Finite Group Databasebeta

← #11722

Subgroups — #11722

Order 256.

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

normal subgroups only

This group has 226 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#2627C211yesmaximal, characteristic
2C2×#343C211yesmaximal, characteristic
3C2×#324C211yesmaximal, characteristic
4#1354C212yesmaximal, complemented
5#1353C212yesmaximal, complemented
6C22×M16—21no—
7C2×C22⋊Q8C411yescharacteristic
8C22×C4⋊C4C2211yescharacteristic
9C2×C22⋊C8—21no—
10#400C411yescharacteristic
11#400C2211yescharacteristic
12#398—21no—
13#398—21no—
14#395C411yescharacteristic
15#395C2211yescharacteristic
16#385—21no—
17#381—21no—
18#380C411yescharacteristic
19#343—21no—
20#343C2212yes—
21#324—21no—
22#324C2212yes—
23C22×Q8C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—24no—
28C2×M16—22no—
29C2×M16—41no—
30C2×M16—42no—
31C22×C8—41no—
32C22×C8—41no—
33C22⋊Q8—41no—
34C22⋊Q8—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—41no—
40C2×C4⋊C4—41no—
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4C2311yescharacteristic
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C22⋊C8—22no—
49C22⋊C8—41no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95C2×C411yescharacteristic
57#95C2×C411yescharacteristic
58C24C22⋊C411yescharacteristic
59C2×Q8—41no—
60C2×Q8—81no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
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×C4SD1612yes—
78C22×C4Q1612yes—
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82M16—24no—
83M16—44no—
84M16—42no—
85C2×C8—44no—
86C2×C8—42no—
87C2×C8—42no—
88C2×C8—82no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—42no—
95C4⋊C4—81no—
96C4⋊C4—81no—
97C4⋊C4—81no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—42no—
101C23—41no—
102C23—41no—
103C23#9911yescharacteristic
104C23#9911yescharacteristic
105C23#10312yes—
106C23#10412yes—
107C23C2×C22⋊C411yescharacteristic
108Q8—82no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
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—42no—
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—42no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4#10312yes—
141C2×C4C2×SD1611yescharacteristic
142C2×C4C2×Q1611yescharacteristic
143C8—44no—
144C8—82no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22#32812yes—
154C22#32812yes—
155C22C2×#9911yescharacteristic
156C22C2×#10311yescharacteristic
157C22C2×#10411yescharacteristic
158C4—21no—
159C4—22no—
160C4—21no—
161C4—41no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—81no—
166C4—81no—
167C4—81no—
168C4—81no—
169C2—21no—
170C2—22no—
171C2—22no—
172C2—21no—
173C2C2×#32811yescharacteristic
174C2#212111yescharacteristic
175C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ 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. #1353C2semidirect

Maximal subgroups

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

subgroupindex
#26272
C2×#3432
C2×#3242
#13542
#13532