Finite Group Databasebeta

← #10028

Subgroups — #10028

Order 256.

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

normal subgroups only

This group has 256 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
2#3112C212yesmaximal, complemented
3#1985C211yesmaximal, characteristic, complemented
4#1530C212yesmaximal, complemented
5#1290C211yesmaximal, characteristic, complemented
6#532C2212yes—
7C2×C4×Q8C2211yescharacteristic
8#480—22no—
9#477—22no—
10#474—22no—
11#466—22no—
12#460—22no—
13#459—22no—
14#448—22no—
15#440—24no—
16#432C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×#97C2211yescharacteristic
19#336—22no—
20#329—24no—
21C2×#41—22no—
22C22×Q8D411yescharacteristic
23C2×SD16—24no—
24C2×D8—22no—
25C22×C8—21no—
26C22×C8D412yes—
27C4⋊Q8D412yes—
28#127D412yes—
29#124—22no—
30#121—22no—
31#121—22no—
32C4×Q8—22no—
33C4×Q8—22no—
34C4×D4—22no—
35C4×D4—22no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4D411yescharacteristic
38C2×C42—21no—
39C2×C42C2311yescharacteristic
40#107—24no—
41#107—22no—
42C4⋊C8—22no—
43C4⋊C8—22no—
44#103—24no—
45#102—24no—
46#102—24no—
47C22⋊C8—22no—
48#97—22no—
49#97D414yes—
50#41—84no—
51C2×Q8—22no—
52C2×Q8—22no—
53C2×Q8—42no—
54C2×D4—22no—
55C2×D4—22no—
56C2×D4—22no—
57C2×D4—42no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—42no—
61C22×C4C2×D411yescharacteristic
62C22×C4C2×D411yescharacteristic
63C22×C4#4111yescharacteristic
64SD16—48no—
65D8—82no—
66C2×C8—22no—
67C2×C8—24no—
68C2×C8—22no—
69C2×C8—41no—
70C2×C8—41no—
71C2×C8—42no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—42no—
76C22⋊C4—42no—
77C22⋊C4—42no—
78C42—41no—
79C42—41no—
80C42C2×D412yes—
81C42C2×D412yes—
82C23—42no—
83C23#12111yescharacteristic
84Q8—24no—
85Q8—42no—
86Q8—82no—
87D4—44no—
88D4—42no—
89D4—42no—
90D4—82no—
91D4—84no—
92C2×C4—22no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4—84no—
103C2×C4#12012yes—
104C2×C4#12011yescharacteristic
105C2×C4#12112yes—
106C2×C4#12711yescharacteristic
107C8—42no—
108C8—44no—
109C8—42no—
110C22—21no—
111C22—21no—
112C22—82no—
113C22—82no—
114C22—82no—
115C22#39211yescharacteristic
116C22#47212yes—
117C4—21no—
118C4—22no—
119C4—22no—
120C4—42no—
121C4—81no—
122C4—82no—
123C4#45511yescharacteristic
124C4#45611yescharacteristic
125C2—22no—
126C2—162no—
127C2#200711yescharacteristic
128C2#201011yescharacteristic
129C2#201811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3245C2
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
2. #3112C2semidirect
3. #1985C2semidirect
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
#31122
#19852
#15302
#12902