Finite Group Databasebeta

← C2×#1508 #17535

Subgroups — C2×#1508 #17535

Order 256.

777 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 777 conjugacy classes of subgroups — too many to draw. Only its 207 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×C22⋊Q8C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic
3C2×C4×C22⋊C4C211yesmaximal, characteristic
4#1508C218yesmaximal, direct product factor
5C2×#328C214yesmaximal, complemented
6C23×Q8C411yescharacteristic, complemented
7C24×C4C2211yescharacteristic
8C22×M16—21no—
9C2×C22⋊Q8—21no—
10C2×C22⋊Q8—22no—
11C2×C22⋊Q8—21no—
12C2×C22⋊Q8—22no—
13C2×C22⋊Q8C412yescomplemented
14C2×C22⋊Q8C414yescomplemented
15C2×C22⋊Q8C2212yes—
16C2×C22⋊Q8C2214yes—
17C22×C4⋊C4—21no—
18C22×C4⋊C4C411yescharacteristic, complemented
19C22×C22⋊C4—21no—
20C22×C22⋊C4—21no—
21C22×C42—21no—
22#407—22no—
23#407C2214yes—
24C2×C22⋊C8—21no—
25C2×C22⋊C8C2212yes—
26C4×C22⋊C4—22no—
27C4×C22⋊C4C2214yes—
28C2×#95C2212yes—
29#328C22116yescomplemented
30C25D411yescharacteristic
31C22×Q8—22no—
32C22×Q8—21no—
33C22×Q8—44no—
34C22×Q8C2×C412yes—
35C22×Q8C2×C414yescomplemented
36C23×C4—21no—
37C23×C4—21no—
38C23×C4—22no—
39C23×C4—44no—
40C23×C4C2×C411yescharacteristic
41C23×C4C2×C412yes—
42C23×C4C2×C411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4D412yes—
46C23×C4D411yescharacteristic
47C23×C4D412yes—
48C23×C4C2311yescharacteristic
49C23×C4C2312yes—
50C2×M16—24no—
51C2×M16—44no—
52C22×C8—22no—
53C22⋊Q8—24no—
54C22⋊Q8—24no—
55C22⋊Q8—24no—
56C22⋊Q8—22no—
57C22⋊Q8—24no—
58C22⋊Q8—22no—
59C22⋊Q8—24no—
60C22⋊Q8C2×C414yes—
61C22⋊Q8C2×C418yescomplemented
62C22⋊Q8C2314yes—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4C2×C412yes—
70C2×C4⋊C4C2×C414yescomplemented
71C2×C22⋊C4—28no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C42—24no—
78C2×C42—44no—
79C22⋊C8—24no—
80C22⋊C8—22no—
81C22⋊C8C2314yes—
82#95—22no—
83#95C2314yes—
84C24—24no—
85C24—24no—
86C24—42no—
87C24C22⋊C411yescharacteristic
88C24C22⋊C412yes—
89C24C22⋊C411yescharacteristic
90C24C2×D412yes—
91C24C2×D411yescharacteristic
92C2×Q8—22no—
93C2×Q8—24no—
94C2×Q8—48no—
95C2×Q8—48no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×Q8C22×C414yes—
99C22×C4—24no—
100C22×C4—24no—
101C22×C4—24no—
102C22×C4—28no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—44no—
109C22×C4—44no—
110C22×C4—44no—
111C22×C4—44no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4C22⋊C412yes—
120C22×C4C22⋊C412yes—
121C22×C4C22⋊C412yes—
122C22×C4C22⋊C411yescharacteristic
123C22×C4C22⋊C412yes—
124C22×C4C22⋊C412yes—
125C22×C4C22⋊C411yescharacteristic
126C22×C4C22×C412yes—
127C22×C4C22×C411yescharacteristic
128C22×C4C22×C411yescharacteristic
129C22×C4C22×C412yes—
130C22×C4C2×D412yes—
131C22×C4C2×D412yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D412yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2411yescharacteristic
137M16—48no—
138C2×C8—24no—
139C2×C8—44no—
140C4⋊C4—24no—
141C4⋊C4—28no—
142C4⋊C4—22no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4C22×C414yes—
146C22⋊C4—216no—
147C22⋊C4—24no—
148C22⋊C4—28no—
149C22⋊C4—24no—
150C42—48no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—24no—
156C23—22no—
157C23—24no—
158C23—22no—
159C23—24no—
160C23—24no—
161C23—22no—
162C23—42no—
163C23—42no—
164C23—416no—
165C23#10414yes—
166C23C2×C22⋊C412yes—
167C23C2×C22⋊C411yescharacteristic
168C23C2×C22⋊C412yes—
169C23C2×C22⋊C411yescharacteristic
170C23#13714yes—
171C23C22×D411yescharacteristic
172Q8—44no—
173Q8—44no—
174Q8—48no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—28no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—44no—
196C2×C4—416no—
197C2×C4—48no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—48no—
201C2×C4—44no—
202C2×C4#9914yes—
203C2×C4C2×C22⋊C411yescharacteristic
204C2×C4C2×C22⋊C412yes—
205C2×C4C2×C22⋊C411yescharacteristic
206C2×C4C2×C22⋊C412yes—
207C2×C4C23×C411yescharacteristic
208C2×C4C22×D411yescharacteristic
209C8—44no—
210C22—22no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—21no—
216C22—28no—
217C22—28no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—48no—
222C22—48no—
223C22—44no—
224C22C2×#9912yes—
225C22#41912yes—
226C22#41912yes—
227C22C2×#10414yes—
228C22C2×#10412yes—
229C22C22×C22⋊C411yescharacteristic
230C22C2×#13712yes—
231C4—21no—
232C4—21no—
233C4—22no—
234C4—21no—
235C4—21no—
236C4—48no—
237C4—44no—
238C4—44no—
239C4C2×#9914yes—
240C2—22no—
241C2—22no—
242C2—24no—
243C2—42no—
244C2#150814yesdirect product factor
245C2C22×#9911yescharacteristic
246C2C2×#41911yescharacteristic
247C2C22×#10411yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC25D4
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
4. #1508C2direct (N × H)
5. C2×#328C2semidirect
6. C23×Q8C4semidirect
13. C2×C22⋊Q8C4semidirect
14. C2×C22⋊Q8C4semidirect
18. C22×C4⋊C4C4semidirect
29. #328C22semidirect
35. C22×Q8C2×C4semidirect
61. C22⋊Q8C2×C4semidirect
70. C2×C4⋊C4C2×C4semidirect
244. C2#1508direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#4072
C2×C4×C22⋊C42
#15082
C2×#3282