Finite Group Databasebeta

← #27620

Subgroups — #27620

Order 256.

335 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 115 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#2316C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2083C212yesmaximal, complemented
5#1921C212yesmaximal, complemented
6#1918C212yesmaximal, complemented
7#1852C212yesmaximal
8#1825C212yesmaximal
9#1818C211yesmaximal, characteristic
10C2×#117C2211yescharacteristic
11C2×#117C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C2×#107C2212yes—
14C2×#107C2212yes—
15C2×C22⋊C8C2212yes—
16#389—22no—
17#389C2212yes—
18#389C2212yes—
19#388—22no—
20#387—22no—
21#384C2211yescharacteristic
22#382C2211yescharacteristic
23#380—22no—
24#380C2212yes—
25C4×C4⋊C4C2212yes—
26#340C2212yes—
27#340C2212yes—
28#340C2212yes—
29#340C2212yes—
30#340C2212yes—
31#340C2212yes—
32#336C2212yes—
33#336C2211yescharacteristic
34#336C2211yescharacteristic
35C23×C4—21no—
36C23×C4C2311yescharacteristic
37C22×C8C2312yes—
38C22×C8C2312yes—
39#117—24no—
40#117—24no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—24no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4D412yes—
47C2×C4⋊C4C2312yes—
48C2×C4⋊C4C2312yes—
49C2×C4⋊C4C2312yes—
50C2×C4⋊C4C2312yes—
51C2×C22⋊C4—21no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4D412yes—
54C2×C42—21no—
55C2×C42—22no—
56C2×C42C2311yescharacteristic
57C2×C42C2311yescharacteristic
58#107—24no—
59#107—24no—
60C22⋊C8—24no—
61#95—22no—
62#95—22no—
63#95—22no—
64#95—22no—
65#95—22no—
66C24C2×D411yescharacteristic
67C22×C4—22no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—22no—
71C22×C4—24no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—82no—
76C22×C4C2×D412yes—
77C22×C4C2×D412yes—
78C22×C4C2×D411yescharacteristic
79C22×C4#4114yes—
80C22×C4C2411yescharacteristic
81C2×C8—22no—
82C2×C8—22no—
83C2×C8—22no—
84C2×C8—22no—
85C2×C8—22no—
86C2×C8—22no—
87C4⋊C4—24no—
88C4⋊C4—24no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—22no—
92C4⋊C4—24no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4—82no—
96C4⋊C4—82no—
97C4⋊C4—82no—
98C4⋊C4#4114yes—
99C22⋊C4—24no—
100C22⋊C4—42no—
101C22⋊C4—82no—
102C42—22no—
103C42—22no—
104C42—42no—
105C42—82no—
106C23—21no—
107C23—21no—
108C23—22no—
109C23—22no—
110C23#12312yes—
111C23C22×D411yescharacteristic
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—24no—
115C2×C4—24no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
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—84no—
135C2×C4#12312yes—
136C2×C4C2×#4112yes—
137C2×C4C2×#4112yes—
138C2×C42−1+411yescharacteristic
139C2×C42−1+411yescharacteristic
140C8—42no—
141C8—42no—
142C22—24no—
143C22—22no—
144C22—41no—
145C22—41no—
146C22—42no—
147C22C2×#12311yescharacteristic
148C22#52511yescharacteristic
149C22#53611yescharacteristic
150C22#57612yes—
151C22#57812yes—
152C4—22no—
153C4—42no—
154C4—42no—
155C4—44no—
156C4—82no—
157C4—82no—
158C4—82no—
159C4#55212yes—
160C4#55212yes—
161C2—42no—
162C2#248411yescharacteristic
163C2#309812yes—
164C2#312211yescharacteristic
165C2#312711yescharacteristic
166C2#324312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2083C2semidirect
5. #1921C2semidirect
6. #1918C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#23162
#23142
#21012
#20832
#19212
#19182
#18522
#18252
#18182