Finite Group Databasebeta

← #10778185

Subgroups — #10778185

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×#3182C211yesmaximal, characteristic, complemented
2#83352C211yesmaximal, characteristic, complemented
3#83296C211yesmaximal, characteristic
4#78512C211yesmaximal, characteristic, complemented
5#71831C211yesmaximal, characteristic, complemented
6#71808C211yesmaximal, characteristic, complemented
7#71377C211yesmaximal, characteristic, complemented
8#71373C211yesmaximal, characteristic, complemented
9#71272C211yesmaximal, characteristic, complemented
10#71268C211yesmaximal, characteristic, complemented
11#71240C211yesmaximal, characteristic, complemented
12#71234C211yesmaximal, characteristic, complemented
13#71209C211yesmaximal, characteristic, complemented
14#71203C211yesmaximal, characteristic, complemented
15#69850C211yesmaximal, characteristic, complemented
16#19182—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×#532C2211yescharacteristic
19C2×#950—21no—
20C4×#235C2211yescharacteristic
21C3×#500C2211yescharacteristic
22C3×#492C2211yescharacteristic
23C3×#487C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#468—21no—
26C3×#468—21no—
27C3×#440—21no—
28C3×#440—21no—
29C3×C4×M16C2211yescharacteristic
30#5233—21no—
31#5128—21no—
32#5128—21no—
33#5094C2211yescharacteristic
34#5090C2211yescharacteristic
35#5090C2211yescharacteristic
36#5087C2211yescharacteristic
37#5083C2211yescharacteristic
38#5083C2211yescharacteristic
39#5075C2211yescharacteristic
40#5072C2211yescharacteristic
41#5057—21no—
42#5054—21no—
43#5047—21no—
44#5045—21no—
45C4×#216—21no—
46C4×#215—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#4547C2211yescharacteristic
59#4547C2211yescharacteristic
60#4547C2211yescharacteristic
61#4547C2211yescharacteristic
62#4544C2211yescharacteristic
63#4544C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3182D311yescharacteristic, complemented
69#3107—31no—
70#3035—31no—
71#2971—31no—
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×#41—21no—
84C2×#235D411yescharacteristic
85C2×C3×SD16—21no—
86C2×C3×SD16—21no—
87C2×C3×M16D411yescharacteristic
88C2×C3×M16D411yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4⋊Q8C2311yescharacteristic
92C3×#127C2311yescharacteristic
93C3×#125—21no—
94C3×#124—21no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8—21no—
97C3×#121—21no—
98C3×#121—21no—
99C3×C4×Q8—21no—
100C3×C4×Q8—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×#117—21no—
104C2×C3×C42C2311yescharacteristic
105#950—44no—
106C2×#216—21no—
107C2×#216—21no—
108C2×#215—21no—
109C2×#215—21no—
110C2×#214—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×#107—21no—
125C3×#107—21no—
126C3×#103—21no—
127C3×#103—21no—
128C3×#102—21no—
129C3×#102—21no—
130C3×#97D411yescharacteristic
131C3×#97D411yescharacteristic
132C3×C4×C8C2311yescharacteristic
133C3×C4×C8C2311yescharacteristic
134#835—21no—
135#835—21no—
136#835—21no—
137#835—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#533C2×D311yescharacteristic
163#532C2×D311yescharacteristic
164C4×#41—31no—
165#500C2×D311yescharacteristic
166#492C2×D311yescharacteristic
167#487C2×D311yescharacteristic
168#486C2×D311yescharacteristic
169#485—61no—
170#475—31no—
171#474—31no—
172#468—21no—
173#468—21no—
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×Q8—31no—
184C8×Q8—31no—
185#440—21no—
186#440—21no—
187C4×Q16—61no—
188C4×SD16—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#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—42no—
213C3×#41—42no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—41no—
221C22×C3×C4—41no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4#4111yescharacteristic
225C2×#67—21no—
226#235—22no—
227#235—41no—
228C2×D12C2×D411yescharacteristic
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×Dic6C2×D411yescharacteristic
233C3×SD16—42no—
234C3×SD16—42no—
235C3×M16—22no—
236C3×M16—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⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—41no—
248C3×C4⋊C4—41no—
249C3×C22⋊C4—41no—
250C3×C22⋊C4—41no—
251C3×C22⋊C4—41no—
252C3×C42—41no—
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42#4111yescharacteristic
256C3×C42C2411yescharacteristic
257#217—21no—
258#216—42no—
259#216—42no—
260#215—42no—
261#215—42no—
262#214—42no—
263#213—42no—
264(C2×D3)⋊C4—21no—
265(C2×D3)⋊C4—21no—
266C4⋊Dic3C2×D411yescharacteristic
267C4⋊Dic3C2×D411yescharacteristic
268Dic3⋊C4—21no—
269Dic3⋊C4—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—42no—
280ΓL(1,𝔽25)—42no—
281ΓL(1,𝔽25)—42no—
282C2×#41—21no—
283C2×#41—31no—
284#135—124no—
285C2×Q16—61no—
286C2×Q16—61no—
287C2×SD16—21no—
288C2×SD16—21no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×D8—61no—
293C2×M16—61no—
294C2×M16—61no—
295C2×M16#6711yescharacteristic
296C2×M16#6711yescharacteristic
297C22×C8—61no—
298C4⋊Q8C22×D311yescharacteristic
299C4⋊Q8C22×D311yescharacteristic
300C4⋊Q8C22×D311yescharacteristic
301#127C22×D311yescharacteristic
302#125—21no—
303#124—21no—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306#121—21no—
307#121—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—31no—
311C4×Q8—31no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—31no—
315C4×D4—31no—
316C4×D4—61no—
317C4×D4—61no—
318#117—21no—
319#117—31no—
320#117—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—21no—
324#107—21no—
325#107—61no—
326#106—61no—
327C4⋊C8—31no—
328C4⋊C8—31no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—31no—
332C4⋊C8—31no—
333C4⋊C8—61no—
334C4⋊C8—61no—
335C4⋊C8—61no—
336#103—21no—
337#103—21no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#103—61no—
343#102—21no—
344#102—21no—
345#102—61no—
346#102—61no—
347#102—61no—
348C22⋊C8—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352#97—61no—
353#97#6711yescharacteristic
354#97#6711yescharacteristic
355C4×C8—61no—
356C4×C8—61no—
357C4×C8—61no—
358C4×C8C22×D311yescharacteristic
359C4×C8C22×D311yescharacteristic
360C23×C3—41no—
361C23×C3#12111yescharacteristic
362C22×D3—21no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—81no—
374C3×D4—82no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—82no—
384C2×C3×C4#12111yescharacteristic
385C2×C3×C4#12111yescharacteristic
386C2×C3×C4#12111yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390#67—42no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394D12—21no—
395D12—21no—
396C4×D3—42no—
397C4×D3—41no—
398C4×D3—41no—
399C4×D3—81no—
400Dic6—21no—
401Dic6AGL(1,ℤ/8)12yes—
402C3×C8—21no—
403C3×C8—21no—
404C3×C8—21no—
405C3×C8#13512yescomplemented
406C3⋊C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412#41—42no—
413#41—42no—
414#41—62no—
415#41—121no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—31no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—31no—
424C2×D4—41no—
425C2×D4—61no—
426C22×C4—41no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4D3×D411yescharacteristic
431C22×C4#23711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433Q16—122no—
434Q16—122no—
435SD16—42no—
436SD16—42no—
437SD16—122no—
438SD16—122no—
439SD16—122no—
440D8—122no—
441M16—22no—
442M16—41no—
443M16—122no—
444M16—122no—
445C2×C8—61no—
446C2×C8—61no—
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—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—31no—
463C4⋊C4—31no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C22⋊C4—61no—
474C42—41no—
475C42—61no—
476C42—61no—
477C42—61no—
478C42D3×D411yescharacteristic
479C42#23711yescharacteristic
480C42C2×#6711yescharacteristic
481C42C23×D311yescharacteristic
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3C2×#12111yescharacteristic
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4C2×#13511yescharacteristic
500C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
501C3×C4#57511yescharacteristic
502C3×C4#57511yescharacteristic
503Dic3—22no—
504Dic3—41no—
505Dic3—41no—
506C23—41no—
507C23—61no—
508C23#93811yescharacteristic
509Q8—32no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—61no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—61no—
522D4—61no—
523D4—81no—
524D4—82no—
525D4—122no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—82no—
538C2×C4—122no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—241no—
542C2×C4#93811yescharacteristic
543C2×C4#93811yescharacteristic
544C2×C4#93811yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×#23711yescharacteristic
547C2×C4C22×#6711yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8—121no—
552C8—121no—
553C8—121no—
554C8—121no—
555C8—121no—
556C8—121no—
557C8#95012yescomplemented
558C2×C3—41no—
559C2×C3—161no—
560C2×C3#303511yescharacteristic
561C2×C3#305511yescharacteristic
562C2×C3#306911yescharacteristic
563D3—81no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—121no—
571C22—121no—
572C22—121no—
573C22C2×#93811yescharacteristic
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—62no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—121no—
583C4—121no—
584C4D3×AGL(1,ℤ/8)11yescharacteristic
585C4#577311yescharacteristic
586C4C2×#95011yescharacteristic
587C4#582211yescharacteristic
588C3#1918211yescharacteristic, complemented, Sylow
589C2—41no—
590C2—161no—
591C2—241no—
592C2#8329611yescharacteristic
593C2#8395011yescharacteristic
594C2#8399811yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3182C2
socleC22×C3C2×#121
prime radical#83296C2
hypercenter#3182D3
hypocenterC3#19182
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3182

Complements

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

NH ≅ G/Ntype
1. C3×#3182C2semidirect
2. #83352C2semidirect
4. #78512C2semidirect
5. #71831C2semidirect
6. #71808C2semidirect
7. #71377C2semidirect
8. #71373C2semidirect
9. #71272C2semidirect
10. #71268C2semidirect
11. #71240C2semidirect
12. #71234C2semidirect
13. #71209C2semidirect
14. #71203C2semidirect
15. #69850C2semidirect
68. #3182D3semidirect
405. 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×#31822
#833522
#832962
#785122
#718312
#718082
#713772
#713732
#712722
#712682
#712402
#712342
#712092
#712032
#698502
#191823