Finite Group Databasebeta

← #11787

Subgroups — #11787

Order 256.

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

normal subgroups only

This group has 230 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
2C4×#99C211yesmaximal, characteristic
3#1465C211yesmaximal, characteristic
4#1351C214yesmaximal, complemented
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×D4C2211yescharacteristic
7C2×#99—21no—
8#407—21no—
9#393C412yescomplemented
10#393C2212yes—
11#390—21no—
12#382—21no—
13#380—21no—
14#380—21no—
15C4×C4⋊C4C411yescharacteristic, complemented
16C4×C22⋊C4—21no—
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×Q8C2×C412yes—
21C22×D4D411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C2×M16—41no—
25C4⋊Q8—41no—
26C4⋊Q8—41no—
27C4×D4—41no—
28C4×D4—41no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4D411yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4C2×C411yescharacteristic
37C2×C22⋊C4C2311yescharacteristic
38C2×C42—21no—
39C2×C42—41no—
40C2×C42C2×C411yescharacteristic
41#99—24no—
42C22⋊C8—22no—
43C22⋊C8—41no—
44#95—22no—
45#95—22no—
46#95—41no—
47#95C2×C412yes—
48C24C22⋊C411yescharacteristic
49C24C2×D411yescharacteristic
50C2×Q8—42no—
51C2×Q8—82no—
52C2×D4—22no—
53C2×D4—22no—
54C2×D4—41no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—42no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22×C411yescharacteristic
68C22×C4C2×D411yescharacteristic
69M16—82no—
70C2×C8—42no—
71C4⋊C4—41no—
72C4⋊C4—42no—
73C4⋊C4—42no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C22⋊C4—24no—
77C22⋊C4—41no—
78C22⋊C4—41no—
79C22⋊C4—42no—
80C22⋊C4—82no—
81C22⋊C4—82no—
82C42—21no—
83C42—21no—
84C42—42no—
85C42—82no—
86C23—21no—
87C23—21no—
88C23—21no—
89C23—21no—
90C23—21no—
91C23—41no—
92C23—41no—
93C23—41no—
94C23—41no—
95C23C2×C22⋊C411yescharacteristic
96C23#13712yes—
97Q8—84no—
98D4—44no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—44no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—81no—
118C2×C4—81no—
119C2×C4—82no—
120C2×C4—82no—
121C2×C4#9911yescharacteristic
122C2×C4#9911yescharacteristic
123C2×C4#10412yes—
124C8—82no—
125C22—21no—
126C22—21no—
127C22—22no—
128C22—42no—
129C22—42no—
130C22—41no—
131C22—41no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22—81no—
136C22—81no—
137C22C2×#9911yescharacteristic
138C22#41911yescharacteristic
139C22C2×#10411yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—41no—
144C4—41no—
145C4—82no—
146C4—84no—
147C4—82no—
148C4#35412yes—
149C2—21no—
150C2—21no—
151C2—42no—
152C2—81no—
153C2#150811yescharacteristic
154C2#212211yescharacteristic
155C2C2×#35411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1351C2semidirect
5. C2×C4⋊Q8C4semidirect
9. #393C4semidirect
15. C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#25632
C4×#992
#14652
#13512