Finite Group Databasebeta

← #20959

Subgroups — #20959

Order 256.

987 conjugacy classes of subgroups; only the 239 normal subgroups are shown.

normal subgroups only

This group has 987 conjugacy classes of subgroups — too many to draw. Only its 239 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
2#3428C211yesmaximal, characteristic, complemented
3C2×#410C211yesmaximal, characteristic, complemented
4#2280C212yesmaximal, complemented
5#1890C214yesmaximal, complemented
6#1889C214yesmaximal, complemented
7#1740C212yesmaximal, complemented
8C22×#41C2211yescharacteristic
9C24×C4C2211yescharacteristic
10#525—24no—
11#525C418yescomplemented
12C2×#123C2212yes—
13C2×C22⋊Q8—21no—
14C2×#121—21no—
15C2×#120C2212yes—
16#520—24no—
17#519—21no—
18#518—21no—
19#518C2214yes—
20C4×#41—24no—
21C2×C4×D4—22no—
22C2×#117—22no—
23C2×#117C2212yes—
24C2×#117C2211yescharacteristic
25C2×#117C2212yes—
26#410—24no—
27C2×#99C2214yes—
28#381C2214yes—
29#380—24no—
30#379C2214yes—
31C4×C22⋊C4—24no—
32#342C2218yes—
33C25C2311yescharacteristic
34C2×#41—22no—
35C2×#41—22no—
36C2×#41—24no—
37C2×#41C2×C414yes—
38C2×#41D414yes—
39C22×Q8D411yescharacteristic
40C22×D4—22no—
41C22×D4D411yescharacteristic
42C22×D4C2312yes—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—24no—
49C23×C4C2×C414yescomplemented
50C23×C4D412yes—
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53#123—24no—
54#123C2×C418yescomplemented
55C22⋊Q8—28no—
56#121—28no—
57#120—24no—
58#120C2×C418yescomplemented
59C4×Q8—28no—
60C4×D4—28no—
61C4×D4—216no—
62C4×D4—216no—
63#117—24no—
64#117—24no—
65#117—22no—
66#117—24no—
67#117—22no—
68#117—22no—
69#117—22no—
70#117—22no—
71#117—24no—
72#117—24no—
73#117C2×C414yescomplemented
74C2×C4⋊C4—24no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4D414yescomplemented
79C2×C22⋊C4—24no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—28no—
83C2×C22⋊C4C2314yes—
84C2×C22⋊C4C2314yes—
85C2×C22⋊C4C2312yes—
86C2×C42—24no—
87C2×C42—21no—
88C2×C42—22no—
89C2×C42D414yescomplemented
90#99—28no—
91#95—28no—
92C24—22no—
93C24—21no—
94C24—21no—
95C24—22no—
96C24—44no—
97C24C22×C412yes—
98C24C2×D412yes—
99C24C2411yescharacteristic
100#41—28no—
101#41—28no—
102#41—28no—
103#41—28no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8C22⋊C414yescomplemented
107C2×D4—24no—
108C2×D4—24no—
109C2×D4—28no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—48no—
113C2×D4C22⋊C414yescomplemented
114C2×D4C22×C414yes—
115C2×D4C2×D414yes—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—28no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—28no—
128C22×C4—28no—
129C22×C4—24no—
130C22×C4—42no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—44no—
134C22×C4—44no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4C22⋊C418yescomplemented
138C22×C4C22×C412yes—
139C22×C4C22×C412yes—
140C22×C4C2×D412yes—
141C22×C4C2×D418yes—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D412yes—
145C22×C4C2×D412yes—
146C22×C4C2×D412yes—
147C22×C4#4114yes—
148C4⋊C4—28no—
149C4⋊C4—24no—
150C4⋊C4—28no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—44no—
154C22⋊C4—28no—
155C22⋊C4—28no—
156C22⋊C4—24no—
157C22⋊C4—24no—
158C22⋊C4—22no—
159C22⋊C4—28no—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—416no—
163C22⋊C4C22×C414yes—
164C42—28no—
165C42—24no—
166C42—24no—
167C42—24no—
168C42—44no—
169C23—28no—
170C23—24no—
171C23—22no—
172C23—24no—
173C23—22no—
174C23—22no—
175C23—44no—
176C23—42no—
177C23—44no—
178C23—42no—
179C23—44no—
180C23—48no—
181C23—44no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23C2×C22⋊C414yes—
186C23C23×C411yescharacteristic
187C23C22×D411yescharacteristic
188C23C22×D412yes—
189C23C22×D411yescharacteristic
190C23C2×#4112yes—
191Q8—28no—
192D4—28no—
193D4—28no—
194D4—28no—
195D4—416no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—28no—
199C2×C4—28no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—28no—
204C2×C4—44no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—48no—
211C2×C4—48no—
212C2×C4—48no—
213C2×C4—44no—
214C2×C4—88no—
215C2×C4C2×C22⋊C414yes—
216C2×C4C2×C22⋊C412yes—
217C2×C4C2×C22⋊C412yes—
218C2×C4C4×D414yes—
219C2×C4C4×D414yes—
220C2×C4#12012yes—
221C2×C4#12011yescharacteristic
222C2×C4#12011yescharacteristic
223C2×C4#12114yes—
224C2×C4#12114yes—
225C2×C4#12312yes—
226C2×C4#12312yes—
227C22—24no—
228C22—22no—
229C22—24no—
230C22—28no—
231C22—24no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—44no—
236C22—42no—
237C22—48no—
238C22—44no—
239C22—44no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—82no—
244C22C22×C22⋊C411yescharacteristic
245C22C2×C4×D412yes—
246C22C2×#12011yescharacteristic
247C22C2×#12112yes—
248C22C2×#12311yescharacteristic
249C4—24no—
250C4—44no—
251C4—48no—
252C4—42no—
253C4—84no—
254C4#38612yes—
255C4#38612yes—
256C4#38614yes—
257C2—21no—
258C2—22no—
259C2—22no—
260C2—21no—
261C2—44no—
262C2—44no—
263C2C2×#38611yescharacteristic
264C2#302512yes—
subgroupG/H
centerC2×C4#120
commutatorC23C23×C4
Frattini, cosocleC24C24
socleC22C2×#120
prime radicalC2×#525C2
LCSG ▷ C23 ▷ 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
2. #3428C2semidirect
3. C2×#410C2semidirect
4. #2280C2semidirect
5. #1890C2semidirect
6. #1889C2semidirect
7. #1740C2semidirect
11. #525C4semidirect
49. C23×C4C2×C4semidirect
54. #123C2×C4semidirect
58. #120C2×C4semidirect
73. #117C2×C4semidirect
78. C2×C4⋊C4D4semidirect
89. C2×C42D4semidirect
106. C2×Q8C22⋊C4semidirect
113. C2×D4C22⋊C4semidirect
137. C22×C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
#34282
C2×#4102
#22802
#18902
#18892
#17402