Finite Group Databasebeta

← C2×#1459 #17346

Subgroups — C2×#1459 #17346

Order 256.

913 conjugacy classes of subgroups; only the 279 normal subgroups are shown.

normal subgroups only

This group has 913 conjugacy classes of subgroups — too many to draw. Only its 279 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×C4×D4C211yesmaximal, characteristic
2C22×C22⋊C8C212yesmaximal, complemented
3#1459C218yesmaximal, direct product factor
4C2×#324C212yesmaximal, complemented
5C2×#323C212yesmaximal, complemented
6C23×D4C411yescharacteristic
7C24×C4C411yescharacteristic
8C24×C4C2211yescharacteristic
9C23×C8—22no—
10C2×C4×D4—22no—
11C2×C4×D4—24no—
12C2×C4×D4—22no—
13C2×C4×D4C414yes—
14C2×C4×D4C2214yes—
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4C411yescharacteristic
17C22×C22⋊C4C2211yescharacteristic
18C22×C42C411yescharacteristic
19C2×C22⋊C8—24no—
20C2×C22⋊C8—22no—
21C2×C22⋊C8C2214yes—
22C2×C22⋊C8C2218yescomplemented
23#324C2218yescomplemented
24#323C2218yescomplemented
25C25C2×C411yescharacteristic
26C25C2×C411yescharacteristic
27C22×D4—22no—
28C22×D4—22no—
29C22×D4—24no—
30C22×D4C818yescomplemented
31C22×D4C2×C414yes—
32C23×C4—22no—
33C23×C4—21no—
34C23×C4—21no—
35C23×C4—44no—
36C23×C4—44no—
37C23×C4C2×C411yescharacteristic
38C23×C4C2×C411yescharacteristic
39C23×C4C2×C411yescharacteristic
40C23×C4C2×C411yescharacteristic
41C23×C4C2×C414yes—
42C23×C4C2×C412yes—
43C23×C4D412yes—
44C23×C4D414yes—
45C23×C4D412yes—
46C23×C4C2311yescharacteristic
47C23×C4C2312yes—
48C22×C8—24no—
49C22×C8—28no—
50C22×C8—48no—
51C4×D4—28no—
52C4×D4—28no—
53C4×D4—28no—
54C4×D4—28no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4C2×C412yes—
58C2×C4⋊C4C2312yes—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4C2×C414yes—
64C2×C22⋊C4C2×C412yes—
65C2×C22⋊C4C2312yes—
66C2×C42—21no—
67C2×C42—22no—
68C2×C42—21no—
69C2×C42C2×C414yes—
70C22⋊C8—24no—
71C22⋊C8—28no—
72C22⋊C8C2318yes—
73C24—21no—
74C24—21no—
75C24—22no—
76C24—42no—
77C24—42no—
78C24—42no—
79C24—42no—
80C24C22⋊C412yes—
81C24C22⋊C412yes—
82C24C2×C814yes—
83C24C2×C814yescomplemented
84C24M1614yes—
85C24C22×C411yescharacteristic
86C24C22×C412yes—
87C24C22×C412yes—
88C2×D4—216no—
89C2×D4—216no—
90C2×D4—28no—
91C2×D4—28no—
92C2×D4C2×C8116yescomplemented
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—28no—
97C22×C4—24no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—44no—
105C22×C4—48no—
106C22×C4—44no—
107C22×C4—44no—
108C22×C4—42no—
109C22×C4—44no—
110C22×C4—48no—
111C22×C4—48no—
112C22×C4—44no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4C22⋊C411yescharacteristic
116C22×C4C22⋊C412yes—
117C22×C4C22⋊C414yes—
118C22×C4C22⋊C414yes—
119C22×C4C22⋊C411yescharacteristic
120C22×C4C2×C814yescomplemented
121C22×C4C22×C412yes—
122C22×C4C22×C412yes—
123C22×C4C22×C412yes—
124C22×C4C22×C412yes—
125C22×C4C22×C411yescharacteristic
126C22×C4C2×D412yes—
127C22×C4C2×D414yes—
128C22×C4C2×D412yes—
129C22×C4C2×D414yes—
130C22×C4C2411yescharacteristic
131C2×C8—28no—
132C2×C8—48no—
133C2×C8—416no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C22⋊C4—28no—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C42—24no—
140C42—24no—
141C23—22no—
142C23—24no—
143C23—22no—
144C23—24no—
145C23—22no—
146C23—22no—
147C23—24no—
148C23—24no—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—48no—
153C23—48no—
154C23—42no—
155C23—42no—
156C23—48no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—48no—
161C23—42no—
162C23—42no—
163C23C22⋊C818yes—
164C23C22⋊C818yes—
165C23C2×C22⋊C414yes—
166C23C2×C22⋊C411yescharacteristic
167C23C2×C22⋊C411yescharacteristic
168C23C22×C812yes—
169C23C22×C814yes—
170C23C22×C814yescomplemented
171C23C2×M1612yes—
172C23C2×M1614yes—
173C23C23×C411yescharacteristic
174D4—232no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—28no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—44no—
188C2×C4—41no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—48no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—48no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4#9914yes—
201C2×C4C2×C22⋊C412yes—
202C2×C4C2×C22⋊C412yes—
203C2×C4C2×C22⋊C412yes—
204C2×C4C22×C814yescomplemented
205C2×C4C22×D412yes—
206C8—48no—
207C22—21no—
208C22—21no—
209C22—24no—
210C22—21no—
211C22—21no—
212C22—24no—
213C22—24no—
214C22—24no—
215C22—216no—
216C22—24no—
217C22—48no—
218C22—44no—
219C22—44no—
220C22—44no—
221C22—44no—
222C22—48no—
223C22—42no—
224C22—42no—
225C22—48no—
226C22C2×C22⋊C812yes—
227C22C2×C22⋊C818yes—
228C22C2×C22⋊C812yes—
229C22C2×#9912yes—
230C22#41312yes—
231C22C22×C22⋊C411yescharacteristic
232C22C23×C811yescharacteristic
233C22C22×M1611yescharacteristic
234C4—24no—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4C2×#9914yes—
241C2—21no—
242C2—21no—
243C2—22no—
244C2—28no—
245C2—44no—
246C2#145914yesdirect product factor
247C2C22×C22⋊C811yescharacteristic
248C2C22×#9911yescharacteristic
249C2C2×#41311yescharacteristic
subgroupG/H
centerC22×C4C22⋊C4
commutatorC22C23×C8
Frattini, cosocleC22×C4C24
socleC23C2×C22⋊C4
prime radicalC23×D4C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊C8C2semidirect
3. #1459C2direct (N × H)
4. C2×#324C2semidirect
5. C2×#323C2semidirect
22. C2×C22⋊C8C22semidirect
23. #324C22semidirect
24. #323C22semidirect
30. C22×D4C8semidirect
83. C24C2×C8semidirect
92. C2×D4C2×C8semidirect
120. C22×C4C2×C8semidirect
170. C23C22×C8semidirect
204. C2×C4C22×C8semidirect
246. C2#1459direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D42
C22×C22⋊C82
#14592
C2×#3242
C2×#3232