Finite Group Databasebeta

← #10212

Subgroups — #10212

Order 256.

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

normal subgroups only

This group has 268 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, complemented
2#3214C212yesmaximal, complemented
3#1963C211yesmaximal, characteristic, complemented
4#1505C212yesmaximal, complemented
5#1307C211yesmaximal, characteristic, complemented
6#532C2211yescharacteristic
7#532C2212yes—
8#496—22no—
9#489—22no—
10#476—22no—
11#469—22no—
12#461—22no—
13#459—22no—
14#459—22no—
15#449—24no—
16#432C2212yes—
17#423C2211yescharacteristic
18C2×#104—22no—
19#376C2211yescharacteristic
20#327—24no—
21C2×#41—21no—
22C2×#41—22no—
23C2×SD16—24no—
24C2×D8—24no—
25C2×D8—22no—
26C2×M16D412yes—
27C4⋊Q8D412yes—
28C4⋊Q8D411yescharacteristic
29#127D412yes—
30#127D411yescharacteristic
31#124—22no—
32#124—21no—
33#121—22no—
34#121—22no—
35#121—22no—
36C4×D4—22no—
37C4×D4—22no—
38C4×D4—22no—
39C2×C42—21no—
40C2×C42C2311yescharacteristic
41#106—22no—
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—
53#41—84no—
54C2×Q8—22no—
55C2×Q8—21no—
56C2×D4—22no—
57C2×D4—22no—
58C2×D4—22no—
59C2×D4—21no—
60C2×D4—22no—
61C2×D4—41no—
62C2×D4—42no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—42no—
66C22×C4C2×D411yescharacteristic
67C22×C4C2×D412yes—
68SD16—84no—
69D8—84no—
70D8—82no—
71M16—42no—
72C2×C8—24no—
73C2×C8—22no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C22⋊C4—42no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C42—22no—
81C42—41no—
82C42C2×D411yescharacteristic
83C42C2×D412yes—
84C42#4111yescharacteristic
85C23—41no—
86C23—42no—
87C23#12711yescharacteristic
88Q8—44no—
89Q8—42no—
90D4—42no—
91D4—44no—
92D4—42no—
93D4—42no—
94D4—42no—
95D4—42no—
96D4—82no—
97D4—82no—
98D4—84no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—84no—
106C2×C4—82no—
107C2×C4—81no—
108C2×C4—81no—
109C2×C4#12012yes—
110C2×C4#12011yescharacteristic
111C2×C4#12111yescharacteristic
112C2×C4#12112yes—
113C8—44no—
114C8—42no—
115C22—21no—
116C22—22no—
117C22—81no—
118C22—82no—
119C22—82no—
120C22—82no—
121C22—82no—
122C22#39211yescharacteristic
123C4—21no—
124C4—22no—
125C4—22no—
126C4—82no—
127C4—81no—
128C4—82no—
129C4#45311yescharacteristic
130C4#45411yescharacteristic
131C2—41no—
132C2—161no—
133C2—162no—
134C2#200611yescharacteristic
135C2#200812yes—
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. #3253C2semidirect
2. #3214C2semidirect
3. #1963C2semidirect
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
#32142
#19632
#15052
#13072