Finite Group Databasebeta

← C8×#1009 #11660845

Subgroups — C8×#1009 #11660845

Order 768.

2422 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1009C211yesmaximal, characteristic
2C3×C8×#41C211yesmaximal, characteristic, complemented
3#82520C211yesmaximal, characteristic, complemented
4#79906C212yesmaximal, complemented
5C8×#239C212yesmaximal, complemented
6#79426C216yesmaximal, complemented
7D3×C8×D4C216yesmaximal, complemented
8#79291C213yesmaximal, complemented
9#79286C213yesmaximal, complemented
10C2×C8×D12C213yesmaximal, complemented
11C8×#235C213yesmaximal, complemented
12C8×H(5,𝔽2)—31nomaximal, Sylow
13C2×#1009C411yescharacteristic
14C3×C4×#41C2211yescharacteristic
15#5827C411yescharacteristic
16C22×D3×C8C2216yescomplemented
17#5574C412yes—
18C4×#239C2212yes—
19#5545C416yes—
20C4×D3×D4C2216yes—
21#5523C413yes—
22#5521C413yes—
23C4×#235C2213yes—
24C2×C4×D12C2213yes—
25C3×C8×Q8C2211yescharacteristic, complemented
26C3×C8×Q8C2211yescharacteristic, complemented
27C3×C8×D4C2213yescomplemented
28C3×C8×D4C2213yescomplemented
29C3×#431C2213yescomplemented
30C2×C3×C4×C8C2213yescomplemented
31C8×#67C2216yescomplemented
32C8×#67C2216yescomplemented
33C2×(C2×D3)⋊C8C2216yescomplemented
34Q8×C3⋊C8C2212yescomplemented
35D4×C3⋊C8C2216yescomplemented
36#4925C2213yescomplemented
37C2×C4⋊(C3⋊C8)C2213yescomplemented
38#4834C2216yescomplemented
39#4834C2216yescomplemented
40#4833C2216yescomplemented
41D3×C4⋊C8C2216yescomplemented
42#67⋊C8C22112yescomplemented
43D3×C22⋊C8C22112yescomplemented
44C8×D12C2216yescomplemented
45C8×D12C2216yescomplemented
46C8×D12C2213yescomplemented
47C8×D12C2213yescomplemented
48C8×D12C2216yescomplemented
49#4685C2216yescomplemented
50C4×D3×C8C2216yescomplemented
51C8×Dic6C2211yescharacteristic, complemented
52C8×Dic6C2213yescomplemented
53C4×H(5,𝔽2)—31no—
54#2988—32no—
55#2988—33no—
56#2988—31no—
57#2972—36no—
58#2972—33no—
59C8×#41—32no—
60C8×#41—33no—
61C8×#41D311yescharacteristic, complemented
62C2×C8×D4—36no—
63C2×C8×D4—33no—
64C2×C3×#41C2×C411yescharacteristic
65#1009C8116yesdirect product factor
66C2×#239C2×C412yes—
67C2×D3×D4C2×C416yes—
68C2×#235C2×C413yes—
69C22×D12C2×C413yes—
70C22×C4×D3C2316yes—
71C22×C3×C8C2313yes—
72C22×C3×C8C2313yes—
73C3×C4×Q8C2×C411yescharacteristic
74C3×C4×Q8C2311yescharacteristic
75C3×C4×D4C2×C413yes—
76C3×C4×D4C2313yes—
77C3×#117C2×C413yes—
78C2×C3×C42C2313yes—
79Q8×Dic3C2×C412yes—
80D4×Dic3C2×C416yes—
81C4×#67C2×C416yes—
82C4×#67C2316yes—
83C2×(C2×D3)⋊C4C2×C416yes—
84#926C2×C413yes—
85C2×C4⋊Dic3C2×C413yes—
86C22×C3⋊C8C2316yes—
87C2×D3×C8—412no—
88C2×D3×C8C2316yes—
89C2×D3×C8C2316yes—
90C2×D3×C8C23112yes—
91#893C2×C416yes—
92#893C2×C416yes—
93#892C2×C416yes—
94D3×C4⋊C4C2×C416yes—
95#67⋊C4C2×C4112yes—
96D3×C22⋊C4C2×C4112yes—
97C4×D12C2×C416yes—
98C4×D12C2×C416yes—
99C4×D12C2×C413yes—
100C4×D12C2313yes—
101C4×D12C2316yes—
102#872C2×C416yes—
103C42×D3C2316yes—
104C4×Dic6C2×C413yes—
105C4×Dic6C2311yescharacteristic
106C3×C4⋊C8C2313yes—
107C3×C4⋊C8C2313yes—
108C3×C22⋊C8C2316yes—
109C3×C4×C8C2311yescharacteristic
110C3×C4×C8C2313yes—
111C3×C4×C8C2313yes—
112C3×C4×C8C2313yes—
113#830C2316yes—
114(C2×D3)⋊C8C2316yes—
115(C2×D3)⋊C8C2316yes—
116(C2×D3)⋊C8C23112yes—
117Dic3⋊C8C2312yes—
118Dic3⋊C8C2316yes—
119C8×Dic3C2312yes—
120C8×Dic3C2316yes—
121C4⋊(C3⋊C8)C2313yes—
122C4⋊(C3⋊C8)C2316yes—
123C4⋊(C3⋊C8)C2311yescharacteristic
124C4×C3⋊C8C2316yes—
125C2×H(5,𝔽2)—31no—
126C23×C8—36no—
127#520—31no—
128#520—32no—
129#520—33no—
130#518—36no—
131#518—33no—
132C4×#41—32no—
133C4×#41—33no—
134C4×#41C2×D311yescharacteristic
135C2×C4×D4—36no—
136C2×C4×D4—33no—
137C8×Q8—32no—
138C8×Q8—31no—
139C8×Q8—33no—
140C8×Q8C2×D311yescharacteristic, complemented
141C8×Q8C2×D311yescharacteristic, complemented
142C8×D4—36no—
143C8×D4—36no—
144C8×D4—36no—
145C8×D4—36no—
146C8×D4—36no—
147C8×D4—312no—
148C8×D4—36no—
149C8×D4—33no—
150C8×D4—36no—
151C8×D4—36no—
152C8×D4—33no—
153C8×D4C2×D313yescomplemented
154C8×D4C2×D313yescomplemented
155#431—33no—
156#431—36no—
157#431—36no—
158#431C2×D313yescomplemented
159C2×C4⋊C8—33no—
160C2×C4⋊C8—36no—
161C2×C22⋊C8—36no—
162C2×C22⋊C8—312no—
163C2×C4×C8—36no—
164C2×C4×C8C2×D313yescomplemented
165C23×D3C22×C416yes—
166C3×#41C2×C818yescomplemented
167C2×C3×Q8C22×C411yescharacteristic
168C2×C3×D4C22×C413yes—
169C22×C3×C4C22×C413yes—
170C22×C3×C4C22×C413yes—
171C22×C3×C4C2413yes—
172C2×#67C22×C416yes—
173C22×Dic3C22×C416yes—
174#239C2×C8116yescomplemented
175D3×D4C2×C8148yescomplemented
176#235C2×C8124yescomplemented
177C2×D12C2×C8124yescomplemented
178C2×D12C22×C413yes—
179C2×D12C22×C416yes—
180C2×C4×D3—46no—
181C2×C4×D3—46no—
182C2×C4×D3C22×C416yes—
183C2×C4×D3C22×C4112yes—
184C2×C4×D3C22×C416yes—
185C2×C4×D3C2416yes—
186C2×Dic6C22×C411yescharacteristic
187C2×C3×C8—26no—
188C2×C3×C8—43no—
189C2×C3×C8C2413yes—
190C2×C3×C8C2411yescharacteristic
191C2×C3×C8C2413yes—
192C2×C3×C8C2411yescharacteristic
193C3×C4⋊C4C22×C413yes—
194C3×C4⋊C4C22×C413yes—
195C3×C22⋊C4C22×C416yes—
196C3×C42C22×C413yes—
197C3×C42C22×C413yes—
198C3×C42C2413yes—
199C3×C42C2411yescharacteristic
200#217C22×C416yes—
201(C2×D3)⋊C4C22×C416yes—
202(C2×D3)⋊C4C22×C416yes—
203(C2×D3)⋊C4C22×C4112yes—
204C4⋊Dic3C22×C413yes—
205C4⋊Dic3C22×C411yescharacteristic
206C4⋊Dic3C22×C416yes—
207Dic3⋊C4C22×C416yes—
208Dic3⋊C4C22×C412yes—
209C4×Dic3C22×C416yes—
210C4×Dic3C22×C416yes—
211C4×Dic3C2412yes—
212C2×C3⋊C8—46no—
213C2×C3⋊C8C2412yes—
214C2×C3⋊C8C2416yes—
215D3×C8—212no—
216D3×C8—412no—
217D3×C8—46no—
218H(5,𝔽2)—316no—
219C2×#41—32no—
220C2×#41—33no—
221C2×#41C4×D311yescharacteristic
222C22×D4—36no—
223C22×D4—33no—
224C23×C4—36no—
225C22×C8—36no—
226C22×C8—36no—
227C22×C8—312no—
228C22×C8—36no—
229C22×C8—1212no—
230C22×C8C22×D313yes—
231C22×C8C22×D313yes—
232C4×Q8—32no—
233C4×Q8—33no—
234C4×Q8—31no—
235C4×Q8C4×D311yescharacteristic
236C4×Q8C22×D311yescharacteristic
237C4×D4—36no—
238C4×D4—36no—
239C4×D4—36no—
240C4×D4—36no—
241C4×D4—36no—
242C4×D4—312no—
243C4×D4—36no—
244C4×D4—36no—
245C4×D4—33no—
246C4×D4—36no—
247C4×D4—33no—
248C4×D4C4×D313yes—
249C4×D4C22×D313yes—
250#117—33no—
251#117—36no—
252#117—36no—
253#117C4×D313yes—
254C2×C4⋊C4—33no—
255C2×C4⋊C4—36no—
256C2×C22⋊C4—36no—
257C2×C22⋊C4—312no—
258C2×C42—36no—
259C2×C42C22×D313yes—
260C4⋊C8—36no—
261C4⋊C8—32no—
262C4⋊C8—36no—
263C4⋊C8—33no—
264C4⋊C8—31no—
265C4⋊C8C22×D313yes—
266C4⋊C8C22×D313yes—
267C22⋊C8—36no—
268C22⋊C8—36no—
269C22⋊C8—312no—
270C22⋊C8—36no—
271C22⋊C8C22×D316yes—
272C4×C8—36no—
273C4×C8—32no—
274C4×C8—36no—
275C4×C8C22×D313yes—
276C4×C8C22×D313yes—
277C4×C8C22×D313yes—
278C4×C8C22×D311yescharacteristic
279C23×C3C23×C413yes—
280C22×D3—46no—
281C22×D3—46no—
282C22×D3C22×C8124yescomplemented
283C22×D3C23×C416yes—
284C3×Q8C22×C814yescomplemented
285C3×D4C22×C8112yescomplemented
286C2×C3×C4—23no—
287C2×C3×C4—23no—
288C2×C3×C4—43no—
289C2×C3×C4C22×C8112yescomplemented
290C2×C3×C4C23×C413yes—
291C2×C3×C4C23×C411yescharacteristic
292C2×C3×C4C23×C413yes—
293C2×C3×C4C23×C413yes—
294C2×C3×C4C23×C411yescharacteristic
295C2×C3×C4C2511yescharacteristic
296#67C22×C8124yescomplemented
297C2×Dic3—46no—
298C2×Dic3C23×C412yes—
299C2×Dic3C23×C416yes—
300C2×Dic3C23×C412yes—
301D12C22×C8112yescomplemented
302D12C22×C8124yescomplemented
303C4×D3—26no—
304C4×D3—26no—
305C4×D3—412no—
306C4×D3—46no—
307C4×D3C22×C8124yescomplemented
308Dic6C22×C814yescomplemented
309C3×C8—23no—
310C3×C8—23no—
311C3×C8—21no—
312C3×C8H(5,𝔽2)12yescomplemented
313C3⋊C8—26no—
314C3⋊C8—22no—
315C24—36no—
316#41—316no—
317#41—324no—
318#41D3×C818yescomplemented
319C2×Q8—31no—
320C2×Q8C2×C4×D311yescharacteristic
321C2×D4—36no—
322C2×D4—348no—
323C2×D4—324no—
324C2×D4—36no—
325C2×D4—33no—
326C2×D4C2×C4×D313yes—
327C22×C4—36no—
328C22×C4—36no—
329C22×C4—36no—
330C22×C4—36no—
331C22×C4—312no—
332C22×C4—126no—
333C22×C4—126no—
334C22×C4C2×C4×D313yes—
335C22×C4C2×C4×D313yes—
336C22×C4C23×D313yes—
337C2×C8—26no—
338C2×C8—32no—
339C2×C8—36no—
340C2×C8—43no—
341C2×C8—612no—
342C2×C8—1212no—
343C2×C8—126no—
344C2×C8—126no—
345C2×C8C23×D313yes—
346C2×C8C23×D311yescharacteristic
347C2×C8C23×D313yes—
348C2×C8C23×D311yescharacteristic
349C4⋊C4—36no—
350C4⋊C4—36no—
351C4⋊C4—33no—
352C4⋊C4—32no—
353C4⋊C4—31no—
354C4⋊C4C2×C4×D313yes—
355C4⋊C4C2×C4×D313yes—
356C22⋊C4—36no—
357C22⋊C4—36no—
358C22⋊C4—312no—
359C22⋊C4—36no—
360C22⋊C4C2×C4×D316yes—
361C42—36no—
362C42—36no—
363C42—32no—
364C42C2×C4×D313yes—
365C42C2×C4×D313yes—
366C42C23×D313yes—
367C42C23×D311yescharacteristic
368C22×C3—23no—
369C22×C3—23no—
370C22×C3C23×C816yes—
371C22×C3C24×C411yescharacteristic
372C2×D3—26no—
373C2×D3—26no—
374C2×D3—424no—
375C2×D3C23×C8112yes—
376C3×C4—23no—
377C3×C4—23no—
378C3×C4—21no—
379C3×C4C23×C812yes—
380C3×C4C23×C816yes—
381C3×C4C2×H(5,𝔽2)11yescharacteristic
382C3×C4#58511yescharacteristic
383Dic3—26no—
384Dic3—22no—
385Dic3C23×C814yes—
386C23—36no—
387C23—324no—
388C23—126no—
389C23—126no—
390C23C22×C4×D313yes—
391Q8—34no—
392Q8C2×D3×C814yescomplemented
393D4—324no—
394D4—324no—
395D4—312no—
396D4C2×D3×C8112yescomplemented
397C2×C4—23no—
398C2×C4—23no—
399C2×C4—32no—
400C2×C4—324no—
401C2×C4—32no—
402C2×C4—36no—
403C2×C4—43no—
404C2×C4—66no—
405C2×C4—66no—
406C2×C4—1212no—
407C2×C4—126no—
408C2×C4—126no—
409C2×C4C2×D3×C8112yescomplemented
410C2×C4C22×C4×D313yes—
411C2×C4C22×C4×D311yescharacteristic
412C2×C4C22×C4×D313yes—
413C2×C4C22×C4×D313yes—
414C2×C4C22×C4×D311yescharacteristic
415C2×C4C24×D311yescharacteristic
416C8—23no—
417C8—21no—
418C8—23no—
419C8—66no—
420C8—62no—
421C8#100912yesdirect product factor
422C2×C3—26no—
423C2×C3C4×H(5,𝔽2)11yescharacteristic
424C2×C3C24×C811yescharacteristic
425C2×C3#357311yescharacteristic
426D3—212no—
427C22—23no—
428C22—23no—
429C22—312no—
430C22—66no—
431C22—66no—
432C22—1224no—
433C22C22×D3×C816yes—
434C22C23×C4×D311yescharacteristic
435C4—23no—
436C4—21no—
437C4—23no—
438C4—34no—
439C4—66no—
440C4—62no—
441C4C22×D3×C816yes—
442C4C22×D3×C812yes—
443C4C2×#100911yescharacteristic
444C4#596411yescharacteristic
445C3C8×H(5,𝔽2)11yescharacteristic, complemented, Sylow
446C2—26no—
447C2—612no—
448C2C4×#100911yescharacteristic
449C2C23×D3×C811yescharacteristic
450C2#8716011yescharacteristic
subgroupG/H
centerC2×C8C23×D3
commutatorC2×C3C24×C8
FrattiniC2×C4C24×D3
FittingC3×C8×#41C2
socleC22×C3C24×C4
prime radicalC2×#1009C4
cosocleC2×C3×C4C25
hypercenterC8×#41D3
hypocenterC3C8×H(5,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. C3×C8×#41C2semidirect
3. #82520C2semidirect
4. #79906C2semidirect
5. C8×#239C2semidirect
6. #79426C2semidirect
7. D3×C8×D4C2semidirect
8. #79291C2semidirect
9. #79286C2semidirect
10. C2×C8×D12C2semidirect
11. C8×#235C2semidirect
16. C22×D3×C8C22semidirect
25. C3×C8×Q8C22semidirect
26. C3×C8×Q8C22semidirect
27. C3×C8×D4C22semidirect
28. C3×C8×D4C22semidirect
29. C3×#431C22semidirect
30. C2×C3×C4×C8C22semidirect
31. C8×#67C22semidirect
32. C8×#67C22semidirect
33. C2×(C2×D3)⋊C8C22semidirect
34. Q8×C3⋊C8C22semidirect
35. D4×C3⋊C8C22semidirect
36. #4925C22semidirect
37. C2×C4⋊(C3⋊C8)C22semidirect
38. #4834C22semidirect
39. #4834C22semidirect
40. #4833C22semidirect
41. D3×C4⋊C8C22semidirect
42. #67⋊C8C22semidirect
43. D3×C22⋊C8C22semidirect
44. C8×D12C22semidirect
45. C8×D12C22semidirect
46. C8×D12C22semidirect
47. C8×D12C22semidirect
48. C8×D12C22semidirect
49. #4685C22semidirect
50. C4×D3×C8C22semidirect
51. C8×Dic6C22semidirect
52. C8×Dic6C22semidirect
61. C8×#41D3semidirect
65. #1009C8direct (N × H)
140. C8×Q8C2×D3semidirect
141. C8×Q8C2×D3semidirect
153. C8×D4C2×D3semidirect
154. C8×D4C2×D3semidirect
158. #431C2×D3semidirect
164. C2×C4×C8C2×D3semidirect
166. C3×#41C2×C8semidirect
174. #239C2×C8semidirect
175. D3×D4C2×C8semidirect
176. #235C2×C8semidirect
177. C2×D12C2×C8semidirect
282. C22×D3C22×C8semidirect
284. C3×Q8C22×C8semidirect
285. C3×D4C22×C8semidirect
289. C2×C3×C4C22×C8semidirect
296. #67C22×C8semidirect
301. D12C22×C8semidirect
302. D12C22×C8semidirect
307. C4×D3C22×C8semidirect
308. Dic6C22×C8semidirect
312. C3×C8H(5,𝔽2)semidirect
318. #41D3×C8semidirect
392. Q8C2×D3×C8semidirect
396. D4C2×D3×C8semidirect
409. C2×C4C2×D3×C8semidirect
421. C8#1009direct (N × H)
445. C3C8×H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C4×#10092
C3×C8×#412
#825202
#799062
C8×#2392
#794262
D3×C8×D42
#792912
#792862
C2×C8×D122
C8×#2352
C8×H(5,𝔽2)3