Finite Group Databasebeta

← #22570039

Subgroups — #22570039

Order 8576.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864026C211yesmaximal, characteristic, complemented
2#16863868C211yesmaximal, characteristic
3#16863698C211yesmaximal, characteristic, complemented
4#16863695C211yesmaximal, characteristic, complemented
5#16863693C211yesmaximal, characteristic, complemented
6#16863687C211yesmaximal, characteristic, complemented
7#16863684C211yesmaximal, characteristic, complemented
8#16863676C211yesmaximal, characteristic, complemented
9#16863675C211yesmaximal, characteristic, complemented
10#16863437C211yesmaximal, characteristic, complemented
11#16863431C211yesmaximal, characteristic, complemented
12#16863421C211yesmaximal, characteristic, complemented
13#16863399C211yesmaximal, characteristic
14#102×D67C211yesmaximal, characteristic
15#478×C67C211yesmaximal, characteristic, complemented
16C2×D4×D67C2211yescharacteristic
17C2×#11761611C2211yescharacteristic
18C2×C8×D67C2211yescharacteristic
19#14524043C2211yescharacteristic, complemented
20#14524042—21no—
21#14524042C2211yescharacteristic, complemented
22#14524034C2211yescharacteristic, complemented
23#14524033—21no—
24#14524023—21no—
25#14524021—21no—
26C4×D268C2211yescharacteristic
27D4×Dic67C2211yescharacteristic
28#14524005C2211yescharacteristic
29C4×C22⋊D67—21no—
30(C22⋊D67)⋊C4—21no—
31#14524001C2211yescharacteristic
32#14524001C2211yescharacteristic
33#14524000C2211yescharacteristic
34C4⋊C4×D67C2211yescharacteristic
35C2×(C2×D67)⋊C4—21no—
36C22⋊C4×D67—21no—
37#14523976C2211yescharacteristic, complemented
38#14523976C2211yescharacteristic, complemented
39#14523976C2211yescharacteristic, complemented
40#14523975C2211yescharacteristic, complemented
41#14523975C2211yescharacteristic, complemented
42#14523974C2211yescharacteristic, complemented
43C4⋊(C67⋊C8)C2211yescharacteristic, complemented
44Dic67⋊C8C2211yescharacteristic, complemented
45Dic67⋊C8C2211yescharacteristic, complemented
46#14523965C2211yescharacteristic
47#14523965C2211yescharacteristic
48#14523964C2211yescharacteristic
49#14523964C2211yescharacteristic
50#14523963C2211yescharacteristic
51#14523963C2211yescharacteristic
52#125×C67C2211yescharacteristic, complemented
53C4×D4×C67C2211yescharacteristic
54#107×C67C2211yescharacteristic, complemented
55#106×C67C2211yescharacteristic, complemented
56C4⋊C8×C67C2211yescharacteristic, complemented
57#102×C67C2211yescharacteristic
58#102×C67C2211yescharacteristic
59C23×D67—21no—
60C2×Dic134—21no—
61C2×Dic134—21no—
62C2×D268C2311yescharacteristic
63C2×C22⋊D67—21no—
64D4×D67—22no—
65D4×D67Q814yescomplemented
66C22×Dic67—21no—
67C2×C4×D67—21no—
68C2×C4×D67—21no—
69C2×C4×D67—21no—
70C2×C4×D67C2311yescharacteristic
71#11761611—22no—
72C2×C67⋊C8C2311yescharacteristic
73C2×C67⋊C8C2311yescharacteristic
74C8×D67—22no—
75C4⋊Dic67—21no—
76C4⋊Dic67—21no—
77C4⋊Dic67C2311yescharacteristic
78C4⋊Dic67C2311yescharacteristic
79C4⋊Dic67C2311yescharacteristic
80Dic67⋊C4—21no—
81Dic67⋊C4—21no—
82Dic67⋊C4—21no—
83Dic67⋊C4D411yescharacteristic
84Dic67⋊C4D411yescharacteristic
85C22⋊Dic67—21no—
86(C2×D67)⋊C4—21no—
87(C2×D67)⋊C4—21no—
88(C2×D67)⋊C4—21no—
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4D411yescharacteristic
91(C2×D67)⋊C4D411yescharacteristic
92C4×Dic67—21no—
93C4×Dic67—21no—
94C4×Dic67C2311yescharacteristic
95C2×D4×C67C2311yescharacteristic
96C22×C4×C67—21no—
97C2×C8×C67C2311yescharacteristic
98C2×C8×C67C2311yescharacteristic
99C4⋊C4×C67—21no—
100C4⋊C4×C67—21no—
101C4⋊C4×C67C2311yescharacteristic
102C4⋊C4×C67C2311yescharacteristic
103C4⋊C4×C67C2311yescharacteristic
104C22⋊C4×C67—21no—
105C42×C67C2311yescharacteristic
106C23×C67—21no—
107C22×D67—21no—
108C22×D67—24no—
109C22×D67—81no—
110C22×D67—81no—
111C22×D67C2×D411yescharacteristic
112D4×C67—21no—
113D4×C67C2×Q812yes—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—81no—
118C2×C4×C67C2×D411yescharacteristic
119C2×C4×C67C2×D411yescharacteristic
120C2×C4×C67C2411yescharacteristic
121C22⋊D67—24no—
122C2×Dic67—21no—
123C2×Dic67—21no—
124C2×Dic67—21no—
125C2×Dic67—81no—
126C2×Dic67C2×D411yescharacteristic
127C2×Dic67C2×D411yescharacteristic
128C2×Dic67C2×D411yescharacteristic
129D268—21no—
130D268C2×Q812yes—
131C4×D67—81no—
132C4×D67—81no—
133C4×D67—81no—
134C4×D67C2×Q812yes—
135C4×D67#4112yes—
136Dic134—81no—
137Dic134—81no—
138C8×C67—21no—
139C8×C67—21no—
140C67⋊C8—21no—
141C67⋊C8—21no—
142C22×C67—22no—
143C22×C67—41no—
144C22×C67—41no—
145C22×C67C22×D411yescharacteristic
146C2×D67—21no—
147C2×D67—21no—
148C2×D67—22no—
149C2×D67—41no—
150C2×D67—41no—
151C2×D67—84no—
152C2×D67C22⋊Q812yes—
153C4×C67—21no—
154C4×C67—21no—
155C4×C67—41no—
156C4×C67—41no—
157C4×C67—41no—
158C4×C67C22×Q811yescharacteristic
159C4×C67C2×#4111yescharacteristic
160Dic67—21no—
161Dic67—21no—
162Dic67—41no—
163Dic67—41no—
164Dic67—41no—
165Dic67C22⋊Q812yes—
166C2×C67—42no—
167C2×C67C2×C22⋊Q811yescharacteristic
168C2×C67#57611yescharacteristic
169C2×C67#57711yescharacteristic
170D67—22no—
171D67—42no—
172#3081—671nomaximal, Sylow
173C67#308111yescharacteristic, complemented, Sylow
174#544—671no—
175#518—671no—
176#478—671no—
177#478—671no—
178#478—671no—
179#478D6711yescharacteristic, complemented
180#476—671no—
181#476—671no—
182#474—671no—
183#474—671no—
184#428—671no—
185#427—671no—
186#424—671no—
187#418—671no—
188C2×#102—671no—
189C22×D4—671no—
190C2×M16—671no—
191C22×C8—671no—
192C4⋊Q8—671no—
193C4⋊Q8—671no—
194C4⋊Q8—1341no—
195#125—671no—
196#125—1341no—
197#125C2×D6711yescharacteristic, complemented
198C22⋊Q8—1341no—
199C22⋊Q8—1341no—
200C4×D4—671no—
201C4×D4—671no—
202C4×D4—671no—
203C4×D4—1341no—
204C4×D4—1341no—
205C4×D4C2×D6711yescharacteristic
206#117—671no—
207#117—671no—
208#117—671no—
209C2×C4⋊C4—671no—
210C2×C22⋊C4—1341no—
211C2×C22⋊C4—1341no—
212#107—671no—
213#107—671no—
214#107—671no—
215#107C2×D6711yescharacteristic, complemented
216#106—671no—
217#106—671no—
218#106—671no—
219#106C2×D6711yescharacteristic, complemented
220C4⋊C8—671no—
221C4⋊C8—671no—
222C4⋊C8—671no—
223C4⋊C8C2×D6711yescharacteristic, complemented
224#102—671no—
225#102—671no—
226#102—671no—
227#102—671no—
228#102—671no—
229#102—671no—
230#102C2×D6711yescharacteristic
231#102C2×D6711yescharacteristic
232C24—1341no—
233C2×Q8—1341no—
234C2×Q8—1341no—
235C2×D4—674no—
236C2×D4—671no—
237C2×D4—1342no—
238C2×D4—1341no—
239C2×D4C22×D6711yescharacteristic
240C22×C4—21no—
241C22×C4—671no—
242C22×C4—1341no—
243C22×C4—1341no—
244C22×C4—1341no—
245C22×C4—1341no—
246M16—1342no—
247C2×C8—671no—
248C2×C8—671no—
249C2×C8—1342no—
250C2×C8C22×D6711yescharacteristic
251C2×C8C22×D6711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—671no—
255C4⋊C4—671no—
256C4⋊C4—671no—
257C4⋊C4—671no—
258C4⋊C4—671no—
259C4⋊C4—1341no—
260C4⋊C4—1341no—
261C4⋊C4—1341no—
262C4⋊C4—1341no—
263C4⋊C4—1341no—
264C4⋊C4C22×D6711yescharacteristic
265C4⋊C4C22×D6711yescharacteristic
266C4⋊C4C22×D6711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—671no—
269C22⋊C4—671no—
270C22⋊C4—1341no—
271C22⋊C4—1341no—
272C22⋊C4—1341no—
273C22⋊C4—1341no—
274C22⋊C4—1341no—
275C42—671no—
276C42—1341no—
277C42—1341no—
278C42C22×D6711yescharacteristic
279C23—21no—
280C23—5361no—
281C23—5361no—
282C23—671no—
283C23—1344no—
284C23—1341no—
285Q8—5361no—
286Q8—5361no—
287D4—21no—
288D4—672no—
289D4—1344no—
290D4—1341no—
291D4Q8×D6712yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—5361no—
297C2×C4—5361no—
298C2×C4—5361no—
299C2×C4—5361no—
300C2×C4—671no—
301C2×C4—672no—
302C2×C4—672no—
303C2×C4—671no—
304C2×C4—671no—
305C2×C4—1341no—
306C2×C4—1341no—
307C2×C4—1341no—
308C2×C4D4×D6711yescharacteristic
309C2×C4D4×D6711yescharacteristic
310C2×C4C23×D6711yescharacteristic
311C8—21no—
312C8—21no—
313C8—1341no—
314C8—1341no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—2681no—
319C22—2681no—
320C22—5364no—
321C22—672no—
322C22—1342no—
323C22—1341no—
324C22—1341no—
325C22C2×D4×D6711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—2681no—
332C4—2681no—
333C4—2681no—
334C4—672no—
335C4—1341no—
336C4—1341no—
337C4C2×Q8×D6711yescharacteristic
338C4#41×D6711yescharacteristic
339C2—42no—
340C2—2682no—
341C2—1342no—
342C2C22⋊Q8×D6711yescharacteristic
343C2#1686420011yescharacteristic
344C2#1686420311yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#478×C67C2
socleC22×C67C22×D4
hypercenter#478D67
hypocenterC67#3081
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #16864026C2semidirect
3. #16863698C2semidirect
4. #16863695C2semidirect
5. #16863693C2semidirect
6. #16863687C2semidirect
7. #16863684C2semidirect
8. #16863676C2semidirect
9. #16863675C2semidirect
10. #16863437C2semidirect
11. #16863431C2semidirect
12. #16863421C2semidirect
15. #478×C67C2semidirect
19. #14524043C22semidirect
21. #14524042C22semidirect
22. #14524034C22semidirect
37. #14523976C22semidirect
38. #14523976C22semidirect
39. #14523976C22semidirect
40. #14523975C22semidirect
41. #14523975C22semidirect
42. #14523974C22semidirect
43. C4⋊(C67⋊C8)C22semidirect
44. Dic67⋊C8C22semidirect
45. Dic67⋊C8C22semidirect
52. #125×C67C22semidirect
54. #107×C67C22semidirect
55. #106×C67C22semidirect
56. C4⋊C8×C67C22semidirect
65. D4×D67Q8semidirect
173. C67#3081semidirect
179. #478D67semidirect
197. #125C2×D67semidirect
215. #107C2×D67semidirect
219. #106C2×D67semidirect
223. C4⋊C8C2×D67semidirect