Finite Group Databasebeta

← #11869

Subgroups — #11869

Order 256.

182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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#3059C211yesmaximal, characteristic
2#2115C211yesmaximal, characteristic
3#1812C211yesmaximal, characteristic
4#1340C214yesmaximal, complemented
5C22×Q16C411yescharacteristic, complemented
6C2×#131C2211yescharacteristic
7#536—21no—
8C2×M32—21no—
9#470C412yescomplemented
10#470C2212yes—
11#467—21no—
12#467—21no—
13#427—21no—
14#427C411yescharacteristic, complemented
15#419—21no—
16#410—21no—
17#349—21no—
18#349C2212yes—
19H(3,ℤ/4)C2212yes—
20C2×#41D411yescharacteristic
21C22×Q8—21no—
22C2×Q16—21no—
23C2×Q16—41no—
24C2×Q16—41no—
25C2×Q16C2×C412yes—
26#131—41no—
27#131—41no—
28C2×M16C2×C411yescharacteristic
29C2×M16D411yescharacteristic
30C2×M16C2311yescharacteristic
31C22×C8C2×C411yescharacteristic
32C22×C8D411yescharacteristic
33C22×C8D411yescharacteristic
34C4⋊Q8—41no—
35#124—41no—
36#123—41no—
37#123—41no—
38C22⋊Q8—22no—
39C22⋊Q8—22no—
40#117—21no—
41#117—22no—
42M32—22no—
43M32—41no—
44C2×C16—41no—
45#107—21no—
46#107—41no—
47#106—41no—
48#106C2×C412yes—
49#103—22no—
50#103—22no—
51#99—41no—
52#99—41no—
53#41—41no—
54#41—41no—
55C2×Q8—22no—
56C2×Q8—42no—
57C2×Q8—42no—
58C2×Q8—41no—
59C2×Q8SD1611yescharacteristic
60C2×D4D811yescharacteristic
61C2×D4D811yescharacteristic
62C2×D4SD1611yescharacteristic
63C22×C4—41no—
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C2×D411yescharacteristic
67Q16—42no—
68Q16—41no—
69Q16—41no—
70M16—41no—
71M16—41no—
72M16—41no—
73C2×C8—21no—
74C2×C8—21no—
75C2×C8—21no—
76C2×C8—21no—
77C2×C8—21no—
78C2×C8—21no—
79C2×C8C22⋊C411yescharacteristic
80C2×C8C22⋊C411yescharacteristic
81C2×C8C22×C411yescharacteristic
82C2×C8C2×D411yescharacteristic
83C4⋊C4—22no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C22⋊C4—41no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C42—42no—
92C42—41no—
93C16—42no—
94C23#10211yescharacteristic
95C23#10211yescharacteristic
96C23C2×D811yescharacteristic
97Q8—41no—
98Q8—42no—
99Q8—42no—
100Q8—82no—
101D4—41no—
102D4—41no—
103D4—41no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—82no—
113C2×C4—82no—
114C2×C4#10211yescharacteristic
115C2×C4#10211yescharacteristic
116C2×C4C2×C22⋊C411yescharacteristic
117C2×C4C2×SD1611yescharacteristic
118C8—21no—
119C8—21no—
120C8—21no—
121C8#9911yescharacteristic
122C8#9911yescharacteristic
123C22—21no—
124C22—21no—
125C22—21no—
126C22—41no—
127C22—41no—
128C22—41no—
129C22C2×#10211yescharacteristic
130C4—21no—
131C4—21no—
132C4—21no—
133C4—82no—
134C4—82no—
135C4—162no—
136C4C2×#9911yescharacteristic
137C4#42111yescharacteristic
138C2—21no—
139C2—41no—
140C2—41no—
141C2—41no—
142C2#151911yescharacteristic
subgroupG/H
center, socleC2#1519
commutatorC2×C8C22×C4
Frattini, cosocleC2×M16C23
prime radicalC2×#41D4
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1340C2semidirect
5. C22×Q16C4semidirect
9. #470C4semidirect
14. #427C4semidirect

Maximal subgroups

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

subgroupindex
#30592
#21152
#18122
#13402