Finite Group Databasebeta

← #10026

Subgroups — #10026

Order 256.

282 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 51 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#3245C211yesmaximal, characteristic, complemented
2#3109C212yesmaximal, complemented
3#1955C211yesmaximal, characteristic, complemented
4#1530C212yesmaximal, complemented
5#1290C211yesmaximal, characteristic, complemented
6#532C2212yes—
7C2×C4×D4C2211yescharacteristic
8#481—22no—
9#476—22no—
10#474—22no—
11#466—22no—
12#459—22no—
13#459—22no—
14#447—22no—
15#442—24no—
16#432C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×C22⋊C8—22no—
19C2×#97C2211yescharacteristic
20#329—24no—
21C2×#41—22no—
22C22×D4D411yescharacteristic
23C23×C4—21no—
24C2×D8—24no—
25C2×D8—22no—
26C22×C8—21no—
27C22×C8D412yes—
28C4⋊Q8D412yes—
29#127D412yes—
30#124—22no—
31#121—22no—
32#121—22no—
33C4×D4—22no—
34C4×D4—22no—
35C4×D4—22no—
36C4×D4—22no—
37C2×C4⋊C4D411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C42C2311yescharacteristic
40#107—22no—
41#106—24no—
42C4⋊C8—22no—
43C4⋊C8—22no—
44#102—24no—
45#102—24no—
46#102—24no—
47C22⋊C8—22no—
48C22⋊C8—44no—
49#97—22no—
50#97D414yes—
51C24—21no—
52#41—84no—
53C2×Q8—22no—
54C2×D4—22no—
55C2×D4—22no—
56C2×D4—22no—
57C2×D4—22no—
58C2×D4—42no—
59C2×D4—42no—
60C22×C4—21no—
61C22×C4—42no—
62C22×C4—42no—
63C22×C4—41no—
64C22×C4—42no—
65C22×C4—41no—
66C22×C4C2×D411yescharacteristic
67C22×C4C2×D411yescharacteristic
68C22×C4#4111yescharacteristic
69D8—48no—
70D8—82no—
71C2×C8—22no—
72C2×C8—24no—
73C2×C8—22no—
74C2×C8—41no—
75C2×C8—41no—
76C2×C8—42no—
77C4⋊C4—22no—
78C4⋊C4—22no—
79C4⋊C4—22no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—42no—
83C42C2×D412yes—
84C42C2×D412yes—
85C23—22no—
86C23—42no—
87C23—41no—
88C23—41no—
89C23—42no—
90C23—42no—
91C23#12111yescharacteristic
92Q8—82no—
93D4—24no—
94D4—42no—
95D4—44no—
96D4—42no—
97D4—42no—
98D4—82no—
99D4—84no—
100C2×C4—22no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—41no—
106C2×C4—42no—
107C2×C4—82no—
108C2×C4—84no—
109C2×C4—81no—
110C2×C4—81no—
111C2×C4#12012yes—
112C2×C4#12011yescharacteristic
113C2×C4#12112yes—
114C2×C4#12711yescharacteristic
115C8—42no—
116C8—44no—
117C8—42no—
118C22—21no—
119C22—21no—
120C22—44no—
121C22—42no—
122C22—82no—
123C22—81no—
124C22—81no—
125C22—82no—
126C22—82no—
127C22—82no—
128C22#39211yescharacteristic
129C22#47212yes—
130C4—21no—
131C4—22no—
132C4—22no—
133C4—81no—
134C4—82no—
135C4#45311yescharacteristic
136C4#45411yescharacteristic
137C2—22no—
138C2—82no—
139C2—162no—
140C2#200611yescharacteristic
141C2#201011yescharacteristic
142C2#201811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3245C2semidirect
2. #3109C2semidirect
3. #1955C2semidirect
4. #1530C2semidirect
5. #1290C2semidirect

Maximal subgroups

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

subgroupindex
#32452
#31092
#19552
#15302
#12902