Finite Group Databasebeta

← #12365

Subgroups — #12365

Order 256.

173 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 173 conjugacy classes of subgroups — too many to draw. Only its 39 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#3291C211yesmaximal, characteristic
2#2230C212yesmaximal, complemented
3#1708C211yesmaximal, characteristic, complemented
4#1359C211yesmaximal, characteristic, complemented
5#1335C212yesmaximal, complemented
6#577—22no—
7#498C2211yescharacteristic, complemented
8#492C2211yescharacteristic
9#487—21no—
10#486—21no—
11#471—22no—
12#456—21no—
13#453—21no—
14#443C2212yes—
15#365—22no—
16#364C2212yescomplemented
17#360—22no—
18C82C2211yescharacteristic, complemented
192−1+4—21no—
20H(5,𝔽2)—21no—
21#137—22no—
22#137—42no—
23AGL(1,ℤ/8)—22no—
24AGL(1,ℤ/8)—42no—
25#135—42no—
26#135D412yes—
27C2×SD16—22no—
28C2×SD16—42no—
29#131—22no—
30C4⋊Q8—21no—
31C4⋊Q8D411yescharacteristic
32#127D411yescharacteristic
33M32—22no—
34#108D412yes—
35#106—22no—
36#104—22no—
37#103—42no—
38#102—42no—
39#101—21no—
40#100—21no—
41C4×C8—21no—
42C4×C8C2311yescharacteristic
43#41—22no—
44#41—42no—
45#41—42no—
46#41—81no—
47C2×Q8—21no—
48C2×Q8—41no—
49C2×Q8—42no—
50C2×D4—21no—
51C2×D4—41no—
52C2×D4—42no—
53C2×D4—81no—
54Q16—42no—
55Q16SD1612yes—
56SD16—22no—
57SD16—42no—
58SD16—42no—
59SD16—42no—
60SD16—42no—
61D8—42no—
62D8SD1612yes—
63M16—22no—
64M16—42no—
65C2×C8—22no—
66C2×C8—42no—
67C2×C8C2×D412yes—
68C4⋊C4—41no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C42C2×D411yescharacteristic
72C16—42no—
73C23—41no—
74C23—81no—
75Q8—22no—
76Q8—42no—
77Q8—82no—
78Q8—81no—
79D4—22no—
80D4—42no—
81D4—42no—
82D4—82no—
83D4—82no—
84D4—81no—
85C2×C4—21no—
86C2×C4—42no—
87C2×C4—41no—
88C2×C4—81no—
89C2×C4—82no—
90C2×C4—81no—
91C2×C4#12011yescharacteristic
92C8—42no—
93C8—42no—
94C8C2×SD1612yes—
95C8AGL(1,ℤ/8)12yes—
96C22—42no—
97C22—81no—
98C22—161no—
99C22—321no—
100C22#45311yescharacteristic
101C4—22no—
102C4—42no—
103C4—81no—
104C4—161no—
105C4#45012yes—
106C2—21no—
107C2—82no—
108C2—161no—
109C2#162911yescharacteristic
subgroupG/H
center, socleC2#1629
commutator, Frattini, cosocleC4×C8C23
prime radical#3291C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2230C2semidirect
3. #1708C2semidirect
4. #1359C2semidirect
5. #1335C2semidirect
7. #498C22semidirect
16. #364C22semidirect
18. C82C22semidirect

Maximal subgroups

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

subgroupindex
#32912
#22302
#17082
#13592
#13352