Finite Group Databasebeta

← #7705

Subgroups — #7705

Order 256.

250 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 65 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#2857C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic
3#1749C211yesmaximal, characteristic
4#1271C214yesmaximal, complemented
5C2×C4⋊Q8C411yescharacteristic
6C2×C22⋊Q8—21no—
7C22×C42C2211yescharacteristic
8C2×C22⋊C8—21no—
9C2×C22⋊C8—21no—
10C2×#97—21no—
11#391—21no—
12#389—21no—
13#389C2212yes—
14#384C411yescharacteristic
15#380—21no—
16#380C412yes—
17#336—21no—
18#336C2212yes—
19#336C2212yes—
20C22×Q8—21no—
21C23×C4C2×C411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C22×C8—22no—
25C22×C8—22no—
26C4⋊Q8—42no—
27C22⋊Q8—44no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4C2×C412yes—
33C2×C4⋊C4C2×C412yes—
34C2×C22⋊C4—22no—
35C2×C42—42no—
36C2×C42C2×C411yescharacteristic
37C2×C42D411yescharacteristic
38C2×C42D411yescharacteristic
39C2×C42C2311yescharacteristic
40C22⋊C8—24no—
41C22⋊C8—42no—
42#97—42no—
43#95—22no—
44#95—22no—
45C24C22⋊C411yescharacteristic
46C2×Q8—42no—
47C2×Q8—42no—
48C2×Q8—42no—
49C22×C4—22no—
50C22×C4—22no—
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—41no—
58C22×C4—41no—
59C22×C4—41no—
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22⋊C411yescharacteristic
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22×C411yescharacteristic
64C22×C4C2×D411yescharacteristic
65C22×C4C2×D411yescharacteristic
66C2×C8—24no—
67C2×C8—42no—
68C2×C8—42no—
69C2×C8—42no—
70C2×C8—42no—
71C2×C8—42no—
72C4⋊C4—24no—
73C4⋊C4—42no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C4⋊C4—42no—
77C4⋊C4—44no—
78C22⋊C4—44no—
79C42—22no—
80C42—22no—
81C42—41no—
82C42—41no—
83C23—21no—
84C23—21no—
85C23—21no—
86C23—21no—
87C23—21no—
88C23—21no—
89C23#10112yes—
90C23C2×C22⋊C411yescharacteristic
91Q8—84no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—44no—
106C2×C4—44no—
107C2×C4—44no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4#9912yes—
115C2×C4#10412yes—
116C2×C4#13112yes—
117C2×C4AGL(1,ℤ/8)12yes—
118C2×C4#13712yes—
119C8—44no—
120C8—82no—
121C22—22no—
122C22—22no—
123C22—22no—
124C22—41no—
125C22—41no—
126C22—41no—
127C22—41no—
128C22#40811yescharacteristic
129C22C2×#9911yescharacteristic
130C22C2×#10111yescharacteristic
131C22#41811yescharacteristic
132C22#41911yescharacteristic
133C22C2×#10411yescharacteristic
134C22#42111yescharacteristic
135C4—22no—
136C4—22no—
137C4—22no—
138C4—22no—
139C4—22no—
140C4—41no—
141C4—82no—
142C4—82no—
143C2—42no—
144C2#147211yescharacteristic
145C2#148511yescharacteristic
146C2#149611yescharacteristic
147C2#150711yescharacteristic
148C2#151111yescharacteristic
149C2#153111yescharacteristic
150C2#155611yescharacteristic
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. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#18172
#17492
#12712