Finite Group Databasebeta

← #11484

Subgroups — #11484

Order 256.

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

normal subgroups only

This group has 184 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#2930C211yesmaximal, characteristic
2#2115C212yesmaximal, complemented
3#1328C212yesmaximal, complemented
4#1320C212yesmaximal, complemented
5C2×#131C2211yescharacteristic
6C22×M16C411yescharacteristic
7#525C411yescharacteristic
8C2×M32—22no—
9#436—22no—
10#435—22no—
11#424C2211yescharacteristic
12#408C411yescharacteristic
13#407C2211yescharacteristic
14#405C411yescharacteristic
15#349—22no—
16#349C2214yes—
17C2×#41C2×C411yescharacteristic
18C23×C4C2×C411yescharacteristic
19#131—42no—
20C2×M16—21no—
21C2×M16—21no—
22C2×M16—42no—
23C2×M16C2×C411yescharacteristic
24C2×M16D412yes—
25C2×M16C2311yescharacteristic
26C22×C8C2×C411yescharacteristic
27C22×C8C2×C411yescharacteristic
28C22×C8D412yes—
29#123C812yes—
30C22⋊Q8—21no—
31#121—21no—
32#120C812yes—
33C4×D4—22no—
34#117C2×C411yescharacteristic
35M32—24no—
36M32—42no—
37C2×C16—42no—
38C4⋊C8—22no—
39C22⋊C8—22no—
40C22⋊C8—22no—
41#97—21no—
42C4×C8—21no—
43C24C2×C811yescharacteristic
44#41—41no—
45#41—41no—
46C2×Q8M1611yescharacteristic
47C2×D4—22no—
48C2×D4C2×C812yes—
49C2×D4M1611yescharacteristic
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—41no—
53C22×C4—41no—
54C22×C4C22⋊C412yes—
55C22×C4C2×C811yescharacteristic
56C22×C4C22×C411yescharacteristic
57M16—42no—
58M16—41no—
59M16—42no—
60M16—42no—
61M16—41no—
62C2×C8—22no—
63C2×C8—22no—
64C2×C8—22no—
65C2×C8—21no—
66C2×C8—21no—
67C2×C8—41no—
68C2×C8C22⋊C411yescharacteristic
69C2×C8C22⋊C411yescharacteristic
70C2×C8C2×D412yes—
71C4⋊C4—22no—
72C4⋊C4—21no—
73C22⋊C4—22no—
74C22⋊C4—22no—
75C22⋊C4C2×C812yes—
76C42—21no—
77C16—44no—
78C23—21no—
79C23—41no—
80C23—81no—
81C23C22⋊C812yes—
82C23C22×C811yescharacteristic
83Q8—41no—
84D4—42no—
85D4—41no—
86D4—82no—
87C2×C4—21no—
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—21no—
93C2×C4—42no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4C22⋊C812yes—
98C2×C4C2×C22⋊C411yescharacteristic
99C2×C4C2×M1611yescharacteristic
100C8—22no—
101C8—21no—
102C8—41no—
103C8—41no—
104C8#9912yes—
105C22—21no—
106C22—22no—
107C22—42no—
108C22—41no—
109C22—41no—
110C22—41no—
111C22—41no—
112C22—82no—
113C22C2×C22⋊C811yescharacteristic
114C4—21no—
115C4—22no—
116C4—42no—
117C4—41no—
118C4C2×#9911yescharacteristic
119C4#41311yescharacteristic
120C2—21no—
121C2—42no—
122C2—41no—
123C2—81no—
124C2#145911yescharacteristic
subgroupG/H
centerC4C2×#99
commutatorC23C22×C8
Frattini, cosocleC2×M16C23
socleC2#1459
prime radical#525C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2115C2semidirect
3. #1328C2semidirect
4. #1320C2semidirect

Maximal subgroups

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

subgroupindex
#29302
#21152
#13282
#13202