Finite Group Databasebeta

← #11528

Subgroups — #11528

Order 256.

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

normal subgroups only

This group has 221 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
1#2322C211yesmaximal, characteristic
2#1466C211yesmaximal, characteristic
3#1465C211yesmaximal, characteristic
4#1326C212yesmaximal, complemented
5#1317C212yesmaximal, complemented
6C2×#123C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C22×C4⋊C4C411yescharacteristic
9#432—21no—
10#407—21no—
11#389—21no—
12#388—21no—
13#387—21no—
14#386C411yescharacteristic
15#385—21no—
16#385C2211yescharacteristic
17#381C2211yescharacteristic
18C4×C22⋊C4C411yescharacteristic
19#324—21no—
20#323—21no—
21#323C2212yes—
22#323C2212yes—
23C22×D4C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4D411yescharacteristic
28C2×M16—41no—
29#123—22no—
30#123—22no—
31C4×D4—41no—
32C4×D4—41no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—41no—
38C2×C4⋊C4—41no—
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4C2×C411yescharacteristic
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C22⋊C4D411yescharacteristic
45C2×C22⋊C4C2311yescharacteristic
46C2×C42—21no—
47C2×C42D411yescharacteristic
48C4⋊C8—41no—
49C22⋊C8—22no—
50C22⋊C8—22no—
51C22⋊C8—41no—
52#97—41no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24C22⋊C411yescharacteristic
59C24C22×C411yescharacteristic
60C2×D4—21no—
61C2×D4—21no—
62C2×D4—41no—
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—21no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—81no—
77C22×C4—81no—
78C22×C4—81no—
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4M1612yescomplemented
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85M16—82no—
86C2×C8—42no—
87C2×C8—42no—
88C4⋊C4—24no—
89C4⋊C4—41no—
90C4⋊C4—42no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—81no—
94C22⋊C4—24no—
95C22⋊C4—21no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—81no—
100C22⋊C4M1612yescomplemented
101C42—21no—
102C42—21no—
103C42—81no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23C2×C22⋊C411yescharacteristic
114C23C2×M1611yescharacteristic
115C23C2×M1611yescharacteristic
116D4—42no—
117D4—42no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—82no—
147C2×C4#9911yescharacteristic
148C2×C4#9911yescharacteristic
149C8—82no—
150C8—82no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—41no—
156C22—41no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—81no—
165C22#40711yescharacteristic
166C22C2×#9911yescharacteristic
167C22#41311yescharacteristic
168C4—21no—
169C4—21no—
170C4—42no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—41no—
175C4—42no—
176C4—81no—
177C4—81no—
178C2—21no—
179C2—21no—
180C2—41no—
181C2—41no—
182C2—81no—
183C2#146511yescharacteristic
184C2#212111yescharacteristic
185C2#212311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1326C2semidirect
5. #1317C2semidirect
82. C22×C4M16semidirect
100. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
#23222
#14662
#14652
#13262
#13172