Finite Group Databasebeta

← #11807

Subgroups — #11807

Order 256.

194 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 49 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#2621C211yesmaximal, characteristic
2#1822C211yesmaximal, characteristic
3#1464C211yesmaximal, characteristic
4#1346C214yesmaximal, complemented
5C2×C4⋊Q8C411yescharacteristic
6C2×C22⋊Q8—21no—
7C2×#117C2211yescharacteristic
8#432—21no—
9#407—21no—
10#398—21no—
11#398C2212yes—
12#397—21no—
13#397C412yes—
14#391—21no—
15#382C411yescharacteristic
16#343—21no—
17#343C2212yes—
18#324C2212yes—
19#323—21no—
20C22×Q8—21no—
21C23×C4D411yescharacteristic
22C2×M16—22no—
23C4⋊Q8—41no—
24C4⋊Q8—41no—
25C22⋊Q8—44no—
26#117—41no—
27#117—41no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4C2×C412yes—
32C2×C4⋊C4D411yescharacteristic
33C2×C4⋊C4C2311yescharacteristic
34C2×C22⋊C4—22no—
35C2×C22⋊C4C2×C411yescharacteristic
36C2×C22⋊C4D411yescharacteristic
37C2×C42C2×C411yescharacteristic
38C2×C42D411yescharacteristic
39C4⋊C8—41no—
40C22⋊C8—22no—
41C22⋊C8—41no—
42C22⋊C8—41no—
43#97—41no—
44#95—21no—
45#95—22no—
46#95—22no—
47#95C2×C412yes—
48C24C22⋊C411yescharacteristic
49C2×Q8—42no—
50C2×Q8—42no—
51C2×Q8—41no—
52C2×Q8—41no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—41no—
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22⋊C411yescharacteristic
62C22×C4C22×C411yescharacteristic
63C22×C4C2×D411yescharacteristic
64C22×C4C2×D411yescharacteristic
65M16—42no—
66M16—42no—
67C2×C8—42no—
68C2×C8—42no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C4⋊C4—42no—
72C4⋊C4—42no—
73C4⋊C4—42no—
74C4⋊C4—44no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C22⋊C4—44no—
78C42—21no—
79C42—21no—
80C42—21no—
81C42—21no—
82C23—21no—
83C23—21no—
84C23—21no—
85C23—41no—
86C23—41no—
87C23C2×C22⋊C411yescharacteristic
88Q8—82no—
89Q8—82no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4#9911yescharacteristic
111C2×C4#9911yescharacteristic
112C2×C4AGL(1,ℤ/8)12yes—
113C8—82no—
114C8—82no—
115C22—21no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—41no—
120C22—41no—
121C22—81no—
122C22C2×#9911yescharacteristic
123C22#41811yescharacteristic
124C22#42111yescharacteristic
125C4—22no—
126C4—21no—
127C4—21no—
128C4—41no—
129C4—41no—
130C4—41no—
131C4—41no—
132C4—82no—
133C4—82no—
134C2—21no—
135C2—21no—
136C2—81no—
137C2#151011yescharacteristic
138C2#212311yescharacteristic
139C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1346C2semidirect

Maximal subgroups

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

subgroupindex
#26212
#18222
#14642
#13462