Finite Group Databasebeta

← #29550

Subgroups — #29550

Order 256.

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

normal subgroups only

This group has 325 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#2320C212yesmaximal
2#2087C211yesmaximal, characteristic
3#2066C212yesmaximal
4#2056C211yesmaximal, characteristic
5#1928C212yesmaximal
6#1920C212yesmaximal
7#1878C212yesmaximal
8#1865C212yesmaximal
9#1832C211yesmaximal, characteristic
10C22×Q16C2211yescharacteristic
11C2×C4×Q8C2212yes—
12C2×#107C2211yescharacteristic
13C2×#106C2212yes—
14C2×C4⋊C8C2211yescharacteristic
15C2×C4⋊C8C2211yescharacteristic
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19#391C2212yes—
20#391C2212yes—
21#389C2212yes—
22#389C2212yes—
23#387—22no—
24#387—22no—
25#383—22no—
26#382—22no—
27C4×C4⋊C4C2212yes—
28#376C2211yescharacteristic
29#340C2211yescharacteristic
30#340C2212yes—
31#340C2211yescharacteristic
32#340C2212yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8C2312yes—
36C2×Q16—24no—
37C2×Q16—41no—
38C22×C8C2311yescharacteristic
39C22×C8C2312yes—
40C22×C8C2311yescharacteristic
41C4×Q8—24no—
42C4×Q8—24no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4C2312yes—
47C2×C4⋊C4C2312yes—
48C2×C4⋊C4C2312yes—
49C2×C42—22no—
50C2×C42—22no—
51C2×C42C2312yes—
52C2×C42C2311yescharacteristic
53#107—41no—
54#106—24no—
55C4⋊C8—41no—
56C4⋊C8—41no—
57#103—24no—
58#103—24no—
59#103—42no—
60#95—22no—
61#95—22no—
62#95—22no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95D411yescharacteristic
68#95D412yes—
69#95D411yescharacteristic
70C2×Q8—22no—
71C2×Q8—22no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—22no—
75C2×Q8#4114yes—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4C2×D412yes—
83C22×C4C2×D411yescharacteristic
84C22×C4C2411yescharacteristic
85Q16—44no—
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—22no—
91C2×C8—21no—
92C2×C8—21no—
93C4⋊C4—28no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—82no—
100C4⋊C4—82no—
101C4⋊C4#4114yes—
102C42—28no—
103C42—22no—
104C42—22no—
105C42—21no—
106C42—21no—
107C42—82no—
108C23C22×D411yescharacteristic
109Q8—24no—
110Q8—24no—
111Q8—42no—
112Q8—42no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—24no—
116C2×C4—24no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—42no—
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×C4C2×#4112yes—
137C2×C4C2×#4112yes—
138C2×C4H(5,𝔽2)11yescharacteristic
139C2×C42−1+411yescharacteristic
140C8—41no—
141C8—41no—
142C8—42no—
143C22#52512yes—
144C22#53711yescharacteristic
145C22#57712yes—
146C22#57811yescharacteristic
147C22#57811yescharacteristic
148C4—42no—
149C4—44no—
150C4—44no—
151C4—42no—
152C4—41no—
153C4—41no—
154C4—82no—
155C4—82no—
156C4#55111yescharacteristic
157C4#55212yes—
158C4#55511yescharacteristic
159C2#260911yescharacteristic
160C2#312012yes—
161C2#312112yes—
162C2#326111yescharacteristic
163C2#326311yescharacteristic
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
#23202
#20872
#20662
#20562
#19282
#19202
#18782
#18652
#18322