Finite Group Databasebeta

← C22×#3504 #10585985

Subgroups — C22×#3504 #10585985

Order 512.

33898 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3504C2196yesmaximal, direct product factor
2C22×#557C214yesmaximal, complemented
3C22×#554C2112yesmaximal, complemented
4C22×#544C216yesmaximal, complemented
5C22×#533C213yesmaximal, complemented
6C23×#125C213yesmaximal, complemented
7C22×#520C212yesmaximal
8C22×C4×#41C211yesmaximal, characteristic
9C23×#41C2211yescharacteristic
10#3504C2211024yesdirect product factor
11C2×#557C221192yescomplemented
12C2×#554C221576yescomplemented
13C2×#544C221288yescomplemented
14C2×#533C221144yescomplemented
15C22×C4⋊Q8C2216yes—
16C22×C4⋊Q8C22112yes—
17C22×#125C2213yes—
18C22×#125C2216yes—
19C22×#125C22112yes—
20C22×#125C2211yescharacteristic
21C22×#125C22112yes—
22C22×#125C221144yescomplemented
23C22×C22⋊Q8C22112yes—
24C22×C22⋊Q8C22124yes—
25C2×#520C22196yes—
26C2×C4×#41C22148yes—
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2214yes—
29C22×C4×Q8C22112yes—
30C22×C4×D4C2216yes—
31C22×C4×D4C22112yes—
32C22×#117C2213yes—
33C22×#117C2216yes—
34C23×C4⋊C4C2216yes—
35C23×C4⋊C4C22112yes—
36C23×C42C2213yes—
37C22×#41Q8132yescomplemented
38C22×#41C23124yes—
39C23×Q8C23112yes—
40C23×Q8C2311yescharacteristic
41C23×D4C2313yes—
42C24×C4C2316yes—
43C24×C4C23112yes—
44C24×C4C2313yes—
45#557C2311024yescomplemented
46#554C2313072yescomplemented
47#544C2311536yescomplemented
48#533C231768yescomplemented
49C2×C4⋊Q8C231144yes—
50C2×C4⋊Q8C231288yes—
51C2×#125C23172yes—
52C2×#125C231144yes—
53C2×#125C231288yes—
54C2×#125C23124yes—
55C2×#125C231288yes—
56C2×#125C231768yescomplemented
57C2×C22⋊Q8C231288yes—
58C2×C22⋊Q8C231576yes—
59#520C231512yes—
60C4×#41C231256yes—
61C2×C4×Q8C23148yes—
62C2×C4×Q8C23196yes—
63C2×C4×Q8C231288yes—
64C2×C4×D4C231144yes—
65C2×C4×D4C231288yes—
66C2×#117C23172yes—
67C2×#117C231144yes—
68C22×C4⋊C4C2316yes—
69C22×C4⋊C4C23112yes—
70C22×C4⋊C4C23112yes—
71C22×C4⋊C4C231144yes—
72C22×C4⋊C4C23124yes—
73C22×C4⋊C4C231288yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2316yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C23112yes—
78C22×C22⋊C4C23112yes—
79C22×C22⋊C4C2316yes—
80C22×C42C2311yescharacteristic
81C22×C42C2313yes—
82C22×C42C2316yes—
83C22×C42C23172yes—
84C22×C42C23112yes—
85C25C2413yes—
86C2×#41C2×Q81384yescomplemented
87C2×#41C24164yes—
88C22×Q8—448no—
89C22×Q8C2×Q8116yes—
90C22×Q8C241144yes—
91C22×Q8C24112yes—
92C22×D4C2×Q8148yes—
93C22×D4C24136yes—
94C23×C4—424no—
95C23×C4—448no—
96C23×C4C2×Q8148yes—
97C23×C4C2414yes—
98C23×C4C2416yes—
99C23×C4C24112yes—
100C23×C4C2412yes—
101C23×C4C241144yes—
102C23×C4C24172yes—
103C23×C4C24136yes—
104C23×C4C2411yescharacteristic
105C23×C4C2413yes—
106C4⋊Q8C241384yes—
107C4⋊Q8C241768yes—
108#125C241192yes—
109#125C241384yes—
110#125C241768yes—
111#125C24164yes—
112#125C241768yes—
113C22⋊Q8C241768yes—
114C22⋊Q8C2411536yes—
115C4×Q8C241128yes—
116C4×Q8C241256yes—
117C4×Q8C241768yes—
118C4×D4C241384yes—
119C4×D4C241768yes—
120#117C241192yes—
121#117C241384yes—
122C2×C4⋊C4C24172yes—
123C2×C4⋊C4C241144yes—
124C2×C4⋊C4C241144yes—
125C2×C4⋊C4C241384yes—
126C2×C4⋊C4C241288yes—
127C2×C4⋊C4C241768yes—
128C2×C4⋊C4C24124yes—
129C2×C4⋊C4C24172yes—
130C2×C4⋊C4C24148yes—
131C2×C4⋊C4C241144yes—
132C2×C22⋊C4C241144yes—
133C2×C22⋊C4C24172yes—
134C2×C42C24136yes—
135C2×C42C241144yes—
136C2×C42C24172yes—
137C2×C42C241192yes—
138C2×C42C24112yes—
139C24—224no—
140C24C22×Q8124yes—
141C24C25118yes—
142C24C2511yescharacteristic
143#41C22×Q81512yescomplemented
144C2×Q8—4288no—
145C2×Q8C22×Q8196yes—
146C2×Q8C251192yes—
147C2×Q8C25116yes—
148C2×D4C22×Q81288yes—
149C2×D4C25148yes—
150C22×C4—28no—
151C22×C4—248no—
152C22×C4—224no—
153C22×C4—216no—
154C22×C4—4144no—
155C22×C4—4288no—
156C22×C4C22×Q818yes—
157C22×C4C22×Q8124yes—
158C22×C4C22×Q81288yes—
159C22×C4C25112yes—
160C22×C4C25124yes—
161C22×C4C25136yes—
162C22×C4C25172yes—
163C22×C4C25196yes—
164C22×C4C251192yes—
165C22×C4C25118yes—
166C22×C4C2516yes—
167C22×C4C25148yes—
168C4⋊C4C25196yes—
169C4⋊C4C251192yes—
170C4⋊C4C251192yes—
171C4⋊C4C251384yes—
172C4⋊C4C25132yes—
173C4⋊C4C25196yes—
174C4⋊C4C25164yes—
175C4⋊C4C251192yes—
176C22⋊C4C251192yes—
177C22⋊C4C25196yes—
178C42C25116yes—
179C42C25148yes—
180C42C25196yes—
181C42C251192yes—
182C23—272no—
183C23—296no—
184C23C23×Q8172yes—
185C23C23×Q814yes—
186C23#58518yes—
187C23C26112yes—
188C23C2613yes—
189Q8—4192no—
190Q8C23×Q8164yes—
191D4C23×Q81192yes—
192C2×C4—224no—
193C2×C4—2144no—
194C2×C4—272no—
195C2×C4—248no—
196C2×C4—496no—
197C2×C4—4192no—
198C2×C4C23×Q8124yes—
199C2×C4C23×Q8172yes—
200C2×C4C23×Q81192yes—
201C2×C4C2618yes—
202C2×C4C26116yes—
203C2×C4C26124yes—
204C2×C4C26148yes—
205C2×C4C26112yes—
206C2×C4C2614yes—
207C22—2144no—
208C22—224no—
209C22#3504116yesdirect product factor
210C22C24×Q8124yes—
211C22C24×Q816yes—
212C22C2×#585112yes—
213C22C2711yescharacteristic
214C4—28no—
215C4—248no—
216C4—224no—
217C4—216no—
218C4C24×Q818yes—
219C4C24×Q8124yes—
220C2—224no—
221C2C2×#3504112yesdirect product factor
222C2C25×Q811yescharacteristic
223C2C22×#58512yes—
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC22C27
prime radicalC23×#41C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3504C2direct (N × H)
2. C22×#557C2semidirect
3. C22×#554C2semidirect
4. C22×#544C2semidirect
5. C22×#533C2semidirect
6. C23×#125C2semidirect
10. #3504C22direct (N × H)
11. C2×#557C22semidirect
12. C2×#554C22semidirect
13. C2×#544C22semidirect
14. C2×#533C22semidirect
22. C22×#125C22semidirect
37. C22×#41Q8semidirect
45. #557C23semidirect
46. #554C23semidirect
47. #544C23semidirect
48. #533C23semidirect
56. C2×#125C23semidirect
86. C2×#41C2×Q8semidirect
143. #41C22×Q8semidirect
209. C22#3504direct (N × H)
221. C2C2×#3504direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#35042
C22×#5572
C22×#5542
C22×#5442
C22×#5332
C23×#1252
C22×#5202
C22×C4×#412