Finite Group Databasebeta

← #12152

Subgroups — #12152

Order 256.

294 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 37 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#2537C211yesmaximal, characteristic, complemented
2#2039C211yesmaximal, characteristic, complemented
3#1598C212yesmaximal, complemented
4#1345C212yesmaximal, complemented
5#1325C211yesmaximal, characteristic, complemented
6#537C2211yescharacteristic
7C2×#121—22no—
8C2×C4×D4—21no—
9#447—22no—
10C2×#104—21no—
11#417—21no—
12#394—21no—
13#394C2212yes—
14#384C2211yescharacteristic
15#380—22no—
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C22×D4—21no—
21C22×D4—21no—
22C22×D4D412yes—
23C23×C4—22no—
24C2×D8—42no—
25C2×M16—21no—
26C22⋊Q8—41no—
27#121—22no—
28#121—41no—
29#121—48no—
30C4×D4—42no—
31C4×D4—42no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4D411yescharacteristic
35C2×C4⋊C4C2311yescharacteristic
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C42D411yescharacteristic
40#104—42no—
41#103—41no—
42#102—42no—
43#102—41no—
44C22⋊C8—22no—
45#95—21no—
46#95D412yes—
47C24—22no—
48#41—41no—
49#41—41no—
50#41—41no—
51#41—41no—
52C2×Q8—21no—
53C2×D4—24no—
54C2×D4—21no—
55C2×D4—22no—
56C2×D4—42no—
57C2×D4—42no—
58C2×D4—42no—
59C2×D4—44no—
60C2×D4—44no—
61C2×D4—41no—
62C2×D4—81no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—44no—
67C22×C4—44no—
68C22×C4—41no—
69C22×C4—44no—
70C22×C4C2×D411yescharacteristic
71C22×C4C2×D412yes—
72D8—84no—
73M16—41no—
74M16—41no—
75C2×C8—42no—
76C2×C8—41no—
77C4⋊C4—22no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4—42no—
81C4⋊C4—44no—
82C22⋊C4—42no—
83C22⋊C4—44no—
84C22⋊C4—44no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C42—21no—
88C42D812yescomplemented
89C23—24no—
90C23—42no—
91C23—42no—
92C23—42no—
93C23—42no—
94C23—42no—
95C23—42no—
96C23—41no—
97C23#12011yescharacteristic
98Q8—41no—
99Q8—41no—
100D4—48no—
101D4—41no—
102D4—41no—
103D4—44no—
104D4—82no—
105D4—82no—
106D4—84no—
107D4—81no—
108D4—81no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—88no—
123C2×C4—81no—
124C2×C4—81no—
125C2×C4C2×D811yescharacteristic
126C2×C4AGL(1,ℤ/8)11yescharacteristic
127C8—81no—
128C8—82no—
129C22—21no—
130C22—21no—
131C22—44no—
132C22—44no—
133C22—44no—
134C22—82no—
135C22—82no—
136C22—82no—
137C22—82no—
138C22—81no—
139C22—81no—
140C22—81no—
141C22#44711yescharacteristic
142C22#45411yescharacteristic
143C22B(4,𝔽2)11yescharacteristic
144C4—24no—
145C4—21no—
146C4—21no—
147C4—42no—
148C4—82no—
149C4—81no—
150C4—81no—
151C2—21no—
152C2—21no—
153C2—84no—
154C2—161no—
155C2#159811yescharacteristic
156C2#219611yescharacteristic
157C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2537C2semidirect
2. #2039C2semidirect
3. #1598C2semidirect
4. #1345C2semidirect
5. #1325C2semidirect
88. C42D8semidirect

Maximal subgroups

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

subgroupindex
#25372
#20392
#15982
#13452
#13252