Finite Group Databasebeta

← C2×#1472 #17378

Subgroups — C2×#1472 #17378

Order 256.

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

normal subgroups only

This group has 713 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
3C22×C22⋊C8C211yesmaximal, characteristic
4#1472C218yesmaximal, direct product factor
5C2×#324C214yesmaximal, complemented
6C23×Q8C411yescharacteristic
7C24×C4C2211yescharacteristic
8C22×M16—21no—
9C23×C8—21no—
10C2×C22⋊Q8—21no—
11C2×C22⋊Q8—21no—
12C2×C22⋊Q8—22no—
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—22no—
16C2×C22⋊Q8C414yes—
17C2×C22⋊Q8C2214yes—
18C22×C4⋊C4C411yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C22⋊C4C412yes—
21#407—22no—
22#407C2214yes—
23C2×C22⋊C8—22no—
24C2×C22⋊C8—21no—
25C2×C22⋊C8—21no—
26C2×C22⋊C8C2212yes—
27C2×C22⋊C8C2214yes—
28C2×C22⋊C8C2212yes—
29#324C22116yescomplemented
30C25C2×C411yescharacteristic
31C22×Q8—22no—
32C22×Q8—21no—
33C22×Q8—21no—
34C22×Q8—44no—
35C22×Q8C2×C414yes—
36C23×C4—44no—
37C23×C4C2×C412yes—
38C23×C4C2×C411yescharacteristic
39C23×C4C2×C412yes—
40C23×C4C2×C412yes—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4D412yes—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4D412yes—
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C2×M16—24no—
50C2×M16—44no—
51C22×C8—24no—
52C22×C8—22no—
53C22×C8—22no—
54C22×C8—44no—
55C22⋊Q8—24no—
56C22⋊Q8—24no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62C22⋊Q8—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4C2×C414yes—
71C2×C4⋊C4C2×C414yes—
72C2×C4⋊C4C2314yes—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4C2×C418yes—
77C22⋊C8—22no—
78C22⋊C8—24no—
79C22⋊C8—24no—
80C22⋊C8—22no—
81C22⋊C8C2314yes—
82C22⋊C8C2314yes—
83C24—24no—
84C24—24no—
85C24—42no—
86C24C22⋊C411yescharacteristic
87C24C22⋊C412yes—
88C24C22⋊C411yescharacteristic
89C24C22×C412yes—
90C24C22×C411yescharacteristic
91C2×Q8—24no—
92C2×Q8—24no—
93C2×Q8—48no—
94C2×Q8—44no—
95C2×Q8—48no—
96C2×Q8—44no—
97C22×C4—24no—
98C22×C4—24no—
99C22×C4—24no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—28no—
105C22×C4—44no—
106C22×C4—44no—
107C22×C4—44no—
108C22×C4—44no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—44no—
113C22×C4C22⋊C412yes—
114C22×C4C22⋊C412yes—
115C22×C4C22⋊C412yes—
116C22×C4C22⋊C411yescharacteristic
117C22×C4C22⋊C412yes—
118C22×C4C22⋊C412yes—
119C22×C4C22⋊C411yescharacteristic
120C22×C4C22×C414yes—
121C22×C4C22×C412yes—
122C22×C4C22×C414yes—
123C22×C4C22×C411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D412yes—
131C22×C4C2×D412yes—
132C22×C4C2411yescharacteristic
133M16—48no—
134C2×C8—24no—
135C2×C8—24no—
136C2×C8—44no—
137C2×C8—48no—
138C2×C8—44no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C22⋊C4—28no—
146C22⋊C4—28no—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—22no—
151C23—24no—
152C23—24no—
153C23—22no—
154C23—22no—
155C23—24no—
156C23—22no—
157C23—24no—
158C23—42no—
159C23—42no—
160C23—416no—
161C23#10114yes—
162C23C2×C22⋊C412yes—
163C23C2×C22⋊C411yescharacteristic
164C23C2×C22⋊C412yes—
165C23C2×C22⋊C411yescharacteristic
166C23#13114yes—
167C23C23×C411yescharacteristic
168Q8—48no—
169Q8—48no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—28no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4#9914yes—
196C2×C4C2×C22⋊C411yescharacteristic
197C2×C4C2×C22⋊C411yescharacteristic
198C2×C4C2×C22⋊C412yes—
199C2×C4C2×C22⋊C412yes—
200C2×C4C22×D411yescharacteristic
201C2×C4C22×D411yescharacteristic
202C8—44no—
203C8—44no—
204C22—22no—
205C22—21no—
206C22—21no—
207C22—22no—
208C22—21no—
209C22—21no—
210C22—28no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—48no—
216C22—48no—
217C22—44no—
218C22#40812yes—
219C22#40812yes—
220C22C2×#9912yes—
221C22C2×#10114yes—
222C22C2×#10112yes—
223C22C22×C22⋊C411yescharacteristic
224C22C2×#13112yes—
225C4—21no—
226C4—21no—
227C4—21no—
228C4—21no—
229C4—22no—
230C4—44no—
231C4—44no—
232C4C2×#9914yes—
233C2—22no—
234C2—22no—
235C2—24no—
236C2—42no—
237C2#147214yesdirect product factor
238C2C2×#40811yescharacteristic
239C2C22×#9911yescharacteristic
240C2C22×#10111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC22×C4C24
prime radicalC25C2×C4
LCSG ▷ C23 ▷ C2 ▷ 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. #1472C2direct (N × H)
5. C2×#324C2semidirect
29. #324C22semidirect
237. C2#1472direct (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
C22×C22⋊C82
#14722
C2×#3242