Finite Group Databasebeta

← #22571025

Subgroups — #22571025

Order 8576.

466 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#16863907C211yesmaximal, characteristic, complemented
2C2×#14524017C211yesmaximal, characteristic, complemented
3#16863727C211yesmaximal, characteristic, complemented
4#16863721C212yesmaximal, complemented
5#16863710C211yesmaximal, characteristic, complemented
6#16863649C212yesmaximal, complemented
7#16863642C212yesmaximal, complemented
8#16863549C211yesmaximal, characteristic, complemented
9#16863543C212yesmaximal, complemented
10#16863532C211yesmaximal, characteristic, complemented
11#407×C67C211yesmaximal, characteristic, complemented
12C2×#11761628C2211yescharacteristic, complemented
13C22×D268C2211yescharacteristic
14C22×C22⋊D67—21no—
15C2×#11761617C2212yescomplemented
16C2×D536C2212yescomplemented
17#14524026—21no—
18#14524022C2211yescharacteristic
19#14524022C2211yescharacteristic, complemented
20#14524017—21no—
21#14524017—21no—
22#14524017C2211yescharacteristic
23#14524017C2212yes—
24#14524017C2212yes—
25#14524017C2211yescharacteristic, complemented
26#14524012—21no—
27C4×C22⋊D67—21no—
28C4×C22⋊D67—21no—
29#14524002C2211yescharacteristic, complemented
30C2×C4⋊Dic67C2211yescharacteristic
31C2×(C2×D67)⋊C4—21no—
32#14523977C2212yescomplemented
33#14523975C2212yescomplemented
34#14523968C2212yescomplemented
35#14523965C2212yescomplemented
36#14523965C2212yescomplemented
37#14523965C2212yescomplemented
38C23×C4×C67C2211yescharacteristic
39C2×M16×C67C2212yescomplemented
40C22⋊C8×C67C2211yescharacteristic
41C22⋊C8×C67C2211yescharacteristic
42C22⋊C8×C67C2212yescomplemented
43C23×D67—21no—
44#11761628—42no—
45C2×Dic134C2311yescharacteristic
46C2×D268—21no—
47C2×D268—42no—
48C2×D268C2312yes—
49C2×D268C2311yescharacteristic
50C2×C22⋊D67—21no—
51C2×C22⋊D67—21no—
52C2×C22⋊D67—21no—
53C2×C22⋊D67—22no—
54C2×C22⋊D67—22no—
55C2×C22⋊D67—44no—
56C22×Dic67—21no—
57C2×C4×D67—21no—
58#11761617—42no—
59D536—42no—
60C4⋊Dic67—21no—
61C4⋊Dic67C2311yescharacteristic
62C4⋊Dic67C2311yescharacteristic
63C4⋊Dic67C2312yes—
64Dic67⋊C4—21no—
65Dic67⋊C4—21no—
66C22⋊Dic67—21no—
67C22⋊Dic67—21no—
68C22⋊Dic67—21no—
69(C2×D67)⋊C4—22no—
70(C2×D67)⋊C4—22no—
71(C2×D67)⋊C4—21no—
72(C2×D67)⋊C4—21no—
73C4×Dic67—21no—
74C24×C67D411yescharacteristic
75C22×C4×C67—42no—
76C22×C4×C67D411yescharacteristic
77C22×C4×C67D411yescharacteristic
78C22×C4×C67D411yescharacteristic
79C22×C4×C67C2311yescharacteristic
80C22×C4×C67C2311yescharacteristic
81C22×C4×C67C2311yescharacteristic
82M16×C67—42no—
83C2×C8×C67C2312yes—
84C2×C8×C67C2312yes—
85C23×C67—22no—
86C23×C67—41no—
87C23×C67—41no—
88C23×C67C2×D411yescharacteristic
89C23×C67C2×D411yescharacteristic
90C23×C67C2×D411yescharacteristic
91C22×D67—21no—
92C22×D67—22no—
93C22×D67—42no—
94C22×D67—42no—
95C22×D67—42no—
96C2×C4×C67—22no—
97C2×C4×C67—22no—
98C2×C4×C67—22no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67C2×D411yescharacteristic
103C2×C4×C67C2×D411yescharacteristic
104C2×C4×C67C2×D411yescharacteristic
105C2×C4×C67C2411yescharacteristic
106C22⋊D67—42no—
107C22⋊D67—44no—
108C22⋊D67—44no—
109C22⋊D67—42no—
110C22⋊D67—42no—
111C2×Dic67—21no—
112C2×Dic67—21no—
113C2×Dic67—22no—
114C2×Dic67—21no—
115C2×Dic67—42no—
116D268—22no—
117D268—22no—
118D268—22no—
119D268—42no—
120C4×D67—42no—
121Dic134—22no—
122C8×C67—22no—
123C8×C67—22no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67—21no—
127C22×C67—21no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67—21no—
131C22×C67—21no—
132C22×C67—44no—
133C22×C67AGL(1,ℤ/8)12yes—
134C22×C67C22×D411yescharacteristic
135C2×D67—41no—
136C2×D67—42no—
137C2×D67—42no—
138C2×D67—41no—
139C2×D67—41no—
140C2×D67—42no—
141C2×D67—84no—
142C4×C67—21no—
143C4×C67—21no—
144C4×C67—21no—
145C4×C67H(5,𝔽2)12yes—
146Dic67—41no—
147Dic67—42no—
148Dic67—41no—
149Dic67—41no—
150C2×C67—22no—
151C2×C67—41no—
152C2×C67—41no—
153C2×C67#53411yescharacteristic
154C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C67#57511yescharacteristic
156D67—81no—
157D67—82no—
158#3194—671nomaximal, Sylow
159C67#319411yescharacteristic, complemented, Sylow
160#525—671no—
161C2×#121—671no—
162#483—671no—
163#482—672no—
164#480—671no—
165#469—672no—
166#468—672no—
167#450—671no—
168#449—672no—
169#447—671no—
170#407D6711yescharacteristic, complemented
171C2×#41—671no—
172C22×D4—671no—
173C22×D4—1341no—
174C23×C4C2×D6711yescharacteristic
175C2×SD16—672no—
176C2×D8—672no—
177C2×M16C2×D6712yescomplemented
178#123—1341no—
179C22⋊Q8—671no—
180C22⋊Q8—671no—
181#121—672no—
182#121—672no—
183#121—671no—
184#121—671no—
185#121—1341no—
186#121—1341no—
187#120—1341no—
188C4×D4—1341no—
189C4×D4—1341no—
190#117—671no—
191C2×C4⋊C4—671no—
192C2×C22⋊C4—1341no—
193#107—672no—
194#106—672no—
195#103—672no—
196#102—672no—
197#102—672no—
198#102—672no—
199C22⋊C8C2×D6711yescharacteristic
200C22⋊C8C2×D6711yescharacteristic
201C22⋊C8C2×D6712yescomplemented
202C24—1341no—
203C24D26811yescharacteristic
204#41—2682no—
205C2×Q8—671no—
206C2×D4—2682no—
207C2×D4—2684no—
208C2×D4—671no—
209C2×D4—672no—
210C2×D4—1341no—
211C2×D4—1341no—
212C2×D4—1341no—
213C2×D4—1342no—
214C2×D4—1342no—
215C2×D4—1341no—
216C22×C4—42no—
217C22×C4—1341no—
218C22×C4—1341no—
219C22×C4D26811yescharacteristic
220C22×C4D26811yescharacteristic
221C22×C4D26811yescharacteristic
222C22×C4C22×D6711yescharacteristic
223C22×C4C22×D6711yescharacteristic
224C22×C4C22×D6711yescharacteristic
225SD16—2682no—
226D8—2682no—
227M16—42no—
228C2×C8C22×D6712yes—
229C2×C8C22×D6712yes—
230C4⋊C4—671no—
231C4⋊C4—671no—
232C4⋊C4—672no—
233C4⋊C4—1341no—
234C4⋊C4—1341no—
235C4⋊C4—1341no—
236C22⋊C4—1341no—
237C22⋊C4—1341no—
238C22⋊C4—1341no—
239C22⋊C4—1342no—
240C22⋊C4—1342no—
241C22⋊C4—1341no—
242C22⋊C4—1341no—
243C42—1341no—
244C23—22no—
245C23—41no—
246C23—41no—
247C23—2682no—
248C23—2682no—
249C23—2682no—
250C23—1341no—
251C23—1342no—
252C23C2×D26811yescharacteristic
253C23C2×D26811yescharacteristic
254C23C2×D26811yescharacteristic
255Q8—1342no—
256D4—2682no—
257D4—2684no—
258D4—2684no—
259D4—2682no—
260D4—2682no—
261D4—2682no—
262D4—1342no—
263D4—1342no—
264D4—1342no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—2682no—
272C2×C4—2682no—
273C2×C4—1341no—
274C2×C4—1341no—
275C2×C4—1342no—
276C2×C4—1341no—
277C2×C4C2×D26811yescharacteristic
278C2×C4C2×D26811yescharacteristic
279C2×C4C2×D26811yescharacteristic
280C2×C4C23×D6711yescharacteristic
281C8—22no—
282C8—22no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—44no—
292C22—2682no—
293C22—2681no—
294C22—2681no—
295C22—2682no—
296C22—2681no—
297C22—2682no—
298C22—5364no—
299C22#1452407112yes—
300C22C22×D26811yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—2682no—
305C4—2681no—
306C4—2681no—
307C4—2681no—
308C4#1452408812yes—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—5362no—
313C2—5361no—
314C2#1686395811yescharacteristic
315C2C2×#1452407111yescharacteristic
316C2#1686419611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#407×C67C2
socleC22×C67C22×D4
hypercenter#407D67
hypocenterC67#3194
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #16863907C2semidirect
2. C2×#14524017C2semidirect
3. #16863727C2semidirect
4. #16863721C2semidirect
5. #16863710C2semidirect
6. #16863649C2semidirect
7. #16863642C2semidirect
8. #16863549C2semidirect
9. #16863543C2semidirect
10. #16863532C2semidirect
11. #407×C67C2semidirect
12. C2×#11761628C22semidirect
15. C2×#11761617C22semidirect
16. C2×D536C22semidirect
19. #14524022C22semidirect
25. #14524017C22semidirect
29. #14524002C22semidirect
32. #14523977C22semidirect
33. #14523975C22semidirect
34. #14523968C22semidirect
35. #14523965C22semidirect
36. #14523965C22semidirect
37. #14523965C22semidirect
39. C2×M16×C67C22semidirect
42. C22⋊C8×C67C22semidirect
159. C67#3194semidirect
170. #407D67semidirect
177. C2×M16C2×D67semidirect
201. C22⋊C8C2×D67semidirect