Finite Group Databasebeta

← #8152

Subgroups — #8152

Order 256.

457 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 111 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×C4×C22⋊C4C211yesmaximal, characteristic
2#1749C212yesmaximal, complemented
3C2×#323C212yesmaximal
4#1280C212yesmaximal, complemented
5C24×C4C411yescharacteristic
6C22×C22⋊C4C411yescharacteristic
7C22×C22⋊C4C2211yescharacteristic
8C22×C42C411yescharacteristic
9C22×C42C2211yescharacteristic
10C2×C22⋊C8C2214yes—
11C2×#97—22no—
12C4×C22⋊C4—22no—
13C4×C22⋊C4—22no—
14C2×#95C411yescharacteristic
15C2×#95C2211yescharacteristic
16#336—22no—
17#323C418yescomplemented
18C25C2×C411yescharacteristic
19C23×C4—22no—
20C23×C4—21no—
21C23×C4—21no—
22C23×C4—22no—
23C23×C4—42no—
24C23×C4—42no—
25C23×C4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—24no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4C2×C414yes—
38C2×C42—21no—
39C2×C42—21no—
40C2×C42—42no—
41C2×C42—42no—
42C2×C42D412yes—
43C2×C42D412yes—
44C22⋊C8—24no—
45C22⋊C8C2×C418yescomplemented
46#97—44no—
47#95—22no—
48#95—22no—
49C24—21no—
50C24—21no—
51C24—21no—
52C24—21no—
53C24—41no—
54C24—41no—
55C24—41no—
56C24—41no—
57C24C4212yes—
58C24M1612yes—
59C24M1612yes—
60C24C22×C411yescharacteristic
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—24no—
66C22×C4—24no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—48no—
80C22×C4—48no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4C4212yes—
92C22×C4C22⋊C412yes—
93C22×C4C22⋊C411yescharacteristic
94C22×C4C22⋊C411yescharacteristic
95C22×C4M1612yes—
96C22×C4M1612yes—
97C22×C4C22×C412yes—
98C22×C4C2×D412yes—
99C22×C4#4112yes—
100C2×C8—28no—
101C2×C8—44no—
102C2×C8—44no—
103C22⋊C4—28no—
104C22⋊C4—28no—
105C42—24no—
106C42—42no—
107C42—42no—
108C42—42no—
109C23—21no—
110C23—21no—
111C23—21no—
112C23—21no—
113C23—22no—
114C23—22no—
115C23—22no—
116C23—22no—
117C23—22no—
118C23—22no—
119C23—42no—
120C23—42no—
121C23—42no—
122C23—42no—
123C23—42no—
124C23—42no—
125C23—42no—
126C23—42no—
127C23—42no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23#9714yescomplemented
136C23C2×C4211yescharacteristic
137C23C2×C22⋊C411yescharacteristic
138C23#11711yescharacteristic
139C23C2×M1611yescharacteristic
140C23C2×M1611yescharacteristic
141C23C2×M1611yescharacteristic
142C23C2×M1611yescharacteristic
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—48no—
160C2×C4—48no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4#9912yes—
176C2×C4#10012yes—
177C2×C4C4×D414yes—
178C8—48no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—24no—
192C22—44no—
193C22—44no—
194C22—44no—
195C22—44no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22C4×C22⋊C411yescharacteristic
203C22C2×#9711yescharacteristic
204C22C4×M1611yescharacteristic
205C22#40711yescharacteristic
206C22#40711yescharacteristic
207C22C2×#9911yescharacteristic
208C22#41011yescharacteristic
209C22C2×#10011yescharacteristic
210C22#41311yescharacteristic
211C22#43211yescharacteristic
212C22#43211yescharacteristic
213C4—22no—
214C4—22no—
215C4—24no—
216C4—42no—
217C4—42no—
218C4—42no—
219C4—42no—
220C2—21no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2—44no—
225C2#146211yescharacteristic
226C2#146411yescharacteristic
227C2#146511yescharacteristic
228C2#146611yescharacteristic
229C2#174911yescharacteristic
230C2C4×#9911yescharacteristic
231C2C4×#10011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C2×C42
Frattini, cosocleC23×C4C23
prime radicalC25C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1749C2semidirect
4. #1280C2semidirect
17. #323C4semidirect
45. C22⋊C8C2×C4semidirect
135. C23#97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊C42
#17492
C2×#3232
#12802