Finite Group Databasebeta

← #11724

Subgroups — #11724

Order 256.

216 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 63 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#2628C211yesmaximal, characteristic
2#1742C211yesmaximal, characteristic
3#1458C211yesmaximal, characteristic
4#1354C214yesmaximal, complemented
5C22×M16—21no—
6C2×C4⋊Q8C411yescharacteristic
7C2×#117C2211yescharacteristic
8#431—21no—
9#400—21no—
10#399—21no—
11#395C2212yes—
12#393C412yes—
13#389—21no—
14#382C411yescharacteristic
15#380—21no—
16#343—21no—
17#343C2212yes—
18#324C2212yes—
19#323—21no—
20C22×Q8C2×C412yes—
21C23×C4D411yescharacteristic
22C2×M16—24no—
23C2×M16—22no—
24C2×M16—41no—
25C2×M16—42no—
26C22×C8—41no—
27C4⋊Q8—41no—
28C4⋊Q8—41no—
29#117—41no—
30#117—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4D411yescharacteristic
36C2×C4⋊C4C2311yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4C2×C411yescharacteristic
39C2×C22⋊C4D411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42D411yescharacteristic
42C4⋊C8—41no—
43C22⋊C8—22no—
44C4×C8—41no—
45#95—21no—
46#95—22no—
47#95—22no—
48#95C2×C412yes—
49C24C22⋊C411yescharacteristic
50C2×Q8—42no—
51C2×Q8—82no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4SD1612yes—
61C22×C4Q1612yes—
62C22×C4C22×C411yescharacteristic
63C22×C4C2×D411yescharacteristic
64C22×C4C2×D411yescharacteristic
65M16—24no—
66M16—44no—
67M16—42no—
68C2×C8—42no—
69C2×C8—44no—
70C2×C8—42no—
71C4⋊C4—41no—
72C4⋊C4—41no—
73C4⋊C4—42no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4—42no—
77C4⋊C4—82no—
78C22⋊C4—41no—
79C22⋊C4—41no—
80C22⋊C4—82no—
81C42—21no—
82C42—21no—
83C42—21no—
84C42—21no—
85C23—21no—
86C23—21no—
87C23—41no—
88C23—41no—
89C23#10312yes—
90C23C2×C22⋊C411yescharacteristic
91Q8—84no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4#9911yescharacteristic
113C2×C4#9911yescharacteristic
114C2×C4#10312yes—
115C2×C4#10412yes—
116C2×C4C2×SD1611yescharacteristic
117C2×C4C2×Q1611yescharacteristic
118C8—44no—
119C8—82no—
120C22—21no—
121C22—21no—
122C22—22no—
123C22—42no—
124C22—41no—
125C22—41no—
126C22—81no—
127C22C2×#9911yescharacteristic
128C22C2×#10311yescharacteristic
129C22C2×#10411yescharacteristic
130C4—21no—
131C4—21no—
132C4—41no—
133C4—41no—
134C4—41no—
135C4—41no—
136C4—82no—
137C4—82no—
138C4#32812yes—
139C4#32812yes—
140C2—21no—
141C2—21no—
142C2—42no—
143C2C2×#32811yescharacteristic
144C2#212211yescharacteristic
145C2#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. #1354C2semidirect

Maximal subgroups

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

subgroupindex
#26282
#17422
#14582
#13542