Finite Group Databasebeta

← #7721

Subgroups — #7721

Order 256.

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

normal subgroups only

This group has 222 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#2855C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic
3#1817C211yesmaximal, characteristic
4#1272C212yesmaximal, complemented
5#1271C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic
7C22×C42C2211yescharacteristic
8C2×C22⋊C8—21no—
9C2×C22⋊C8—21no—
10#391C411yescharacteristic
11#391C2211yescharacteristic
12#389C411yescharacteristic
13#389C2211yescharacteristic
14#387—21no—
15#387—21no—
16#385—21no—
17#381—21no—
18#380C411yescharacteristic
19#336—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×Q8C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C22×C8—22no—
28C22×C8—22no—
29C22⋊Q8—42no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C42—42no—
39C2×C42C2×C411yescharacteristic
40C2×C42D411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42C2311yescharacteristic
43C22⋊C8—42no—
44C22⋊C8—42no—
45#95—21no—
46#95—21no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—21no—
52C24C22⋊C411yescharacteristic
53C2×Q8—22no—
54C2×Q8—41no—
55C2×Q8—41no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22×C411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C2×C8—42no—
76C2×C8—42no—
77C2×C8—42no—
78C2×C8—42no—
79C2×C8—42no—
80C2×C8—42no—
81C4⋊C4—22no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C22⋊C4—42no—
88C22⋊C4—42no—
89C42—41no—
90C42—41no—
91C42—41no—
92C42—41no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23#9912yes—
100C23C2×C22⋊C411yescharacteristic
101Q8—82no—
102C2×C4—21no—
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—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—44no—
118C2×C4—44no—
119C2×C4—44no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4#13112yes—
133C2×C4#13512yes—
134C2×C4#13712yes—
135C8—82no—
136C8—82no—
137C22—22no—
138C22—22no—
139C22—22no—
140C22—41no—
141C22—41no—
142C22—41no—
143C22—41no—
144C22#40811yescharacteristic
145C22C2×#9911yescharacteristic
146C22#41311yescharacteristic
147C22#41611yescharacteristic
148C22#41911yescharacteristic
149C22#42111yescharacteristic
150C22#42111yescharacteristic
151C4—22no—
152C4—22no—
153C4—22no—
154C4—41no—
155C4—41no—
156C4—41no—
157C4—81no—
158C4—81no—
159C4—81no—
160C4—81no—
161C2—42no—
162C2#146811yescharacteristic
163C2#148211yescharacteristic
164C2#149711yescharacteristic
165C2#151111yescharacteristic
166C2#151611yescharacteristic
167C2#153511yescharacteristic
168C2#155411yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1272C2semidirect
5. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#28552
#18172
#12722
#12712