Finite Group Databasebeta

← #7251

Subgroups — #7251

Order 256.

232 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 59 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×#343C211yesmaximal, characteristic
2C2×#324C211yesmaximal, characteristic
3#1441C211yesmaximal, characteristic
4C22×M16—21no—
5C22×C4⋊C4C411yescharacteristic
6C22×C4⋊C4C2211yescharacteristic
7C2×C22⋊C8—21no—
8C2×#95—21no—
9C2×#95—21no—
10C2×#95C411yescharacteristic
11#343—21no—
12#343C412yescomplemented
13#324—21no—
14#324C412yescomplemented
15C23×C4—21no—
16C23×C4—21no—
17C23×C4C2×C411yescharacteristic
18C23×C4D411yescharacteristic
19C23×C4D411yescharacteristic
20C2×M16—22no—
21C2×M16—22no—
22C2×M16—22no—
23C2×M16—41no—
24C2×M16—41no—
25C2×M16—41no—
26C22×C8—41no—
27C22×C8—41no—
28C2×C4⋊C4—41no—
29C2×C4⋊C4—41no—
30C2×C4⋊C4—41no—
31C2×C4⋊C4—42no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4D411yescharacteristic
35C2×C4⋊C4Q811yescharacteristic
36C22⋊C8—22no—
37C22⋊C8—41no—
38#95—41no—
39#95—42no—
40#95—41no—
41#95—41no—
42C24C22⋊C411yescharacteristic
43C22×C4—21no—
44C22×C4—21no—
45C22×C4—21no—
46C22×C4—21no—
47C22×C4—21no—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—21no—
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—41no—
54C22×C4—41no—
55C22×C4—41no—
56C22×C4—41no—
57C22×C4—41no—
58C22×C4—41no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4C4211yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C4⋊C411yescharacteristic
73C22×C4C4⋊C411yescharacteristic
74C22×C4C4⋊C411yescharacteristic
75C22×C4D811yescharacteristic
76C22×C4SD1611yescharacteristic
77C22×C4SD1611yescharacteristic
78C22×C4Q1611yescharacteristic
79M16—24no—
80M16—42no—
81M16—42no—
82M16—42no—
83C2×C8—42no—
84C2×C8—42no—
85C2×C8—42no—
86C2×C8—42no—
87C2×C8—81no—
88C2×C8—81no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23#9511yescharacteristic
100C23#9911yescharacteristic
101C23#9911yescharacteristic
102C23#10211yescharacteristic
103C23#10311yescharacteristic
104C23#10411yescharacteristic
105C23#10411yescharacteristic
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4#10211yescharacteristic
147C2×C4#10311yescharacteristic
148C2×C4#10611yescharacteristic
149C2×C4#10711yescharacteristic
150C8—44no—
151C8—82no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22#32711yescharacteristic
165C22#32711yescharacteristic
166C22#32811yescharacteristic
167C22#32811yescharacteristic
168C22#33911yescharacteristic
169C22#34011yescharacteristic
170C22#34211yescharacteristic
171C22AΣL(1,𝔽16)11yescharacteristic
172C22#35211yescharacteristic
173C22#35511yescharacteristic
174C22#35611yescharacteristic
175C4—21no—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—41no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—82no—
184C4—82no—
185C2—21no—
186C2—21no—
187C2—21no—
188C2—21no—
189C2#129411yescharacteristic
190C2#134511yescharacteristic
191C2#134611yescharacteristic
192C2#135311yescharacteristic
193C2#135411yescharacteristic
194C2#139011yescharacteristic
195C2#139411yescharacteristic
subgroupG/H
center, socleC23#99
commutatorC22×C4C42
Frattini, cosocleC22×C4⋊C4C22
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #343C4semidirect
14. #324C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3432
C2×#3242
#14412