471C4⋊C4—31no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—61no—
479C22⋊C4—61no—
480C22⋊C4—61no—
481C42—61no—
482C42—61no—
483C42—61no—
484C42D3×D411yescharacteristic
485C42#23711yescharacteristic
486C42C2×#6711yescharacteristic
487C42C23×D311yescharacteristic
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3C2×#12111yescharacteristic
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—41no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4C2×#13511yescharacteristic
508C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
509C3×C4#57511yescharacteristic
510C3×C4#57511yescharacteristic
511Dic3—22no—
512Dic3—41no—
513Dic3—41no—
514C23—41no—
515C23—41no—
516C23—61no—
517C23#93811yescharacteristic
518Q8—32no—
519Q8—41no—
520Q8—41no—
521Q8—61no—
522Q8—81no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—61no—
532D4—61no—
533D4—81no—
534D4—81no—
535D4—82no—
536D4—82no—
537D4—122no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—122no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—241no—
554C2×C4#93811yescharacteristic
555C2×C4#93811yescharacteristic
556C2×C4#93811yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×#23711yescharacteristic
559C2×C4C22×#6711yescharacteristic
560C8—21no—
561C8—21no—
562C8—21no—
563C8—121no—
564C8—121no—
565C8—121no—
566C8—121no—
567C8—121no—
568C8—121no—
569C8#95012yescomplemented
570C2×C3—41no—
571C2×C3—161no—
572C2×C3—161no—
573C2×C3#303511yescharacteristic
574C2×C3#305511yescharacteristic
575C2×C3#306811yescharacteristic
576D3—81no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—121no—
587C22—121no—
588C22—121no—
589C22C2×#93811yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—62no—
595C4—81no—
596C4—81no—
597C4—121no—
598C4—121no—
599C4D3×AGL(1,ℤ/8)11yescharacteristic
600C4#577311yescharacteristic
601C4C2×#95011yescharacteristic
602C4#582211yescharacteristic
603C3#1918411yescharacteristic, complemented, Sylow
604C2—41no—
605C2—161no—
606C2—161no—
607C2—241no—
608C2#8329611yescharacteristic
609C2#8395011yescharacteristic
610C2#8396311yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3184C2
socleC22×C3C2×#121
hypercenter#3184D3
hypocenterC3#19184
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3184

Complements

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

NH ≅ G/Ntype
1. C3×#3184C2semidirect
2. #83299C2semidirect
3. #83296C2semidirect
4. #78512C2semidirect
5. #71808C2semidirect
6. #71807C2semidirect
7. #71348C2semidirect
8. #71345C2semidirect
9. #71286C2semidirect
10. #71283C2semidirect
11. #71209C2semidirect
12. #71208C2semidirect
13. #71203C2semidirect
14. #71202C2semidirect
15. #69850C2semidirect
68. #3184D3semidirect
410. 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×#31842
#832992
#832962
#785122
#718082
#718072
#713482
#713452
#712862
#712832
#712092
#712082
#712032
#712022
#698502
#191843