Finite Group Databasebeta

← #8153

Subgroups — #8153

Order 256.

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

normal subgroups only

This group has 349 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×C4⋊C4C211yesmaximal, characteristic
2#1749C212yesmaximal
3C2×#324C212yesmaximal
4#1280C212yesmaximal
5C22×C4⋊C4C411yescharacteristic
6C22×C4⋊C4C2211yescharacteristic
7C22×C42C411yescharacteristic
8C22×C42C411yescharacteristic
9C22×C42C2211yescharacteristic
10C2×C22⋊C8C2214yes—
11C2×#97—22no—
12C4×C4⋊C4—22no—
13C4×C4⋊C4—22no—
14C2×#95C411yescharacteristic
15C2×#95C2211yescharacteristic
16#336—22no—
17#324C418yescomplemented
18C23×C4C2×C411yescharacteristic
19C23×C4C2×C411yescharacteristic
20C23×C4C2×C411yescharacteristic
21C23×C4C2×C411yescharacteristic
22C23×C4C2×C411yescharacteristic
23C23×C4C2×C411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—24no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4C2×C414yes—
32C2×C42—21no—
33C2×C42—21no—
34C2×C42—22no—
35C2×C42—22no—
36C2×C42—21no—
37C2×C42—21no—
38C2×C42—42no—
39C2×C42—42no—
40C2×C42D412yes—
41C2×C42D412yes—
42C22⋊C8—24no—
43C22⋊C8C2×C418yes—
44#97—44no—
45#95—22no—
46#95—22no—
47C24C22×C411yescharacteristic
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4C4212yes—
75C22×C4C4212yes—
76C22×C4C22⋊C412yes—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4M1612yes—
80C22×C4M1612yes—
81C22×C4M1612yes—
82C22×C4M1612yes—
83C22×C4C22×C412yes—
84C22×C4C2×D412yes—
85C22×C4#4112yes—
86C2×C8—28no—
87C2×C8—44no—
88C2×C8—44no—
89C4⋊C4—28no—
90C4⋊C4—28no—
91C42—24no—
92C42—24no—
93C42—24no—
94C42—42no—
95C42—42no—
96C42—42no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23C2×C4211yescharacteristic
102C23C2×C22⋊C411yescharacteristic
103C23#11711yescharacteristic
104C23C2×M1611yescharacteristic
105C23C2×M1611yescharacteristic
106C23C2×M1611yescharacteristic
107C23C2×M1611yescharacteristic
108C2×C4—24no—
109C2×C4—24no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4#9714yescomplemented
145C2×C4#9912yes—
146C2×C4#10112yes—
147C2×C4C4×D414yes—
148C8—48no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22C4×C22⋊C411yescharacteristic
162C22C2×#9711yescharacteristic
163C22C4×M1611yescharacteristic
164C22#40711yescharacteristic
165C22#40711yescharacteristic
166C22C2×#9911yescharacteristic
167C22#41011yescharacteristic
168C22C2×#10111yescharacteristic
169C22#41311yescharacteristic
170C22#43211yescharacteristic
171C22#43211yescharacteristic
172C4—22no—
173C4—22no—
174C4—24no—
175C4—24no—
176C4—42no—
177C4—42no—
178C4—42no—
179C4—42no—
180C2—21no—
181C2—21no—
182C2—21no—
183C2—21no—
184C2#146311yescharacteristic
185C2#146411yescharacteristic
186C2#146511yescharacteristic
187C2#146711yescharacteristic
188C2#174911yescharacteristic
189C2C4×#9911yescharacteristic
190C2C4×#10111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C2×C42
Frattini, cosocleC23×C4C23
prime radicalC24C22×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
17. #324C4semidirect
144. C2×C4#97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊C42
#17492
C2×#3242
#12802