Finite Group Databasebeta

← #8154

Subgroups — #8154

Order 256.

301 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 99 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×#387C211yesmaximal, characteristic
2#1749C211yesmaximal, characteristic
3C4×C22⋊C8C211yesmaximal, characteristic
4C2×#324C212yesmaximal
5#1280C212yesmaximal
6C22×C4⋊C4C411yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8C22×C42C2211yescharacteristic
9C2×C22⋊C8C2212yes—
10C2×C22⋊C8C2212yes—
11C2×#97—21no—
12C2×C4×C8—21no—
13#387—21no—
14#387—21no—
15#387—21no—
16#387—21no—
17C2×#95C411yescharacteristic
18C2×#95C411yescharacteristic
19C2×#95C411yescharacteristic
20C2×#95C2211yescharacteristic
21#336—21no—
22#336—21no—
23#324C418yescomplemented
24C23×C4C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32C22×C8—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—42no—
39C2×C4⋊C4C2×C414yes—
40C2×C42—42no—
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42D411yescharacteristic
45C22⋊C8—22no—
46C22⋊C8—22no—
47C22⋊C8C2×C414yes—
48C22⋊C8C2×C414yes—
49#97—42no—
50C4×C8—42no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59C24C22×C411yescharacteristic
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4C4212yes—
85C22×C4C4212yes—
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22×C412yes—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4#4112yes—
94C2×C8—24no—
95C2×C8—24no—
96C2×C8—42no—
97C2×C8—42no—
98C2×C8—42no—
99C2×C8—42no—
100C4⋊C4—24no—
101C4⋊C4—24no—
102C4⋊C4—42no—
103C4⋊C4—42no—
104C42—22no—
105C42—22no—
106C42—41no—
107C42—41no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—21no—
112C23C2×C4211yescharacteristic
113C23C2×C22⋊C411yescharacteristic
114C23#11711yescharacteristic
115C23#13112yes—
116C23#13112yes—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—24no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—44no—
137C2×C4—44no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—82no—
152C2×C4—82no—
153C2×C4#9912yes—
154C2×C4#10112yes—
155C2×C4C4×D412yes—
156C2×C4C4×D412yes—
157C8—44no—
158C8—44no—
159C22—22no—
160C22—22no—
161C22—22no—
162C22—22no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22C4×C22⋊C411yescharacteristic
168C22#40512yes—
169C22#40811yescharacteristic
170C22#40811yescharacteristic
171C22C2×#9911yescharacteristic
172C22#41011yescharacteristic
173C22C2×#10111yescharacteristic
174C22#41311yescharacteristic
175C22#43312yes—
176C4—22no—
177C4—22no—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—42no—
183C4—42no—
184C4—81no—
185C4—81no—
186C2—22no—
187C2—22no—
188C2#146811yescharacteristic
189C2#147011yescharacteristic
190C2#147211yescharacteristic
191C2#147311yescharacteristic
192C2#175011yescharacteristic
193C2C4×#9911yescharacteristic
194C2C4×#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
23. #324C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3872
#17492
C4×C22⋊C82
C2×#3242
#12802