Finite Group Databasebeta

← #12124

Subgroups — #12124

Order 256.

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

normal subgroups only

This group has 271 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#2842C211yesmaximal, characteristic, complemented
2#2021C211yesmaximal, characteristic
3#1613C212yesmaximal, complemented
4#1352C212yesmaximal, complemented
5#1316C211yesmaximal, characteristic, complemented
6#534C2211yescharacteristic
7C2×#120—21no—
8#481—22no—
9C2×#99—21no—
10#399—21no—
11#399C2212yes—
12#396—22no—
13#396—22no—
14#379—21no—
15#379C2211yescharacteristic
16#342C2211yescharacteristic
17#327—22no—
18#323C2212yes—
19C25D411yescharacteristic
20C22×D4—21no—
21C22×D4—41no—
22#123—41no—
23#121—22no—
24#120—42no—
25#120—41no—
26#120—42no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4D412yescomplemented
31C2×C22⋊C4—21no—
32C2×C22⋊C4—21no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—41no—
38C2×C22⋊C4D411yescharacteristic
39C2×C22⋊C4C2311yescharacteristic
40#106—42no—
41#102—42no—
42#99—42no—
43C22⋊C8—22no—
44#95—21no—
45#95D412yescomplemented
46C24—21no—
47C24—21no—
48C24—21no—
49C24—41no—
50C24—41no—
51C24—82no—
52C24SD1612yes—
53C24C2×D411yescharacteristic
54C2×D4—22no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4—41no—
58C2×D4—44no—
59C2×D4—41no—
60C2×D4—41no—
61C2×D4—41no—
62C2×D4—84no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—41no—
68C22×C4C2×D412yes—
69C2×C8—42no—
70C4⋊C4—22no—
71C4⋊C4—42no—
72C4⋊C4—82no—
73C4⋊C4—82no—
74C4⋊C4—81no—
75C22⋊C4—22no—
76C22⋊C4—41no—
77C22⋊C4—42no—
78C22⋊C4—44no—
79C22⋊C4—41no—
80C22⋊C4—41no—
81C22⋊C4—82no—
82C22⋊C4—82no—
83C22⋊C4—82no—
84C22⋊C4—81no—
85C22⋊C4—81no—
86C23—21no—
87C23—21no—
88C23—22no—
89C23—42no—
90C23—42no—
91C23—42no—
92C23—42no—
93C23—41no—
94C23—41no—
95C23—44no—
96C23—41no—
97C23—41no—
98C23—41no—
99C23—81no—
100C23—82no—
101C23—82no—
102C23—82no—
103C23—84no—
104C23—82no—
105C23#12011yescharacteristic
106C23C2×SD1611yescharacteristic
107C23#13711yescharacteristic
108D4—82no—
109D4—81no—
110D4—81no—
111D4—88no—
112D4—81no—
113D4—81no—
114C2×C4—22no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—42no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—82no—
129C8—82no—
130C22—22no—
131C22—21no—
132C22—21no—
133C22—41no—
134C22—41no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—44no—
139C22—42no—
140C22—81no—
141C22—82no—
142C22—82no—
143C22—84no—
144C22—82no—
145C22—82no—
146C22—81no—
147C22—82no—
148C22—161no—
149C22#44811yescharacteristic
150C22#45311yescharacteristic
151C22B(4,𝔽2)11yescharacteristic
152C4—42no—
153C4—81no—
154C4—82no—
155C4—82no—
156C4—81no—
157C4—82no—
158C2—21no—
159C2—21no—
160C2—41no—
161C2—41no—
162C2—82no—
163C2—161no—
164C2#159611yescharacteristic
165C2#220012yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#2021C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2842C2semidirect
3. #1613C2semidirect
4. #1352C2semidirect
5. #1316C2semidirect
30. C2×C4⋊C4D4semidirect
45. #95D4semidirect

Maximal subgroups

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

subgroupindex
#28422
#20212
#16132
#13522
#13162