Finite Group Databasebeta

← #8864

Subgroups — #8864

Order 256.

209 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 63 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#2973C211yesmaximal, characteristic
2#1955C211yesmaximal, characteristic
3#1573C212yesmaximal, complemented
4#1460C211yesmaximal, characteristic
5#1302C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7#435—22no—
8#435—22no—
9#432C2212yes—
10#431C2212yes—
11C2×C4⋊C8C2211yescharacteristic
12#407—21no—
13C2×C22⋊C8—21no—
14C2×C22⋊C8—21no—
15C2×C22⋊C8—21no—
16C2×#97C2211yescharacteristic
17#335—22no—
18#334—22no—
19#323C412yes—
20#323C412yes—
21C22×D4Q811yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C2×M16—21no—
25C2×M16—21no—
26C22×C8—21no—
27C22×C8—21no—
28C22×C8—21no—
29C22×C8—21no—
30C4×D4—22no—
31C4×D4—22no—
32C2×C4⋊C4Q811yescharacteristic
33C2×C22⋊C4C2×C411yescharacteristic
34C2×C22⋊C4C2×C411yescharacteristic
35C2×C42C2311yescharacteristic
36C4⋊C8—22no—
37C4⋊C8—22no—
38C4⋊C8D412yes—
39C4⋊C8D412yes—
40C22⋊C8—22no—
41C22⋊C8—22no—
42C22⋊C8—22no—
43C22⋊C8—42no—
44C22⋊C8—42no—
45C22⋊C8C2×C414yes—
46#97—22no—
47#97—22no—
48C4×C8—22no—
49C24C4⋊C411yescharacteristic
50C24C4⋊C411yescharacteristic
51C2×D4—22no—
52C2×D4—22no—
53C2×D4—41no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—41no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4—41no—
62C22×C4C4⋊C411yescharacteristic
63C22×C4C4⋊C411yescharacteristic
64C22×C4C22×C411yescharacteristic
65C22×C4C2×D411yescharacteristic
66C22×C4C2×Q811yescharacteristic
67M16—81no—
68M16—81no—
69C2×C8—22no—
70C2×C8—24no—
71C2×C8—22no—
72C2×C8—22no—
73C2×C8—41no—
74C2×C8—41no—
75C2×C8—41no—
76C2×C8—41no—
77C2×C8—41no—
78C2×C8—81no—
79C2×C8—81no—
80C4⋊C4—22no—
81C22⋊C4—22no—
82C22⋊C4—22no—
83C42C2×D412yes—
84C42#4112yes—
85C23—21no—
86C23—21no—
87C23—21no—
88C23—21no—
89C23—21no—
90C23—41no—
91C23—41no—
92C23—41no—
93C23—41no—
94C23#10812yes—
95C23C2×C4⋊C411yescharacteristic
96D4—44no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—81no—
111C2×C4—81no—
112C2×C4C4×D412yes—
113C2×C4#12011yescharacteristic
114C2×C4C22⋊Q812yes—
115C2×C4#12311yescharacteristic
116C8—44no—
117C8—42no—
118C8—42no—
119C8—42no—
120C8—81no—
121C22—21no—
122C22—21no—
123C22—22no—
124C22—42no—
125C22—42no—
126C22—41no—
127C22—41no—
128C22—41no—
129C22—41no—
130C22—41no—
131C22—81no—
132C22—81no—
133C22#38511yescharacteristic
134C22C2×#10811yescharacteristic
135C22#43011yescharacteristic
136C4—22no—
137C4—21no—
138C4—22no—
139C4—41no—
140C4—41no—
141C4#45311yescharacteristic
142C4#45511yescharacteristic
143C2—21no—
144C2—21no—
145C2—42no—
146C2—81no—
147C2#185511yescharacteristic
148C2#185611yescharacteristic
149C2#186811yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#385
prime radicalC22×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1573C2semidirect
5. #1302C2semidirect

Maximal subgroups

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

subgroupindex
#29732
#19552
#15732
#14602
#13022