Finite Group Databasebeta

← #32821

Subgroups — #32821

Order 256.

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

normal subgroups only

This group has 326 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
1#2862C211yesmaximal, characteristic
2#2859C211yesmaximal, characteristic
3#2095C212yesmaximal
4#2072C212yesmaximal
5#2068C214yesmaximal
6#2057C212yesmaximal
7#2043C212yesmaximal
8#1831C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#125—21no—
12C2×#107C2214yes—
13C2×C4⋊C8C2212yes—
14C2×#103C2214yes—
15C2×#103C2214yes—
16#391—21no—
17#391—21no—
18#391C2212yes—
19#391C2212yes—
20#389—21no—
21#389—21no—
22#389C2212yes—
23#389C2212yes—
24#384C2211yescharacteristic
25#384C2211yescharacteristic
26#382—21no—
27#382—21no—
28#382—21no—
29C43C2211yescharacteristic
30#340C2214yes—
31#336C2212yes—
32#336C2212yes—
33C22×Q8—21no—
34C22×Q8C2312yes—
35C22×C8C2314yes—
36C4⋊Q8—42no—
37C4⋊Q8—42no—
38#125—44no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—21no—
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×C42D411yescharacteristic
52C2×C42D411yescharacteristic
53C2×C42D411yescharacteristic
54C2×C42D411yescharacteristic
55C2×C42C2311yescharacteristic
56C2×C42C2311yescharacteristic
57C2×C42C2311yescharacteristic
58#107—44no—
59C4⋊C8—42no—
60#103—44no—
61#103—44no—
62#95—22no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95—22no—
68C2×Q8—22no—
69C2×Q8—24no—
70C2×Q8—24no—
71C2×Q8—22no—
72C2×Q8—42no—
73C2×Q8—42no—
74C2×Q8—42no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2411yescharacteristic
86C2×C8—24no—
87C2×C8—24no—
88C2×C8—24no—
89C4⋊C4—22no—
90C4⋊C4—24no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C4⋊C4—44no—
97C4⋊C4—42no—
98C4⋊C4—42no—
99C4⋊C4—42no—
100C4⋊C4—44no—
101C4⋊C4—42no—
102C42—22no—
103C42—22no—
104C42—21no—
105C42—22no—
106C42—22no—
107C42—21no—
108C42—41no—
109C42Q1614yes—
110C23C22×D411yescharacteristic
111Q8—44no—
112Q8—44no—
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—21no—
125C2×C4—21no—
126C2×C4—21no—
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—42no—
138C2×C4—42no—
139C2×C4C2×Q1612yes—
140C2×C4C2×Q1612yes—
141C2×C4C2×Q1612yes—
142C2×C4AGL(1,ℤ/8)12yes—
143C2×C4#13712yes—
144C2×C4H(5,𝔽2)11yescharacteristic
145C2×C4H(5,𝔽2)12yes—
146C2×C42−1+412yes—
147C2×C42−1+411yescharacteristic
148C8—44no—
149C22#53511yescharacteristic
150C22#53611yescharacteristic
151C22#53711yescharacteristic
152C22C22×Q1611yescharacteristic
153C22C2×AGL(1,ℤ/8)11yescharacteristic
154C22C2×#13711yescharacteristic
155C22#57511yescharacteristic
156C4—22no—
157C4—22no—
158C4—22no—
159C4—22no—
160C4—22no—
161C4—41no—
162C4—82no—
163C4—82no—
164C4—82no—
165C4—82no—
166C4#56212yes—
167C4#56312yes—
168C2#282011yescharacteristic
169C2#320911yescharacteristic
170C2#321311yescharacteristic
171C2#323011yescharacteristic
172C2#323611yescharacteristic
173C2#324911yescharacteristic
174C2#325411yescharacteristic
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
#28592
#20952
#20722
#20682
#20572
#20432
#18312