Finite Group Databasebeta

← #11472

Subgroups — #11472

Order 256.

244 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C4×#123C211yesmaximal, characteristic
2#1460C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1326C212yesmaximal, complemented
5#1317C212yesmaximal, complemented
6C2×#123C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C22×C42C411yescharacteristic
9#431—21no—
10C2×C22⋊C8—21no—
11#388—21no—
12#387—21no—
13#386C411yescharacteristic
14#385C2211yescharacteristic
15#381C2211yescharacteristic
16C4×C4⋊C4—21no—
17C4×C22⋊C4—21no—
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
28C22×C8—41no—
29#123—22no—
30#123C814yescomplemented
31C4×D4—41no—
32C4×D4—41no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4C2×C411yescharacteristic
39C2×C22⋊C4C2×C411yescharacteristic
40C2×C22⋊C4D411yescharacteristic
41C2×C22⋊C4C2311yescharacteristic
42C2×C42—21no—
43C2×C42—21no—
44C2×C42—21no—
45C2×C42—41no—
46C2×C42—41no—
47C2×C42D411yescharacteristic
48C4⋊C8—41no—
49C22⋊C8—22no—
50C22⋊C8—22no—
51C22⋊C8—41no—
52C4×C8—41no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24C22⋊C411yescharacteristic
59C24C22×C411yescharacteristic
60C2×D4—41no—
61C2×D4—41no—
62C2×D4C2×C812yes—
63C2×D4M1612yes—
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—42no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4C2×C812yescomplemented
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C2×C8—42no—
86C2×C8—42no—
87C2×C8—82no—
88C4⋊C4—24no—
89C4⋊C4—41no—
90C4⋊C4—42no—
91C22⋊C4—24no—
92C22⋊C4—21no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—42no—
96C22⋊C4—41no—
97C22⋊C4C2×C812yescomplemented
98C42—21no—
99C42—21no—
100C42—42no—
101C42—42no—
102C42—41no—
103C42—41no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23C22⋊C812yes—
113C23C2×C22⋊C411yescharacteristic
114C23C22×C811yescharacteristic
115C23C2×M1611yescharacteristic
116D4—42no—
117D4—42no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—81no—
147C2×C4C22⋊C812yes—
148C2×C4#9911yescharacteristic
149C2×C4#9911yescharacteristic
150C8—82no—
151C8—82no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—22no—
157C22—42no—
158C22—41no—
159C22—41no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22—81no—
166C22C2×C22⋊C811yescharacteristic
167C22C2×#9911yescharacteristic
168C22#41311yescharacteristic
169C4—21no—
170C4—22no—
171C4—42no—
172C4—41no—
173C4—41no—
174C4—41no—
175C4—42no—
176C4—42no—
177C4—81no—
178C4#35212yes—
179C2—21no—
180C2—21no—
181C2—41no—
182C2—41no—
183C2—42no—
184C2#145911yescharacteristic
185C2C2×#35211yescharacteristic
186C2#212011yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC23C22×C8
Frattini, cosocleC2×C22⋊C4C23
socleC22C2×#99
prime radicalC22×D4C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
30. #123C8semidirect
82. C22×C4C2×C8semidirect
97. C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#1232
#14602
#14592
#13262
#13172