Finite Group Databasebeta

← #11551

Subgroups — #11551

Order 256.

199 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 51 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#2503C211yesmaximal, characteristic
2#1463C211yesmaximal, characteristic
3C2×#324C211yesmaximal, characteristic
4#1324C212yesmaximal, complemented
5#1319C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8#407—21no—
9C2×C22⋊C8—21no—
10#400—21no—
11#400—21no—
12#391C411yescharacteristic
13#391C2211yescharacteristic
14#382C411yescharacteristic
15#382C2211yescharacteristic
16#381—21no—
17#381—21no—
18C4×C22⋊C4C411yescharacteristic
19#324—21no—
20#324—21no—
21#324C2212yes—
22#324C2212yes—
23C22×Q8C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—41no—
28C22×C8—41no—
29C22⋊Q8—41no—
30C22⋊Q8—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—41no—
33C2×C4⋊C4—41no—
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4D411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4C2311yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C42C2×C411yescharacteristic
42C2×C42C2×C411yescharacteristic
43C22⋊C8—22no—
44C22⋊C8—22no—
45C22⋊C8—41no—
46C22⋊C8—41no—
47#95—21no—
48#95—21no—
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—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—41no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22×C411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80M16—82no—
81C2×C8—42no—
82C2×C8—42no—
83C2×C8—82no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C42—22no—
94C42—41no—
95C23—22no—
96C23—21no—
97C23—21no—
98C23—41no—
99C23#9911yescharacteristic
100C23#9911yescharacteristic
101C23C2×C22⋊C411yescharacteristic
102Q8—82no—
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—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
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—42no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—81no—
132C2×C4—81no—
133C2×C4#13112yes—
134C8—82no—
135C8—82no—
136C22—21no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—42no—
144C22—42no—
145C22#35412yes—
146C22#40811yescharacteristic
147C22C2×#9911yescharacteristic
148C22#41311yescharacteristic
149C4—22no—
150C4—41no—
151C4—41no—
152C4—41no—
153C4—41no—
154C4—41no—
155C4—81no—
156C4—81no—
157C4—81no—
158C4—81no—
159C2—21no—
160C2—21no—
161C2—22no—
162C2—41no—
163C2#146811yescharacteristic
164C2C2×#35411yescharacteristic
165C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1324C2semidirect
5. #1319C2semidirect

Maximal subgroups

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

subgroupindex
#25032
#14632
C2×#3242
#13242
#13192