Finite Group Databasebeta

← #8583

Subgroups — #8583

Order 256.

270 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C4×#124C211yesmaximal, characteristic
2#1767C212yesmaximal, complemented
3C2×#329C212yesmaximal
4#1295C212yesmaximal, complemented
5C2×#124C2211yescharacteristic
6C2×C4×Q8C411yescharacteristic
7C2×C4×D4C411yescharacteristic
8C2×C4⋊C8—22no—
9C2×#97C2214yes—
10#391C411yescharacteristic
11#390C411yescharacteristic
12#388—22no—
13#382C2211yescharacteristic
14C4×C22⋊C4—22no—
15C43C2211yescharacteristic
16#336—22no—
17#329—24no—
18C22×Q8C2×C411yescharacteristic
19C22×D4C2×C411yescharacteristic
20C23×C4—21no—
21C22×C8—24no—
22#124—22no—
23#124D412yes—
24#124Q812yes—
25C4×Q8—42no—
26C4×D4—42no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C22⋊C4—22no—
31C2×C22⋊C4—21no—
32C2×C22⋊C4—22no—
33C2×C42—21no—
34C2×C42C2×C411yescharacteristic
35C2×C42C2×C411yescharacteristic
36C2×C42D412yes—
37C2×C42D412yes—
38C2×C42C2311yescharacteristic
39C4⋊C8—28no—
40#97—24no—
41#97—24no—
42#95—22no—
43#95—22no—
44C24—21no—
45C2×Q8—21no—
46C2×Q8—41no—
47C2×Q8—41no—
48C2×Q8C4⋊C412yes—
49C2×D4—21no—
50C2×D4—41no—
51C2×D4—41no—
52C2×D4C4⋊C412yes—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—44no—
57C22×C4—42no—
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C412yes—
61C22×C4C22×C411yescharacteristic
62C22×C4C2×D412yes—
63C2×C8—28no—
64C2×C8—44no—
65C2×C8—44no—
66C4⋊C4—41no—
67C4⋊C4—41no—
68C4⋊C4—42no—
69C22⋊C4—24no—
70C22⋊C4—24no—
71C22⋊C4—42no—
72C22⋊C4—44no—
73C42—22no—
74C42—24no—
75C42—41no—
76C42—42no—
77C42—42no—
78C42C2×D411yescharacteristic
79C42C2×Q811yescharacteristic
80C42#4112yes—
81C23—22no—
82C23—41no—
83C23—41no—
84C23—42no—
85C23—42no—
86C23C2×C22⋊C411yescharacteristic
87Q8—42no—
88Q8—42no—
89D4—42no—
90D4—42no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—21no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—21no—
100C2×C4—22no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—84no—
108C2×C4#10012yes—
109C2×C4#10414yes—
110C2×C4C2×C4⋊C411yescharacteristic
111C2×C4#11711yescharacteristic
112C2×C4#12112yes—
113C2×C4C22⋊Q812yes—
114C8—48no—
115C22—42no—
116C22—44no—
117C22—81no—
118C22—82no—
119C22—81no—
120C22#38011yescharacteristic
121C22C2×#10011yescharacteristic
122C22#41311yescharacteristic
123C22C2×#10412yes—
124C22#42112yes—
125C4—21no—
126C4—21no—
127C4—22no—
128C4—22no—
129C4—24no—
130C4—22no—
131C4—42no—
132C4—41no—
133C4—81no—
134C4—81no—
135C2—82no—
136C2#152511yescharacteristic
137C2#153212yes—
138C2#153311yescharacteristic
139C2#180011yescharacteristic
140C2#180612yes—
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1767C2semidirect
4. #1295C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1242
#17672
C2×#3292
#12952