Finite Group Databasebeta

← #30060

Subgroups — #30060

Order 256.

354 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 121 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#2862C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic
3#2096C212yesmaximal
4#2058C212yesmaximal
5#2043C212yesmaximal
6#2024C212yesmaximal
7#1927C214yesmaximal
8#1839C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4×Q8—21no—
13C2×#106C2214yes—
14C2×#103C2214yes—
15C2×#103C2214yes—
16#391—21no—
17#391—21no—
18#391—21no—
19#391C2212yes—
20#391C2212yes—
21#389—21no—
22#389C2212yes—
23#384C2211yescharacteristic
24#382—21no—
25#382—21no—
26C4×C4⋊C4—21no—
27C4×C4⋊C4C2212yes—
28C43C2211yescharacteristic
29#340C2214yes—
30#336C2212yes—
31#336C2212yes—
32#336C2212yes—
33C22×Q8—21no—
34C22×Q8—21no—
35C22×Q8C2312yes—
36C22×C8C2314yes—
37C4⋊Q8—24no—
38C4⋊Q8—42no—
39C4⋊Q8—42no—
40C4×Q8—44no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4C2312yes—
49C2×C4⋊C4C2312yes—
50C2×C4⋊C4C2312yes—
51C2×C42—22no—
52C2×C42D411yescharacteristic
53C2×C42D411yescharacteristic
54C2×C42D411yescharacteristic
55C2×C42D411yescharacteristic
56C2×C42C2311yescharacteristic
57C2×C42C2311yescharacteristic
58C2×C42C2311yescharacteristic
59#106—44no—
60#103—28no—
61#103—44no—
62#95—22no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95—22no—
68C2×Q8—24no—
69C2×Q8—24no—
70C2×Q8—24no—
71C2×Q8—24no—
72C2×Q8—42no—
73C2×Q8—42no—
74C2×Q8—42no—
75C2×Q8—42no—
76C2×Q8—42no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2411yescharacteristic
88C2×C8—24no—
89C2×C8—24no—
90C2×C8—24no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—24no—
94C4⋊C4—24no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C4⋊C4—44no—
99C4⋊C4—42no—
100C4⋊C4—42no—
101C4⋊C4#4114yes—
102C42—22no—
103C42—22no—
104C42—22no—
105C42—22no—
106C42—22no—
107C42—41no—
108C42—44no—
109C42SD1614yes—
110C23C22×D411yescharacteristic
111Q8—48no—
112Q8—48no—
113Q8—84no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—24no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
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—42no—
138C2×C4—42no—
139C2×C4C2×SD1612yes—
140C2×C4C2×SD1612yes—
141C2×C4C2×SD1612yes—
142C2×C4#13712yes—
143C2×C4#13712yes—
144C2×C4C2×#4112yes—
145C2×C4H(5,𝔽2)12yes—
146C2×C42−1+412yes—
147C8—44no—
148C22#52511yescharacteristic
149C22#53511yescharacteristic
150C22#53611yescharacteristic
151C22C22×SD1611yescharacteristic
152C22C2×#13711yescharacteristic
153C22C2×#13711yescharacteristic
154C22#57511yescharacteristic
155C4—22no—
156C4—22no—
157C4—22no—
158C4—22no—
159C4—22no—
160C4—44no—
161C4—41no—
162C4—82no—
163C4—82no—
164C4—82no—
165C4#54214yes—
166C2#263211yescharacteristic
167C2#310111yescharacteristic
168C2#311311yescharacteristic
169C2#320611yescharacteristic
170C2#321511yescharacteristic
171C2#322911yescharacteristic
172C2#323711yescharacteristic
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
#28622
C4×C4⋊Q82
#20962
#20582
#20432
#20242
#19272
#18392