Finite Group Databasebeta

← #7601

Subgroups — #7601

Order 256.

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

normal subgroups only

This group has 246 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#1831C212yesmaximal, complemented
3#1272C212yesmaximal, complemented
4#1270C212yesmaximal
5C2×#124C411yescharacteristic
6C2×C4×Q8C411yescharacteristic
7C2×C4×D4C411yescharacteristic
8C2×C4⋊C8—22no—
9#391C2211yescharacteristic
10#390C2211yescharacteristic
11#388—22no—
12#382C411yescharacteristic
13C4×C22⋊C4—22no—
14C43C2211yescharacteristic
15#336—22no—
16#336C2214yes—
17C22×Q8C2×C411yescharacteristic
18C22×D4C2×C411yescharacteristic
19C23×C4—21no—
20C22×C8—24no—
21#124—22no—
22#124C814yescomplemented
23C4×Q8—42no—
24C4×D4—42no—
25C2×C4⋊C4—21no—
26C2×C4⋊C4C2×C411yescharacteristic
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C22⋊C4—22no—
29C2×C22⋊C4—21no—
30C2×C22⋊C4—22no—
31C2×C42—21no—
32C2×C42C2×C411yescharacteristic
33C2×C42C2×C411yescharacteristic
34C2×C42D412yes—
35C2×C42D412yes—
36C2×C42C2311yescharacteristic
37C4⋊C8—44no—
38#95—22no—
39#95—22no—
40C24—21no—
41C2×Q8—21no—
42C2×Q8—41no—
43C2×Q8—41no—
44C2×Q8C2×C812yes—
45C2×D4—21no—
46C2×D4—41no—
47C2×D4—41no—
48C2×D4C2×C812yes—
49C22×C4—21no—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—44no—
53C22×C4—42no—
54C22×C4C22⋊C411yescharacteristic
55C22×C4C22⋊C412yes—
56C22×C4C22⋊C411yescharacteristic
57C22×C4C22×C411yescharacteristic
58C22×C4C2×D412yes—
59C2×C8—44no—
60C2×C8—44no—
61C2×C8—44no—
62C4⋊C4—41no—
63C4⋊C4—41no—
64C4⋊C4—42no—
65C22⋊C4—28no—
66C22⋊C4—42no—
67C22⋊C4—42no—
68C22⋊C4—42no—
69C42—22no—
70C42—22no—
71C42—22no—
72C42—42no—
73C42—41no—
74C42—42no—
75C42C2×C812yes—
76C42M1612yes—
77C23—22no—
78C23—41no—
79C23—41no—
80C23—42no—
81C23—42no—
82C23C2×C22⋊C411yescharacteristic
83Q8—42no—
84Q8—42no—
85D4—42no—
86D4—42no—
87C2×C4—22no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—21no—
96C2×C4—22no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—84no—
104C2×C4C22⋊C814yes—
105C2×C4#10012yes—
106C2×C4C22×C811yescharacteristic
107C2×C4C2×M1611yescharacteristic
108C2×C4AGL(1,ℤ/8)12yes—
109C2×C4#13712yes—
110C8—84no—
111C22—42no—
112C22—44no—
113C22—81no—
114C22—82no—
115C22—81no—
116C22C2×C22⋊C811yescharacteristic
117C22#41011yescharacteristic
118C22C2×#10011yescharacteristic
119C22#41712yes—
120C22#42112yes—
121C4—21no—
122C4—21no—
123C4—24no—
124C4—22no—
125C4—22no—
126C4—22no—
127C4—41no—
128C4—42no—
129C4—81no—
130C4—81no—
131C2—82no—
132C2#146011yescharacteristic
133C2#147712yes—
134C2#151212yes—
135C2#153311yescharacteristic
136C2#156011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1831C2semidirect
3. #1272C2semidirect
22. #124C8semidirect

Maximal subgroups

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

subgroupindex
C4×#1242
#18312
#12722
#12702