Finite Group Databasebeta

← #12095

Subgroups — #12095

Order 256.

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

normal subgroups only

This group has 229 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#2817C211yesmaximal, characteristic, complemented
2#2022C211yesmaximal, characteristic
3#1607C212yesmaximal
4#1344C212yesmaximal, complemented
5#1317C211yesmaximal, characteristic, complemented
6#534C2211yescharacteristic
7C2×#123—21no—
8C2×C22⋊Q8—21no—
9#480—22no—
10C2×#99—21no—
11#400—22no—
12#397—21no—
13#397C2212yes—
14#395—22no—
15#381C2211yescharacteristic
16#342C2211yescharacteristic
17#327—22no—
18#323C2212yes—
19C22×Q8—21no—
20C22×D4—21no—
21C23×C4D411yescharacteristic
22#123—42no—
23#123—41no—
24#123—42no—
25C22⋊Q8—42no—
26C22⋊Q8—42no—
27#121—22no—
28#120—41no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—41no—
33C2×C4⋊C4D412yes—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—41no—
37C2×C22⋊C4D411yescharacteristic
38C2×C22⋊C4C2311yescharacteristic
39#107—42no—
40#102—42no—
41#99—42no—
42C22⋊C8—22no—
43#95—22no—
44#95—21no—
45#95—22no—
46#95—22no—
47#95D412yes—
48C24—21no—
49C24C2×D411yescharacteristic
50C2×Q8—42no—
51C2×Q8—81no—
52C2×Q8—81no—
53C2×D4—22no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4—41no—
58C2×D4—41no—
59C2×D4—41no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4D812yes—
70C22×C4C2×D412yes—
71C2×C8—42no—
72C4⋊C4—22no—
73C4⋊C4—42no—
74C4⋊C4—42no—
75C4⋊C4—44no—
76C4⋊C4—44no—
77C4⋊C4—81no—
78C22⋊C4—22no—
79C22⋊C4—41no—
80C22⋊C4—42no—
81C22⋊C4—44no—
82C22⋊C4—42no—
83C22⋊C4—41no—
84C22⋊C4—42no—
85C22⋊C4—41no—
86C23—21no—
87C23—21no—
88C23—42no—
89C23—41no—
90C23—41no—
91C23—41no—
92C23—82no—
93C23#12011yescharacteristic
94C23C2×D811yescharacteristic
95C23AGL(1,ℤ/8)11yescharacteristic
96Q8—84no—
97D4—82no—
98D4—81no—
99D4—81no—
100D4—81no—
101D4—81no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—41no—
110C2×C4—42no—
111C2×C4—44no—
112C2×C4—41no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—81no—
120C2×C4—82no—
121C8—82no—
122C22—22no—
123C22—21no—
124C22—21no—
125C22—41no—
126C22—41no—
127C22—42no—
128C22—81no—
129C22—82no—
130C22—81no—
131C22—82no—
132C22—161no—
133C22#44711yescharacteristic
134C22#45311yescharacteristic
135C22B(4,𝔽2)11yescharacteristic
136C4—42no—
137C4—42no—
138C4—82no—
139C4—81no—
140C4—82no—
141C4—81no—
142C4—82no—
143C2—21no—
144C2—21no—
145C2—41no—
146C2—41no—
147C2—161no—
148C2#159511yescharacteristic
149C2#220112yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#534C22
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. #2817C2semidirect
4. #1344C2semidirect
5. #1317C2semidirect

Maximal subgroups

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

subgroupindex
#28172
#20222
#16072
#13442
#13172