Finite Group Databasebeta

← #7731

Subgroups — #7731

Order 256.

225 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#2851C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic
3#1797C211yesmaximal, characteristic
4#1271C212yesmaximal, complemented
5#1271C212yesmaximal, complemented
6C2×#125C411yescharacteristic
7C22×C42C2211yescharacteristic
8C2×C4⋊C8—21no—
9C2×C22⋊C8—21no—
10C2×C22⋊C8—21no—
11#389C2211yescharacteristic
12#389C2211yescharacteristic
13#387—21no—
14#387—21no—
15#385—21no—
16#385—21no—
17#382C411yescharacteristic
18#380C411yescharacteristic
19#380C411yescharacteristic
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C23×C4C2×C411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4D411yescharacteristic
26C22×C8—22no—
27C22×C8—22no—
28#125—42no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C42—42no—
40C2×C42C2×C411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C2×C42C2311yescharacteristic
44C4⋊C8—42no—
45C22⋊C8—42no—
46C22⋊C8—42no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53C24C22⋊C411yescharacteristic
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—22no—
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×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22×C411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C2×C8—42no—
75C2×C8—42no—
76C2×C8—42no—
77C2×C8—42no—
78C2×C8—42no—
79C2×C8—42no—
80C4⋊C4—22no—
81C4⋊C4—22no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C42—22no—
92C42—22no—
93C42—41no—
94C42—41no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23#10112yes—
102C23C2×C22⋊C411yescharacteristic
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—44no—
118C2×C4—44no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4#9912yes—
134C2×C4#13112yes—
135C2×C4#13512yes—
136C2×C4AGL(1,ℤ/8)11yescharacteristic
137C2×C4#13711yescharacteristic
138C8—82no—
139C8—82no—
140C22—22no—
141C22—22no—
142C22—22no—
143C22—41no—
144C22—41no—
145C22—41no—
146C22—41no—
147C22#40811yescharacteristic
148C22C2×#9911yescharacteristic
149C22C2×#10111yescharacteristic
150C22#41611yescharacteristic
151C22#41711yescharacteristic
152C22#42111yescharacteristic
153C22#42111yescharacteristic
154C4—22no—
155C4—21no—
156C4—21no—
157C4—22no—
158C4—22no—
159C4—41no—
160C4—41no—
161C4—81no—
162C4—81no—
163C4—81no—
164C4—81no—
165C2—42no—
166C2#147211yescharacteristic
167C2#148211yescharacteristic
168C2#149511yescharacteristic
169C2#151311yescharacteristic
170C2#151611yescharacteristic
171C2#153411yescharacteristic
172C2#155211yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1271C2semidirect
5. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#28512
#18172
#17972
#12712