Finite Group Databasebeta

← #10213

Subgroups — #10213

Order 256.

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

normal subgroups only

This group has 249 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#3253C211yesmaximal, characteristic
2#3216C212yesmaximal, complemented
3#1995C211yesmaximal, characteristic, complemented
4#1505C212yesmaximal, complemented
5#1307C211yesmaximal, characteristic, complemented
6#533C2211yescharacteristic
7#532C2212yes—
8#497—22no—
9#489—22no—
10#476—22no—
11#469—22no—
12#462—22no—
13#460—22no—
14#459—22no—
15#452—24no—
16#432C2212yes—
17#423C2211yescharacteristic
18#376C2211yescharacteristic
19#339—22no—
20#327—24no—
21C2×#41—22no—
22C2×Q16—24no—
23C2×SD16—24no—
24C2×D8—22no—
25C2×M16D412yes—
26C4⋊Q8D412yes—
27C4⋊Q8D412yes—
28#127D412yes—
29#125—21no—
30#124—22no—
31C22⋊Q8—22no—
32#121—22no—
33#121—22no—
34C4×Q8—22no—
35C4×D4—22no—
36C4×D4—22no—
37#117—21no—
38C2×C42—21no—
39C2×C42C2311yescharacteristic
40#106—22no—
41C4⋊C8—22no—
42C4⋊C8—22no—
43#103—24no—
44#102—24no—
45#102—24no—
46C22⋊C8D414yes—
47#97—22no—
48#95—22no—
49#41—84no—
50C2×Q8—22no—
51C2×Q8—22no—
52C2×D4—22no—
53C2×D4—22no—
54C2×D4—22no—
55C2×D4—42no—
56C22×C4—41no—
57C22×C4—42no—
58C22×C4C2×D411yescharacteristic
59C22×C4C2×D412yes—
60Q16—84no—
61SD16—84no—
62D8—82no—
63M16—42no—
64C2×C8—24no—
65C2×C8—22no—
66C4⋊C4—22no—
67C4⋊C4—22no—
68C4⋊C4—22no—
69C4⋊C4—22no—
70C4⋊C4—42no—
71C22⋊C4—41no—
72C22⋊C4—42no—
73C22⋊C4—42no—
74C42—22no—
75C42—41no—
76C42—41no—
77C42C2×D411yescharacteristic
78C42C2×D412yes—
79C42#4111yescharacteristic
80C23—42no—
81C23#12711yescharacteristic
82Q8—42no—
83Q8—42no—
84Q8—44no—
85D4—44no—
86D4—42no—
87D4—42no—
88D4—82no—
89D4—84no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—42no—
93C2×C4—42no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—84no—
97C2×C4—82no—
98C2×C4#12012yes—
99C2×C4#12011yescharacteristic
100C2×C4#12111yescharacteristic
101C2×C4#12112yes—
102C8—44no—
103C8—42no—
104C22—21no—
105C22—22no—
106C22—82no—
107C22—82no—
108C22—82no—
109C22#39211yescharacteristic
110C4—21no—
111C4—22no—
112C4—22no—
113C4—82no—
114C4—82no—
115C4—82no—
116C4#45511yescharacteristic
117C4#45611yescharacteristic
118C2—41no—
119C2—162no—
120C2#200711yescharacteristic
121C2#200812yes—
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3253C2
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. #3216C2semidirect
3. #1995C2semidirect
4. #1505C2semidirect
5. #1307C2semidirect

Maximal subgroups

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

subgroupindex
#32532
#32162
#19952
#15052
#13072