Finite Group Databasebeta

← #10584

Subgroups — #10584

Order 256.

187 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 57 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#2844C211yesmaximal, characteristic
2#1847C211yesmaximal, characteristic
3#1836C212yesmaximal
4#1295C212yesmaximal
5#1295C211yesmaximal, characteristic
6C2×#107—21no—
7C2×#106—21no—
8C2×#97C2212yes—
9C2×#97C2211yescharacteristic
10#400—22no—
11#400—21no—
12#395—22no—
13#395—21no—
14#384C2211yescharacteristic
15#382C2212yes—
16#382C2211yescharacteristic
17#340—22no—
18#340—22no—
19C22×Q8—21no—
20C22×C8—21no—
21C22×C8—22no—
22C2×C4⋊C4—22no—
23C2×C4⋊C4—21no—
24C2×C4⋊C4—21no—
25C2×C4⋊C4—22no—
26C2×C4⋊C4D412yes—
27C2×C4⋊C4D412yes—
28C2×C4⋊C4D411yescharacteristic
29C2×C4⋊C4D411yescharacteristic
30C2×C42C2311yescharacteristic
31#107—42no—
32#106—42no—
33#97—21no—
34#97—22no—
35#97Q812yescomplemented
36#97Q814yescomplemented
37#95—22no—
38#95—21no—
39#95—21no—
40#95—22no—
41#95—22no—
42#95—22no—
43C2×Q8—81no—
44C2×Q8—82no—
45C22×C4—21no—
46C22×C4—22no—
47C22×C4—21no—
48C22×C4—22no—
49C22×C4—41no—
50C22×C4C2×D411yescharacteristic
51C22×C4C2×D412yes—
52C2×C8—22no—
53C2×C8—22no—
54C2×C8—22no—
55C2×C8—24no—
56C2×C8—42no—
57C2×C8—42no—
58C4⋊C4—41no—
59C4⋊C4—41no—
60C4⋊C4—42no—
61C4⋊C4—42no—
62C4⋊C4—41no—
63C4⋊C4—41no—
64C4⋊C4—44no—
65C4⋊C4—81no—
66C4⋊C4—82no—
67C42C2×Q811yescharacteristic
68C42C2×Q812yes—
69C42#4111yescharacteristic
70C23#12011yescharacteristic
71Q8—84no—
72C2×C4—21no—
73C2×C4—22no—
74C2×C4—22no—
75C2×C4—21no—
76C2×C4—41no—
77C2×C4—41no—
78C2×C4—42no—
79C2×C4—42no—
80C2×C4—42no—
81C2×C4—42no—
82C2×C4—42no—
83C2×C4—42no—
84C2×C4—42no—
85C2×C4—42no—
86C2×C4—81no—
87C2×C4—82no—
88C2×C4C22⋊Q811yescharacteristic
89C2×C4C22⋊Q812yes—
90C2×C4C4⋊Q811yescharacteristic
91C2×C4C4⋊Q812yes—
92C8—24no—
93C8—44no—
94C22#39511yescharacteristic
95C22#45311yescharacteristic
96C22#45412yes—
97C22#45512yes—
98C22#45611yescharacteristic
99C4—21no—
100C4—21no—
101C4—22no—
102C4—22no—
103C4—81no—
104C4—81no—
105C4—82no—
106C4—82no—
107C4—161no—
108C2#165711yescharacteristic
109C2#165911yescharacteristic
110C2#166412yes—
111C2#206011yescharacteristic
112C2#206112yes—
subgroupG/H
center, socle, prime radicalC23#120
commutator, Frattini, cosocleC2×C42C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
35. #97Q8semidirect
36. #97Q8semidirect

Maximal subgroups

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

subgroupindex
#28442
#18472
#18362
#12952