Finite Group Databasebeta

← #22571032

Subgroups — #22571032

Order 8576.

410 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524023C211yesmaximal, characteristic
2C22⋊Q8×D67C211yesmaximal, characteristic
3#16863728C212yesmaximal, complemented
4#16863726C211yesmaximal, characteristic, complemented
5#16863655C212yesmaximal, complemented
6#16863653C212yesmaximal, complemented
7#16863554C211yesmaximal, characteristic, complemented
8#16863553C212yesmaximal, complemented
9#16863551C211yesmaximal, characteristic, complemented
10#16863358C211yesmaximal, characteristic
11#483×C67C211yesmaximal, characteristic, complemented
12C22×Dic134C2211yescharacteristic
13C2×Q8×D67C2211yescharacteristic
14C22×C4×D67C2211yescharacteristic
15C2×Dic268C2212yescomplemented
16C2×#11761618C2212yescomplemented
17C2×#11761611C2212yes—
18#14524023—21no—
19#14524023—21no—
20#14524023C2212yes—
21#14524023C2211yescharacteristic
22#14524023C2212yes—
23#14524022C2211yescharacteristic
24#14524021—21no—
25#14524020—21no—
26#14524019C2211yescharacteristic
27C2×C4⋊Dic67—21no—
28C4⋊C4×D67—21no—
29C4⋊C4×D67C2211yescharacteristic
30C2×(C2×D67)⋊C4—21no—
31C22⋊C4×D67—21no—
32#14523974C2212yescomplemented
33#14523968C2212yescomplemented
34#14523966C2212yescomplemented
35#14523966C2212yescomplemented
36C22⋊(C67⋊C8)C2211yescharacteristic
37(C2×D67)⋊C8C2212yes—
38C22⋊Q8×C67C2211yescharacteristic
39C2×C4⋊C4×C67C2211yescharacteristic
40#106×C67C2212yescomplemented
41#103×C67C2212yescomplemented
42C22⋊C8×C67C2211yescharacteristic
43C23×D67D411yescharacteristic
44C2×Dic134—21no—
45C2×Dic134—21no—
46C2×Dic134—42no—
47C2×Dic134C2311yescharacteristic
48C2×Dic134C2312yes—
49Q8×D67—42no—
50C22×Dic67—21no—
51C22×Dic67D411yescharacteristic
52C2×C4×D67—21no—
53C2×C4×D67—21no—
54C2×C4×D67—42no—
55C2×C4×D67D411yescharacteristic
56C2×C4×D67D411yescharacteristic
57C2×C4×D67C2311yescharacteristic
58C2×C4×D67C2311yescharacteristic
59Dic268—42no—
60#11761618—42no—
61#11761611—42no—
62C2×C67⋊C8C2312yes—
63C4⋊Dic67—21no—
64C4⋊Dic67—22no—
65C4⋊Dic67—22no—
66C4⋊Dic67C2311yescharacteristic
67Dic67⋊C4—21no—
68Dic67⋊C4—21no—
69Dic67⋊C4—21no—
70C22⋊Dic67—21no—
71(C2×D67)⋊C4—21no—
72(C2×D67)⋊C4—21no—
73(C2×D67)⋊C4—22no—
74(C2×D67)⋊C4—22no—
75C2×Q8×C67C2311yescharacteristic
76C22×C4×C67—21no—
77C22×C4×C67C2311yescharacteristic
78C2×C8×C67C2312yes—
79C4⋊C4×C67—21no—
80C4⋊C4×C67—21no—
81C4⋊C4×C67C2312yes—
82C4⋊C4×C67C2311yescharacteristic
83C22⋊C4×C67—21no—
84C23×C67C2×D411yescharacteristic
85C22×D67—22no—
86C22×D67—22no—
87C22×D67—41no—
88C22×D67C2×D411yescharacteristic
89C22×D67C2×D411yescharacteristic
90Q8×C67—22no—
91C2×C4×C67—22no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—22no—
95C2×C4×C67—42no—
96C2×C4×C67C2×D411yescharacteristic
97C2×C4×C67C2411yescharacteristic
98C2×Dic67—22no—
99C2×Dic67—21no—
100C2×Dic67—21no—
101C2×Dic67—41no—
102C2×Dic67—42no—
103C2×Dic67C2×D411yescharacteristic
104C2×Dic67C2×D411yescharacteristic
105C4×D67—22no—
106C4×D67—22no—
107C4×D67—41no—
108C4×D67—41no—
109C4×D67—42no—
110C4×D67—42no—
111Dic134—22no—
112Dic134—22no—
113Dic134—22no—
114Dic134—42no—
115Dic134—42no—
116C8×C67—22no—
117C67⋊C8—22no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67#13712yes—
121C22×C67C22×D411yescharacteristic
122C2×D67—21no—
123C2×D67—21no—
124C2×D67—21no—
125C2×D67—21no—
126C2×D67—21no—
127C2×D67—44no—
128C2×D67#13712yes—
129C4×C67—21no—
130C4×C67—41no—
131C4×C67—41no—
132C4×C67—42no—
133C4×C67H(5,𝔽2)12yes—
134Dic67—21no—
135Dic67—21no—
136Dic67—41no—
137Dic67—42no—
138Dic67—41no—
139C2×C67—22no—
140C2×C67#53411yescharacteristic
141C2×C67C2×#13711yescharacteristic
142C2×C67C2×#13711yescharacteristic
143D67—22no—
144D67—41no—
145#3195—671nomaximal, Sylow
146C67#319511yescharacteristic, complemented, Sylow
147C2×C22⋊Q8—671no—
148C2×C22⋊Q8—671no—
149#483—671no—
150#483—672no—
151#483D6711yescharacteristic, complemented
152#470—672no—
153#470—672no—
154#451—671no—
155#451—672no—
156#451—671no—
157#407—671no—
158C22×Q8—671no—
159C22×Q8—671no—
160C23×C4—671no—
161C2×Q16—672no—
162C2×Q16—672no—
163C2×M16—672no—
164C22⋊Q8—671no—
165C22⋊Q8—671no—
166C22⋊Q8—672no—
167C22⋊Q8—671no—
168C22⋊Q8—672no—
169C22⋊Q8—1341no—
170C22⋊Q8—1341no—
171C22⋊Q8—1341no—
172C22⋊Q8—1341no—
173C22⋊Q8C2×D6711yescharacteristic
174C2×C4⋊C4—671no—
175C2×C4⋊C4—1341no—
176C2×C4⋊C4—1341no—
177C2×C4⋊C4C2×D6711yescharacteristic
178C2×C22⋊C4—1341no—
179C2×C22⋊C4—1341no—
180#106—672no—
181#106C2×D6712yescomplemented
182#103—672no—
183#103—672no—
184#103—672no—
185#103C2×D6712yescomplemented
186C22⋊C8—671no—
187C22⋊C8—672no—
188C22⋊C8C2×D6711yescharacteristic
189C24—671no—
190C2×Q8—2682no—
191C2×Q8—2682no—
192C2×Q8—672no—
193C2×Q8—671no—
194C2×Q8—1341no—
195C2×Q8—1341no—
196C2×Q8C22×D6711yescharacteristic
197C22×C4—21no—
198C22×C4—2682no—
199C22×C4—671no—
200C22×C4—671no—
201C22×C4—671no—
202C22×C4—671no—
203C22×C4—671no—
204C22×C4—1341no—
205C22×C4—1341no—
206C22×C4—1341no—
207C22×C4C22×D6711yescharacteristic
208Q16—2682no—
209Q16—2682no—
210M16—2682no—
211C2×C8—672no—
212C2×C8C22×D6712yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—671no—
216C4⋊C4—1341no—
217C4⋊C4—1341no—
218C4⋊C4—1341no—
219C4⋊C4—1342no—
220C4⋊C4—1342no—
221C4⋊C4—1341no—
222C4⋊C4C22×D6711yescharacteristic
223C4⋊C4C22×D6712yes—
224C22⋊C4—21no—
225C22⋊C4—1341no—
226C22⋊C4—1341no—
227C22⋊C4—1342no—
228C22⋊C4—1342no—
229C22⋊C4—1341no—
230C23—2681no—
231C23—671no—
232C23—671no—
233C23—1342no—
234C23—1342no—
235C23D4×D6711yescharacteristic
236Q8—22no—
237Q8—2682no—
238Q8—2682no—
239Q8—1342no—
240Q8—1342no—
241Q8—1342no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—2681no—
248C2×C4—2681no—
249C2×C4—2682no—
250C2×C4—2682no—
251C2×C4—2682no—
252C2×C4—2681no—
253C2×C4—671no—
254C2×C4—671no—
255C2×C4—1341no—
256C2×C4—1342no—
257C2×C4—1341no—
258C2×C4—1342no—
259C2×C4—1342no—
260C2×C4D4×D6711yescharacteristic
261C2×C4C23×D6711yescharacteristic
262C8—22no—
263C8—1342no—
264C22—21no—
265C22—21no—
266C22—2684no—
267C22—672no—
268C22—1341no—
269C22—1341no—
270C22—1341no—
271C22—1341no—
272C22—1341no—
273C22#1452407212yes—
274C22C2×D4×D6711yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—2681no—
280C4—2681no—
281C4—2682no—
282C4—1341no—
283C4—1341no—
284C4#1452408912yes—
285C2—22no—
286C2—2681no—
287C2—1342no—
288C2#1686395711yescharacteristic
289C2#137×D6711yescharacteristic
290C2C2×#1452407211yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#483×C67C2
socleC22×C67C22×D4
hypercenter#483D67
hypocenterC67#3195
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #16863728C2semidirect
4. #16863726C2semidirect
5. #16863655C2semidirect
6. #16863653C2semidirect
7. #16863554C2semidirect
8. #16863553C2semidirect
9. #16863551C2semidirect
11. #483×C67C2semidirect
15. C2×Dic268C22semidirect
16. C2×#11761618C22semidirect
32. #14523974C22semidirect
33. #14523968C22semidirect
34. #14523966C22semidirect
35. #14523966C22semidirect
40. #106×C67C22semidirect
41. #103×C67C22semidirect
146. C67#3195semidirect
151. #483D67semidirect
181. #106C2×D67semidirect
185. #103C2×D67semidirect