Finite Group Databasebeta

← #11808

Subgroups — #11808

Order 256.

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

normal subgroups only

This group has 340 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
1C2×#393C211yesmaximal, characteristic
2#1819C211yesmaximal, characteristic
3#1462C211yesmaximal, characteristic
4#1351C214yesmaximal, complemented
5C23×Q8C411yescharacteristic, complemented
6C22×C22⋊C4—21no—
7C22×C22⋊C4C2211yescharacteristic
8#407—21no—
9#393—21no—
10#393—21no—
11#393C412yescomplemented
12#393C2212yes—
13#379—21no—
14C2×#95—21no—
15C2×#95C411yescharacteristic, complemented
16#342—21no—
17#342C2212yes—
18#323—21no—
19#323C2212yes—
20C25D411yescharacteristic
21C22×Q8—21no—
22C22×Q8—44no—
23C22×Q8—41no—
24C22×Q8—41no—
25C22×Q8C2×C412yes—
26C23×C4—21no—
27C23×C4—21no—
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C2×M16—41no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—41no—
35C2×C22⋊C4—41no—
36C2×C22⋊C4—44no—
37C2×C22⋊C4—41no—
38C2×C22⋊C4—41no—
39C2×C22⋊C4C2×C411yescharacteristic
40C2×C22⋊C4D411yescharacteristic
41C2×C22⋊C4D411yescharacteristic
42C2×C22⋊C4C2311yescharacteristic
43C22⋊C8—22no—
44C22⋊C8—41no—
45#95—22no—
46#95—21no—
47#95—22no—
48#95C2×C412yes—
49C24—22no—
50C24—21no—
51C24—21no—
52C24—41no—
53C24—42no—
54C24—42no—
55C24C22⋊C411yescharacteristic
56C24C22⋊C411yescharacteristic
57C24C2×D411yescharacteristic
58C2×Q8—44no—
59C2×Q8—44no—
60C2×Q8—42no—
61C2×Q8—42no—
62C2×Q8—48no—
63C2×Q8—44no—
64C2×Q8—44no—
65C22×C4—24no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—42no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22×C411yescharacteristic
84C22×C4C2×D411yescharacteristic
85M16—82no—
86C2×C8—42no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—44no—
92C22⋊C4—44no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—82no—
96C22⋊C4—82no—
97C23—22no—
98C23—22no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23—21no—
105C23—21no—
106C23—42no—
107C23—42no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—48no—
113C23—42no—
114C23—42no—
115C23—81no—
116C23—81no—
117C23—84no—
118C23#9911yescharacteristic
119C23#9911yescharacteristic
120C23C2×C22⋊C411yescharacteristic
121C23#13712yes—
122Q8—44no—
123Q8—44no—
124Q8—48no—
125C2×C4—24no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—42no—
131C2×C4—44no—
132C2×C4—42no—
133C2×C4—44no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—81no—
145C2×C4—84no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—82no—
149C8—82no—
150C22—24no—
151C22—24no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—22no—
159C22—42no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22—81no—
167C22—84no—
168C22—84no—
169C22—81no—
170C22AΣL(1,𝔽16)12yes—
171C22#35412yes—
172C22C2×#9911yescharacteristic
173C22#41911yescharacteristic
174C22#42111yescharacteristic
175C4—44no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4—82no—
181C4—162no—
182C2—21no—
183C2—21no—
184C2—22no—
185C2—22no—
186C2—42no—
187C2—81no—
188C2#151111yescharacteristic
189C2C2×AΣL(1,𝔽16)11yescharacteristic
190C2C2×#35411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1351C2semidirect
5. C23×Q8C4semidirect
11. #393C4semidirect
15. C2×#95C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3932
#18192
#14622
#13512