Finite Group Databasebeta

← #11542

Subgroups — #11542

Order 256.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#2482C211yesmaximal, characteristic
2#1464C211yesmaximal, characteristic, complemented
3#1458C211yesmaximal, characteristic, complemented
4#1326C212yesmaximal, complemented
5#1316C212yesmaximal
6C2×#117C2211yescharacteristic
7C22×C22⋊C4C411yescharacteristic
8#432—21no—
9#431—21no—
10#399—21no—
11#397—21no—
12#388—21no—
13#388—21no—
14#385C411yescharacteristic
15#385C2211yescharacteristic
16#381C411yescharacteristic
17#379C411yescharacteristic
18#379C2211yescharacteristic
19#324—21no—
20#324—21no—
21#323C2212yes—
22#323C2212yes—
23C25C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26#117—41no—
27#117—41no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4D411yescharacteristic
31C2×C4⋊C4D411yescharacteristic
32C2×C22⋊C4—21no—
33C2×C22⋊C4—21no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—21no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—41no—
39C2×C22⋊C4—41no—
40C2×C22⋊C4C2×C411yescharacteristic
41C2×C22⋊C4C2×C411yescharacteristic
42C2×C22⋊C4C2×C411yescharacteristic
43C2×C22⋊C4C2311yescharacteristic
44C2×C42D411yescharacteristic
45C2×C42D411yescharacteristic
46C4⋊C8—41no—
47C4⋊C8—41no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50#97—41no—
51C4×C8—41no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56C24—21no—
57C24—22no—
58C24—41no—
59C24—42no—
60C24—41no—
61C24—81no—
62C24C22×C411yescharacteristic
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—81no—
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C2×C8—42no—
79C2×C8—42no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—81no—
83C4⋊C4—81no—
84C22⋊C4—22no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—81no—
94C22⋊C4—81no—
95C42—21no—
96C42—21no—
97C42—21no—
98C42—21no—
99C23—21no—
100C23—21no—
101C23—22no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23—42no—
108C23—42no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—82no—
113C23—84no—
114C23—84no—
115C23—82no—
116C23C2×C22⋊C411yescharacteristic
117C23#13112yes—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—82no—
143C2×C4#9911yescharacteristic
144C2×C4#9911yescharacteristic
145C8—82no—
146C8—82no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22—41no—
155C22—41no—
156C22—42no—
157C22—81no—
158C22—81no—
159C22—84no—
160C22—84no—
161C22—81no—
162C22—81no—
163C22#40811yescharacteristic
164C22C2×#9911yescharacteristic
165C22#41311yescharacteristic
166C4—21no—
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—81no—
174C4—81no—
175C4—81no—
176C2—21no—
177C2—21no—
178C2—42no—
179C2—82no—
180C2#146811yescharacteristic
181C2#212011yescharacteristic
182C2#212211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC25C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1464C2semidirect
3. #1458C2semidirect
4. #1326C2semidirect

Maximal subgroups

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

subgroupindex
#24822
#14642
#14582
#13262
#13162