Finite Group Databasebeta

← #13303

Subgroups — #13303

Order 256.

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

normal subgroups only

This group has 198 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#2565C211yesmaximal, characteristic
2#1801C211yesmaximal, characteristic
3C4×#101C211yesmaximal, characteristic
4#1394C212yesmaximal
5#1394C212yesmaximal
6C2×#125C411yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8#423—21no—
9C2×#101C2212yes—
10C2×#101C2212yes—
11C4×M16—21no—
12#398—21no—
13#398—21no—
14#391C2211yescharacteristic
15#391C2211yescharacteristic
16#387—21no—
17#387—21no—
18#384C411yescharacteristic
19#382C411yescharacteristic
20C4×C4⋊C4C411yescharacteristic
21#343—21no—
22#343—21no—
23C22×Q8C2311yescharacteristic
24C2×M16—22no—
25C2×M16—22no—
26#125—41no—
27#125—41no—
28C4×Q8—41no—
29C4×Q8—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42C2×C411yescharacteristic
42C2×C42C2×C411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42D411yescharacteristic
45C4⋊C8—41no—
46C4⋊C8—41no—
47#101—22no—
48#101—24no—
49#101Q814yes—
50#97—41no—
51C4×C8—41no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59C2×Q8—41no—
60C2×Q8—41no—
61C2×Q8C2×Q812yes—
62C2×Q8#4112yes—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22×C411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74M16—24no—
75M16—42no—
76M16—42no—
77M16—42no—
78C2×C8—42no—
79C2×C8—42no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C42—21no—
92C42—21no—
93C42—21no—
94C42—21no—
95C42—21no—
96C42—21no—
97C42—41no—
98C42—41no—
99C23C2×C22⋊C411yescharacteristic
100Q8—42no—
101Q8—42no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—22no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4#9911yescharacteristic
127C2×C4#9911yescharacteristic
128C2×C4C4×Q812yes—
129C2×C4C22⋊Q812yes—
130C2×C4#12411yescharacteristic
131C2×C4C4⋊Q811yescharacteristic
132C8—44no—
133C8—82no—
134C22—21no—
135C22—21no—
136C22#39111yescharacteristic
137C22C2×#9911yescharacteristic
138C22#41011yescharacteristic
139C4—21no—
140C4—21no—
141C4—22no—
142C4—21no—
143C4—22no—
144C4—21no—
145C4—42no—
146C4—41no—
147C4—41no—
148C4—41no—
149C4—81no—
150C4—81no—
151C4—81no—
152C2—21no—
153C2—21no—
154C2#199611yescharacteristic
155C2#213211yescharacteristic
156C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC23C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#25652
#18012
C4×#1012
#13942