Finite Group Databasebeta

← #22565979

Subgroups — #22565979

Order 8576.

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

normal subgroups only

This group has 528 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×#14524041C211yesmaximal, characteristic, complemented
2C2×#14524022C211yesmaximal, characteristic
3C2×#14524002C211yesmaximal, characteristic
4#16863329C214yesmaximal, complemented
5#16863324C214yesmaximal
6#16863303C211yesmaximal, characteristic, complemented
7#16863301C211yesmaximal, characteristic, complemented
8#16863261C211yesmaximal, characteristic, complemented
9C2×C22⋊Q8×C67C211yesmaximal, characteristic
10C22×Dic134C2211yescharacteristic
11#14524041—24no—
12#14524022—24no—
13#14524002—24no—
14C2×C4⋊Dic67C2211yescharacteristic
15C2×C4⋊Dic67C2212yes—
16C2×Dic67⋊C4C2214yes—
17C2×Dic67⋊C4C2212yes—
18C2×C22⋊Dic67C2212yes—
19C2×C22⋊Dic67C2212yes—
20C2×C4×Dic67C2211yescharacteristic
21C2×C4×Dic67C2211yescharacteristic
22C67⋊#95C2214yes—
23C67⋊#95C2214yes—
24C67⋊#95C2214yes—
25C22×Q8×C67C2211yescharacteristic
26C23×C4×C67C2211yescharacteristic
27C22⋊Q8×C67—24no—
28C2×C4⋊C4×C67C2211yescharacteristic
29C2×C4⋊C4×C67C2212yes—
30C2×C22⋊C4×C67C2212yes—
31C2×Dic134—22no—
32C2×Dic134—22no—
33C2×Dic134—22no—
34C22×Dic67C2312yes—
35C22×Dic67C2312yes—
36C22×Dic67C2312yes—
37C22×Dic67C2312yes—
38C4⋊Dic67—22no—
39C4⋊Dic67—24no—
40Dic67⋊C4—24no—
41Dic67⋊C4—28no—
42C22⋊Dic67—24no—
43C22⋊Dic67—24no—
44C4×Dic67—22no—
45C4×Dic67D414yes—
46C24×C67C2311yescharacteristic
47C2×Q8×C67—22no—
48C2×Q8×C67—22no—
49C2×Q8×C67—22no—
50C22×C4×C67—22no—
51C22×C4×C67—22no—
52C22×C4×C67D414yes—
53C22×C4×C67C2312yes—
54C22×C4×C67C2312yes—
55C22×C4×C67C2311yescharacteristic
56C22×C4×C67C2311yescharacteristic
57C4⋊C4×C67—22no—
58C4⋊C4×C67—24no—
59C22⋊C4×C67—24no—
60C23×C67—21no—
61C23×C67—21no—
62C23×C67—22no—
63C23×C67—22no—
64C23×C67C2×D412yes—
65C23×C67C2411yescharacteristic
66Q8×C67—44no—
67C2×C4×C67—21no—
68C2×C4×C67—21no—
69C2×C4×C67—28no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—24no—
75C2×C4×C67C2×D412yes—
76C2×C4×C67C2×D412yes—
77C2×C4×C67C2×D412yes—
78C2×C4×C67#4112yes—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—22no—
86C2×Dic67—22no—
87C2×Dic67—22no—
88C2×Dic67—24no—
89C2×Dic67C2×D414yes—
90Dic134—44no—
91C22×C67—22no—
92C22×C67—22no—
93C22×C67—22no—
94C22×C67—42no—
95C22×C67—42no—
96C22×C67C22×D411yescharacteristic
97C22×C67C22×D411yescharacteristic
98C22×C67C2×#4111yescharacteristic
99C22×C672−1+412yes—
100C22×C672−1+412yes—
101C4×C67—22no—
102C4×C67—42no—
103C4×C67—42no—
104C4×C67#12114yes—
105Dic67—24no—
106Dic67—42no—
107Dic67—42no—
108Dic67—42no—
109C2×C67—42no—
110C2×C67C2×#12111yescharacteristic
111C2×C67#53611yescharacteristic
112C2×C67#53612yes—
113C2×C67#53611yescharacteristic
114C2×C67#54112yes—
115#2621—671nomaximal, Sylow
116C67#262111yescharacteristic, complemented, Sylow
117C2×C4⋊Q8—671no—
118C2×C22⋊Q8—671no—
119C2×C22⋊Q8D6711yescharacteristic
120C2×#117—671no—
121#398—674no—
122#397—674no—
123#391—671no—
124#391—671no—
125#382—671no—
126C22×Q8—671no—
127C22×Q8C2×D6711yescharacteristic
128C23×C4C2×D6711yescharacteristic
129C4⋊Q8—1344no—
130C22⋊Q8—24no—
131C22⋊Q8—1344no—
132#117—1344no—
133C2×C4⋊C4—672no—
134C2×C4⋊C4—672no—
135C2×C4⋊C4—671no—
136C2×C4⋊C4—674no—
137C2×C4⋊C4C2×D6711yescharacteristic
138C2×C4⋊C4C2×D6712yes—
139C2×C22⋊C4—672no—
140C2×C22⋊C4—672no—
141C2×C22⋊C4C2×D6712yes—
142C2×C42—671no—
143C2×C42—671no—
144#95—674no—
145#95—674no—
146#95—674no—
147C24C22×D6711yescharacteristic
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—1342no—
152C2×Q8—1342no—
153C2×Q8—1342no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—672no—
157C22×C4—672no—
158C22×C4—672no—
159C22×C4—672no—
160C22×C4C22⋊D6714yes—
161C22×C4C22×D6712yes—
162C22×C4C22×D6712yes—
163C22×C4C22×D6711yescharacteristic
164C22×C4C22×D6711yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—24no—
167C4⋊C4—1344no—
168C4⋊C4—1348no—
169C4⋊C4—1344no—
170C4⋊C4—1342no—
171C22⋊C4—24no—
172C22⋊C4—1344no—
173C22⋊C4—1344no—
174C42—674no—
175C42—1342no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23—22no—
180C23C2×C22⋊D6712yes—
181C23C23×D6711yescharacteristic
182Q8—44no—
183Q8—2684no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—28no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—674no—
193C2×C4—1342no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4—1342no—
197C2×C4—1342no—
198C2×C4—1342no—
199C2×C4—1342no—
200C2×C4—1344no—
201C2×C4—1342no—
202C2×C4—1342no—
203C2×C4D4×D6712yes—
204C2×C4C2×C22⋊D6712yes—
205C2×C4C2×C22⋊D6712yes—
206C2×C4#1176162712yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C2×D4×D6711yescharacteristic
213C22C22×C22⋊D6711yescharacteristic
214C22C2×#1176162711yescharacteristic
215C22#1452409012yes—
216C22#1452409112yes—
217C4—22no—
218C4—42no—
219C4—42no—
220C4—2682no—
221C4—2682no—
222C4—2682no—
223C4—1344no—
224C4#1452401514yes—
225C2—42no—
226C2C2×#1452401511yescharacteristic
227C2#1686396711yescharacteristic
228C2#1686396912yes—
229C2#1686397011yescharacteristic
230C2#1686400612yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×C22⋊Q8×C67C2
hypercenterC2×C22⋊Q8D67
hypocenterC67#2621
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14524041C2semidirect
4. #16863329C2semidirect
6. #16863303C2semidirect
7. #16863301C2semidirect
8. #16863261C2semidirect
116. C67#2621semidirect