Finite Group Databasebeta

← #14291

Subgroups — #14291

Order 256.

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

normal subgroups only

This group has 805 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
1C22×#126C211yesmaximal, characteristic
2C2×#382C211yesmaximal, characteristic, complemented
3C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4#1441C212yesmaximal, complemented
5#1441C212yesmaximal, complemented
6#1439C212yesmaximal
7#1439C212yesmaximal
8#1437C212yesmaximal
9#1436C212yesmaximal, complemented
10C2×#126—24no—
11C2×#126—24no—
12C2×#126C418yescomplemented
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2212yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2214yes—
22C22×C42C2212yes—
23C22×C42C2211yescharacteristic
24#382—24no—
25C4×C4⋊C4—24no—
26C2×#95C2212yes—
27C2×#95C2214yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2214yes—
31C2×#95C2214yes—
32C25C2311yescharacteristic
33C23×C4C2314yes—
34C23×C4C2311yescharacteristic
35C23×C4C2311yescharacteristic
36C23×C4C2311yescharacteristic
37C23×C4C2314yes—
38C23×C4C2311yescharacteristic
39C23×C4C2311yescharacteristic
40C23×C4C2311yescharacteristic
41#126—232no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—28no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—44no—
52C2×C4⋊C4C2×C414yes—
53C2×C4⋊C4C2×C414yes—
54C2×C4⋊C4C2×C414yes—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—28no—
62C2×C22⋊C4—44no—
63C2×C22⋊C4—44no—
64C2×C22⋊C4C2×C414yes—
65C2×C22⋊C4C2×C414yes—
66C2×C22⋊C4C2×C414yes—
67C2×C42—24no—
68C2×C42—44no—
69C2×C42C2×C414yes—
70C2×C42D414yes—
71C2×C42D414yes—
72#95—24no—
73#95—24no—
74#95—24no—
75#95—28no—
76#95—44no—
77#95—44no—
78C24—21no—
79C24—21no—
80C24—24no—
81C24—22no—
82C24—22no—
83C24—22no—
84C24—22no—
85C24C22×C412yes—
86C24C2411yescharacteristic
87C22×C4—24no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—24no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—24no—
100C22×C4—24no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—24no—
116C22×C4C22×C412yes—
117C22×C4C22×C412yes—
118C22×C4C22×C412yes—
119C22×C4C22×C412yes—
120C22×C4C22×C412yes—
121C22×C4C22×C412yes—
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2×D412yes—
127C22×C4C2×D412yes—
128C22×C4#4118yes—
129C4⋊C4—216no—
130C4⋊C4—28no—
131C4⋊C4—28no—
132C4⋊C4—28no—
133C4⋊C4—48no—
134C22⋊C4—28no—
135C22⋊C4—28no—
136C22⋊C4—28no—
137C22⋊C4—416no—
138C42—48no—
139C42C22⋊C4116yescomplemented
140C23—22no—
141C23—24no—
142C23—24no—
143C23—24no—
144C23—42no—
145C23—42no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—42no—
151C23—42no—
152C23—44no—
153C23—44no—
154C23—44no—
155C23—44no—
156C23C23×C411yescharacteristic
157C23C22×D411yescharacteristic
158C23C22×D411yescharacteristic
159C23C2×#4114yes—
160C23H(5,𝔽2)12yes—
161C23H(5,𝔽2)12yes—
162C232−1+412yes—
163C232−1+412yes—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—28no—
182C2×C4—28no—
183C2×C4—28no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4C2×C22⋊C414yes—
204C2×C4C2×C22⋊C414yes—
205C2×C4C2×C22⋊C414yes—
206C22—42no—
207C22—42no—
208C22—48no—
209C22—42no—
210C22—82no—
211C22—82no—
212C22—82no—
213C22—82no—
214C22C22×C22⋊C411yescharacteristic
215C22C4×#4112yes—
216C22#52012yes—
217C22#52012yes—
218C22#52512yes—
219C22#53212yes—
220C22#53511yescharacteristic
221C22#53511yescharacteristic
222C22#53611yescharacteristic
223C22#53611yescharacteristic
224C22#53712yes—
225C22#53712yes—
226C22#53814yes—
227C22#53914yes—
228C22#54114yes—
229C22#54214yes—
230C4—24no—
231C4—24no—
232C4—24no—
233C4—42no—
234C4—42no—
235C4—48no—
236C4—42no—
237C4—84no—
238C2—82no—
239C2#229711yescharacteristic
240C2#231111yescharacteristic
241C2#231111yescharacteristic
242C2#233414yes—
243C2#263612yes—
244C2#263712yes—
245C2#266212yes—
246C2#266412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC25C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#382C2semidirect
3. C2×C4×C4⋊C4C2semidirect
4. #1441C2semidirect
5. #1441C2semidirect
9. #1436C2semidirect
12. C2×#126C4semidirect
139. C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#1262
C2×#3822
C2×C4×C4⋊C42
#14412
#14392
#14372
#14362