Finite Group Databasebeta

← C2×#1458 #17345

Subgroups — C2×#1458 #17345

Order 256.

695 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 271 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×#117C211yesmaximal, characteristic
2C2×#431C212yesmaximal, complemented
3#1458C218yesmaximal, direct product factor
4C2×#324C212yesmaximal, complemented
5C2×#323C212yesmaximal
6C24×C4C411yescharacteristic
7C2×#117—22no—
8C2×#117—24no—
9C2×#117—22no—
10C2×#117C414yes—
11C2×#117C2214yes—
12C22×C4⋊C4C411yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4C411yescharacteristic
15C22×C22⋊C4C2211yescharacteristic
16C22×C42C411yescharacteristic
17C22×C42C2211yescharacteristic
18#431—24no—
19#431C2218yescomplemented
20C2×C4⋊C8—22no—
21C2×C22⋊C8C2214yes—
22C2×C4×C8—22no—
23#324C2218yescomplemented
24#323C2218yes—
25C25C2×C411yescharacteristic
26C23×C4—22no—
27C23×C4—21no—
28C23×C4—21no—
29C23×C4—24no—
30C23×C4C818yescomplemented
31C23×C4C2×C411yescharacteristic
32C23×C4C2×C414yes—
33C23×C4C2×C411yescharacteristic
34C23×C4C2×C411yescharacteristic
35C23×C4C2×C411yescharacteristic
36C23×C4C2×C411yescharacteristic
37C23×C4C2311yescharacteristic
38C22×C8—24no—
39#117—28no—
40#117—28no—
41#117—28no—
42#117—28no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4C2×C414yes—
48C2×C4⋊C4C2×C412yes—
49C2×C4⋊C4C2312yes—
50C2×C22⋊C4—22no—
51C2×C22⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4C2×C414yes—
55C2×C22⋊C4C2×C412yes—
56C2×C22⋊C4C2312yes—
57C2×C42—21no—
58C2×C42—22no—
59C2×C42—21no—
60C2×C42C2×C414yes—
61C2×C42C2×C412yes—
62C2×C42D412yes—
63C2×C42D414yes—
64C2×C42D412yes—
65C2×C42C2312yes—
66C4⋊C8—28no—
67C22⋊C8—24no—
68C22⋊C8C2318yes—
69C4×C8—28no—
70C24—21no—
71C24—21no—
72C24—22no—
73C24—42no—
74C24—42no—
75C24C2×C814yes—
76C24C22×C411yescharacteristic
77C24C22×C412yes—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—216no—
82C22×C4—28no—
83C22×C4—28no—
84C22×C4—24no—
85C22×C4—24no—
86C22×C4—28no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4C22⋊C412yes—
96C22×C4C22⋊C412yes—
97C22×C4C22⋊C411yescharacteristic
98C22×C4C22⋊C412yes—
99C22×C4C22⋊C411yescharacteristic
100C22×C4C2×C814yes—
101C22×C4C2×C814yes—
102C22×C4C2×C8116yescomplemented
103C22×C4M1614yes—
104C22×C4C22×C412yes—
105C22×C4C22×C412yes—
106C22×C4C22×C412yes—
107C22×C4C22×C411yescharacteristic
108C22×C4C22×C412yes—
109C22×C4C22×C412yes—
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4C2411yescharacteristic
113C2×C8—28no—
114C2×C8—48no—
115C4⋊C4—24no—
116C4⋊C4—28no—
117C4⋊C4—24no—
118C22⋊C4—24no—
119C22⋊C4—28no—
120C22⋊C4—24no—
121C42—24no—
122C42—24no—
123C42C22⋊C414yes—
124C42C22⋊C414yes—
125C42C2×D414yes—
126C42C2×D414yes—
127C23—22no—
128C23—22no—
129C23—22no—
130C23—22no—
131C23—24no—
132C23—24no—
133C23—48no—
134C23—42no—
135C23—42no—
136C23—42no—
137C23—42no—
138C23—48no—
139C23—42no—
140C23—42no—
141C23C2×C22⋊C411yescharacteristic
142C23C2×C22⋊C411yescharacteristic
143C23C22×C812yes—
144C23C22×C814yes—
145C23C2×M1612yes—
146C23C23×C411yescharacteristic
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—216no—
154C2×C4—216no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4C22⋊C818yes—
169C2×C4C22⋊C814yes—
170C2×C4C22⋊C814yes—
171C2×C4C2×C22⋊C412yes—
172C2×C4C2×C22⋊C414yes—
173C2×C4C2×C22⋊C412yes—
174C2×C4C2×C22⋊C412yes—
175C2×C4C22×C814yes—
176C2×C4C22×C814yes—
177C2×C4C2×M1614yes—
178C2×C4C22×D412yes—
179C8—48no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—24no—
185C22—24no—
186C22—24no—
187C22—44no—
188C22—44no—
189C22—48no—
190C22—42no—
191C22—42no—
192C22—48no—
193C22C2×C22⋊C812yes—
194C22C2×C22⋊C812yes—
195C22#41012yes—
196C22#41312yes—
197C22C22×C22⋊C411yescharacteristic
198C22C23×C811yescharacteristic
199C22C22×M1611yescharacteristic
200C4—24no—
201C4—22no—
202C4—22no—
203C4—24no—
204C4—42no—
205C4—42no—
206C4C2×C22⋊C814yes—
207C4C2×C22⋊C814yes—
208C2—21no—
209C2—21no—
210C2—22no—
211C2—44no—
212C2#145814yesdirect product factor
213C2C22×C22⋊C811yescharacteristic
214C2C2×#41011yescharacteristic
215C2C2×#41311yescharacteristic
subgroupG/H
centerC22×C4C22⋊C4
commutatorC22C23×C8
Frattini, cosocleC22×C4C24
socleC23C2×C22⋊C4
prime radicalC25C2×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#431C2semidirect
3. #1458C2direct (N × H)
4. C2×#324C2semidirect
19. #431C22semidirect
23. #324C22semidirect
30. C23×C4C8semidirect
102. C22×C4C2×C8semidirect
212. C2#1458direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#1172
C2×#4312
#14582
C2×#3242
C2×#3232