Finite Group Databasebeta

← #22565969

Subgroups — #22565969

Order 8576.

612 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 157 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
2C2×#14524023C211yesmaximal, characteristic, complemented
3C2×#14524018C211yesmaximal, characteristic
4C2×#14524000C211yesmaximal, characteristic, complemented
5#16863318C214yesmaximal, complemented
6#16863313C214yesmaximal, complemented
7#16863301C211yesmaximal, characteristic, complemented
8#16863297C211yesmaximal, characteristic, complemented
9#384×C67C211yesmaximal, characteristic, complemented
10C22×Dic134C2211yescharacteristic
11C22×D268C2211yescharacteristic
12C22×D268C2212yes—
13C22×C4×D67C2211yescharacteristic
14#14524031—24no—
15#14524023—24no—
16#14524018—24no—
17#14524000—24no—
18C2×C4⋊Dic67C2212yes—
19C2×Dic67⋊C4C2212yes—
20C2×(C2×D67)⋊C4C2212yes—
21C2×(C2×D67)⋊C4C2212yes—
22C2×(C2×D67)⋊C4C2214yes—
23C2×(C2×D67)⋊C4C2212yes—
24C2×(C2×D67)⋊C4C2214yes—
25C2×C4×Dic67C2211yescharacteristic
26C67⋊#95C2214yes—
27C2×C4⋊C4×C67C2211yescharacteristic
28C2×C4⋊C4×C67C2211yescharacteristic
29C2×C4⋊C4×C67C2214yes—
30C2×C42×C67C2211yescharacteristic
31C23×D67C2312yes—
32C23×D67C2311yescharacteristic
33C2×Dic134—22no—
34C2×Dic134—22no—
35C2×Dic134—22no—
36C2×D268—24no—
37C2×D268—22no—
38C2×D268—22no—
39C2×D268—22no—
40C2×D268—44no—
41C22×Dic67C2312yes—
42C22×Dic67C2312yes—
43C22×Dic67C2311yescharacteristic
44C2×C4×D67—22no—
45C2×C4×D67—22no—
46C2×C4×D67D414yes—
47C4⋊Dic67—24no—
48Dic67⋊C4—24no—
49(C2×D67)⋊C4—24no—
50(C2×D67)⋊C4—24no—
51(C2×D67)⋊C4—28no—
52(C2×D67)⋊C4—24no—
53(C2×D67)⋊C4—44no—
54C4×Dic67—22no—
55C22×C4×C67C2312yes—
56C22×C4×C67C2312yes—
57C22×C4×C67C2312yes—
58C22×C4×C67C2311yescharacteristic
59C4⋊C4×C67—22no—
60C4⋊C4×C67—22no—
61C4⋊C4×C67—44no—
62C42×C67D414yescomplemented
63C23×C67C2411yescharacteristic
64C22×D67—21no—
65C22×D67—21no—
66C22×D67—22no—
67C22×D67—22no—
68C22×D67—22no—
69C22×D67—22no—
70C22×D67—22no—
71C22×D67—24no—
72C22×D67—24no—
73C22×D67C2×D412yes—
74C2×C4×C67—22no—
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—24no—
79C2×C4×C67—24no—
80C2×C4×C67C2×D412yes—
81C2×C4×C67C2×D414yes—
82C2×C4×C67C2×D412yes—
83C2×C4×C67#4112yes—
84C2×Dic67—21no—
85C2×Dic67—21no—
86C2×Dic67—22no—
87C2×Dic67—22no—
88C2×Dic67—22no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—22no—
92C2×Dic67C2×D412yes—
93D268—48no—
94D268—44no—
95C4×D67—28no—
96Dic134—44no—
97C22×C67C22×D411yescharacteristic
98C22×C67C22×D411yescharacteristic
99C22×C67C2×#4111yescharacteristic
100C22×C67H(5,𝔽2)12yes—
101C22×C672−1+412yes—
102C2×D67—22no—
103C2×D67—22no—
104C2×D67—22no—
105C2×D67—42no—
106C2×D67—42no—
107C2×D67—42no—
108C2×D67—42no—
109C2×D67—42no—
110C2×D67—44no—
111C2×D67—44no—
112C4×C67—24no—
113C4×C67—42no—
114C4×C67—42no—
115C4×C67#12114yes—
116Dic67—22no—
117Dic67—42no—
118Dic67—42no—
119C2×C67C2×#12111yescharacteristic
120C2×C67#53511yescharacteristic
121C2×C67#53611yescharacteristic
122C2×C67#53712yes—
123C2×C67#54212yes—
124D67—42no—
125D67—82no—
126#2620—671nomaximal, Sylow
127C67#262011yescharacteristic, complemented, Sylow
128C2×#124—671no—
129C2×C22⋊Q8—671no—
130C2×#121—671no—
131C2×#117—671no—
132#396—674no—
133#394—674no—
134#391—671no—
135#390—671no—
136#384D6711yescharacteristic, complemented
137C22×Q8—671no—
138C22×D4—672no—
139C22×D4—671no—
140C23×C4—671no—
141#124—1344no—
142C22⋊Q8—1344no—
143#121—1344no—
144#117—1344no—
145C2×C4⋊C4—672no—
146C2×C4⋊C4—672no—
147C2×C4⋊C4C2×D6711yescharacteristic
148C2×C4⋊C4C2×D6711yescharacteristic
149C2×C4⋊C4C2×D6714yes—
150C2×C22⋊C4—674no—
151C2×C22⋊C4—672no—
152C2×C22⋊C4—672no—
153C2×C22⋊C4—672no—
154C2×C22⋊C4—674no—
155C2×C42—671no—
156C2×C42C2×D6711yescharacteristic
157#95—674no—
158C24—671no—
159C24—672no—
160C2×Q8—1342no—
161C2×Q8—1342no—
162C2×Q8—1342no—
163C2×D4—2684no—
164C2×D4—1344no—
165C2×D4—1342no—
166C2×D4—1342no—
167C2×D4—1342no—
168C22×C4—671no—
169C22×C4—674no—
170C22×C4—672no—
171C22×C4—672no—
172C22×C4—1342no—
173C22×C4—1342no—
174C22×C4C22×D6712yes—
175C22×C4C22×D6712yes—
176C22×C4C22×D6712yes—
177C22×C4C22×D6711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—44no—
181C4⋊C4—1344no—
182C4⋊C4—1344no—
183C22⋊C4—2684no—
184C22⋊C4—1344no—
185C22⋊C4—1344no—
186C22⋊C4—1344no—
187C22⋊C4—1348no—
188C42—1342no—
189C42D26814yescomplemented
190C23—672no—
191C23—1342no—
192C23—1342no—
193C23—1342no—
194C23—1344no—
195C23—1344no—
196C23—1341no—
197C23—1341no—
198C23—1342no—
199C23—1342no—
200C23C23×D6711yescharacteristic
201Q8—2684no—
202D4—2688no—
203D4—2684no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—672no—
211C2×C4—1342no—
212C2×C4—1342no—
213C2×C4—1342no—
214C2×C4—1342no—
215C2×C4—1342no—
216C2×C4—1342no—
217C2×C4—1341no—
218C2×C4—1341no—
219C2×C4—1348no—
220C2×C4D4×D6712yes—
221C2×C4C2×D26814yes—
222C2×C4C2×D26812yes—
223C2×C4#1176162912yes—
224C22—2682no—
225C22—2682no—
226C22—2684no—
227C22—2682no—
228C22—2682no—
229C22—2682no—
230C22—2684no—
231C22—1342no—
232C22—1342no—
233C22—1342no—
234C22C2×D4×D6711yescharacteristic
235C22C22×D26811yescharacteristic
236C22C2×#1176162911yescharacteristic
237C22#1452408812yes—
238C22#1452409112yes—
239C4—24no—
240C4—42no—
241C4—42no—
242C4—2682no—
243C4—2682no—
244C4—1342no—
245C4#1452401814yes—
246C2—2682no—
247C2—5362no—
248C2C2×#1452401811yescharacteristic
249C2#1686395911yescharacteristic
250C2#1686397111yescharacteristic
251C2#1686397712yes—
252C2#1686402012yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#384×C67C2
hypercenter#384D67
hypocenterC67#2620
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#14524031C2semidirect
2. C2×#14524023C2semidirect
4. C2×#14524000C2semidirect
5. #16863318C2semidirect
6. #16863313C2semidirect
7. #16863301C2semidirect
8. #16863297C2semidirect
9. #384×C67C2semidirect
62. C42×C67D4semidirect
127. C67#2620semidirect
136. #384D67semidirect
189. C42D268semidirect