Finite Group Databasebeta

← #22566014

Subgroups — #22566014

Order 8576.

668 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14524022C211yesmaximal, characteristic, complemented
2#16863332C212yesmaximal, complemented
3#16863319C214yesmaximal, complemented
4#16863308C212yesmaximal, complemented
5#16863269C212yesmaximal, complemented
6#16863256C211yesmaximal, characteristic, complemented
7#16863252C212yesmaximal
8C22×C22⋊C4×C67C211yesmaximal, characteristic
9C22×Dic134C2211yescharacteristic, complemented
10#14524022—24no—
11C2×C4⋊Dic67C2211yescharacteristic, complemented
12C2×C4⋊Dic67C2212yescomplemented
13C2×Dic67⋊C4C2212yescomplemented
14C2×Dic67⋊C4C2214yescomplemented
15C2×C22⋊Dic67C2212yes—
16C2×C22⋊Dic67C2212yes—
17C2×C22⋊Dic67C2214yes—
18C2×C22⋊Dic67C2214yes—
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—
27C2×Dic134—22no—
28C2×Dic134—42no—
29C22×Dic67C2314yes—
30C22×Dic67C2312yes—
31C22×Dic67C2312yes—
32C4⋊Dic67—22no—
33C4⋊Dic67—42no—
34Dic67⋊C4—24no—
35Dic67⋊C4—44no—
36C22⋊Dic67—24no—
37C22⋊Dic67—24no—
38C22⋊Dic67—44no—
39C22⋊Dic67—44no—
40C24×C67—22no—
41C24×C67—22no—
42C24×C67—42no—
43C24×C67—42no—
44C24×C67Q814yescomplemented
45C24×C67C2312yes—
46C24×C67C2311yescharacteristic
47C22×C4×C67—22no—
48C22×C4×C67—42no—
49C22×C4×C67—42no—
50C22×C4×C67D414yes—
51C22×C4×C67C2312yes—
52C22×C4×C67C2311yescharacteristic
53C22×C4×C67C2311yescharacteristic
54C22⋊C4×C67—24no—
55C22⋊C4×C67—24no—
56C23×C67—22no—
57C23×C67—22no—
58C23×C67—28no—
59C23×C67—22no—
60C23×C67—22no—
61C23×C67—24no—
62C23×C67—24no—
63C23×C67—48no—
64C23×C67—44no—
65C23×C67—44no—
66C23×C67—48no—
67C23×C67C2×D412yes—
68C23×C67C2×Q814yes—
69C23×C67C2×Q812yes—
70C23×C67#4112yes—
71C23×C67C2411yescharacteristic
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—24no—
76C2×C4×C67—44no—
77C2×C4×C67—44no—
78C2×C4×C67C2×D412yes—
79C2×C4×C67C2×D412yes—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—24no—
83C2×Dic67—24no—
84C2×Dic67—24no—
85C2×Dic67—24no—
86C2×Dic67—24no—
87Dic134—44no—
88C22×C67—22no—
89C22×C67—22no—
90C22×C67—24no—
91C22×C67—24no—
92C22×C67—24no—
93C22×C67—28no—
94C22×C67—416no—
95C22×C67—42no—
96C22×C67—42no—
97C22×C67—44no—
98C22×C67C22⋊Q814yes—
99C22×C67C22×D411yescharacteristic
100C22×C67C22×Q811yescharacteristic
101C22×C67C2×#4111yescharacteristic
102C22×C67H(5,𝔽2)12yes—
103C22×C67H(5,𝔽2)12yes—
104C4×C67—22no—
105C4×C67—22no—
106C4×C67—42no—
107Dic67—42no—
108Dic67—42no—
109Dic67—44no—
110C2×C67—24no—
111C2×C67—44no—
112C2×C67C2×C22⋊Q811yescharacteristic
113C2×C67#53411yescharacteristic
114C2×C67#53511yescharacteristic
115C2×C67#53812yes—
116C2×C67#54312yes—
117#2626—671nomaximal, Sylow
118C67#262611yescharacteristic, complemented, Sylow
119C2×C22⋊Q8—671no—
120C22×C22⋊C4D6711yescharacteristic
121#399—672no—
122#396—674no—
123#393—672no—
124#385—672no—
125#380—671no—
126#379—672no—
127C25C2×D6711yescharacteristic
128C22×Q8—671no—
129C23×C4C2×D6711yescharacteristic
130C23×C4C2×D6711yescharacteristic
131C22⋊Q8—1344no—
132C2×C4⋊C4—674no—
133C2×C4⋊C4—672no—
134C2×C4⋊C4—672no—
135C2×C4⋊C4—671no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—674no—
138C2×C22⋊C4—674no—
139C2×C22⋊C4—672no—
140C2×C22⋊C4—672no—
141C2×C22⋊C4C2×D6712yes—
142C2×C22⋊C4C2×D6712yes—
143#95—672no—
144#95—674no—
145C24—22no—
146C24—22no—
147C24—42no—
148C24—42no—
149C24Dic13414yescomplemented
150C24C22×D6712yes—
151C24C22×D6711yescharacteristic
152C2×Q8—2682no—
153C2×Q8—1342no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—674no—
158C22×C4—672no—
159C22×C4—672no—
160C22×C4C22⋊D6714yes—
161C22×C4C22×D6712yes—
162C22×C4C22×D6711yescharacteristic
163C22×C4C22×D6711yescharacteristic
164C4⋊C4—2684no—
165C4⋊C4—2682no—
166C4⋊C4—1344no—
167C4⋊C4—1342no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—2684no—
171C22⋊C4—2684no—
172C22⋊C4—1344no—
173C22⋊C4—1344no—
174C23—22no—
175C23—22no—
176C23—28no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23—48no—
182C23—44no—
183C23—44no—
184C23—48no—
185C23C2×C22⋊D6712yes—
186C23C2×Dic13414yes—
187C23C2×Dic13412yes—
188C23#1176162812yes—
189C23C23×D6711yescharacteristic
190Q8—2684no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—1342no—
198C2×C4—1342no—
199C2×C4—1344no—
200C2×C4—1344no—
201C2×C4—1344no—
202C2×C4—1344no—
203C2×C4—1344no—
204C2×C4C2×C22⋊D6712yes—
205C2×C4C2×C22⋊D6712yes—
206C22—24no—
207C22—22no—
208C22—22no—
209C22—24no—
210C22—28no—
211C22—24no—
212C22—416no—
213C22—42no—
214C22—42no—
215C22—44no—
216C22#1452402214yes—
217C22C22×C22⋊D6711yescharacteristic
218C22C22×Dic13411yescharacteristic
219C22C2×#1176162811yescharacteristic
220C22#1452408812yes—
221C22#1452408812yes—
222C4—22no—
223C4—22no—
224C4—42no—
225C4—2682no—
226C4—2682no—
227C4—2684no—
228C2—24no—
229C2—44no—
230C2C2×#1452402211yescharacteristic
231C2#1686395511yescharacteristic
232C2#1686396311yescharacteristic
233C2#1686398412yes—
234C2#1686402212yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC22×C22⋊C4×C67C2
hypercenterC22×C22⋊C4D67
hypocenterC67#2626
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×#14524022C2semidirect
2. #16863332C2semidirect
3. #16863319C2semidirect
4. #16863308C2semidirect
5. #16863269C2semidirect
6. #16863256C2semidirect
9. C22×Dic134C22semidirect
11. C2×C4⋊Dic67C22semidirect
12. C2×C4⋊Dic67C22semidirect
13. C2×Dic67⋊C4C22semidirect
14. C2×Dic67⋊C4C22semidirect
19. C67⋊#95C22semidirect
20. C67⋊#95C22semidirect
44. C24×C67Q8semidirect
118. C67#2626semidirect
149. C24Dic134semidirect