Finite Group Databasebeta

← C2×#1473 #17379

Subgroups — C2×#1473 #17379

Order 256.

533 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 191 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×#126C211yesmaximal, characteristic
2C2×#432C211yesmaximal, characteristic, complemented
3C2×#431C211yesmaximal, characteristic, complemented
4#1473C218yesmaximal, direct product factor
5C2×#324C212yesmaximal, complemented
6C2×#323C212yesmaximal
7C2×#126—21no—
8C2×#126—21no—
9C2×#126—21no—
10C2×#126—21no—
11C2×#126—22no—
12C2×#126—22no—
13C2×#126C414yes—
14C2×#126C2214yes—
15C22×C4⋊C4C411yescharacteristic
16C22×C4⋊C4C411yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4C411yescharacteristic
19C22×C22⋊C4C411yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C42C2211yescharacteristic
22#432—22no—
23#432C2214yescomplemented
24#431—22no—
25#431C2214yescomplemented
26C2×C4⋊C8—21no—
27C2×C4⋊C8—21no—
28C2×C22⋊C8C2212yes—
29C2×C22⋊C8C2212yes—
30C2×#97—21no—
31C2×C4×C8—21no—
32#324C2218yescomplemented
33#323C2218yes—
34C25C2×C411yescharacteristic
35C23×C4C2×C411yescharacteristic
36C23×C4C2×C411yescharacteristic
37C23×C4C2×C411yescharacteristic
38C23×C4C2×C411yescharacteristic
39C23×C4C2×C411yescharacteristic
40C23×C4C2311yescharacteristic
41C22×C8—22no—
42C22×C8—22no—
43#126—24no—
44#126—24no—
45#126—24no—
46#126—24no—
47#126—24no—
48#126—24no—
49#126—24no—
50#126—24no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4C2×C414yes—
62C2×C4⋊C4C2×C414yes—
63C2×C4⋊C4C2×C412yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4C2×C414yes—
76C2×C22⋊C4C2×C414yes—
77C2×C22⋊C4C2×C412yes—
78C2×C22⋊C4C2312yes—
79C2×C42C2×C412yes—
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42D412yes—
85C2×C42D412yes—
86C2×C42C2312yes—
87C4⋊C8—24no—
88C4⋊C8—24no—
89C22⋊C8—22no—
90C22⋊C8—22no—
91C22⋊C8C2314yes—
92C22⋊C8C2314yes—
93#97—24no—
94C4×C8—24no—
95C24—21no—
96C24—21no—
97C24—24no—
98C24—42no—
99C24—42no—
100C24C22×C411yescharacteristic
101C24C22×C412yes—
102C22×C4—24no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—24no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4C22⋊C412yes—
115C22×C4C22⋊C411yescharacteristic
116C22×C4C22⋊C411yescharacteristic
117C22×C4C22⋊C411yescharacteristic
118C22×C4C22⋊C411yescharacteristic
119C22×C4C22⋊C412yes—
120C22×C4C22×C412yes—
121C22×C4C22×C412yes—
122C22×C4C22×C412yes—
123C22×C4C22×C412yes—
124C22×C4C22×C411yescharacteristic
125C22×C4C22×C412yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2411yescharacteristic
131C2×C8—24no—
132C2×C8—24no—
133C2×C8—44no—
134C2×C8—44no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C42C22⋊C412yes—
152C42C22⋊C412yes—
153C42C22⋊C412yes—
154C42C22⋊C412yes—
155C42C2×D412yes—
156C42C2×D412yes—
157C42C2×D412yes—
158C42C2×D412yes—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—24no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—88no—
172C23C2×C22⋊C411yescharacteristic
173C23C2×C22⋊C411yescharacteristic
174C23#13114yes—
175C23C23×C411yescharacteristic
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4C2×C22⋊C411yescharacteristic
203C2×C4C2×C22⋊C411yescharacteristic
204C2×C4C2×C22⋊C412yes—
205C2×C4C2×C22⋊C412yes—
206C2×C4C2×C22⋊C412yes—
207C2×C4C2×C22⋊C412yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C8—44no—
211C8—44no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—28no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—84no—
221C22—84no—
222C22—84no—
223C22#40812yes—
224C22#40812yes—
225C22#41012yes—
226C22#41312yes—
227C22C22×C22⋊C411yescharacteristic
228C22C2×#13112yes—
229C4—22no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—42no—
239C4—42no—
240C2—22no—
241C2—22no—
242C2—82no—
243C2#147314yesdirect product factor
244C2C2×#40811yescharacteristic
245C2C2×#41011yescharacteristic
246C2C2×#41311yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC22×C4C24
prime radicalC25C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#432C2semidirect
3. C2×#431C2semidirect
4. #1473C2direct (N × H)
5. C2×#324C2semidirect
23. #432C22semidirect
25. #431C22semidirect
32. #324C22semidirect
243. C2#1473direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#1262
C2×#4322
C2×#4312
#14732
C2×#3242
C2×#3232