Finite Group Databasebeta

← C2×#1510 #17551

Subgroups — C2×#1510 #17551

Order 256.

895 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 895 conjugacy classes of subgroups — too many to draw. Only its 199 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×#121C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic, complemented
3C2×#381C211yesmaximal, characteristic, complemented
4#1510C218yesmaximal, direct product factor
5C2×#327C214yesmaximal, complemented
6C23×D4—21no—
7C23×D4C411yescharacteristic, complemented
8C24×C4C2211yescharacteristic
9C22×M16—21no—
10C2×#121—21no—
11C2×#121—22no—
12C2×#121—21no—
13C2×#121—22no—
14C2×#121C412yescomplemented
15C2×#121C414yescomplemented
16C2×#121C2212yes—
17C2×#121C2214yes—
18C22×C4⋊C4C411yescharacteristic, complemented
19C22×C22⋊C4—21no—
20C22×C22⋊C4—21no—
21#407—22no—
22#407C2214yescomplemented
23C2×C22⋊C8—21no—
24C2×C22⋊C8C2212yescomplemented
25#381—22no—
26#381C2214yescomplemented
27C2×#95—21no—
28C2×#95C2212yescomplemented
29#327C22116yescomplemented
30C25—21no—
31C25D411yescharacteristic
32C22×D4—24no—
33C22×D4—22no—
34C22×D4—24no—
35C22×D4—22no—
36C22×D4—22no—
37C22×D4—21no—
38C22×D4—44no—
39C22×D4—48no—
40C22×D4C2×C412yes—
41C22×D4C2×C414yescomplemented
42C23×C4—21no—
43C23×C4—22no—
44C23×C4—44no—
45C23×C4C2×C411yescharacteristic
46C23×C4C2×C412yes—
47C23×C4C2×C411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4D412yes—
51C23×C4D411yescharacteristic
52C23×C4D412yes—
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C2×M16—24no—
56C2×M16—44no—
57C22×C8—22no—
58#121—24no—
59#121—24no—
60#121—24no—
61#121—22no—
62#121—24no—
63#121—22no—
64#121—24no—
65#121C2×C414yes—
66#121C2×C418yescomplemented
67#121C2314yes—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2×C412yes—
71C2×C4⋊C4C2×C414yescomplemented
72C2×C22⋊C4—24no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—44no—
78C22⋊C8—24no—
79C22⋊C8—22no—
80C22⋊C8C2314yescomplemented
81#95—24no—
82#95—22no—
83#95C2314yescomplemented
84C24—24no—
85C24—24no—
86C24—24no—
87C24—22no—
88C24—42no—
89C24—44no—
90C24—44no—
91C24—44no—
92C24C22⋊C411yescharacteristic
93C24C22⋊C412yes—
94C24C22⋊C411yescharacteristic
95C24C2×D412yes—
96C24C2×D411yescharacteristic
97C2×D4—24no—
98C2×D4—28no—
99C2×D4—24no—
100C2×D4—22no—
101C2×D4—24no—
102C2×D4—44no—
103C2×D4—48no—
104C2×D4—48no—
105C2×D4—44no—
106C2×D4—416no—
107C2×D4—416no—
108C2×D4—48no—
109C2×D4—48no—
110C2×D4C22×C414yes—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—44no—
120C22×C4—42no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4C22⋊C412yes—
129C22×C4C22⋊C412yes—
130C22×C4C22⋊C411yescharacteristic
131C22×C4C22⋊C411yescharacteristic
132C22×C4C22⋊C412yes—
133C22×C4C22⋊C412yes—
134C22×C4C22⋊C412yes—
135C22×C4C22×C412yes—
136C22×C4C22×C411yescharacteristic
137C22×C4C22×C411yescharacteristic
138C22×C4C22×C412yes—
139C22×C4C2×D412yes—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D412yes—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D412yes—
144C22×C4C2×D411yescharacteristic
145C22×C4C2411yescharacteristic
146M16—48no—
147C2×C8—24no—
148C2×C8—44no—
149C4⋊C4—22no—
150C4⋊C4—24no—
151C4⋊C4C22×C414yes—
152C22⋊C4—24no—
153C22⋊C4—28no—
154C22⋊C4—24no—
155C22⋊C4—48no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—24no—
161C23—24no—
162C23—24no—
163C23—22no—
164C23—24no—
165C23—22no—
166C23—22no—
167C23—24no—
168C23—42no—
169C23—42no—
170C23—416no—
171C23—48no—
172C23—48no—
173C23—48no—
174C23—44no—
175C23—44no—
176C23—44no—
177C23—816no—
178C23#9914yes—
179C23C2×C22⋊C412yes—
180C23C2×C22⋊C412yes—
181C23C2×C22⋊C411yescharacteristic
182C23C2×C22⋊C411yescharacteristic
183C23AGL(1,ℤ/8)14yes—
184C23C22×D411yescharacteristic
185D4—44no—
186D4—44no—
187D4—48no—
188D4—416no—
189D4—48no—
190D4—48no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—48no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—88no—
216C2×C4C2×C22⋊C411yescharacteristic
217C2×C4C2×C22⋊C412yes—
218C2×C4C2×C22⋊C411yescharacteristic
219C2×C4C2×C22⋊C412yes—
220C2×C4C23×C411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C8—44no—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—21no—
227C22—22no—
228C22—21no—
229C22—28no—
230C22—28no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—48no—
235C22—48no—
236C22—44no—
237C22—44no—
238C22—44no—
239C22—44no—
240C22—816no—
241C22—88no—
242C22C2×#9914yes—
243C22C2×#9912yes—
244C22#41812yes—
245C22#41812yes—
246C22#42112yes—
247C22C22×C22⋊C411yescharacteristic
248C22C2×AGL(1,ℤ/8)12yes—
249C4—21no—
250C4—21no—
251C4—22no—
252C4—21no—
253C4—22no—
254C4—21no—
255C4—44no—
256C4—84no—
257C2—22no—
258C2—24no—
259C2—22no—
260C2—42no—
261C2—84no—
262C2#151014yesdirect product factor
263C2C22×#9911yescharacteristic
264C2C2×#41811yescharacteristic
265C2C2×#42111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#121C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#407C2semidirect
3. C2×#381C2semidirect
4. #1510C2direct (N × H)
5. C2×#327C2semidirect
7. C23×D4C4semidirect
14. C2×#121C4semidirect
15. C2×#121C4semidirect
18. C22×C4⋊C4C4semidirect
22. #407C22semidirect
24. C2×C22⋊C8C22semidirect
26. #381C22semidirect
28. C2×#95C22semidirect
29. #327C22semidirect
41. C22×D4C2×C4semidirect
66. #121C2×C4semidirect
71. C2×C4⋊C4C2×C4semidirect
80. C22⋊C8C23semidirect
83. #95C23semidirect
262. C2#1510direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#1212
C2×#4072
C2×#3812
#15102
C2×#3272