Finite Group Databasebeta

← #11580

Subgroups — #11580

Order 256.

276 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 53 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#2409C211yesmaximal, characteristic
2#1465C211yesmaximal, characteristic
3#1460C211yesmaximal, characteristic
4#1326C214yesmaximal, complemented
5C23×Q8C411yescharacteristic
6C2×C22⋊Q8—21no—
7C2×C4×D4C2211yescharacteristic
8#431—21no—
9#407—21no—
10#395—21no—
11#393—21no—
12#391—21no—
13#386C412yes—
14#385C2212yes—
15#380C411yescharacteristic
16#324—21no—
17#323—21no—
18#323C2212yes—
19#323C2212yes—
20C22×Q8—21no—
21C22×Q8—21no—
22C22×Q8—41no—
23C22×Q8—41no—
24C22×Q8—44no—
25C22×D4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C412yes—
28C23×C4D411yescharacteristic
29C2×M16—41no—
30C22⋊Q8—44no—
31C4×D4—41no—
32C4×D4—41no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—22no—
37C2×C22⋊C4C2×C412yes—
38C2×C22⋊C4D411yescharacteristic
39C2×C22⋊C4C2311yescharacteristic
40C2×C42D411yescharacteristic
41C4⋊C8—41no—
42C22⋊C8—22no—
43C22⋊C8—22no—
44C22⋊C8—41no—
45C4×C8—41no—
46#95—21no—
47#95—22no—
48#95—22no—
49C24C22⋊C411yescharacteristic
50C24C22×C411yescharacteristic
51C2×Q8—24no—
52C2×Q8—42no—
53C2×Q8—44no—
54C2×Q8—42no—
55C2×Q8—44no—
56C2×Q8—48no—
57C2×Q8—44no—
58C2×Q8—41no—
59C2×Q8—41no—
60C2×D4—22no—
61C2×D4—22no—
62C2×D4—41no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—42no—
70C22×C4—44no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78M16—82no—
79C2×C8—42no—
80C2×C8—42no—
81C4⋊C4—41no—
82C4⋊C4—42no—
83C4⋊C4—44no—
84C22⋊C4—24no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—44no—
88C42—21no—
89C42—21no—
90C23—21no—
91C23—21no—
92C23—21no—
93C23—21no—
94C23—21no—
95C23—41no—
96C23—41no—
97C23—41no—
98C23—41no—
99C23C2×C22⋊C411yescharacteristic
100C23#13112yes—
101Q8—44no—
102Q8—48no—
103Q8—44no—
104D4—44no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—24no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—44no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—44no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—81no—
126C2×C4#9911yescharacteristic
127C2×C4#9911yescharacteristic
128C2×C4#10112yes—
129C8—82no—
130C8—82no—
131C22—21no—
132C22—21no—
133C22—22no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—41no—
138C22—41no—
139C22—41no—
140C22—41no—
141C22—41no—
142C22—81no—
143C22#40811yescharacteristic
144C22C2×#9911yescharacteristic
145C22C2×#10111yescharacteristic
146C4—21no—
147C4—22no—
148C4—41no—
149C4—44no—
150C4—41no—
151C4—41no—
152C4—82no—
153C4AΣL(1,𝔽16)12yes—
154C2—21no—
155C2—21no—
156C2—42no—
157C2—81no—
158C2#147211yescharacteristic
159C2C2×AΣL(1,𝔽16)11yescharacteristic
160C2#212311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1326C2semidirect

Maximal subgroups

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

subgroupindex
#24092
#14652
#14602
#13262