Finite Group Databasebeta

← #28216

Subgroups — #28216

Order 256.

359 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 123 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×#479C212yesmaximal
2#2330C211yesmaximal, characteristic
3#2327C211yesmaximal, characteristic
4#2085C212yesmaximal
5#2043C212yesmaximal
6#1928C212yesmaximal
7#1927C212yesmaximal
8#1832C211yesmaximal, characteristic
9#1803C212yesmaximal
10C2×C4⋊Q8C2211yescharacteristic
11C2×#125C2211yescharacteristic
12C2×C4×Q8C2212yes—
13#479—28no—
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21#391C2212yes—
22#389—22no—
23#389C2212yes—
24#387—22no—
25#387—22no—
26#387—22no—
27#384C2211yescharacteristic
28#382C2211yescharacteristic
29C4×C4⋊C4C2212yes—
30#376C2211yescharacteristic
31#340C2212yes—
32#340C2212yes—
33#336C2212yes—
34#336C2211yescharacteristic
35#336C2211yescharacteristic
36C22×Q8C2312yes—
37C22×C8C2312yes—
38C22×C8C2312yes—
39C4⋊Q8—24no—
40#125—22no—
41#125D414yes—
42C4×Q8—24no—
43C4×Q8—24no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4C2312yes—
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C42—22no—
54C2×C42—22no—
55C2×C42C2311yescharacteristic
56C2×C42C2311yescharacteristic
57C2×C42C2311yescharacteristic
58#107—24no—
59#106—24no—
60C4⋊C8—24no—
61#103—24no—
62#103—24no—
63#103—24no—
64#103—24no—
65#95—22no—
66#95—21no—
67#95—21no—
68#95—22no—
69#95—24no—
70#95D412yes—
71#95D412yes—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—22no—
75C2×Q8—22no—
76C2×Q8—22no—
77C2×Q8#4114yes—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4C2×D412yes—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D412yes—
86C22×C4C2411yescharacteristic
87C2×C8—22no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—22no—
93C4⋊C4—24no—
94C4⋊C4—24no—
95C4⋊C4—28no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—44no—
102C4⋊C4—82no—
103C4⋊C4—82no—
104C4⋊C4C2×D414yes—
105C42—22no—
106C42—22no—
107C42—44no—
108C42—82no—
109C42C2×D412yes—
110C42#4112yes—
111C23C22×D411yescharacteristic
112Q8—24no—
113Q8—24no—
114Q8—42no—
115Q8—42no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—24no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4#12114yes—
138C2×C4C22×D411yescharacteristic
139C2×C4C2×#4112yes—
140C2×C4C2×#4111yescharacteristic
141C2×C42−1+412yes—
142C8—42no—
143C8—42no—
144C22C2×#12111yescharacteristic
145C22#52511yescharacteristic
146C22#53611yescharacteristic
147C22#57712yes—
148C22#57812yes—
149C4—24no—
150C4—41no—
151C4—41no—
152C4—42no—
153C4—44no—
154C4—44no—
155C4—82no—
156C4—82no—
157C4#54812yes—
158C4#55212yes—
159C2#253611yescharacteristic
160C2#304612yes—
161C2#312611yescharacteristic
162C2#312711yescharacteristic
163C2#324212yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ 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
C2×#4792
#23302
#23272
#20852
#20432
#19282
#19272
#18322
#18032