Finite Group Databasebeta

← #11660897

Subgroups — #11660897

Order 768.

2142 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1004C211yesmaximal, characteristic
2C2×C3×C8×D4C211yesmaximal, characteristic, complemented
3C2×#5111C214yesmaximal, complemented
4#82301C211yesmaximal, characteristic, complemented
5#79427C214yesmaximal, complemented
6D4×ΓL(1,𝔽25)C214yesmaximal, complemented
7#79406C214yesmaximal, complemented
8#79404C214yesmaximal, complemented
9#79014C212yesmaximal, complemented
10#78602C214yesmaximal, complemented
11#78447C212yesmaximal, complemented
12#60309—31nomaximal, Sylow
13C2×#1004C411yescharacteristic
14C23×C3×C8C2212yescomplemented
15C2×C3×C4×D4C2211yescharacteristic
16#5798C411yescharacteristic
17C2×C4×#67C2214yes—
18C22×ΓL(1,𝔽25)C2214yescomplemented
19#5546C414yes—
20#5545C414yes—
21C4×D3×D4C2214yes—
22C4×#237C2214yes—
23#5504C412yes—
24#5486C414yes—
25C4×#235C2212yes—
26C3×C8×D4C2214yescomplemented
27C3×C8×D4C2214yescomplemented
28C2×C3×C4⋊C8C2211yescharacteristic, complemented
29C2×C3×C22⋊C8C2212yescomplemented
30C2×C3×C4×C8C2211yescharacteristic, complemented
31C2×#830C2214yescomplemented
32#5111C2214yescomplemented
33#5111C2218yescomplemented
34#5111C2214yescomplemented
35#5111C2218yescomplemented
36#5111C2218yescomplemented
37C2×(C2×D3)⋊C8C2214yescomplemented
38C2×#815C2214yescomplemented
39C2×Dic3⋊C8C2214yescomplemented
40D4×C3⋊C8C2214yescomplemented
41D4×C3⋊C8C2214yescomplemented
42#4925C2212yescomplemented
43#4838C2214yescomplemented
44#4837C2214yescomplemented
45#4835C2214yescomplemented
46#4831C2214yescomplemented
47#4730C2218yescomplemented
48#4730C2218yescomplemented
49#4726C2218yescomplemented
50#4719C2218yescomplemented
51#4688C2212yescomplemented
52#4688C2212yescomplemented
53C4×ΓL(1,𝔽25)C2214yescomplemented
54#4685C2214yescomplemented
55#4679C2212yescomplemented
56#4679C2212yescomplemented
57C4×H(5,𝔽2)—31no—
58#2990—34no—
59#2989—32no—
60#2974—34no—
61#2973—34no—
62#2972—31no—
63#2970—34no—
64#2965—32no—
65D4×M16—34no—
66C2×#435—34no—
67C2×C8×D4D311yescharacteristic, complemented
68C22×C3×D4C2×C411yescharacteristic
69C23×C3×C4C2312yes—
70C22×#67C2×C414yes—
71#1004—28no—
72C2×#237C2×C414yes—
73C2×D3×D4C2×C414yes—
74C2×#235C2×C412yes—
75C22×C4×D3C2314yes—
76C22×C3×C8—44no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2314yes—
79C22×C3×C8C2312yes—
80C22×C3×C8C2314yes—
81C22×C3×C8C2311yescharacteristic
82C3×C4×D4C2×C414yes—
83C3×C4×D4C2314yes—
84C2×C3×C4⋊C4C2×C411yescharacteristic
85C2×C3×C22⋊C4C2×C412yes—
86C2×C3×C42C2311yescharacteristic
87C2×#217C2×C414yes—
88D4×Dic3C2×C414yes—
89D4×Dic3C2×C414yes—
90C4×#67C2×C414yes—
91C4×#67C2×C418yes—
92C4×#67C2×C418yes—
93C4×#67C2314yes—
94C4×#67C2318yes—
95C2×(C2×D3)⋊C4C2×C414yes—
96#926C2×C412yes—
97C2×Dic3⋊C4C2×C414yes—
98C2×C4×Dic3C2314yes—
99C22×C3⋊C8C2314yes—
100C22×C3⋊C8C2314yes—
101C2×ΓL(1,𝔽25)—48no—
102C2×ΓL(1,𝔽25)C2314yes—
103C2×ΓL(1,𝔽25)C2314yes—
104C2×ΓL(1,𝔽25)C2318yes—
105#893C2×C414yes—
106#892C2×C414yes—
107D3×C4⋊C4C2×C414yes—
108#887C2×C414yes—
109#67⋊C4C2×C418yes—
110#67⋊C4C2×C418yes—
111D3×C22⋊C4C2×C418yes—
112#877C2×C418yes—
113C4×D12C2×C412yes—
114C4×D12C2312yes—
115#872C2×C414yes—
116C42×D3C2314yes—
117C4×Dic6C2×C412yes—
118C4×Dic6C2312yes—
119C3×C4⋊C8C2314yes—
120C3×C22⋊C8C2318yes—
121C3×C4×C8C2312yes—
122C3×C4×C8C2312yes—
123#830C2314yes—
124#830C2318yes—
125(C2×D3)⋊C8C2314yes—
126(C2×D3)⋊C8C2314yes—
127(C2×D3)⋊C8C2318yes—
128#815C2314yes—
129#815C2314yes—
130#815C2318yes—
131Dic3⋊C8C2314yes—
132Dic3⋊C8C2314yes—
133Dic3⋊C8C2318yes—
134C4⋊(C3⋊C8)C2312yes—
135C4⋊(C3⋊C8)C2312yes—
136C4×C3⋊C8C2314yes—
137C2×H(5,𝔽2)—31no—
138C22×M16—34no—
139C23×C8C2×D312yescomplemented
140#520—34no—
141#520—32no—
142#518—31no—
143#518—34no—
144#518—34no—
145C4×#41—34no—
146C4×#41—32no—
147C2×C4×D4—34no—
148C2×C4×D4—34no—
149C2×C4×D4C2×D311yescharacteristic
150#446—34no—
151#446—32no—
152#446—32no—
153#436—34no—
154#436—32no—
155#436—32no—
156#435—34no—
157#435—38no—
158#435—38no—
159#435—38no—
160#435—34no—
161#435—38no—
162#435—38no—
163C8×D4—34no—
164C8×D4—34no—
165C8×D4C2×D314yescomplemented
166C8×D4C2×D314yescomplemented
167#432—34no—
168#432—38no—
169#431—32no—
170#431—34no—
171#423—34no—
172C2×C4⋊C8—34no—
173C2×C4⋊C8C2×D311yescharacteristic, complemented
174#407—38no—
175C2×C22⋊C8—34no—
176C2×C22⋊C8—34no—
177C2×C22⋊C8C2×D312yescomplemented
178C4×M16—34no—
179C2×#97—34no—
180C2×C4×C8C2×D311yescharacteristic, complemented
181C24×C3C22×C412yes—
182C23×D3C22×C414yes—
183C2×C3×D4M1618yescomplemented
184C2×C3×D4C22×C414yes—
185C22×C3×C4—42no—
186C22×C3×C4—42no—
187C22×C3×C4C22×C414yes—
188C22×C3×C4C22×C411yescharacteristic
189C22×C3×C4C22×C412yes—
190C22×C3×C4C22×C411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×C3×C4C2414yes—
193C2×#67—216no—
194C2×#67C22×C418yes—
195C2×#67C22×C414yes—
196C22×Dic3C22×C414yes—
197C22×Dic3C22×C414yes—
198C22×Dic3C22×C414yes—
199#237—216no—
200D3×D4—216no—
201#235—28no—
202C2×D12C22×C412yes—
203C2×C4×D3—44no—
204C2×C4×D3—44no—
205C2×C4×D3C22×C414yes—
206C2×C4×D3C22×C414yes—
207C2×C4×D3C22×C418yes—
208C2×C4×D3C2414yes—
209C2×Dic6C22×C412yes—
210C2×C3×C8—28no—
211C2×C3×C8—22no—
212C2×C3×C8—42no—
213C2×C3×C8—44no—
214C2×C3×C8—41no—
215C2×C3×C8C2411yescharacteristic
216C2×C3×C8C2412yes—
217C2×C3×C8C2414yes—
218C2×C3×C8C2411yescharacteristic
219C3×C4⋊C4C22×C414yes—
220C3×C22⋊C4C22×C418yes—
221C3×C42C22×C412yes—
222C3×C42C2412yes—
223#217C22×C418yes—
224#217C22×C414yes—
225(C2×D3)⋊C4C22×C414yes—
226(C2×D3)⋊C4C22×C414yes—
227(C2×D3)⋊C4C22×C418yes—
228C4⋊Dic3C22×C412yes—
229C4⋊Dic3C22×C412yes—
230Dic3⋊C4C22×C414yes—
231Dic3⋊C4C22×C414yes—
232Dic3⋊C4C22×C418yes—
233C4×Dic3C22×C414yes—
234C4×Dic3C22×C414yes—
235C4×Dic3C22×C418yes—
236C4×Dic3C2414yes—
237C2×C3⋊C8—44no—
238C2×C3⋊C8—44no—
239C2×C3⋊C8C2414yes—
240C2×C3⋊C8C2414yes—
241ΓL(1,𝔽25)—28no—
242ΓL(1,𝔽25)—48no—
243ΓL(1,𝔽25)—44no—
244H(5,𝔽2)—68no—
245C2×#41—34no—
246C2×#41—32no—
247C22×D4—34no—
248C22×D4—34no—
249C22×D4C4×D311yescharacteristic
250C23×C4—34no—
251C23×C4C22×D312yes—
252C2×M16—34no—
253C2×M16—38no—
254C2×M16—34no—
255C2×M16—128no—
256C22×C8—34no—
257C22×C8—34no—
258C22×C8—44no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D314yes—
261C22×C8C22×D314yes—
262C22×C8C22×D312yes—
263C22×C8C22×D311yescharacteristic
264C4×Q8—34no—
265C4×Q8—32no—
266C4×Q8—32no—
267C4×D4—34no—
268C4×D4—34no—
269C4×D4—38no—
270C4×D4—38no—
271C4×D4—34no—
272C4×D4—34no—
273C4×D4—38no—
274C4×D4—34no—
275C4×D4—38no—
276C4×D4—38no—
277C4×D4—32no—
278C4×D4—32no—
279C4×D4C4×D314yes—
280C4×D4C22×D314yes—
281#117—32no—
282#117—34no—
283#117—38no—
284#117—34no—
285C2×C4⋊C4—34no—
286C2×C4⋊C4—34no—
287C2×C4⋊C4C4×D311yescharacteristic
288C2×C22⋊C4—34no—
289C2×C22⋊C4—34no—
290C2×C22⋊C4—38no—
291C2×C22⋊C4C4×D312yes—
292C2×C42—34no—
293C2×C42—34no—
294C2×C42C22×D311yescharacteristic
295C4⋊C8—34no—
296C4⋊C8—38no—
297C4⋊C8—34no—
298C4⋊C8—32no—
299C4⋊C8—32no—
300C4⋊C8C22×D314yes—
301C22⋊C8—34no—
302C22⋊C8—38no—
303C22⋊C8—34no—
304C22⋊C8—38no—
305C22⋊C8—34no—
306C22⋊C8C22×D318yes—
307#97—34no—
308#97—38no—
309#97—34no—
310C4×C8—34no—
311C4×C8C22×D312yes—
312C4×C8C22×D312yes—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3C2×M1618yescomplemented
316C23×C3C23×C411yescharacteristic
317C23×C3C23×C414yes—
318C22×D3—28no—
319C22×D3—44no—
320C22×D3—44no—
321C22×D3C23×C414yes—
322C3×D4C2×M16116yescomplemented
323C2×C3×C4—24no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—21no—
327C2×C3×C4—42no—
328C2×C3×C4—44no—
329C2×C3×C4—41no—
330C2×C3×C4C2×M1614yescomplemented
331C2×C3×C4C23×C412yes—
332C2×C3×C4C23×C414yes—
333C2×C3×C4C23×C411yescharacteristic
334C2×C3×C4C23×C412yes—
335C2×C3×C4C2511yescharacteristic
336#67—28no—
337#67—216no—
338C2×Dic3—28no—
339C2×Dic3—44no—
340C2×Dic3—44no—
341C2×Dic3C23×C414yes—
342C2×Dic3C23×C414yes—
343C2×Dic3C23×C414yes—
344D12—24no—
345C4×D3—28no—
346C4×D3—24no—
347C4×D3—24no—
348C4×D3—48no—
349C4×D3—44no—
350Dic6—24no—
351C3×C8—24no—
352C3×C8—21no—
353C3×C8—22no—
354C3×C8H(5,𝔽2)12yescomplemented
355C3⋊C8—24no—
356C3⋊C8—24no—
357C24—34no—
358C24C2×C4×D312yes—
359#41—616no—
360#41—68no—
361C2×Q8—32no—
362C2×D4—34no—
363C2×D4—38no—
364C2×D4—32no—
365C2×D4—616no—
366C2×D4—616no—
367C2×D4ΓL(1,𝔽25)18yescomplemented
368C2×D4C2×C4×D314yes—
369C22×C4—34no—
370C22×C4—34no—
371C22×C4—34no—
372C22×C4—34no—
373C22×C4—34no—
374C22×C4—34no—
375C22×C4—38no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—124no—
379C22×C4—124no—
380C22×C4C2×C4×D311yescharacteristic
381C22×C4C2×C4×D314yes—
382C22×C4C2×C4×D312yes—
383C22×C4C2×C4×D311yescharacteristic
384C22×C4C23×D311yescharacteristic
385C22×C4C23×D314yes—
386M16—68no—
387M16—128no—
388M16—124no—
389C2×C8—22no—
390C2×C8—28no—
391C2×C8—34no—
392C2×C8—34no—
393C2×C8—44no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—124no—
397C2×C8—124no—
398C2×C8C23×D312yes—
399C2×C8C23×D311yescharacteristic
400C2×C8C23×D314yes—
401C2×C8C23×D311yescharacteristic
402C4⋊C4—38no—
403C4⋊C4—34no—
404C4⋊C4—32no—
405C4⋊C4—34no—
406C4⋊C4—32no—
407C4⋊C4C2×C4×D314yes—
408C22⋊C4—34no—
409C22⋊C4—38no—
410C22⋊C4—38no—
411C22⋊C4—34no—
412C22⋊C4—34no—
413C22⋊C4C2×C4×D318yes—
414C42—38no—
415C42—34no—
416C42—34no—
417C42—34no—
418C42C2×C4×D312yes—
419C42C23×D312yes—
420C22×C3—24no—
421C22×C3—21no—
422C22×C3—24no—
423C22×C3—21no—
424C22×C3—48no—
425C22×C3C22×M1618yes—
426C22×C3C22×M1612yes—
427C22×C3C24×C411yescharacteristic
428C2×D3—24no—
429C2×D3—24no—
430C2×D3—24no—
431C2×D3—88no—
432C3×C4—24no—
433C3×C4—21no—
434C3×C4—22no—
435C3×C4C22×M1614yes—
436C3×C4C2×H(5,𝔽2)11yescharacteristic
437C3×C4#58511yescharacteristic
438Dic3—24no—
439Dic3—24no—
440Dic3—24no—
441C23—34no—
442C23—42no—
443C23—42no—
444C23—68no—
445C23—124no—
446C23—124no—
447C23C2×ΓL(1,𝔽25)18yescomplemented
448C23C22×C4×D311yescharacteristic
449C23C22×C4×D314yes—
450Q8—64no—
451D4—616no—
452D4—68no—
453D4—64no—
454D4C2×ΓL(1,𝔽25)116yescomplemented
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—34no—
460C2×C4—34no—
461C2×C4—34no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—68no—
466C2×C4—64no—
467C2×C4—64no—
468C2×C4—68no—
469C2×C4—128no—
470C2×C4—124no—
471C2×C4—124no—
472C2×C4—124no—
473C2×C4C2×ΓL(1,𝔽25)14yescomplemented
474C2×C4C22×C4×D312yes—
475C2×C4C22×C4×D314yes—
476C2×C4C22×C4×D311yescharacteristic
477C2×C4C22×C4×D312yes—
478C2×C4C24×D311yescharacteristic
479C8—21no—
480C8—24no—
481C8—22no—
482C8—64no—
483C8—64no—
484C8#100412yescomplemented
485C2×C3—28no—
486C2×C3—22no—
487C2×C3C4×H(5,𝔽2)11yescharacteristic
488C2×C3C23×M1611yescharacteristic
489C2×C3#357311yescharacteristic
490D3—44no—
491C22—21no—
492C22—24no—
493C22—21no—
494C22—24no—
495C22—48no—
496C22—64no—
497C22—64no—
498C22—64no—
499C22—248no—
500C22C22×ΓL(1,𝔽25)18yes—
501C22C22×ΓL(1,𝔽25)12yes—
502C22C23×C4×D311yescharacteristic
503C4—22no—
504C4—21no—
505C4—24no—
506C4—64no—
507C4—64no—
508C4—64no—
509C4C22×ΓL(1,𝔽25)14yes—
510C4C2×#100411yescharacteristic
511C4#596411yescharacteristic
512C3#6030911yescharacteristic, complemented, Sylow
513C2—22no—
514C2—28no—
515C2—124no—
516C2C4×#100411yescharacteristic
517C2C23×ΓL(1,𝔽25)11yescharacteristic
518C2#8716011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC2×C3×C8×D4C2
prime radicalC2×#1004C4
cosocleC2×C3×C4C25
hypercenterC2×C8×D4D3
hypocenterC3#60309
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×D4C2semidirect
3. C2×#5111C2semidirect
4. #82301C2semidirect
5. #79427C2semidirect
6. D4×ΓL(1,𝔽25)C2semidirect
7. #79406C2semidirect
8. #79404C2semidirect
9. #79014C2semidirect
10. #78602C2semidirect
11. #78447C2semidirect
14. C23×C3×C8C22semidirect
18. C22×ΓL(1,𝔽25)C22semidirect
26. C3×C8×D4C22semidirect
27. C3×C8×D4C22semidirect
28. C2×C3×C4⋊C8C22semidirect
29. C2×C3×C22⋊C8C22semidirect
30. C2×C3×C4×C8C22semidirect
31. C2×#830C22semidirect
32. #5111C22semidirect
33. #5111C22semidirect
34. #5111C22semidirect
35. #5111C22semidirect
36. #5111C22semidirect
37. C2×(C2×D3)⋊C8C22semidirect
38. C2×#815C22semidirect
39. C2×Dic3⋊C8C22semidirect
40. D4×C3⋊C8C22semidirect
41. D4×C3⋊C8C22semidirect
42. #4925C22semidirect
43. #4838C22semidirect
44. #4837C22semidirect
45. #4835C22semidirect
46. #4831C22semidirect
47. #4730C22semidirect
48. #4730C22semidirect
49. #4726C22semidirect
50. #4719C22semidirect
51. #4688C22semidirect
52. #4688C22semidirect
53. C4×ΓL(1,𝔽25)C22semidirect
54. #4685C22semidirect
55. #4679C22semidirect
56. #4679C22semidirect
67. C2×C8×D4D3semidirect
139. C23×C8C2×D3semidirect
165. C8×D4C2×D3semidirect
166. C8×D4C2×D3semidirect
173. C2×C4⋊C8C2×D3semidirect
177. C2×C22⋊C8C2×D3semidirect
180. C2×C4×C8C2×D3semidirect
183. C2×C3×D4M16semidirect
315. C23×C3C2×M16semidirect
322. C3×D4C2×M16semidirect
330. C2×C3×C4C2×M16semidirect
354. C3×C8H(5,𝔽2)semidirect
367. C2×D4ΓL(1,𝔽25)semidirect
447. C23C2×ΓL(1,𝔽25)semidirect
454. D4C2×ΓL(1,𝔽25)semidirect
473. C2×C4C2×ΓL(1,𝔽25)semidirect
484. C8#1004semidirect
512. C3#60309semidirect

Maximal subgroups

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

subgroupindex
C4×#10042
C2×C3×C8×D42
C2×#51112
#823012
#794272
D4×ΓL(1,𝔽25)2
#794062
#794042
#790142
#786022
#784472
#603093