Finite Group Databasebeta

← #30205

Subgroups — #30205

Order 256.

375 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 143 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#2868C211yesmaximal, characteristic
2#2862C211yesmaximal, characteristic
3#2072C214yesmaximal
4#2024C214yesmaximal
5#1990C212yesmaximal
6#1956C212yesmaximal
7#1744C211yesmaximal, characteristic
8C2×C4⋊Q8—21no—
9C2×C4⋊Q8—21no—
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4⋊Q8C2212yes—
13C2×#103C2218yes—
14C2×C4×C8C2212yes—
15#391—21no—
16#391—21no—
17#391C2212yes—
18#391C2212yes—
19#384—21no—
20#384—21no—
21#384C2212yes—
22#384C2212yes—
23#382—21no—
24#382—21no—
25C43C2211yescharacteristic
26#340C2218yes—
27#336C2212yes—
28#336C2212yes—
29C22×Q8—22no—
30C22×Q8—21no—
31C22×Q8C2312yes—
32C22×C8C2314yes—
33C4⋊Q8—24no—
34C4⋊Q8—24no—
35C4⋊Q8—44no—
36C4⋊Q8—44no—
37C4⋊Q8—44no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4—22no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4C2312yes—
47C2×C4⋊C4C2314yes—
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42D411yescharacteristic
51C2×C42D411yescharacteristic
52C2×C42C2311yescharacteristic
53C2×C42C2311yescharacteristic
54C2×C42C2311yescharacteristic
55#103—48no—
56C4×C8—24no—
57#95—22no—
58#95—22no—
59#95—22no—
60#95—22no—
61C2×Q8—24no—
62C2×Q8—24no—
63C2×Q8—24no—
64C2×Q8—44no—
65C2×Q8—44no—
66C2×Q8—42no—
67C2×Q8—44no—
68C2×Q8—42no—
69C2×Q8—42no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—24no—
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C22×C4C2411yescharacteristic
80C2×C8—24no—
81C2×C8—24no—
82C2×C8—24no—
83C4⋊C4—24no—
84C4⋊C4—28no—
85C4⋊C4—44no—
86C4⋊C4—44no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—42no—
90C4⋊C4—44no—
91C4⋊C4—44no—
92C4⋊C4—44no—
93C42—22no—
94C42—22no—
95C42—22no—
96C42—22no—
97C42SD1614yes—
98C42Q1614yes—
99C42#4114yes—
100C23C22×D411yescharacteristic
101Q8—48no—
102Q8—88no—
103Q8—84no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—44no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—44no—
118C2×C4—44no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4#12412yes—
122C2×C4#12412yes—
123C2×C4C2×SD1612yes—
124C2×C4C2×SD1612yes—
125C2×C4C2×SD1612yes—
126C2×C4C2×Q1612yes—
127C2×C4C2×Q1612yes—
128C2×C4C2×Q1612yes—
129C2×C4AGL(1,ℤ/8)12yes—
130C2×C4#13712yes—
131C2×C4C2×#4112yes—
132C2×C4H(5,𝔽2)12yes—
133C2×C42−1+412yes—
134C8—44no—
135C22C2×#12411yescharacteristic
136C22#53511yescharacteristic
137C22#53611yescharacteristic
138C22C22×SD1611yescharacteristic
139C22C22×Q1611yescharacteristic
140C22C2×AGL(1,ℤ/8)11yescharacteristic
141C22C2×#13711yescharacteristic
142C4—22no—
143C4—22no—
144C4—22no—
145C4—22no—
146C4—82no—
147C4—82no—
148C4—84no—
149C4#48714yes—
150C4#48914yes—
151C4#54214yes—
152C2#264011yescharacteristic
153C2C2×#48711yescharacteristic
154C2C2×#48911yescharacteristic
155C2#320811yescharacteristic
156C2#320911yescharacteristic
157C2#322911yescharacteristic
158C2#323011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#28682
#28622
#20722
#20242
#19902
#19562
#17442