Finite Group Databasebeta

← C2×#1897 #20978

Subgroups — C2×#1897 #20978

Order 256.

977 conjugacy classes of subgroups; only the 247 normal subgroups are shown.

normal subgroups only

This group has 977 conjugacy classes of subgroups — too many to draw. Only its 247 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
1C2×#525C211yesmaximal, characteristic
2C22×#117C211yesmaximal, characteristic, complemented
3C22×#104C212yesmaximal, complemented
4C2×#407C211yesmaximal, characteristic, complemented
5#1897C218yesmaximal, direct product factor
6C2×#339C212yesmaximal, complemented
7C22×#41C2211yescharacteristic
8C24×C4C2211yescharacteristic
9C22×M16C2212yes—
10#525—22no—
11#525—22no—
12#525C414yescomplemented
13#525C2214yes—
14C2×#123—21no—
15C2×C22⋊Q8C412yescomplemented
16C2×#121C412yescomplemented
17C2×#120—21no—
18C2×C4×D4—22no—
19C2×#117—22no—
20C2×#117—28no—
21C2×#117C2211yescharacteristic
22C2×#117C2214yescomplemented
23C22×C4⋊C4—21no—
24C22×C22⋊C4—21no—
25C22×C42C2212yes—
26C2×#104—28no—
27C2×#104C2218yescomplemented
28#407—22no—
29#407C2214yescomplemented
30C2×C22⋊C8—22no—
31#339C2218yescomplemented
32C25D411yescharacteristic
33C2×#41—21no—
34C2×#41—21no—
35C2×#41—24no—
36C2×#41—24no—
37C2×#41C2×C412yes—
38C2×#41C2312yes—
39C22×Q8C2×C411yescharacteristic
40C22×D4—22no—
41C22×D4—21no—
42C22×D4C2×C411yescharacteristic
43C23×C4—24no—
44C23×C4—24no—
45C23×C4—22no—
46C23×C4—21no—
47C23×C4C2×C412yes—
48C23×C4C2×C412yes—
49C23×C4D411yescharacteristic
50C23×C4D412yes—
51C23×C4D412yes—
52C23×C4D412yes—
53C23×C4C2312yes—
54C23×C4C2311yescharacteristic
55C2×M16—22no—
56C2×M16—24no—
57C2×M16—24no—
58C2×M16C2314yes—
59C22×C8—22no—
60#123—24no—
61#123—24no—
62C22⋊Q8—24no—
63C22⋊Q8C2×C418yescomplemented
64#121—24no—
65#121C2×C418yescomplemented
66#120—24no—
67#120—24no—
68C4×D4—28no—
69C4×D4—28no—
70#117—216no—
71#117—21no—
72#117—21no—
73#117—216no—
74#117C2×C412yes—
75#117C2312yes—
76C2×C4⋊C4—28no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4C2×C412yes—
80C2×C22⋊C4—28no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C42—22no—
86C2×C42—24no—
87C2×C42—21no—
88C2×C42D418yescomplemented
89C2×C42C2314yes—
90#104—216no—
91C22⋊C8—28no—
92C24—22no—
93C24—22no—
94C24—21no—
95C24—44no—
96C24C22⋊C412yes—
97C24C22⋊C412yes—
98C24C2×D411yescharacteristic
99C24C2×D412yes—
100#41—28no—
101#41—24no—
102#41—24no—
103#41—42no—
104#41—44no—
105#41—42no—
106C2×Q8—21no—
107C2×Q8—24no—
108C2×Q8C22×C412yes—
109C2×D4—28no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—24no—
114C2×D4—48no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4C22×C412yes—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—28no—
121C22×C4—28no—
122C22×C4—24no—
123C22×C4—28no—
124C22×C4—28no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—28no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—48no—
135C22×C4—44no—
136C22×C4C22⋊C412yes—
137C22×C4C22⋊C414yes—
138C22×C4C22⋊C414yes—
139C22×C4C22⋊C412yes—
140C22×C4C22×C414yes—
141C22×C4C22×C411yescharacteristic
142C22×C4C22×C411yescharacteristic
143C22×C4C2×D412yes—
144C22×C4C2×D414yes—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D414yes—
148C22×C4C2×D411yescharacteristic
149C22×C4#4114yes—
150C22×C4C2411yescharacteristic
151M16—24no—
152M16—24no—
153M16—44no—
154C2×C8—24no—
155C2×C8—44no—
156C4⋊C4—216no—
157C4⋊C4—28no—
158C4⋊C4—22no—
159C4⋊C4C22×C414yes—
160C22⋊C4—216no—
161C22⋊C4—22no—
162C22⋊C4—28no—
163C22⋊C4—22no—
164C22⋊C4—28no—
165C42—22no—
166C42—28no—
167C42—24no—
168C42—22no—
169C42C2×D418yescomplemented
170C23—24no—
171C23—22no—
172C23—22no—
173C23—24no—
174C23—24no—
175C23—24no—
176C23—24no—
177C23—22no—
178C23—44no—
179C23—44no—
180C23—48no—
181C23—48no—
182C23—42no—
183C23—44no—
184C23C2×C22⋊C414yes—
185C23C2×C22⋊C411yescharacteristic
186C23C2×C22⋊C411yescharacteristic
187C23#12012yes—
188C23#12312yes—
189C23C22×D411yescharacteristic
190Q8—24no—
191Q8—42no—
192D4—24no—
193D4—416no—
194D4—42no—
195D4—42no—
196D4—44no—
197D4—42no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—28no—
203C2×C4—28no—
204C2×C4—28no—
205C2×C4—28no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—28no—
209C2×C4—48no—
210C2×C4—416no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4C2×C22⋊C414yes—
216C2×C4C2×C22⋊C411yescharacteristic
217C2×C4C2×C22⋊C411yescharacteristic
218C2×C4C4×D414yes—
219C2×C4C4×D414yes—
220C2×C4#12012yes—
221C2×C4#12114yes—
222C2×C4#12114yes—
223C2×C4#12312yes—
224C2×C4C23×C411yescharacteristic
225C2×C4C22×D412yes—
226C2×C4C22×D411yescharacteristic
227C2×C4C2×#4112yes—
228C8—44no—
229C22—22no—
230C22—22no—
231C22—24no—
232C22—24no—
233C22—24no—
234C22—24no—
235C22—24no—
236C22—44no—
237C22—44no—
238C22—48no—
239C22—48no—
240C22—44no—
241C22—42no—
242C22—84no—
243C22#38612yes—
244C22#38612yes—
245C22#42114yes—
246C22C22×C22⋊C411yescharacteristic
247C22C2×#12011yescharacteristic
248C22C2×#12311yescharacteristic
249C4—24no—
250C4—48no—
251C4—44no—
252C4—42no—
253C4#38612yes—
254C4#38612yes—
255C4C2×C4×D412yes—
256C4C2×#12112yes—
257C2—21no—
258C2—21no—
259C2—22no—
260C2—44no—
261C2—82no—
262C2#189714yesdirect product factor
263C2C2×#38611yescharacteristic
264C2C2×#42112yes—
subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC23C2×C22⋊C4
prime radicalC2×#525C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#117C2semidirect
3. C22×#104C2semidirect
4. C2×#407C2semidirect
5. #1897C2direct (N × H)
6. C2×#339C2semidirect
12. #525C4semidirect
15. C2×C22⋊Q8C4semidirect
16. C2×#121C4semidirect
22. C2×#117C22semidirect
27. C2×#104C22semidirect
29. #407C22semidirect
31. #339C22semidirect
63. C22⋊Q8C2×C4semidirect
65. #121C2×C4semidirect
88. C2×C42D4semidirect
169. C42C2×D4semidirect
262. C2#1897direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5252
C22×#1172
C22×#1042
C2×#4072
#18972
C2×#3392