Finite Group Databasebeta

← #26925

Subgroups — #26925

Order 256.

357 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 111 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#3082C212yesmaximal
2#2950C212yesmaximal
3#2047C212yesmaximal
4#2028C211yesmaximal, characteristic
5#2016C211yesmaximal, characteristic
6#1930C212yesmaximal
7#1922C212yesmaximal
8#1886C212yesmaximal
9#1857C211yesmaximal, characteristic
10C22×Q16C2212yes—
11#544C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13#519C2212yes—
14#519C2211yescharacteristic
15#479—22no—
16#479—22no—
17#477—22no—
18#475—22no—
19#441—22no—
20#441—22no—
21C4×Q16—22no—
22C4×Q16—22no—
23#428C2212yes—
24#427C2212yes—
25#424C2212yes—
26#419C2212yes—
27#419C2212yes—
28#419C2212yes—
29C2×#103C2212yes—
30C2×#101C2211yescharacteristic
31C2×#101C2212yes—
32#405C2212yes—
33#391C2211yescharacteristic
34#343C2211yescharacteristic
35#341C2212yes—
36#339C2212yes—
37#339C2212yes—
38#336C2211yescharacteristic
39C22×Q8C2312yes—
40C22×Q8C2311yescharacteristic
41C2×Q16—24no—
42C2×Q16—24no—
43C2×Q16—22no—
44C2×Q16—22no—
45C2×M16C2312yes—
46C2×M16C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—22no—
49C4⋊Q8—24no—
50#125—22no—
51C22⋊Q8—41no—
52C4×Q8—22no—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8—22no—
58#117—41no—
59#117—42no—
60#117C2312yes—
61#117C2312yes—
62C2×C4⋊C4—41no—
63C2×C4⋊C4C2311yescharacteristic
64C2×C42C2311yescharacteristic
65#107—22no—
66#107D412yes—
67#107D412yes—
68#106—22no—
69#106D412yes—
70#106D412yes—
71C4⋊C8—22no—
72C4⋊C8—22no—
73#103—22no—
74#103—22no—
75#103—22no—
76#103—22no—
77#103—22no—
78#103—22no—
79#103—22no—
80#103—22no—
81#101—24no—
82#101—22no—
83#97—22no—
84C4×C8—22no—
85#95—41no—
86C2×Q8—22no—
87C2×Q8—21no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—41no—
95C2×Q8#4114yes—
96C2×Q8#4112yes—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4C2411yescharacteristic
102Q16—28no—
103Q16—28no—
104M16—22no—
105M16—22no—
106M16—22no—
107M16—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8C2×D412yes—
113C2×C8C2×D412yes—
114C4⋊C4—22no—
115C4⋊C4—44no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—41no—
121C4⋊C4—42no—
122C4⋊C4C2×D412yes—
123C4⋊C4C2×D412yes—
124C4⋊C4C2×D412yes—
125C4⋊C4C2×D412yes—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C42—22no—
129C42—22no—
130C42—22no—
131C42—42no—
132C42—42no—
133C42—42no—
134C23H(5,𝔽2)11yescharacteristic
135Q8—24no—
136Q8—24no—
137Q8—24no—
138Q8—42no—
139Q8—42no—
140Q8—42no—
141Q8—161no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—81no—
157C2×C4C22×D412yes—
158C2×C4C2×#4111yescharacteristic
159C2×C4C2×#4112yes—
160C2×C42−1+411yescharacteristic
161C8—22no—
162C8—22no—
163C8—42no—
164C8—42no—
165C22—21no—
166C22—21no—
167C22#53911yescharacteristic
168C22D4211yescharacteristic
169C22#55111yescharacteristic
170C4—42no—
171C4—42no—
172C4—44no—
173C4—42no—
174C4—42no—
175C4—42no—
176C4—42no—
177C4—81no—
178C4#52512yes—
179C4#54812yes—
180C2—21no—
181C2—21no—
182C2#246311yescharacteristic
183C2#328911yescharacteristic
184C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC23H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#30822
#29502
#20472
#20282
#20162
#19302
#19222
#18862
#18572