Finite Group Databasebeta

← #7709

Subgroups — #7709

Order 256.

186 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 61 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#1831C211yesmaximal, characteristic
3#1745C211yesmaximal, characteristic
4#1272C212yesmaximal
5#1271C212yesmaximal
6C2×C4⋊C8—21no—
7C2×#97—21no—
8#391C411yescharacteristic
9#391C411yescharacteristic
10#391C2211yescharacteristic
11#389C411yescharacteristic
12#389C411yescharacteristic
13#389C2211yescharacteristic
14#387—21no—
15#387—21no—
16#387—21no—
17#383—21no—
18C43C2211yescharacteristic
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
39C4⋊C8—42no—
40#97—24no—
41#95—21no—
42#95—21no—
43#95—21no—
44#95—21no—
45#95—21no—
46#95—21no—
47#95—21no—
48#95—21no—
49#95—21no—
50C2×Q8—22no—
51C2×Q8—41no—
52C2×Q8—41no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22×C411yescharacteristic
62C22×C4C2×D411yescharacteristic
63C22×C4C2×D411yescharacteristic
64C2×C8—24no—
65C2×C8—42no—
66C2×C8—42no—
67C2×C8—42no—
68C2×C8—42no—
69C2×C8—42no—
70C4⋊C4—22no—
71C4⋊C4—41no—
72C4⋊C4—41no—
73C4⋊C4—42no—
74C4⋊C4—42no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C42—22no—
78C42—22no—
79C42—21no—
80C42—21no—
81C42—22no—
82C42—41no—
83C42—41no—
84C42—41no—
85C23C2×C22⋊C411yescharacteristic
86Q8—82no—
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—21no—
102C2×C4—21no—
103C2×C4—21no—
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—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4#10412yes—
117C2×C4#13112yes—
118C2×C4AGL(1,ℤ/8)11yescharacteristic
119C2×C4#13711yescharacteristic
120C2×C4#13712yes—
121C8—44no—
122C8—82no—
123C22#40811yescharacteristic
124C22#41011yescharacteristic
125C22#41311yescharacteristic
126C22#41711yescharacteristic
127C22#41911yescharacteristic
128C22C2×#10411yescharacteristic
129C22#42111yescharacteristic
130C4—22no—
131C4—22no—
132C4—22no—
133C4—21no—
134C4—21no—
135C4—41no—
136C4—41no—
137C4—41no—
138C4—81no—
139C4—81no—
140C4—81no—
141C4—81no—
142C2#147311yescharacteristic
143C2#148311yescharacteristic
144C2#149711yescharacteristic
145C2#150611yescharacteristic
146C2#151211yescharacteristic
147C2#152911yescharacteristic
148C2#155911yescharacteristic
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
#18312
#17452
#12722
#12712