Finite Group Databasebeta

← #9490

Subgroups — #9490

Order 256.

261 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 67 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#2983C211yesmaximal, characteristic, complemented
2#2536C211yesmaximal, characteristic
3#1801C211yesmaximal, characteristic, complemented
4#1486C212yesmaximal
5#1286C212yesmaximal, complemented
6C2×C22⋊Q8—21no—
7C2×C4×Q8C2211yescharacteristic
8C2×C4×D4C2211yescharacteristic
9#446—24no—
10#432—22no—
11#431C2212yes—
12#423C2212yes—
13C2×#101—21no—
14#398C412yes—
15#394C412yes—
16#388—22no—
17#382C2211yescharacteristic
18#380—21no—
19#343—21no—
20D4⋊C8—24no—
21C22×Q8D411yescharacteristic
22C22×D4C2×C411yescharacteristic
23C23×C4—21no—
24C2×M16—22no—
25C22⋊Q8—44no—
26C4×Q8—22no—
27C4×Q8—22no—
28C4×D4—22no—
29C4×D4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C412yes—
34C2×C4⋊C4D411yescharacteristic
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C42D412yes—
39C2×C42C2311yescharacteristic
40C4⋊C8—24no—
41C4⋊C8—22no—
42C4⋊C8—22no—
43C4⋊C8—22no—
44#101—42no—
45C22⋊C8—22no—
46#97—24no—
47C4×C8D414yes—
48#95—22no—
49#95—21no—
50#95C2×C412yes—
51C24—21no—
52C2×Q8—22no—
53C2×Q8—21no—
54C2×Q8—22no—
55C2×Q8—21no—
56C2×D4—22no—
57C2×D4—42no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4C22⋊C412yes—
65C22×C4C22⋊C412yes—
66C22×C4C22×C411yescharacteristic
67C22×C4C2×D411yescharacteristic
68C22×C4C2×D411yescharacteristic
69M16—42no—
70M16—42no—
71C2×C8—24no—
72C2×C8—42no—
73C4⋊C4—24no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—44no—
77C4⋊C4—42no—
78C22⋊C4—44no—
79C22⋊C4—44no—
80C22⋊C4—41no—
81C22⋊C4—41no—
82C42—22no—
83C42—22no—
84C42C2×D412yes—
85C42#4112yes—
86C23—22no—
87C23—41no—
88C23—41no—
89C23—41no—
90C23—41no—
91C23—41no—
92C23—41no—
93C23C2×C22⋊C411yescharacteristic
94Q8—42no—
95Q8—42no—
96D4—24no—
97D4—42no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—82no—
112C2×C4—82no—
113C2×C4C4×D412yes—
114C2×C4#12011yescharacteristic
115C2×C4#12112yes—
116C2×C4#12311yescharacteristic
117C2×C4AGL(1,ℤ/8)12yes—
118C2×C4AGL(1,ℤ/8)11yescharacteristic
119C2×C4#13711yescharacteristic
120C8—44no—
121C8—82no—
122C22—21no—
123C22—21no—
124C22—42no—
125C22—42no—
126C22—42no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22—81no—
131C22#38611yescharacteristic
132C22#41711yescharacteristic
133C22#41811yescharacteristic
134C4—22no—
135C4—21no—
136C4—21no—
137C4—42no—
138C4—42no—
139C4—82no—
140C4—81no—
141C4#46812yes—
142C4#46912yes—
143C2—21no—
144C2—21no—
145C2—82no—
146C2#189111yescharacteristic
147C2#189511yescharacteristic
148C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2983C2semidirect
3. #1801C2semidirect
5. #1286C2semidirect

Maximal subgroups

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

subgroupindex
#29832
#25362
#18012
#14862
#12862