Finite Group Databasebeta

← #10216

Subgroups — #10216

Order 256.

234 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 49 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#3255C211yesmaximal, characteristic, complemented
2#3235C212yesmaximal
3#1963C211yesmaximal, characteristic
4#1506C212yesmaximal, complemented
5#1307C211yesmaximal, characteristic, complemented
6#533C2212yes—
7#532C2211yescharacteristic
8#501—22no—
9#490—22no—
10#482—24no—
11#476—22no—
12#475—22no—
13#474—22no—
14#470—22no—
15#462—22no—
16#432C2212yes—
17#423C2211yescharacteristic
18C2×#104—22no—
19#376C2211yescharacteristic
20#328—24no—
21C2×#41—21no—
22C2×Q16—22no—
23C2×M16D412yes—
24C4⋊Q8D411yescharacteristic
25C4⋊Q8D412yes—
26C4⋊Q8D412yes—
27#127D411yescharacteristic
28#125—22no—
29#124—21no—
30C22⋊Q8—22no—
31C22⋊Q8—22no—
32#121—22no—
33C4×Q8—22no—
34C4×Q8—22no—
35C4×D4—22no—
36#117—22no—
37C2×C42—21no—
38C2×C42C2311yescharacteristic
39#107—24no—
40#106—22no—
41#106—24no—
42C4⋊C8—22no—
43C4⋊C8—22no—
44#104—44no—
45#103—24no—
46#102—24no—
47#102—24no—
48C22⋊C8D414yes—
49#97—22no—
50#95—22no—
51#41—42no—
52#41—42no—
53C2×Q8—22no—
54C2×Q8—21no—
55C2×Q8—22no—
56C2×D4—22no—
57C2×D4—21no—
58C2×D4—41no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4C2×D411yescharacteristic
62C22×C4C2×D412yes—
63Q16—82no—
64M16—42no—
65C2×C8—24no—
66C2×C8—22no—
67C4⋊C4—22no—
68C4⋊C4—22no—
69C4⋊C4—22no—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4—42no—
73C4⋊C4—42no—
74C22⋊C4—42no—
75C22⋊C4—42no—
76C42—22no—
77C42—41no—
78C42—42no—
79C42C2×D411yescharacteristic
80C42C2×D412yes—
81C42#4111yescharacteristic
82C23—41no—
83C23#12711yescharacteristic
84Q8—42no—
85Q8—42no—
86Q8—42no—
87Q8—82no—
88D4—44no—
89D4—42no—
90D4—82no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—42no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—41no—
98C2×C4—82no—
99C2×C4—81no—
100C2×C4—81no—
101C2×C4#12012yes—
102C2×C4#12011yescharacteristic
103C2×C4#12112yes—
104C2×C4#12111yescharacteristic
105C8—44no—
106C8—42no—
107C22—21no—
108C22—22no—
109C22—81no—
110C22—82no—
111C22#39211yescharacteristic
112C4—21no—
113C4—22no—
114C4—22no—
115C4—82no—
116C4—81no—
117C4—82no—
118C4—82no—
119C4#45311yescharacteristic
120C4#45411yescharacteristic
121C2—41no—
122C2—161no—
123C2#200611yescharacteristic
124C2#200912yes—
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
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. #3255C2semidirect
4. #1506C2semidirect
5. #1307C2semidirect

Maximal subgroups

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

subgroupindex
#32552
#32352
#19632
#15062
#13072