Finite Group Databasebeta

← #23443

Subgroups — #23443

Order 256.

1129 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1129 conjugacy classes of subgroups — too many to draw. Only its 183 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
1#3449C211yesmaximal, characteristic, complemented
2C2×#525C212yesmaximal, complemented
3C2×#410C212yesmaximal, complemented
4#2280C211yesmaximal, characteristic, complemented
5#2022C214yesmaximal, complemented
6#2021C214yesmaximal, complemented
7#1740C211yesmaximal, characteristic, complemented
8C22×#41C2212yes—
9C24×C4C2211yescharacteristic
10#537—22no—
11#536—22no—
12#535—22no—
13#534C2214yes—
14#532—22no—
15#525—28no—
16#525—24no—
17#525—28no—
18C2×#123C2214yes—
19C2×C22⋊Q8—22no—
20C2×#121—22no—
21C2×#120C2214yes—
22C2×C4×D4—24no—
23C2×#117C2211yescharacteristic
24C2×#117C2212yes—
25C2×#117C2211yescharacteristic
26#410—28no—
27C2×#99C2218yes—
28#381C2212yes—
29#380—22no—
30#379C2212yes—
31C4×C22⋊C4—22no—
32#342C2214yes—
33C25C2311yescharacteristic
34C2×#41—24no—
35C2×#41—24no—
36C2×#41—22no—
37C2×#41—22no—
38C2×#41—22no—
39C2×#41—22no—
40C2×#41—24no—
41C2×#41—24no—
42C22×Q8D412yescomplemented
43C22×D4—24no—
44C22×D4D412yescomplemented
45C22×D4C2314yes—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—24no—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4—22no—
52C23×C4D414yescomplemented
53C23×C4D414yescomplemented
54C23×C4C2311yescharacteristic
55C23×C4C2311yescharacteristic
56C4⋊Q8—22no—
57#127—22no—
58#124—24no—
59#123—28no—
60#123—28no—
61#123—28no—
62C22⋊Q8—216no—
63C22⋊Q8—28no—
64#121—216no—
65#121—28no—
66#121—28no—
67#120—28no—
68#120—28no—
69#120—28no—
70C4×D4—232no—
71C4×D4—28no—
72#117—24no—
73#117—22no—
74#117—22no—
75#117—24no—
76#117D418yescomplemented
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D412yescomplemented
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4C2312yes—
85C2×C22⋊C4C2312yes—
86C2×C22⋊C4C2314yes—
87C2×C42—22no—
88C2×C42—21no—
89C2×C42D412yescomplemented
90#99—216no—
91#95—24no—
92C24—21no—
93C24—22no—
94C24—22no—
95C24—21no—
96C24—44no—
97C24C2×D414yes—
98C24C2×D412yes—
99C24C2411yescharacteristic
100#41—44no—
101#41—44no—
102#41—44no—
103#41—44no—
104#41—44no—
105#41—44no—
106#41—44no—
107#41—44no—
108C2×Q8—22no—
109C2×Q8—24no—
110C2×Q8—22no—
111C2×Q8—24no—
112C2×D4—216no—
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—24no—
116C2×D4—24no—
117C2×D4—24no—
118C2×D4—24no—
119C2×D4—28no—
120C2×D4—22no—
121C2×D4—24no—
122C2×D4—22no—
123C2×D4—24no—
124C2×D4—416no—
125C22×C4—21no—
126C22×C4—24no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—28no—
131C22×C4—28no—
132C22×C4—28no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—28no—
138C22×C4—22no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4C2×D414yes—
147C22×C4C2×D412yes—
148C22×C4C2×D414yes—
149C22×C4C2×D412yes—
150C22×C4C2×D412yes—
151C22×C4C2×D412yes—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D414yes—
154C22×C4C2×D411yescharacteristic
155C22×C4#4112yes—
156C4⋊C4—216no—
157C4⋊C4—28no—
158C4⋊C4—28no—
159C4⋊C4—24no—
160C4⋊C4—42no—
161C22⋊C4—216no—
162C22⋊C4—216no—
163C22⋊C4—216no—
164C22⋊C4—24no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—48no—
168C22⋊C4C2×D418yes—
169C42—28no—
170C42—22no—
171C42—22no—
172C42—42no—
173C23—24no—
174C23—28no—
175C23—24no—
176C23—22no—
177C23—24no—
178C23—24no—
179C23—24no—
180C23—44no—
181C23—42no—
182C23—42no—
183C23—44no—
184C23—48no—
185C23—44no—
186C23—48no—
187C23—41no—
188C23—42no—
189C23—41no—
190C23C22×D412yes—
191C23C22×D411yescharacteristic
192C23C22×D411yescharacteristic
193C23C22×D412yes—
194C23C2×#4111yescharacteristic
195Q8—44no—
196Q8—44no—
197D4—432no—
198D4—44no—
199D4—44no—
200D4—44no—
201D4—44no—
202D4—44no—
203D4—44no—
204D4—88no—
205C2×C4—24no—
206C2×C4—28no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—48no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—44no—
217C2×C4—48no—
218C2×C4—44no—
219C2×C4—48no—
220C2×C4—48no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4#12014yes—
225C2×C4#12011yescharacteristic
226C2×C4#12012yes—
227C2×C4#12014yes—
228C2×C4#12011yescharacteristic
229C2×C4#12114yes—
230C2×C4#12112yes—
231C2×C4#12112yes—
232C2×C4#12114yes—
233C2×C4#12712yes—
234C2×C4#12712yes—
235C22—22no—
236C22—24no—
237C22—28no—
238C22—24no—
239C22—41no—
240C22—42no—
241C22—41no—
242C22—44no—
243C22—44no—
244C22—42no—
245C22—48no—
246C22—48no—
247C22—42no—
248C22—44no—
249C22—44no—
250C22—48no—
251C22—84no—
252C22C2×#12012yes—
253C22C2×#12011yescharacteristic
254C22C2×#12112yes—
255C22C2×#12111yescharacteristic
256C22C2×#12711yescharacteristic
257C4—48no—
258C4—42no—
259C4—44no—
260C4—44no—
261C4—82no—
262C4#39212yes—
263C4#39214yes—
264C4#39212yes—
265C2—22no—
266C2—21no—
267C2—21no—
268C2—22no—
269C2—44no—
270C2—84no—
271C2C2×#39211yescharacteristic
272C2#302512yes—
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC24C24
socleC22C2×#120
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3449C2semidirect
2. C2×#525C2semidirect
3. C2×#410C2semidirect
4. #2280C2semidirect
5. #2022C2semidirect
6. #2021C2semidirect
7. #1740C2semidirect
42. C22×Q8D4semidirect
44. C22×D4D4semidirect
52. C23×C4D4semidirect
53. C23×C4D4semidirect
76. #117D4semidirect
80. C2×C4⋊C4D4semidirect
89. C2×C42D4semidirect

Maximal subgroups

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

subgroupindex
#34492
C2×#5252
C2×#4102
#22802
#20222
#20212
#17402