Finite Group Databasebeta

← #22565775

Subgroups — #22565775

Order 8576.

604 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 145 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×C4×D268C211yesmaximal, characteristic, complemented
2#16863335C211yesmaximal, characteristic, complemented
3#16863332C211yesmaximal, characteristic, complemented
4#16863313C211yesmaximal, characteristic, complemented
5#16863305C211yesmaximal, characteristic
6#16863287C212yesmaximal, complemented
7#16863275C212yesmaximal, complemented
8#16863272C212yesmaximal, complemented
9#16863267C212yesmaximal, complemented
10#16863264C211yesmaximal, characteristic, complemented
11C2×#117×C67C211yesmaximal, characteristic, complemented
12C22×D268C2211yescharacteristic
13C22×C22⋊D67C2212yes—
14C22×C4×D67C2212yes—
15C4×D268—24no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×Dic67⋊C4C2212yes—
18C2×C22⋊Dic67C2212yes—
19C2×(C2×D67)⋊C4C2212yes—
20C2×(C2×D67)⋊C4C2212yes—
21C2×(C2×D67)⋊C4C2212yes—
22C2×(C2×D67)⋊C4C2212yes—
23C67⋊#95C2212yes—
24C67⋊#95C2212yes—
25C67⋊#95C2211yescharacteristic
26C67⋊#95C2211yescharacteristic
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29C23×C4×C67C2211yescharacteristic
30#117×C67—24no—
31C2×C4⋊C4×C67C2211yescharacteristic
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C22⋊C4×C67C2211yescharacteristic
34C2×C22⋊C4×C67C2211yescharacteristic
35C2×C42×C67C2211yescharacteristic
36C2×C42×C67C2211yescharacteristic
37C23×D67C2312yes—
38C2×D268—24no—
39C2×D268—22no—
40C2×C22⋊D67—24no—
41C2×C22⋊D67—24no—
42C2×C22⋊D67—42no—
43C22×Dic67C2312yes—
44C22×Dic67C2312yes—
45C22×Dic67C2312yes—
46C2×C4×D67—24no—
47C2×C4×D67—42no—
48C2×C4×D67—42no—
49C4⋊Dic67—22no—
50Dic67⋊C4—42no—
51C22⋊Dic67—42no—
52(C2×D67)⋊C4—24no—
53(C2×D67)⋊C4—24no—
54(C2×D67)⋊C4—24no—
55(C2×D67)⋊C4—42no—
56C24×C67C2311yescharacteristic
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—22no—
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2311yescharacteristic
63C22×C4×C67C2311yescharacteristic
64C22×C4×C67C2311yescharacteristic
65C22×C4×C67C2311yescharacteristic
66C4⋊C4×C67—22no—
67C4⋊C4×C67—22no—
68C22⋊C4×C67—22no—
69C22⋊C4×C67—22no—
70C42×C67—22no—
71C42×C67D414yescomplemented
72C23×C67—21no—
73C23×C67—21no—
74C23×C67—22no—
75C23×C67—22no—
76C23×C67#4112yes—
77C23×C67C2411yescharacteristic
78C22×D67—22no—
79C22×D67—22no—
80C22×D67—22no—
81C22×D67—22no—
82C22×D67—22no—
83C22×D67—22no—
84C22×D67#4114yes—
85C2×C4×C67—28no—
86C2×C4×C67—21no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—22no—
91C2×C4×C67—22no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67C2×D412yes—
97C2×C4×C67C2×D412yes—
98C2×C4×C67C2×D412yes—
99C2×C4×C67#4112yes—
100C22⋊D67—48no—
101C2×Dic67—22no—
102C2×Dic67—22no—
103C2×Dic67—22no—
104C2×Dic67—22no—
105C2×Dic67—22no—
106C2×Dic67—22no—
107C2×Dic67—22no—
108C2×Dic67—22no—
109C2×Dic67—22no—
110D268—28no—
111C4×D67—48no—
112C22×C67—22no—
113C22×C67—24no—
114C22×C67—41no—
115C22×C67—41no—
116C22×C67—42no—
117C22×C67C22×D411yescharacteristic
118C22×C67C2×#4112yes—
119C22×C67C2×#4111yescharacteristic
120C22×C67C2×#4111yescharacteristic
121C22×C67H(5,𝔽2)12yes—
122C2×D67—24no—
123C2×D67—24no—
124C2×D67—24no—
125C2×D67—42no—
126C2×D67—42no—
127C2×D67—42no—
128C2×D67—42no—
129C4×C67—22no—
130C4×C67—22no—
131C4×C67—22no—
132C4×C67—22no—
133C4×C67—41no—
134C4×C67—41no—
135Dic67—42no—
136Dic67—42no—
137Dic67—42no—
138C2×C67—42no—
139C2×C67#52511yescharacteristic
140C2×C67#52511yescharacteristic
141C2×C67#53511yescharacteristic
142C2×C67#55112yes—
143C2×C67#55312yes—
144D67—44no—
145#2598—671nomaximal, Sylow
146C67#259811yescharacteristic, complemented, Sylow
147C2×C4×D4—671no—
148C2×#117D6711yescharacteristic, complemented
149#400—671no—
150#399—671no—
151#394—671no—
152#392—671no—
153#388—672no—
154#386—672no—
155#386—672no—
156#385—672no—
157#383—671no—
158C22×D4—672no—
159C22×D4—671no—
160C23×C4—672no—
161C23×C4C2×D6711yescharacteristic
162C4×D4—1344no—
163#117—24no—
164C2×C4⋊C4—672no—
165C2×C4⋊C4—671no—
166C2×C4⋊C4C2×D6711yescharacteristic
167C2×C4⋊C4C2×D6711yescharacteristic
168C2×C22⋊C4—672no—
169C2×C22⋊C4—672no—
170C2×C22⋊C4—672no—
171C2×C22⋊C4—672no—
172C2×C22⋊C4—672no—
173C2×C22⋊C4C2×D6711yescharacteristic
174C2×C22⋊C4C2×D6711yescharacteristic
175C2×C42C2×D6711yescharacteristic
176C2×C42C2×D6711yescharacteristic
177#95—672no—
178#95—672no—
179#95—671no—
180#95—671no—
181#95—672no—
182#95—672no—
183C24—672no—
184C24C22×D6711yescharacteristic
185C2×D4—2682no—
186C2×D4—1344no—
187C2×D4—1344no—
188C2×D4—1344no—
189C2×D4—1342no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—2682no—
194C22×C4—2682no—
195C22×C4—672no—
196C22×C4—672no—
197C22×C4—672no—
198C22×C4—1344no—
199C22×C4C22×D6711yescharacteristic
200C22×C4C22×D6711yescharacteristic
201C22×C4C22×D6711yescharacteristic
202C22×C4C22×D6711yescharacteristic
203C22×C4C22×D6711yescharacteristic
204C22×C4C22×D6711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—2682no—
208C4⋊C4—1342no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—2682no—
212C22⋊C4—2682no—
213C22⋊C4—1344no—
214C22⋊C4—1344no—
215C22⋊C4—1344no—
216C42—22no—
217C42C22⋊D6714yescomplemented
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—674no—
223C23—1342no—
224C23—1342no—
225C23—1342no—
226C23—1342no—
227C23—1342no—
228C23—1342no—
229C23#1176162812yes—
230C23C23×D6711yescharacteristic
231D4—2688no—
232D4—1348no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—2688no—
245C2×C4—1342no—
246C2×C4—1342no—
247C2×C4—1342no—
248C2×C4—1342no—
249C2×C4—1342no—
250C2×C4—1342no—
251C2×C4—1342no—
252C2×C4—1342no—
253C2×C4—1342no—
254C2×C4C2×C22⋊D6712yes—
255C2×C4C2×C22⋊D6712yes—
256C2×C4C2×C22⋊D6712yes—
257C2×C4#1176162812yes—
258C22—24no—
259C22—22no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—2682no—
264C22—2682no—
265C22—2682no—
266C22—2682no—
267C22—1344no—
268C22—1344no—
269C22—1344no—
270C22C22×C22⋊D6711yescharacteristic
271C22#41×D6712yes—
272C22C2×#1176162811yescharacteristic
273C22C2×#1176162811yescharacteristic
274C22#1452408912yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—2682no—
282C4—2682no—
283C4—2682no—
284C2—42no—
285C2—2684no—
286C2#1686390311yescharacteristic
287C2#1686390711yescharacteristic
288C2#1686396011yescharacteristic
289C2#1686407012yes—
290C2#1686409012yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#117×C67C2
hypercenterC2×#117D67
hypocenterC67#2598
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C4×D268C2semidirect
2. #16863335C2semidirect
3. #16863332C2semidirect
4. #16863313C2semidirect
6. #16863287C2semidirect
7. #16863275C2semidirect
8. #16863272C2semidirect
9. #16863267C2semidirect
10. #16863264C2semidirect
11. C2×#117×C67C2semidirect
71. C42×C67D4semidirect
146. C67#2598semidirect
148. C2×#117D67semidirect
217. C42C22⋊D67semidirect