Finite Group Databasebeta

← C2×#30632075 #44560798

Subgroups — C2×#30632075 #44560798

Order 32128.

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
1C22×#16860799C211yesmaximal, characteristic, complemented
2C22×#16860798C211yesmaximal, characteristic, complemented
3C22×#16860795C211yesmaximal, characteristic, complemented
4C2×#22548711C211yesmaximal, characteristic, complemented
5C2×#22548702C211yesmaximal, characteristic, complemented
6#30632075C218yesmaximal, direct product factor
7C2×#22548625C211yesmaximal, characteristic, complemented
8C22×D8×C251C211yesmaximal, characteristic, complemented
9C22×Dic502C2211yescharacteristic
10C22×D1004C2211yescharacteristic
11C22×C22⋊D251—22no—
12C2×#16860799—24no—
13C2×#16860799C2214yescomplemented
14C2×#16860798—24no—
15C2×#16860798C2214yescomplemented
16C2×#16860795—24no—
17C2×#16860795C2214yescomplemented
18C22×C251⋊C8C2211yescharacteristic
19#22548711—22no—
20#22548711C2214yescomplemented
21#22548702—22no—
22#22548702C2214yescomplemented
23C2×C22⋊Dic251—22no—
24C2×C4×Dic251C2211yescharacteristic
25#22548625C2214yescomplemented
26C22×D4×C251C2211yescharacteristic
27C22×D4×C251C2211yescharacteristic
28C2×D8×C251—24no—
29C2×D8×C251C2214yescomplemented
30C22×C8×C251C2211yescharacteristic
31C23×D251—21no—
32C2×Dic502—21no—
33C2×Dic502—22no—
34C2×Dic502—22no—
35C2×Dic502C2312yes—
36C2×D1004—22no—
37C2×D1004—22no—
38C2×D1004—21no—
39C2×D1004C2312yes—
40C2×C22⋊D251—28no—
41C2×C22⋊D251—48no—
42C22×Dic251—21no—
43C22×Dic251D412yes—
44#16860799—28no—
45#16860798—28no—
46#16860795—28no—
47C2×C251⋊C8D414yescomplemented
48C2×C251⋊C8C2312yes—
49C22⋊Dic251—28no—
50C4×Dic251D412yes—
51C4×Dic251C2312yes—
52C24×C251—21no—
53C24×C251—21no—
54C2×D4×C251—22no—
55C2×D4×C251—22no—
56C2×D4×C251—22no—
57C2×D4×C251—22no—
58C2×D4×C251—21no—
59C2×D4×C251—21no—
60C2×D4×C251C2312yes—
61C2×D4×C251C2312yes—
62C22×C4×C251C2311yescharacteristic
63D8×C251—28no—
64C2×C8×C251D414yescomplemented
65C2×C8×C251C2312yes—
66C22×D251—22no—
67C22×D251—42no—
68C22×D251—44no—
69Dic502—22no—
70Dic502—22no—
71Dic502—42no—
72D1004—22no—
73D1004—22no—
74D1004—42no—
75C22⋊D251—416no—
76C2×Dic251—24no—
77C2×Dic251—22no—
78C2×Dic251—42no—
79C2×Dic251C2×D414yes—
80C251⋊C8C2×D414yescomplemented
81C23×C251—22no—
82C23×C251—22no—
83C23×C251—42no—
84C23×C251—42no—
85C23×C251—44no—
86C23×C251—44no—
87C23×C251C2×D411yescharacteristic
88D4×C251—22no—
89D4×C251—22no—
90D4×C251—22no—
91D4×C251—22no—
92D4×C251—42no—
93D4×C251—42no—
94C2×C4×C251C2×D411yescharacteristic
95C2×C4×C251C2×D412yes—
96C2×C4×C251C2×D412yes—
97C2×C4×C251C2411yescharacteristic
98C8×C251C2×D414yescomplemented
99C22×C251—42no—
100C22×C251—42no—
101C22×C251—44no—
102C22×C251—44no—
103C22×C251—84no—
104C22×C251—84no—
105C22×C251#12712yes—
106C22×C251AGL(1,ℤ/8)14yes—
107C22×C251C22×D411yescharacteristic
108C2×D251—42no—
109C2×D251—44no—
110C2×D251—84no—
111C4×C251#12712yes—
112C4×C251C22×D411yescharacteristic
113C4×C251C22×D411yescharacteristic
114Dic251—24no—
115Dic251—42no—
116C2×C251—82no—
117C2×C251—82no—
118C2×C251#49614yescomplemented
119C2×C251C2×#12711yescharacteristic
120C2×C251C2×AGL(1,ℤ/8)12yes—
121D251—82no—
122C251C2×#49611yescharacteristic, complemented, Sylow
123C2×#496—2511nomaximal, Sylow
124C22×SD16—2511no—
125C22×SD16—2511no—
126C22×D8—2511no—
127C22×D8D25111yescharacteristic, complemented
128C2×#127—2511no—
129C2×#124—2511no—
130#496—2518no—
131C2×#97—2511no—
132C22×Q8—2511no—
133C22×D4—5022no—
134C22×D4—2511no—
135C22×D4C2×D25111yescharacteristic
136C22×D4C2×D25111yescharacteristic
137C2×SD16—5024no—
138C2×SD16—5024no—
139C2×SD16—2514no—
140C2×SD16—2514no—
141C2×D8—24no—
142C2×D8—5024no—
143C2×D8—2514no—
144C2×D8C2×D25114yescomplemented
145C22×C8—2511no—
146C22×C8C2×D25111yescharacteristic
147#127—5022no—
148#127—2514no—
149#124—5022no—
150#124—2514no—
151C2×C22⋊C4—5022no—
152C2×C42—2511no—
153#97—2514no—
154C24—21no—
155C24—21no—
156C24—5021no—
157C2×Q8—5022no—
158C2×Q8—5022no—
159C2×Q8—5021no—
160C2×Q8—2512no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—10048no—
168C2×D4—5028no—
169C2×D4—5021no—
170C2×D4—5022no—
171C2×D4—5022no—
172C2×D4—2512no—
173C2×D4C22×D25112yes—
174C2×D4C22×D25112yes—
175C22×C4—5021no—
176C22×C4—2512no—
177C22×C4C22×D25111yescharacteristic
178SD16—5028no—
179SD16—5028no—
180D8—28no—
181D8—5028no—
182C2×C8—2514no—
183C2×C8—2512no—
184C2×C8C22⋊D25114yescomplemented
185C2×C8C22×D25112yes—
186C22⋊C4—5028no—
187C42—2512no—
188C42—2512no—
189C23—22no—
190C23—22no—
191C23—44no—
192C23—44no—
193C23—42no—
194C23—42no—
195C23—10044no—
196C23—10042no—
197C23—5022no—
198C23D4×D25111yescharacteristic
199Q8—10042no—
200Q8—5022no—
201Q8—5022no—
202D4—22no—
203D4—22no—
204D4—22no—
205D4—22no—
206D4—42no—
207D4—42no—
208D4—100416no—
209D4—10042no—
210D4—5022no—
211D4—5022no—
212C2×C4—10042no—
213C2×C4—5022no—
214C2×C4—5024no—
215C2×C4—2514no—
216C2×C4D4×D25112yes—
217C2×C4D4×D25111yescharacteristic
218C2×C4C2×C22⋊D25112yes—
219C2×C4C23×D25111yescharacteristic
220C8—2514no—
221C8C2×C22⋊D25114yescomplemented
222C22—44no—
223C22—44no—
224C22—42no—
225C22—42no—
226C22—84no—
227C22—84no—
228C22—20084no—
229C22—10044no—
230C22—10042no—
231C22#2254871112yes—
232C22#2254873914yes—
233C22C2×D4×D25111yescharacteristic
234C4—10042no—
235C4—5024no—
236C4#2254871112yes—
237C4C2×D4×D25111yescharacteristic
238C4C22×C22⋊D25111yescharacteristic
239C2—82no—
240C2—82no—
241C2—20082no—
242C2#3063207514yesdirect product factor
243C2C2×#2254871111yescharacteristic
244C2C2×#2254873912yes—
subgroupG/H
centerC23D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×D8×C251C2
socleC23×C251C2×D4
hypercenterC22×D8D251
hypocenterC251C2×#496
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#16860799C2semidirect
2. C22×#16860798C2semidirect
3. C22×#16860795C2semidirect
4. C2×#22548711C2semidirect
5. C2×#22548702C2semidirect
6. #30632075C2direct (N × H)
7. C2×#22548625C2semidirect
8. C22×D8×C251C2semidirect
13. C2×#16860799C22semidirect
15. C2×#16860798C22semidirect
17. C2×#16860795C22semidirect
20. #22548711C22semidirect
22. #22548702C22semidirect
25. #22548625C22semidirect
29. C2×D8×C251C22semidirect
47. C2×C251⋊C8D4semidirect
64. C2×C8×C251D4semidirect
80. C251⋊C8C2×D4semidirect
98. C8×C251C2×D4semidirect
118. C2×C251#496semidirect
122. C251C2×#496semidirect
127. C22×D8D251semidirect
144. C2×D8C2×D251semidirect
184. C2×C8C22⋊D251semidirect
221. C8C2×C22⋊D251semidirect
242. C2#30632075direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168607992
C22×#168607982
C22×#168607952
C2×#225487112
C2×#225487022
#306320752
C2×#225486252
C22×D8×C2512
C2×#496251