Finite Group Databasebeta

← #12156

Subgroups — #12156

Order 256.

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

normal subgroups only

This group has 242 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#2819C211yesmaximal, characteristic, complemented
2#2040C211yesmaximal, characteristic, complemented
3#1598C212yesmaximal, complemented
4#1345C212yesmaximal, complemented
5#1324C211yesmaximal, characteristic, complemented
6#537C2211yescharacteristic
7C2×#124—21no—
8#447—22no—
9C2×#104—21no—
10#417—21no—
11#394—21no—
12#394C2212yes—
13#393—22no—
14#392—22no—
15#382C2211yescharacteristic
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C22×Q8—21no—
21C22×D4—21no—
22C22×D4—21no—
23C22×D4D412yes—
24C2×D8—42no—
25C2×M16—21no—
26#124—42no—
27#124—42no—
28C22⋊Q8—41no—
29#121—22no—
30#121—41no—
31C2×C4⋊C4D411yescharacteristic
32C2×C4⋊C4C2311yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C42D411yescharacteristic
38#104—42no—
39#103—41no—
40#102—42no—
41#102—41no—
42C22⋊C8—22no—
43#95—21no—
44#95—22no—
45#95D412yes—
46C24—22no—
47#41—41no—
48#41—41no—
49#41—41no—
50#41—41no—
51C2×Q8—21no—
52C2×Q8—42no—
53C2×Q8—81no—
54C2×Q8—81no—
55C2×D4—21no—
56C2×D4—22no—
57C2×D4—42no—
58C2×D4—42no—
59C2×D4—41no—
60C2×D4—82no—
61C2×D4—81no—
62C2×D4—81no—
63C2×D4—82no—
64C2×D4—81no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—22no—
68C22×C4—41no—
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D412yes—
71D8—84no—
72M16—41no—
73M16—41no—
74C2×C8—42no—
75C2×C8—41no—
76C4⋊C4—22no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C22⋊C4—44no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—42no—
83C22⋊C4—82no—
84C22⋊C4—82no—
85C42—21no—
86C42D812yescomplemented
87C23—42no—
88C23—42no—
89C23—42no—
90C23—42no—
91C23—42no—
92C23—42no—
93C23—41no—
94C23—42no—
95C23#12011yescharacteristic
96Q8—41no—
97Q8—41no—
98Q8—84no—
99D4—41no—
100D4—41no—
101D4—44no—
102D4—82no—
103D4—82no—
104D4—84no—
105D4—84no—
106D4—81no—
107D4—81no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—81no—
122C2×C4—81no—
123C2×C4C2×D811yescharacteristic
124C2×C4AGL(1,ℤ/8)11yescharacteristic
125C8—81no—
126C8—82no—
127C22—21no—
128C22—21no—
129C22—82no—
130C22—82no—
131C22—82no—
132C22—82no—
133C22—81no—
134C22—82no—
135C22—82no—
136C22—81no—
137C22—82no—
138C22—81no—
139C22#44711yescharacteristic
140C22#45411yescharacteristic
141C22B(4,𝔽2)11yescharacteristic
142C4—21no—
143C4—21no—
144C4—42no—
145C4—42no—
146C4—82no—
147C4—81no—
148C4—81no—
149C2—21no—
150C2—21no—
151C2—161no—
152C2—162no—
153C2#159811yescharacteristic
154C2#219911yescharacteristic
155C2#220211yescharacteristic
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. #2819C2semidirect
2. #2040C2semidirect
3. #1598C2semidirect
4. #1345C2semidirect
5. #1324C2semidirect
86. C42D8semidirect

Maximal subgroups

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

subgroupindex
#28192
#20402
#15982
#13452
#13242