Finite Group Databasebeta

← #11889

Subgroups — #11889

Order 256.

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

normal subgroups only

This group has 206 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
1C2×#467C211yesmaximal, characteristic
2#2112C211yesmaximal, characteristic
3#1807C211yesmaximal, characteristic
4#1348C214yesmaximal, complemented
5C22×Q16C411yescharacteristic, complemented
6C23×C8C2211yescharacteristic
7C2×C22⋊Q8—21no—
8C2×M32—21no—
9#467—21no—
10#467—21no—
11#467C412yescomplemented
12#467C2212yes—
13C2×#107C411yescharacteristic, complemented
14C2×#106—21no—
15C2×#103—21no—
16#380—21no—
17C22⋊C16—21no—
18C22⋊C16C2212yes—
19#340C2212yes—
20C22×Q8—21no—
21C23×C4D411yescharacteristic
22C2×Q16—21no—
23C2×Q16—42no—
24C2×Q16C2×C412yes—
25C22×C8—42no—
26C22×C8C2×C411yescharacteristic
27C22×C8C2×C411yescharacteristic
28C22×C8D411yescharacteristic
29C22×C8D411yescharacteristic
30C22×C8D411yescharacteristic
31C22×C8C2311yescharacteristic
32C22⋊Q8—22no—
33C22⋊Q8—22no—
34C22⋊Q8—41no—
35C22⋊Q8—41no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—41no—
39C2×C22⋊C4—41no—
40C2×C22⋊C4—41no—
41M32—42no—
42C2×C16—22no—
43#107—21no—
44#107C2×C412yes—
45#106—42no—
46#103—22no—
47#103—22no—
48#95—41no—
49C24D811yescharacteristic
50C2×Q8—22no—
51C2×Q8—42no—
52C2×Q8—42no—
53C2×Q8—41no—
54C22×C4—41no—
55C22×C4—41no—
56C22×C4—41no—
57C22×C4—41no—
58C22×C4—42no—
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4D811yescharacteristic
62C22×C4SD1611yescharacteristic
63C22×C4SD1611yescharacteristic
64C22×C4C2×D411yescharacteristic
65Q16—42no—
66Q16—42no—
67C2×C8—22no—
68C2×C8—22no—
69C2×C8—22no—
70C2×C8—41no—
71C2×C8—41no—
72C2×C8—41no—
73C2×C8C22⋊C411yescharacteristic
74C2×C8C22⋊C411yescharacteristic
75C2×C8C22×C411yescharacteristic
76C2×C8C2×D411yescharacteristic
77C4⋊C4—22no—
78C4⋊C4—42no—
79C4⋊C4—42no—
80C4⋊C4—41no—
81C4⋊C4—42no—
82C4⋊C4—42no—
83C22⋊C4—42no—
84C22⋊C4—42no—
85C22⋊C4—82no—
86C16—42no—
87C23—22no—
88C23—22no—
89C23—41no—
90C23#10211yescharacteristic
91C23#10211yescharacteristic
92C23C2×D811yescharacteristic
93Q8—42no—
94Q8—42no—
95Q8—82no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—82no—
108C2×C4—82no—
109C2×C4—82no—
110C2×C4—82no—
111C2×C4—82no—
112C2×C4#10211yescharacteristic
113C2×C4#10211yescharacteristic
114C2×C4C2×C22⋊C411yescharacteristic
115C2×C4C2×SD1611yescharacteristic
116C8—21no—
117C8—21no—
118C8—21no—
119C8#9912yes—
120C22—21no—
121C22—21no—
122C22—21no—
123C22—21no—
124C22—21no—
125C22—21no—
126C22—21no—
127C22—44no—
128C22#36012yes—
129C22C2×#10211yescharacteristic
130C22#51012yes—
131C4—21no—
132C4—21no—
133C4—21no—
134C4—82no—
135C4—82no—
136C4—162no—
137C4C2×#9911yescharacteristic
138C4#42111yescharacteristic
139C2—22no—
140C2—22no—
141C2—41no—
142C2#151911yescharacteristic
143C2#214111yescharacteristic
144C2C2×#36011yescharacteristic
subgroupG/H
center, socleC22C2×#102
commutatorC2×C8C22×C4
Frattini, cosocleC22×C8C23
prime radicalC24D8
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1348C2semidirect
5. C22×Q16C4semidirect
11. #467C4semidirect
13. C2×#107C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
#21122
#18072
#13482