Finite Group Databasebeta

← #22564483

Subgroups — #22564483

Order 8576.

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

normal subgroups only

This group has 672 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×#14524002C211yesmaximal, characteristic, complemented
2#16863332C212yesmaximal, complemented
3#16863321C212yesmaximal, complemented
4#16863284C214yesmaximal, complemented
5#16863269C212yesmaximal, complemented
6#16863259C211yesmaximal, characteristic, complemented
7#16863252C212yesmaximal
8C22×C22⋊C4×C67C211yesmaximal, characteristic
9#14524002—24no—
10C2×C4⋊Dic67C2211yescharacteristic, complemented
11C2×C4⋊Dic67C2211yescharacteristic, complemented
12C2×Dic67⋊C4C2214yescomplemented
13C2×C22⋊Dic67C2212yes—
14C2×C22⋊Dic67C2212yes—
15C2×C22⋊Dic67C2214yes—
16C2×C22⋊Dic67C2214yes—
17C2×C4×Dic67C2212yescomplemented
18C67⋊#95C2212yescomplemented
19C67⋊#95C2212yescomplemented
20C67⋊#95C2214yescomplemented
21C25×C67C2211yescharacteristic
22C23×C4×C67C2211yescharacteristic
23C23×C4×C67C2211yescharacteristic
24C2×C22⋊C4×C67—24no—
25C2×C22⋊C4×C67C2212yes—
26C2×C22⋊C4×C67C2212yes—
27C22×Dic67C2314yes—
28C22×Dic67C2312yes—
29C22×Dic67C2312yes—
30C4⋊Dic67—22no—
31C4⋊Dic67—22no—
32Dic67⋊C4—44no—
33C22⋊Dic67—24no—
34C22⋊Dic67—24no—
35C22⋊Dic67—28no—
36C22⋊Dic67—44no—
37C4×Dic67—24no—
38C24×C67—22no—
39C24×C67—24no—
40C24×C67—24no—
41C24×C67—42no—
42C24×C67C2312yes—
43C24×C67C2311yescharacteristic
44C22×C4×C67—24no—
45C22×C4×C67—22no—
46C22×C4×C67—42no—
47C22×C4×C67D414yes—
48C22×C4×C67C2312yes—
49C22×C4×C67C2311yescharacteristic
50C22×C4×C67C2311yescharacteristic
51C22⋊C4×C67—24no—
52C22⋊C4×C67—24no—
53C23×C67—22no—
54C23×C67—22no—
55C23×C67—28no—
56C23×C67—22no—
57C23×C67—24no—
58C23×C67—22no—
59C23×C67—24no—
60C23×C67—416no—
61C23×C67—44no—
62C23×C67—44no—
63C23×C67C2×D412yes—
64C23×C67#4114yes—
65C23×C67#4114yes—
66C23×C67C2411yescharacteristic
67C2×C4×C67—28no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—24no—
72C2×C4×C67—44no—
73C2×C4×C67C2×D412yes—
74C2×C4×C67C2×D412yes—
75C2×Dic67—22no—
76C2×Dic67—22no—
77C2×Dic67—24no—
78C2×Dic67—24no—
79C2×Dic67—24no—
80C2×Dic67—24no—
81C2×Dic67—24no—
82C22×C67—22no—
83C22×C67—22no—
84C22×C67—28no—
85C22×C67—24no—
86C22×C67—28no—
87C22×C67—416no—
88C22×C67—42no—
89C22×C67—42no—
90C22×C67—44no—
91C22×C67#12314yes—
92C22×C67C22×D411yescharacteristic
93C22×C67C2×#4112yes—
94C22×C67C2×#4112yes—
95C22×C67H(5,𝔽2)12yes—
96C4×C67—22no—
97C4×C67—22no—
98C4×C67—42no—
99Dic67—42no—
100Dic67—42no—
101Dic67—44no—
102C2×C67—24no—
103C2×C67—44no—
104C2×C67C2×#12311yescharacteristic
105C2×C67#52511yescharacteristic
106C2×C67#53511yescharacteristic
107C2×C67#55114yes—
108#2482—671nomaximal, Sylow
109C67#248211yescharacteristic, complemented, Sylow
110C2×#117—671no—
111C22×C22⋊C4D6711yescharacteristic
112#399—672no—
113#397—672no—
114#388—674no—
115#385—672no—
116#381—671no—
117#379—672no—
118C25C2×D6711yescharacteristic
119C23×C4C2×D6711yescharacteristic
120C23×C4C2×D6711yescharacteristic
121#117—1344no—
122C2×C4⋊C4—674no—
123C2×C4⋊C4—671no—
124C2×C4⋊C4—671no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—674no—
127C2×C22⋊C4—672no—
128C2×C22⋊C4—672no—
129C2×C22⋊C4—674no—
130C2×C22⋊C4C2×D6712yes—
131C2×C22⋊C4C2×D6712yes—
132C2×C42—672no—
133#95—672no—
134#95—672no—
135#95—674no—
136C24—22no—
137C24—24no—
138C24—24no—
139C24—42no—
140C24C22×D6712yes—
141C24C22×D6711yescharacteristic
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—674no—
146C22×C4—672no—
147C22×C4—672no—
148C22×C4C22⋊D6714yes—
149C22×C4C22×D6712yes—
150C22×C4C22×D6711yescharacteristic
151C22×C4C22×D6711yescharacteristic
152C4⋊C4—2684no—
153C4⋊C4—1342no—
154C4⋊C4—1342no—
155C22⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—2684no—
158C22⋊C4—1344no—
159C22⋊C4—1344no—
160C22⋊C4—1348no—
161C42—1344no—
162C23—22no—
163C23—22no—
164C23—28no—
165C23—22no—
166C23—24no—
167C23—22no—
168C23—24no—
169C23—416no—
170C23—44no—
171C23—44no—
172C23C2×C22⋊D6712yes—
173C23#1176162714yes—
174C23#1176162814yes—
175C23C23×D6711yescharacteristic
176C2×C4—28no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—44no—
182C2×C4—1342no—
183C2×C4—1342no—
184C2×C4—1344no—
185C2×C4—1344no—
186C2×C4—1344no—
187C2×C4—1344no—
188C2×C4—1344no—
189C2×C4C2×C22⋊D6712yes—
190C2×C4C2×C22⋊D6712yes—
191C22—28no—
192C22—22no—
193C22—22no—
194C22—28no—
195C22—24no—
196C22—416no—
197C22—44no—
198C22—42no—
199C22—42no—
200C22#1452402414yes—
201C22C22×C22⋊D6711yescharacteristic
202C22C2×#1176162712yes—
203C22C2×#1176162812yes—
204C22#1452408812yes—
205C4—22no—
206C4—22no—
207C4—42no—
208C4—2684no—
209C4—2682no—
210C4—2682no—
211C2—24no—
212C2—44no—
213C2C2×#1452402411yescharacteristic
214C2#1686390711yescharacteristic
215C2#1686396311yescharacteristic
216C2#1686407214yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC22×C22⋊C4×C67C2
hypercenterC22×C22⋊C4D67
hypocenterC67#2482
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14524002C2semidirect
2. #16863332C2semidirect
3. #16863321C2semidirect
4. #16863284C2semidirect
5. #16863269C2semidirect
6. #16863259C2semidirect
10. C2×C4⋊Dic67C22semidirect
11. C2×C4⋊Dic67C22semidirect
12. C2×Dic67⋊C4C22semidirect
17. C2×C4×Dic67C22semidirect
18. C67⋊#95C22semidirect
19. C67⋊#95C22semidirect
20. C67⋊#95C22semidirect
109. C67#2482semidirect