Finite Group Databasebeta

← #7729

Subgroups — #7729

Order 256.

173 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 173 conjugacy classes of subgroups — too many to draw. Only its 59 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#2863C211yesmaximal, characteristic
2#1839C211yesmaximal, characteristic
3#1839C211yesmaximal, characteristic
4#1272C212yesmaximal
5#1271C212yesmaximal
6#391C411yescharacteristic
7#391C411yescharacteristic
8#391C2211yescharacteristic
9#389C411yescharacteristic
10#389C411yescharacteristic
11#389C2211yescharacteristic
12#387—21no—
13#387—21no—
14#387—21no—
15#383—21no—
16C43C2211yescharacteristic
17#336—21no—
18#336—21no—
19#336—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×Q8C2×C411yescharacteristic
24C22×C8—22no—
25C22×C8—22no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C42C2×C411yescharacteristic
33C2×C42C2×C411yescharacteristic
34C2×C42D411yescharacteristic
35C2×C42D411yescharacteristic
36C2×C42D411yescharacteristic
37C2×C42D411yescharacteristic
38C2×C42C2311yescharacteristic
39#95—21no—
40#95—21no—
41#95—21no—
42#95—21no—
43#95—21no—
44#95—21no—
45#95—21no—
46#95—21no—
47#95—21no—
48C2×Q8—22no—
49C2×Q8—41no—
50C2×Q8—41no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4C22⋊C411yescharacteristic
56C22×C4C22⋊C411yescharacteristic
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22×C411yescharacteristic
60C22×C4C2×D411yescharacteristic
61C22×C4C2×D411yescharacteristic
62C2×C8—42no—
63C2×C8—42no—
64C2×C8—42no—
65C2×C8—42no—
66C2×C8—42no—
67C2×C8—42no—
68C4⋊C4—22no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C4⋊C4—42no—
72C4⋊C4—42no—
73C4⋊C4—42no—
74C42—22no—
75C42—22no—
76C42—22no—
77C42—41no—
78C42—41no—
79C42—41no—
80C42—41no—
81C23C2×C22⋊C411yescharacteristic
82Q8—82no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4#13112yes—
114C2×C4#13512yes—
115C2×C4#13712yes—
116C8—82no—
117C8—82no—
118C22#40811yescharacteristic
119C22#41011yescharacteristic
120C22#41311yescharacteristic
121C22#41611yescharacteristic
122C22#41911yescharacteristic
123C22#42111yescharacteristic
124C22#42111yescharacteristic
125C4—22no—
126C4—22no—
127C4—22no—
128C4—41no—
129C4—41no—
130C4—41no—
131C4—41no—
132C4—81no—
133C4—81no—
134C4—81no—
135C4—81no—
136C2#147311yescharacteristic
137C2#148211yescharacteristic
138C2#149711yescharacteristic
139C2#151511yescharacteristic
140C2#152111yescharacteristic
141C2#153511yescharacteristic
142C2#155411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#28632
#18392
#12722
#12712