Finite Group Databasebeta

← #11560

Subgroups — #11560

Order 256.

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

normal subgroups only

This group has 267 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#2534C211yesmaximal, characteristic
2#1466C211yesmaximal, characteristic, complemented
3#1459C211yesmaximal, characteristic, complemented
4#1317C212yesmaximal, complemented
5#1316C212yesmaximal, complemented
6C2×#123C411yescharacteristic
7C2×#120C411yescharacteristic
8C2×C4×D4C2211yescharacteristic
9C22×C22⋊C4C411yescharacteristic
10#432—21no—
11C2×C22⋊C8—21no—
12#396—21no—
13#394—21no—
14#388—21no—
15#386—21no—
16#381C2211yescharacteristic
17#380C411yescharacteristic
18#379C2211yescharacteristic
19#324—21no—
20#323—21no—
21#323C2212yes—
22#323C2212yes—
23C25C2×C411yescharacteristic
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29C22×C8—41no—
30#123—42no—
31#120—42no—
32C4×D4—41no—
33C4×D4—41no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4D411yescharacteristic
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—41no—
44C2×C22⋊C4—41no—
45C2×C22⋊C4C2×C411yescharacteristic
46C2×C22⋊C4C2×C411yescharacteristic
47C2×C22⋊C4D411yescharacteristic
48C2×C22⋊C4C2311yescharacteristic
49C2×C42D411yescharacteristic
50C4⋊C8—41no—
51C22⋊C8—22no—
52C22⋊C8—22no—
53C22⋊C8—41no—
54#97—41no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24—21no—
59C24—22no—
60C24—41no—
61C24—42no—
62C24—41no—
63C24—81no—
64C24C22⋊C411yescharacteristic
65C24C22×C411yescharacteristic
66C2×D4—22no—
67C2×D4—22no—
68C2×D4—41no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—81no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—81no—
83C22×C4—81no—
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C2×C8—42no—
90C2×C8—42no—
91C2×C8—82no—
92C4⋊C4—41no—
93C4⋊C4—42no—
94C4⋊C4—81no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—81no—
106C42—21no—
107C42—21no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—21no—
112C23—22no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—42no—
121C23—42no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—82no—
126C23—84no—
127C23—84no—
128C23—82no—
129C23#10012yes—
130C23C2×C22⋊C411yescharacteristic
131C23#13112yes—
132D4—44no—
133D4—84no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—82no—
158C2×C4#9911yescharacteristic
159C2×C4#9911yescharacteristic
160C8—82no—
161C8—82no—
162C22—21no—
163C22—21no—
164C22—22no—
165C22—22no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—42no—
170C22—42no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—42no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—84no—
180C22—84no—
181C22—81no—
182C22—81no—
183C22#40811yescharacteristic
184C22C2×#9911yescharacteristic
185C22C2×#10011yescharacteristic
186C4—21no—
187C4—21no—
188C4—41no—
189C4—41no—
190C4—41no—
191C4—41no—
192C4—81no—
193C4—81no—
194C4—81no—
195C2—21no—
196C2—21no—
197C2—42no—
198C2—42no—
199C2—82no—
200C2#147111yescharacteristic
201C2#212011yescharacteristic
202C2#212111yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC2×#120C4
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
2. #1466C2semidirect
3. #1459C2semidirect
4. #1317C2semidirect
5. #1316C2semidirect

Maximal subgroups

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

subgroupindex
#25342
#14662
#14592
#13172
#13162