Finite Group Databasebeta

← #11739

Subgroups — #11739

Order 256.

210 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 51 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#2638C211yesmaximal, characteristic
2C4×#100C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1354C212yesmaximal, complemented
5#1346C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C2×#100—21no—
9C2×C22⋊C8—21no—
10C4×M16—21no—
11#398C2211yescharacteristic
12#397—21no—
13#397C411yescharacteristic
14#395C2211yescharacteristic
15#394—21no—
16#393C411yescharacteristic
17#387—21no—
18#385—21no—
19#381C411yescharacteristic
20#343C2212yes—
21#324C2212yes—
22#323—21no—
23C22×Q8C2×C411yescharacteristic
24C22×D4D411yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—22no—
28C22×C8—41no—
29C22⋊Q8—41no—
30C22⋊Q8—41no—
31C4×D4—41no—
32C4×D4—41no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2311yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4C2×C411yescharacteristic
43C2×C22⋊C4D411yescharacteristic
44C2×C42D411yescharacteristic
45#100—24no—
46C22⋊C8—22no—
47C22⋊C8—41no—
48#97—41no—
49C4×C8—41no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24C22⋊C411yescharacteristic
58C24C22⋊C411yescharacteristic
59C2×Q8—41no—
60C2×Q8—81no—
61C2×D4—22no—
62C2×D4—22no—
63C2×D4—41no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—81no—
73C22×C4—81no—
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22×C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79M16—24no—
80M16—42no—
81C2×C8—42no—
82C2×C8—42no—
83C2×C8—82no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—42no—
89C4⋊C4—81no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—42no—
93C22⋊C4—42no—
94C22⋊C4—81no—
95C22⋊C4—81no—
96C42—21no—
97C42—21no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23#10412yes—
108C23C2×C22⋊C411yescharacteristic
109Q8—82no—
110D4—44no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
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—41no—
132C2×C4—41no—
133C2×C4—81no—
134C2×C4—81no—
135C2×C4—81no—
136C2×C4#9911yescharacteristic
137C2×C4#9911yescharacteristic
138C2×C4#13512yes—
139C8—44no—
140C8—82no—
141C22—21no—
142C22—21no—
143C22—22no—
144C22—42no—
145C22—41no—
146C22—41no—
147C22—41no—
148C22—41no—
149C22—41no—
150C22—81no—
151C22—81no—
152C22—81no—
153C22C2×#9911yescharacteristic
154C22#41611yescharacteristic
155C22C2×#10411yescharacteristic
156C4—21no—
157C4—21no—
158C4—22no—
159C4—41no—
160C4—41no—
161C4—41no—
162C4—81no—
163C4—81no—
164C4—81no—
165C4—81no—
166C2—21no—
167C2—21no—
168C2—42no—
169C2—81no—
170C2#150111yescharacteristic
171C2#212111yescharacteristic
172C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1354C2semidirect
5. #1346C2semidirect

Maximal subgroups

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

subgroupindex
#26382
C4×#1002
#14592
#13542
#13462