Finite Group Databasebeta

← #11552

Subgroups — #11552

Order 256.

183 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 49 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#2499C211yesmaximal, characteristic
2#1464C211yesmaximal, characteristic
3#1458C211yesmaximal, characteristic
4#1325C212yesmaximal, complemented
5#1319C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic
7C2×#117C2211yescharacteristic
8#432—21no—
9#431—21no—
10#397—21no—
11#397—21no—
12#391C411yescharacteristic
13#391C2211yescharacteristic
14#387—21no—
15#387—21no—
16#384C2211yescharacteristic
17#382C411yescharacteristic
18C4×C22⋊C4C411yescharacteristic
19#324C2212yes—
20#324C2212yes—
21#323—21no—
22#323—21no—
23C22×Q8C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C22⋊Q8—41no—
26C22⋊Q8—41no—
27#117—41no—
28#117—41no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2311yescharacteristic
35C2×C22⋊C4—21no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4D411yescharacteristic
38C2×C22⋊C4D411yescharacteristic
39C2×C42C2×C411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C4⋊C8—41no—
44C4⋊C8—41no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47#97—41no—
48C4×C8—41no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56C24C22⋊C411yescharacteristic
57C2×Q8—22no—
58C2×Q8—41no—
59C2×Q8—41no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
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—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—42no—
80C4⋊C4—42no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C22⋊C4—41no—
84C22⋊C4—41no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C42—21no—
88C42—21no—
89C42—21no—
90C42—21no—
91C42—22no—
92C42—41no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23C2×C22⋊C411yescharacteristic
99Q8—82no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—22no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—41no—
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—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4#9911yescharacteristic
127C2×C4#9911yescharacteristic
128C2×C4#13112yes—
129C8—82no—
130C8—82no—
131C22—21no—
132C22—21no—
133C22—41no—
134C22—41no—
135C22—41no—
136C22—81no—
137C22—81no—
138C22#40811yescharacteristic
139C22C2×#9911yescharacteristic
140C22#41311yescharacteristic
141C4—21no—
142C4—21no—
143C4—22no—
144C4—42no—
145C4—41no—
146C4—41no—
147C4—41no—
148C4—41no—
149C4—81no—
150C4—81no—
151C4—81no—
152C2—21no—
153C2—21no—
154C2—81no—
155C2#146811yescharacteristic
156C2#212611yescharacteristic
157C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1325C2semidirect
5. #1319C2semidirect

Maximal subgroups

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

subgroupindex
#24992
#14642
#14582
#13252
#13192