Finite Group Databasebeta

← #18148

Subgroups — #18148

Order 256.

465 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×#544C211yesmaximal, characteristic
2#3095C212yesmaximal
3#2484C211yesmaximal, characteristic
4#1617C214yesmaximal, complemented
5#1611C214yesmaximal, complemented
6#1511C212yesmaximal
7#1463C211yesmaximal, characteristic
8#544—24no—
9#544—22no—
10#544C2214yes—
11C2×C4⋊Q8—22no—
12C2×#125—22no—
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C2212yes—
15C2×#117—21no—
16C2×#117C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#485—24no—
19#484C2214yes—
20#483C2214yes—
21#428—22no—
22#419—22no—
23#407C2212yes—
24#400C2212yes—
25#398C2212yes—
26#387—22no—
27#385—21no—
28#381—21no—
29#381C2211yescharacteristic
30#328C2218yes—
31#324C2214yes—
32C22×Q8—21no—
33C22×Q8D412yes—
34C23×C4D411yescharacteristic
35C23×C4D411yescharacteristic
36C23×C4C2311yescharacteristic
37C2×M16—22no—
38C4⋊Q8—28no—
39C4⋊Q8—28no—
40#125—28no—
41#125—28no—
42C22⋊Q8—24no—
43C22⋊Q8—24no—
44C22⋊Q8—22no—
45C22⋊Q8—22no—
46C22⋊Q8D414yes—
47C22⋊Q8C2314yes—
48#117—24no—
49#117—24no—
50#117—22no—
51#117—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D412yes—
61C2×C4⋊C4D412yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C42—22no—
69C2×C42—21no—
70#107—24no—
71#106—24no—
72#103—24no—
73#103—24no—
74C22⋊C8—22no—
75C22⋊C8C2314yes—
76#95—22no—
77#95—22no—
78#95—21no—
79#95—22no—
80#95C2312yes—
81C24C2×D411yescharacteristic
82C2×Q8—24no—
83C2×Q8—22no—
84C2×Q8—44no—
85C2×Q8—82no—
86C2×Q8C2×D414yes—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—24no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4C2×D412yes—
100C22×C4C2×D412yes—
101C22×C4C2×D412yes—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D412yes—
104C22×C4C2411yescharacteristic
105M16—44no—
106C2×C8—24no—
107C4⋊C4—24no—
108C4⋊C4—24no—
109C4⋊C4—24no—
110C4⋊C4—28no—
111C4⋊C4—24no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—44no—
119C4⋊C4C2×D414yes—
120C22⋊C4—24no—
121C22⋊C4—24no—
122C22⋊C4—42no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C42—28no—
126C42—42no—
127C23—22no—
128C23—22no—
129C23—41no—
130C23#12011yescharacteristic
131C23#12011yescharacteristic
132C23C22×D411yescharacteristic
133Q8—44no—
134Q8—82no—
135Q8—84no—
136C2×C4—24no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—24no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—44no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—81no—
159C2×C4#12012yes—
160C2×C4#13714yes—
161C2×C4C22×D412yes—
162C8—44no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22#45812yes—
172C22C2×#12011yescharacteristic
173C22C2×#13712yes—
174C4—24no—
175C4—22no—
176C4—22no—
177C4—44no—
178C4—42no—
179C4—44no—
180C4—82no—
181C4—81no—
182C4—81no—
183C2—22no—
184C2—22no—
185C2—41no—
186C2#300411yescharacteristic
187C2#302011yescharacteristic
188C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1617C2semidirect
5. #1611C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#30952
#24842
#16172
#16112
#15112
#14632