Finite Group Databasebeta

← #11569

Subgroups — #11569

Order 256.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#2563C211yesmaximal, characteristic
2#1465C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1324C214yesmaximal, complemented
5C2×C4⋊Q8C411yescharacteristic
6C2×C4×D4C2211yescharacteristic
7#407—21no—
8C2×C22⋊C8—21no—
9#393—21no—
10#393—21no—
11#390C412yes—
12#382C2212yes—
13#380—21no—
14#380—21no—
15C4×C4⋊C4C411yescharacteristic
16#324C2212yes—
17#324C2212yes—
18#323—21no—
19#323—21no—
20C22×Q8—21no—
21C22×D4C2×C411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C2×M16—41no—
25C22×C8—41no—
26C4⋊Q8—41no—
27C4⋊Q8—41no—
28C4×D4—41no—
29C4×D4—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4C2×C412yes—
33C2×C4⋊C4C2311yescharacteristic
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4D411yescharacteristic
37C2×C22⋊C4D411yescharacteristic
38C2×C42C2×C412yes—
39C2×C42C2×C411yescharacteristic
40C22⋊C8—22no—
41C22⋊C8—22no—
42C22⋊C8—41no—
43C22⋊C8—41no—
44#95—21no—
45#95—22no—
46#95—22no—
47C24C22⋊C411yescharacteristic
48C24C22⋊C411yescharacteristic
49C2×Q8—42no—
50C2×Q8—82no—
51C2×D4—22no—
52C2×D4—22no—
53C2×D4—41no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22×C411yescharacteristic
65C22×C4C2×D411yescharacteristic
66C22×C4C2×D411yescharacteristic
67M16—82no—
68C2×C8—42no—
69C2×C8—42no—
70C2×C8—82no—
71C4⋊C4—41no—
72C4⋊C4—42no—
73C4⋊C4—42no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C22⋊C4—41no—
77C22⋊C4—41no—
78C22⋊C4—82no—
79C22⋊C4—82no—
80C42—21no—
81C42—21no—
82C42—24no—
83C23—21no—
84C23—21no—
85C23—21no—
86C23—21no—
87C23—21no—
88C23—41no—
89C23—41no—
90C23—41no—
91C23—41no—
92C23#10112yes—
93C23C2×C22⋊C411yescharacteristic
94Q8—84no—
95D4—44no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—22no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—81no—
114C2×C4—81no—
115C2×C4#9911yescharacteristic
116C2×C4#9911yescharacteristic
117C2×C4#13112yes—
118C8—82no—
119C8—82no—
120C22—21no—
121C22—21no—
122C22—22no—
123C22—42no—
124C22—42no—
125C22—41no—
126C22—41no—
127C22—41no—
128C22—41no—
129C22—41no—
130C22—81no—
131C22—81no—
132C22#40811yescharacteristic
133C22C2×#9911yescharacteristic
134C22C2×#10111yescharacteristic
135C4—21no—
136C4—22no—
137C4—41no—
138C4—41no—
139C4—41no—
140C4—82no—
141C4—82no—
142C4#35412yes—
143C2—21no—
144C2—21no—
145C2—42no—
146C2—81no—
147C2#147211yescharacteristic
148C2C2×#35411yescharacteristic
149C2#213411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1324C2semidirect

Maximal subgroups

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

subgroupindex
#25632
#14652
#14592
#13242