Finite Group Databasebeta

← #22566349

Subgroups — #22566349

Order 8576.

534 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14524031C211yesmaximal, characteristic, complemented
2#16863318C212yesmaximal, complemented
3#16863316C211yesmaximal, characteristic, complemented
4#16863313C211yesmaximal, characteristic, complemented
5#16863312C212yesmaximal, complemented
6#16863308C211yesmaximal, characteristic, complemented
7#16863305C211yesmaximal, characteristic
8#16863296C212yesmaximal, complemented
9#16863286C212yesmaximal, complemented
10C4×C4⋊Dic67C211yesmaximal, characteristic, complemented
11C2×#126×C67C211yesmaximal, characteristic, complemented
12C22×Dic134C2211yescharacteristic
13C22×D268C2211yescharacteristic
14C22×C22⋊D67C2212yes—
15#14524031—24no—
16C2×C4⋊Dic67C2212yes—
17C2×Dic67⋊C4C2212yes—
18C2×C22⋊Dic67C2212yes—
19C2×(C2×D67)⋊C4C2212yes—
20C2×(C2×D67)⋊C4C2212yes—
21C2×(C2×D67)⋊C4C2212yes—
22C2×(C2×D67)⋊C4C2212yes—
23C2×(C2×D67)⋊C4C2212yes—
24C2×C4×Dic67C2212yes—
25C67⋊#95C2211yescharacteristic
26C67⋊#95C2211yescharacteristic
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29#126×C67—24no—
30C2×C4⋊C4×C67C2211yescharacteristic
31C2×C4⋊C4×C67C2211yescharacteristic
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C22⋊C4×C67C2211yescharacteristic
34C2×C22⋊C4×C67C2211yescharacteristic
35C2×C22⋊C4×C67C2211yescharacteristic
36C2×C42×C67C2211yescharacteristic
37C23×D67C2312yes—
38C2×Dic134—22no—
39C2×Dic134—42no—
40C2×D268—22no—
41C2×D268—42no—
42C2×C22⋊D67—24no—
43C2×C22⋊D67—42no—
44C2×C22⋊D67—42no—
45C22×Dic67C2312yes—
46C22×Dic67C2312yes—
47C22×Dic67C2312yes—
48C4⋊Dic67—24no—
49Dic67⋊C4—42no—
50C22⋊Dic67—42no—
51(C2×D67)⋊C4—24no—
52(C2×D67)⋊C4—24no—
53(C2×D67)⋊C4—24no—
54(C2×D67)⋊C4—42no—
55(C2×D67)⋊C4—42no—
56C4×Dic67—42no—
57C24×C67C2311yescharacteristic
58C22×C4×C67C2311yescharacteristic
59C22×C4×C67C2311yescharacteristic
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2311yescharacteristic
63C22×C4×C67C2311yescharacteristic
64C4⋊C4×C67—22no—
65C4⋊C4×C67—22no—
66C4⋊C4×C67—22no—
67C22⋊C4×C67—22no—
68C22⋊C4×C67—22no—
69C22⋊C4×C67—22no—
70C42×C67D414yescomplemented
71C23×C67—21no—
72C23×C67—22no—
73C23×C67—22no—
74C23×C67—21no—
75C23×C67—21no—
76C23×C67C2411yescharacteristic
77C22×D67—22no—
78C22×D67—22no—
79C22×D67—22no—
80C22×D67—22no—
81C22×D67—22no—
82C22×D67—22no—
83C22×D67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—21no—
86C2×C4×C67—22no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—22no—
92C2×C4×C67—21no—
93C2×C4×C67—22no—
94C2×C4×C67—21no—
95C2×C4×C67C2×D412yes—
96C2×C4×C67C2×D412yes—
97C2×C4×C67C2×D412yes—
98C22⋊D67—48no—
99C2×Dic67—22no—
100C2×Dic67—22no—
101C2×Dic67—22no—
102C2×Dic67—22no—
103C2×Dic67—22no—
104C2×Dic67—22no—
105C2×Dic67—22no—
106C2×Dic67—22no—
107C2×Dic67#4114yes—
108D268—44no—
109Dic134—44no—
110C22×C67—41no—
111C22×C67—41no—
112C22×C67—41no—
113C22×C67—42no—
114C22×C67—42no—
115C22×C67C22×D411yescharacteristic
116C22×C67C2×#4112yes—
117C22×C67H(5,𝔽2)12yes—
118C22×C67H(5,𝔽2)11yescharacteristic
119C22×C672−1+411yescharacteristic
120C2×D67—42no—
121C2×D67—42no—
122C2×D67—42no—
123C2×D67—42no—
124C2×D67—42no—
125C2×D67—42no—
126C2×D67—42no—
127C4×C67—22no—
128C4×C67—22no—
129C4×C67—22no—
130C4×C67—41no—
131C4×C67—41no—
132C4×C67—41no—
133Dic67—24no—
134Dic67—42no—
135Dic67—42no—
136C2×C67—81no—
137C2×C67#53211yescharacteristic
138C2×C67#53511yescharacteristic
139C2×C67#53711yescharacteristic
140C2×C67#53812yes—
141C2×C67#54212yes—
142D67—82no—
143#2662—671nomaximal, Sylow
144C67#266211yescharacteristic, complemented, Sylow
145C2×#126D6711yescharacteristic, complemented
146C2×#124—671no—
147#396—672no—
148#395—671no—
149#394—671no—
150#394—672no—
151#393—671no—
152#392—671no—
153#390—672no—
154#388—672no—
155C4×C4⋊C4—671no—
156C22×Q8—671no—
157C22×D4—672no—
158C22×D4—671no—
159#126—24no—
160#124—1344no—
161C2×C4⋊C4—672no—
162C2×C4⋊C4—672no—
163C2×C4⋊C4C2×D6711yescharacteristic
164C2×C4⋊C4C2×D6711yescharacteristic
165C2×C4⋊C4C2×D6711yescharacteristic
166C2×C22⋊C4—672no—
167C2×C22⋊C4—672no—
168C2×C22⋊C4—672no—
169C2×C22⋊C4—672no—
170C2×C22⋊C4—672no—
171C2×C22⋊C4—672no—
172C2×C22⋊C4C2×D6711yescharacteristic
173C2×C22⋊C4C2×D6711yescharacteristic
174C2×C22⋊C4C2×D6711yescharacteristic
175C2×C42—672no—
176C2×C42C2×D6711yescharacteristic
177#95—671no—
178#95—671no—
179#95—672no—
180#95—672no—
181C24—672no—
182C24C22×D6711yescharacteristic
183C2×Q8—2682no—
184C2×Q8—1342no—
185C2×D4—2682no—
186C2×D4—2682no—
187C2×D4—2682no—
188C2×D4—1344no—
189C2×D4—1342no—
190C22×C4—672no—
191C22×C4—672no—
192C22×C4—672no—
193C22×C4C22×D6711yescharacteristic
194C22×C4C22×D6711yescharacteristic
195C22×C4C22×D6711yescharacteristic
196C22×C4C22×D6711yescharacteristic
197C22×C4C22×D6711yescharacteristic
198C22×C4C22×D6711yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—2682no—
203C4⋊C4—1344no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2682no—
208C22⋊C4—2682no—
209C22⋊C4—2682no—
210C22⋊C4—1344no—
211C22⋊C4—1344no—
212C22⋊C4—1344no—
213C42—2682no—
214C42C22⋊D6714yescomplemented
215C23—21no—
216C23—22no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—1342no—
221C23—1342no—
222C23—1342no—
223C23—1342no—
224C23—1342no—
225C23—1342no—
226C23—1342no—
227C23C23×D6711yescharacteristic
228Q8—2684no—
229D4—2684no—
230D4—2688no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—674no—
243C2×C4—1342no—
244C2×C4—1342no—
245C2×C4—1342no—
246C2×C4—1342no—
247C2×C4—1342no—
248C2×C4—1342no—
249C2×C4—1342no—
250C2×C4—1342no—
251C2×C4C2×C22⋊D6712yes—
252C2×C4C2×C22⋊D6712yes—
253C2×C4C2×C22⋊D6712yes—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—42no—
259C22—2682no—
260C22—2682no—
261C22—2682no—
262C22—2682no—
263C22—2682no—
264C22—2682no—
265C22—2682no—
266C22C22×C22⋊D6711yescharacteristic
267C22#41×D6712yes—
268C22#1452408912yes—
269C22#1452408911yescharacteristic
270C22#1452409011yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—2682no—
278C4—2682no—
279C4—1344no—
280C2—81no—
281C2—5362no—
282C2#1686394211yescharacteristic
283C2#1686396011yescharacteristic
284C2#1686397411yescharacteristic
285C2#1686398512yes—
286C2#1686401212yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#126×C67C2
hypercenterC2×#126D67
hypocenterC67#2662
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#14524031C2semidirect
2. #16863318C2semidirect
3. #16863316C2semidirect
4. #16863313C2semidirect
5. #16863312C2semidirect
6. #16863308C2semidirect
8. #16863296C2semidirect
9. #16863286C2semidirect
10. C4×C4⋊Dic67C2semidirect
11. C2×#126×C67C2semidirect
70. C42×C67D4semidirect
144. C67#2662semidirect
145. C2×#126D67semidirect
214. C42C22⋊D67semidirect