Finite Group Databasebeta

← #22566327

Subgroups — #22566327

Order 8576.

448 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14524041C211yesmaximal, characteristic
2C2×#14524033C211yesmaximal, characteristic
3#16863330C212yesmaximal
4#16863329C212yesmaximal
5#16863317C211yesmaximal, characteristic
6#16863316C211yesmaximal, characteristic
7#16863315C211yesmaximal, characteristic
8#16863265C212yesmaximal
9#16863260C212yesmaximal
10#16863241C211yesmaximal, characteristic
11#395×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13#14524041—24no—
14#14524033—24no—
15C2×C4⋊Dic67C2211yescharacteristic
16C2×C4⋊Dic67C2212yes—
17C2×Dic67⋊C4C2211yescharacteristic
18C2×Dic67⋊C4C2212yes—
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2212yes—
22C2×Dic67⋊C4C2211yescharacteristic
23C2×Dic67⋊C4C2212yes—
24C2×Dic67⋊C4C2212yes—
25C2×C4×Dic67C2211yescharacteristic
26C2×C4×Dic67C2212yes—
27C67⋊#95C2211yescharacteristic
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C22×Q8×C67C2211yescharacteristic
33C2×C4⋊C4×C67C2211yescharacteristic
34C2×C4⋊C4×C67C2212yes—
35#95×C67C2211yescharacteristic
36#95×C67C2212yes—
37C2×Dic134—22no—
38C2×Dic134—42no—
39C22×Dic67C2311yescharacteristic
40C22×Dic67C2312yes—
41C22×Dic67C2311yescharacteristic
42C22×Dic67C2311yescharacteristic
43C22×Dic67C2312yes—
44C22×Dic67C2311yescharacteristic
45C4⋊Dic67—22no—
46C4⋊Dic67—42no—
47Dic67⋊C4—22no—
48Dic67⋊C4—22no—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51Dic67⋊C4—24no—
52Dic67⋊C4—24no—
53Dic67⋊C4—42no—
54Dic67⋊C4—42no—
55C4×Dic67D414yes—
56C4×Dic67Q818yescomplemented
57C2×Q8×C67—22no—
58C2×Q8×C67—42no—
59C22×C4×C67C2311yescharacteristic
60C22×C4×C67C2312yes—
61C22×C4×C67C2311yescharacteristic
62C22×C4×C67C2311yescharacteristic
63C22×C4×C67C2312yes—
64C4⋊C4×C67—22no—
65C4⋊C4×C67—42no—
66C23×C67C2411yescharacteristic
67Q8×C67—44no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—21no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67C2×D412yes—
80C2×C4×C67C2×Q814yes—
81C2×Dic67—22no—
82C2×Dic67—21no—
83C2×Dic67—21no—
84C2×Dic67—22no—
85C2×Dic67—21no—
86C2×Dic67—21no—
87C2×Dic67—22no—
88C2×Dic67—22no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—21no—
92C2×Dic67—21no—
93C2×Dic67C2×D412yes—
94C2×Dic67C2×D412yes—
95C2×Dic67C2×Q814yes—
96C2×Dic67C2×Q814yes—
97Dic134—44no—
98C22×C67C22×D411yescharacteristic
99C22×C67C22×Q812yes—
100C22×C67H(5,𝔽2)12yes—
101C22×C672−1+411yescharacteristic
102C22×C672−1+411yescharacteristic
103C4×C67—22no—
104C4×C67—24no—
105C4×C67—41no—
106C4×C67—42no—
107C4×C67—41no—
108Dic67—22no—
109Dic67—24no—
110Dic67—42no—
111Dic67—41no—
112Dic67—41no—
113Dic67C4⋊Q814yes—
114C2×C67C2×C4⋊Q811yescharacteristic
115C2×C67#53511yescharacteristic
116C2×C67#53611yescharacteristic
117C2×C67#54412yes—
118C2×C67#54412yes—
119#2660—671nomaximal, Sylow
120C67#266011yescharacteristic, complemented, Sylow
121C2×C4⋊Q8—671no—
122C2×#125—671no—
123#398—672no—
124#398—672no—
125#395—671no—
126#395—671no—
127#395—671no—
128#395D6711yescharacteristic
129#384—672no—
130#382—672no—
131#376—671no—
132C22×Q8—671no—
133C22×Q8C2×D6711yescharacteristic
134C4⋊Q8—1344no—
135#125—1344no—
136C2×C4⋊C4—672no—
137C2×C4⋊C4—672no—
138C2×C4⋊C4—672no—
139C2×C4⋊C4—671no—
140C2×C4⋊C4—671no—
141C2×C4⋊C4—671no—
142C2×C4⋊C4—671no—
143C2×C4⋊C4—671no—
144C2×C4⋊C4—672no—
145C2×C4⋊C4—672no—
146C2×C4⋊C4C2×D6711yescharacteristic
147C2×C4⋊C4C2×D6712yes—
148C2×C42—671no—
149C2×C42—672no—
150#95—671no—
151#95—672no—
152#95—672no—
153#95—672no—
154#95—672no—
155#95C2×D6711yescharacteristic
156#95C2×D6712yes—
157C2×Q8—22no—
158C2×Q8—42no—
159C2×Q8—2682no—
160C2×Q8—1342no—
161C22×C4—671no—
162C22×C4—672no—
163C22×C4—671no—
164C22×C4—671no—
165C22×C4—672no—
166C22×C4—671no—
167C22×C4C22×D6711yescharacteristic
168C22×C4C22×D6712yes—
169C22×C4C22×D6711yescharacteristic
170C22×C4C22×D6711yescharacteristic
171C22×C4C22×D6712yes—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—2682no—
175C4⋊C4—2682no—
176C4⋊C4—2682no—
177C4⋊C4—1342no—
178C4⋊C4—1342no—
179C4⋊C4—1342no—
180C4⋊C4—1342no—
181C4⋊C4—1344no—
182C4⋊C4—1344no—
183C4⋊C4—1342no—
184C42—674no—
185C42—678no—
186C23C23×D6711yescharacteristic
187Q8—44no—
188Q8—2684no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—672no—
201C2×C4—674no—
202C2×C4—672no—
203C2×C4—674no—
204C2×C4—1341no—
205C2×C4—1341no—
206C2×C4—1342no—
207C2×C4—1342no—
208C2×C4—1342no—
209C2×C4—1342no—
210C2×C4—1341no—
211C2×C4—1341no—
212C2×C4—1342no—
213C2×C4—1341no—
214C2×C4—1342no—
215C2×C4—1341no—
216C2×C4D4×D6712yes—
217C2×C4Q8×D6714yes—
218C22C2×D4×D6711yescharacteristic
219C22C2×Q8×D6712yes—
220C22#1452408812yes—
221C22#1452409011yescharacteristic
222C22#1452409111yescharacteristic
223C4—24no—
224C4—22no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—2682no—
229C4—2681no—
230C4—2681no—
231C4—674no—
232C4—1344no—
233C4—1342no—
234C2C4⋊Q8×D6711yescharacteristic
235C2#1686396111yescharacteristic
236C2#1686396911yescharacteristic
237C2#1686402312yes—
238C2#1686402512yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#395×C67C2
hypercenter#395D67
hypocenterC67#2660
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
56. C4×Dic67Q8semidirect
120. C67#2660semidirect