Finite Group Databasebeta

← #22564511

Subgroups — #22564511

Order 8576.

624 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14524026C212yesmaximal
2C2×C4×C22⋊D67C211yesmaximal, characteristic
3#16863327C212yesmaximal, complemented
4#16863322C212yesmaximal, complemented
5#16863292C212yesmaximal, complemented
6#16863259C212yesmaximal, complemented
7#16863256C211yesmaximal, characteristic, complemented
8#16863253C212yesmaximal, complemented
9C22×C4⋊C4×C67C211yesmaximal, characteristic, complemented
10C22×C22⋊D67C2211yescharacteristic
11C22×C4×D67C2211yescharacteristic
12#14524026—28no—
13C4×C22⋊D67—24no—
14C2×C4⋊Dic67C2212yes—
15C2×Dic67⋊C4C2214yes—
16C2×Dic67⋊C4C2211yescharacteristic
17C2×C22⋊Dic67C2212yes—
18C2×C22⋊Dic67C2211yescharacteristic
19C2×(C2×D67)⋊C4C2214yes—
20C2×(C2×D67)⋊C4C2211yescharacteristic
21C2×C4×Dic67C2211yescharacteristic
22C67⋊#95C2214yes—
23C67⋊#95C2212yes—
24C67⋊#95C2214yes—
25C23×C4×C67C2212yes—
26C23×C4×C67C2211yescharacteristic
27C2×C4⋊C4×C67—24no—
28C2×C4⋊C4×C67C2212yes—
29C2×C4⋊C4×C67C2212yes—
30C23×D67C2311yescharacteristic
31C2×C22⋊D67—22no—
32C2×C22⋊D67—22no—
33C2×C22⋊D67D414yes—
34C22×Dic67C2311yescharacteristic
35C22×Dic67C2311yescharacteristic
36C22×Dic67C2314yes—
37C22×Dic67C2311yescharacteristic
38C2×C4×D67—22no—
39C2×C4×D67—24no—
40C4⋊Dic67—24no—
41Dic67⋊C4—22no—
42Dic67⋊C4—28no—
43C22⋊Dic67—22no—
44C22⋊Dic67—24no—
45(C2×D67)⋊C4—28no—
46(C2×D67)⋊C4—22no—
47C4×Dic67—22no—
48C24×C67C2311yescharacteristic
49C22×C4×C67—24no—
50C22×C4×C67—24no—
51C22×C4×C67—24no—
52C22×C4×C67—24no—
53C22×C4×C67D414yes—
54C22×C4×C67C2314yes—
55C22×C4×C67C2311yescharacteristic
56C22×C4×C67C2311yescharacteristic
57C4⋊C4×C67—24no—
58C4⋊C4×C67—24no—
59C23×C67—22no—
60C23×C67—22no—
61C23×C67C2×D412yes—
62C23×C67C2×D412yes—
63C23×C67#4112yes—
64C23×C67C2411yescharacteristic
65C22×D67—21no—
66C22×D67—22no—
67C22×D67—21no—
68C22×D67—22no—
69C22×D67C2×D412yes—
70C2×C4×C67—22no—
71C2×C4×C67—28no—
72C2×C4×C67—24no—
73C2×C4×C67—24no—
74C2×C4×C67—24no—
75C2×C4×C67—48no—
76C2×C4×C67C2×D412yes—
77C2×C4×C67C2×D412yes—
78C2×C4×C67#4114yes—
79C22⋊D67—28no—
80C2×Dic67—21no—
81C2×Dic67—21no—
82C2×Dic67—21no—
83C2×Dic67—24no—
84C2×Dic67—21no—
85C2×Dic67—24no—
86C2×Dic67—22no—
87C2×Dic67—22no—
88C2×Dic67—24no—
89C2×Dic67C2×D412yes—
90C4×D67—44no—
91C22×C67—22no—
92C22×C67—22no—
93C22×C67—24no—
94C22×C67—24no—
95C22×C67#12114yes—
96C22×C67C22×D411yescharacteristic
97C22×C67C22×D411yescharacteristic
98C22×C67C2×#4111yescharacteristic
99C22×C67C2×#4112yes—
100C22×C672−1+412yes—
101C2×D67—22no—
102C2×D67—24no—
103C2×D67—41no—
104C2×D67—41no—
105C2×D67—42no—
106C4×C67—22no—
107C4×C67—44no—
108C4×C67#12314yescomplemented
109Dic67—22no—
110Dic67—41no—
111Dic67—41no—
112Dic67—44no—
113C2×C67—24no—
114C2×C67C2×#12111yescharacteristic
115C2×C67C2×#12311yescharacteristic
116C2×C67#53611yescharacteristic
117C2×C67#54712yes—
118C2×C67#55212yes—
119D67—42no—
120#2486—671nomaximal, Sylow
121C67#248611yescharacteristic, complemented, Sylow
122C2×#123—672no—
123C2×C4×D4—671no—
124C22×C4⋊C4D6711yescharacteristic, complemented
125#398—672no—
126#397—672no—
127#389—672no—
128#381—672no—
129#380—671no—
130#380—672no—
131C22×D4—671no—
132C23×C4—671no—
133C23×C4C2×D6712yes—
134C23×C4C2×D6711yescharacteristic
135#123—1348no—
136C4×D4—1344no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—671no—
139C2×C4⋊C4—674no—
140C2×C4⋊C4—672no—
141C2×C4⋊C4C2×D6712yes—
142C2×C4⋊C4C2×D6712yes—
143C2×C22⋊C4—671no—
144C2×C22⋊C4—672no—
145C2×C22⋊C4—671no—
146C2×C22⋊C4—674no—
147C2×C42—671no—
148#95—674no—
149#95—672no—
150#95—674no—
151C24—671no—
152C24C22×D6711yescharacteristic
153C2×D4—674no—
154C2×D4—1342no—
155C2×D4—1342no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—671no—
161C22×C4—671no—
162C22×C4—671no—
163C22×C4—674no—
164C22×C4—1342no—
165C22×C4—1344no—
166C22×C4C22⋊D6714yes—
167C22×C4C22×D6714yes—
168C22×C4C22×D6711yescharacteristic
169C22×C4C22×D6711yescharacteristic
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—1348no—
173C4⋊C4—1342no—
174C4⋊C4—1344no—
175C22⋊C4—1348no—
176C22⋊C4—1344no—
177C22⋊C4—1342no—
178C22⋊C4—1342no—
179C42—1342no—
180C23—22no—
181C23—22no—
182C23—672no—
183C23—1341no—
184C23—1341no—
185C23—1342no—
186C23—1342no—
187C23D4×D6712yes—
188C23C2×C22⋊D6712yes—
189C23#1176162712yes—
190C23C23×D6711yescharacteristic
191D4—1348no—
192C2×C4—22no—
193C2×C4—28no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—48no—
198C2×C4—2684no—
199C2×C4—672no—
200C2×C4—1344no—
201C2×C4—1341no—
202C2×C4—1341no—
203C2×C4—1344no—
204C2×C4—1342no—
205C2×C4—1344no—
206C2×C4—1342no—
207C2×C4—1341no—
208C2×C4—1341no—
209C2×C4C2×C22⋊D6712yes—
210C2×C4C2×C22⋊D6712yes—
211C2×C4#1176162814yes—
212C22—24no—
213C22—24no—
214C22—22no—
215C22—22no—
216C22—2682no—
217C22—2681no—
218C22—2681no—
219C22—1344no—
220C22—1342no—
221C22#1452401414yes—
222C22C2×D4×D6711yescharacteristic
223C22C22×C22⋊D6711yescharacteristic
224C22C2×#1176162711yescharacteristic
225C22C2×#1176162812yes—
226C22#1452409112yes—
227C4—22no—
228C4—44no—
229C4—2684no—
230C4—2681no—
231C4—2681no—
232C4—1342no—
233C4#1452402614yescomplemented
234C2—24no—
235C2—2682no—
236C2C2×#1452401411yescharacteristic
237C2C2×#1452402611yescharacteristic
238C2#1686397011yescharacteristic
239C2#1686404712yes—
240C2#1686408112yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC22×C4⋊C4×C67C2
hypercenterC22×C4⋊C4D67
hypocenterC67#2486
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #16863327C2semidirect
4. #16863322C2semidirect
5. #16863292C2semidirect
6. #16863259C2semidirect
7. #16863256C2semidirect
8. #16863253C2semidirect
9. C22×C4⋊C4×C67C2semidirect
108. C4×C67#123semidirect
121. C67#2486semidirect
124. C22×C4⋊C4D67semidirect
233. C4#14524026semidirect