Finite Group Databasebeta

← #7597

Subgroups — #7597

Order 256.

327 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 83 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#2297C211yesmaximal, characteristic
2#1797C212yesmaximal, complemented
3#1272C212yesmaximal, complemented
4#1270C212yesmaximal, complemented
5C22×#41C411yescharacteristic
6C2×#117C411yescharacteristic
7C22×C42C2211yescharacteristic
8C2×C4⋊C8—22no—
9C2×C22⋊C8—22no—
10#391C411yescharacteristic
11#391C2211yescharacteristic
12#390C411yescharacteristic
13#390C2211yescharacteristic
14#386—22no—
15C4×C22⋊C4—22no—
16#336C2214yes—
17C2×#41—22no—
18C2×#41—42no—
19C2×#41—42no—
20C2×#41C814yescomplemented
21C22×Q8C2×C411yescharacteristic
22C22×D4—21no—
23C22×D4C2×C411yescharacteristic
24C23×C4—21no—
25C23×C4C2×C411yescharacteristic
26C23×C4D412yes—
27C22×C8—24no—
28#117—42no—
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C22⋊C4—21no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C42—21no—
35C2×C42—42no—
36C2×C42C2×C411yescharacteristic
37C2×C42D412yes—
38C2×C42C2311yescharacteristic
39C4⋊C8—44no—
40C22⋊C8—44no—
41#95—22no—
42#95—22no—
43C24—21no—
44C24C22⋊C411yescharacteristic
45#41—48no—
46#41—48no—
47C2×Q8—21no—
48C2×Q8—41no—
49C2×Q8—41no—
50C2×Q8C2×C812yes—
51C2×D4—24no—
52C2×D4—21no—
53C2×D4—41no—
54C2×D4—41no—
55C2×D4—42no—
56C2×D4—42no—
57C2×D4C2×C812yes—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—24no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4C22⋊C412yes—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C2×C812yes—
73C22×C4M1612yes—
74C22×C4C22×C411yescharacteristic
75C22×C4C2×D412yes—
76C2×C8—44no—
77C2×C8—44no—
78C2×C8—44no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C22⋊C4—44no—
82C22⋊C4—42no—
83C22⋊C4—42no—
84C22⋊C4—42no—
85C42—24no—
86C42—41no—
87C42—41no—
88C42—42no—
89C23—21no—
90C23—21no—
91C23—22no—
92C23—22no—
93C23—22no—
94C23—41no—
95C23—41no—
96C23—42no—
97C23—42no—
98C23C22⋊C812yes—
99C23C2×C22⋊C411yescharacteristic
100Q8—42no—
101Q8—42no—
102D4—42no—
103D4—42no—
104D4—48no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—22no—
109C2×C4—24no—
110C2×C4—24no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—44no—
116C2×C4—44no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—48no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4C22⋊C812yes—
125C2×C4#9912yes—
126C2×C4#10012yes—
127C2×C4C22×C811yescharacteristic
128C2×C4C2×M1611yescharacteristic
129C2×C4AGL(1,ℤ/8)12yes—
130C2×C4#13712yes—
131C8—84no—
132C22—22no—
133C22—24no—
134C22—41no—
135C22—41no—
136C22—42no—
137C22—42no—
138C22—44no—
139C22—81no—
140C22—81no—
141C22—82no—
142C22C2×C22⋊C811yescharacteristic
143C22C2×#9911yescharacteristic
144C22C2×#10011yescharacteristic
145C22#41712yes—
146C22#42112yes—
147C4—21no—
148C4—21no—
149C4—22no—
150C4—22no—
151C4—42no—
152C4—42no—
153C4—81no—
154C4—81no—
155C4#32314yes—
156C2—42no—
157C2—82no—
158C2C2×#32311yescharacteristic
159C2#147712yes—
160C2#151312yes—
161C2#153611yescharacteristic
162C2#156011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC22×#41C4
LCSG ▷ C2×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
2. #1797C2semidirect
3. #1272C2semidirect
4. #1270C2semidirect
20. C2×#41C8semidirect

Maximal subgroups

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

subgroupindex
#22972
#17972
#12722
#12702