Finite Group Databasebeta

← #14166

Subgroups — #14166

Order 256.

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

normal subgroups only

This group has 845 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
1C22×#126C211yesmaximal, characteristic
2C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#376C211yesmaximal, characteristic, complemented
4#1441C212yesmaximal, complemented
5#1439C212yesmaximal
6#1437C212yesmaximal
7#1437C212yesmaximal
8#1436C212yesmaximal, complemented
9#1436C212yesmaximal, complemented
10C2×#126—24no—
11C2×#126—24no—
12C2×#126C418yescomplemented
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4C2214yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C42C2212yes—
22C22×C42C2212yes—
23C22×C42C2211yescharacteristic
24C4×C4⋊C4—24no—
25#376—24no—
26C2×#95C2214yes—
27C2×#95C2214yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2214yes—
32C25C2311yescharacteristic
33C23×C4C2311yescharacteristic
34C23×C4C2311yescharacteristic
35C23×C4C2311yescharacteristic
36C23×C4C2314yes—
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⋊C4C2×C414yes—
51C2×C4⋊C4C2×C414yes—
52C2×C4⋊C4C2×C414yes—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—28no—
60C2×C22⋊C4—28no—
61C2×C22⋊C4—44no—
62C2×C22⋊C4C2×C414yes—
63C2×C22⋊C4C2×C414yes—
64C2×C22⋊C4C2×C414yes—
65C2×C42—24no—
66C2×C42—28no—
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—28no—
77#95—44no—
78C24—21no—
79C24—21no—
80C24—24no—
81C24—24no—
82C24—22no—
83C24—22no—
84C24C22×C412yes—
85C24C2411yescharacteristic
86C22×C4—24no—
87C22×C4—24no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—24no—
91C22×C4—24no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—24no—
99C22×C4—24no—
100C22×C4—24no—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4C22×C412yes—
113C22×C4C22×C412yes—
114C22×C4C22×C412yes—
115C22×C4C22×C412yes—
116C22×C4C22×C412yes—
117C22×C4C22×C412yes—
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C22×C4C2×D412yes—
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4#4118yes—
125C22×C4#4118yes—
126C4⋊C4—216no—
127C4⋊C4—28no—
128C4⋊C4—28no—
129C4⋊C4—28no—
130C22⋊C4—28no—
131C22⋊C4—28no—
132C22⋊C4—28no—
133C22⋊C4—416no—
134C42—216no—
135C42—48no—
136C42C22⋊C4116yescomplemented
137C23—22no—
138C23—24no—
139C23—24no—
140C23—24no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—41no—
146C23—41no—
147C23—42no—
148C23—44no—
149C23—44no—
150C23—44no—
151C23—44no—
152C23C23×C411yescharacteristic
153C23C22×D411yescharacteristic
154C23C22×D411yescharacteristic
155C23C2×#4114yes—
156C23C2×#4114yes—
157C23H(5,𝔽2)12yes—
158C232−1+412yes—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—28no—
176C2×C4—28no—
177C2×C4—28no—
178C2×C4—28no—
179C2×C4—28no—
180C2×C4—28no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4C2×C22⋊C414yes—
197C2×C4C2×C22⋊C414yes—
198C2×C4C2×C22⋊C414yes—
199C22—42no—
200C22—48no—
201C22—42no—
202C22—42no—
203C22—84no—
204C22—82no—
205C22—82no—
206C22C22×C22⋊C411yescharacteristic
207C22C4×#4112yes—
208C22C4×#4112yes—
209C22#52012yes—
210C22#52512yes—
211C22#52512yes—
212C22#53212yes—
213C22#53212yes—
214C22#53511yescharacteristic
215C22#53611yescharacteristic
216C22#53712yes—
217C22#55114yes—
218C22#55214yes—
219C22#55314yes—
220C22#55614yes—
221C4—24no—
222C4—24no—
223C4—24no—
224C4—42no—
225C4—48no—
226C4—48no—
227C4—42no—
228C4—42no—
229C2—82no—
230C2#229711yescharacteristic
231C2#229711yescharacteristic
232C2#231111yescharacteristic
233C2#237314yes—
234C2#260114yes—
235C2#260312yes—
236C2#260412yes—
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×C4×C4⋊C4C2semidirect
3. C2×#376C2semidirect
4. #1441C2semidirect
8. #1436C2semidirect
9. #1436C2semidirect
12. C2×#126C4semidirect
136. 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×C4×C4⋊C42
C2×#3762
#14412
#14392
#14372
#14362