Finite Group Databasebeta

← #11718

Subgroups — #11718

Order 256.

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

normal subgroups only

This group has 224 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#1740C211yesmaximal, characteristic
3#1458C211yesmaximal, characteristic
4#1352C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×#117—21no—
7C2×#117C2211yescharacteristic
8#431—21no—
9#400C412yescomplemented
10#399C2212yes—
11#395—21no—
12#393—21no—
13#389—21no—
14#382C411yescharacteristic, complemented
15#380—21no—
16#342—21no—
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×Q8—21no—
21C23×C4D411yescharacteristic
22C4⋊Q8—41no—
23C4⋊Q8—41no—
24#117—24no—
25#117—41no—
26#117—41no—
27#117—42no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—41no—
32C2×C4⋊C4C2×C412yes—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4D411yescharacteristic
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4D411yescharacteristic
38C2×C22⋊C4C2311yescharacteristic
39C2×C42—41no—
40C2×C42C2×C411yescharacteristic
41C2×C42D411yescharacteristic
42C4⋊C8—41no—
43C22⋊C8—22no—
44C4×C8—41no—
45#95—22no—
46#95—21no—
47#95—22no—
48#95C2×C412yes—
49C24C2×D411yescharacteristic
50C2×Q8—42no—
51C2×Q8—82no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—42no—
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22⋊C411yescharacteristic
62C22×C4SD1612yes—
63C22×C4Q1612yes—
64C22×C4C22×C411yescharacteristic
65C22×C4C2×D411yescharacteristic
66C2×C8—42no—
67C4⋊C4—41no—
68C4⋊C4—41no—
69C4⋊C4—42no—
70C4⋊C4—44no—
71C4⋊C4—41no—
72C4⋊C4—41no—
73C4⋊C4—42no—
74C4⋊C4—82no—
75C22⋊C4—24no—
76C22⋊C4—41no—
77C22⋊C4—41no—
78C22⋊C4—42no—
79C22⋊C4—82no—
80C42—21no—
81C42—21no—
82C42—21no—
83C42—21no—
84C42—44no—
85C23—21no—
86C23—21no—
87C23—41no—
88C23—41no—
89C23C2×C22⋊C411yescharacteristic
90C23C2×SD1611yescharacteristic
91C23C2×Q1611yescharacteristic
92Q8—84no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—44no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—82no—
115C2×C4—82no—
116C2×C4#9911yescharacteristic
117C2×C4#9911yescharacteristic
118C2×C4#10312yes—
119C2×C4#10312yes—
120C2×C4#10412yes—
121C8—82no—
122C22—21no—
123C22—21no—
124C22—21no—
125C22—21no—
126C22—41no—
127C22—41no—
128C22—41no—
129C22—41no—
130C22—81no—
131C22C2×#9911yescharacteristic
132C22C2×#10311yescharacteristic
133C22C2×#10411yescharacteristic
134C4—22no—
135C4—41no—
136C4—41no—
137C4—41no—
138C4—41no—
139C4—82no—
140C4—84no—
141C4—82no—
142C4#32812yes—
143C4#32812yes—
144C2—21no—
145C2—21no—
146C2—41no—
147C2—41no—
148C2C2×#32811yescharacteristic
149C2#212311yescharacteristic
150C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
9. #400C4semidirect
14. #382C4semidirect

Maximal subgroups

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

subgroupindex
#26282
#17402
#14582
#13522