Finite Group Databasebeta

← #11697

Subgroups — #11697

Order 256.

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

normal subgroups only

This group has 268 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#2392C211yesmaximal, characteristic
2#1742C211yesmaximal, characteristic, complemented
3#1458C211yesmaximal, characteristic, complemented
4#1346C212yesmaximal, complemented
5#1345C212yesmaximal, complemented
6C22×M16—21no—
7C2×C4×D4—21no—
8C2×C4×D4C411yescharacteristic
9C2×#117C2211yescharacteristic
10#431—21no—
11#398C2211yescharacteristic
12#397C411yescharacteristic
13#396—21no—
14#394—21no—
15#394C2211yescharacteristic
16#392C411yescharacteristic
17C4×C4⋊C4C411yescharacteristic
18C4×C22⋊C4—21no—
19#343—21no—
20#343C2212yes—
21#324C2212yes—
22#323—21no—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4—21no—
26C23×C4D411yescharacteristic
27C2×M16—24no—
28C2×M16—22no—
29C2×M16—41no—
30C2×M16—42no—
31C22×C8—41no—
32C4×D4—41no—
33C4×D4—41no—
34C4×D4—42no—
35C4×D4—42no—
36#117—41no—
37#117—41no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4C2311yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4C2×C411yescharacteristic
49C2×C22⋊C4D411yescharacteristic
50C2×C42—21no—
51C2×C42C2×C411yescharacteristic
52C2×C42D411yescharacteristic
53C4⋊C8—41no—
54C22⋊C8—22no—
55C4×C8—41no—
56#95—21no—
57#95—21no—
58#95C2×C411yescharacteristic
59#95C2×C411yescharacteristic
60C24—21no—
61C24C22⋊C411yescharacteristic
62C2×D4—41no—
63C2×D4—42no—
64C2×D4—42no—
65C2×D4—42no—
66C2×D4—42no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4D812yes—
81C22×C4SD1612yes—
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85M16—24no—
86M16—44no—
87M16—42no—
88C2×C8—42no—
89C2×C8—44no—
90C2×C8—42no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—42no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—42no—
102C22⋊C4—42no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C42—21no—
106C42—21no—
107C42—21no—
108C42—21no—
109C42—42no—
110C23—21no—
111C23—21no—
112C23—22no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23#10212yes—
122C23C2×C22⋊C411yescharacteristic
123D4—42no—
124D4—48no—
125D4—42no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4#9911yescharacteristic
152C2×C4#9911yescharacteristic
153C2×C4#10212yes—
154C2×C4#10412yes—
155C2×C4C2×D811yescharacteristic
156C2×C4C2×SD1611yescharacteristic
157C8—44no—
158C8—82no—
159C22—21no—
160C22—21no—
161C22—22no—
162C22—42no—
163C22—42no—
164C22—42no—
165C22—41no—
166C22—41no—
167C22—42no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22C2×#9911yescharacteristic
174C22C2×#10211yescharacteristic
175C22C2×#10411yescharacteristic
176C4—21no—
177C4—21no—
178C4—42no—
179C4—41no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—81no—
184C4—81no—
185C4#32712yes—
186C4#32712yes—
187C2—21no—
188C2—21no—
189C2—42no—
190C2—82no—
191C2C2×#32711yescharacteristic
192C2#212011yescharacteristic
193C2#213211yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
socleC22C2×#99
prime radical#392C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
2. #1742C2semidirect
3. #1458C2semidirect
4. #1346C2semidirect
5. #1345C2semidirect

Maximal subgroups

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

subgroupindex
#23922
#17422
#14582
#13462
#13452