Finite Group Databasebeta

← #27139890

Subgroups — #27139890

Order 12416.

106 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 25 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#20260534C211yesmaximal, characteristic, complemented
2#20260484C211yesmaximal, characteristic, complemented
3#456×C97C211yesmaximal, characteristic
4C4⋊#11715895—21no—
5#15690729—21no—
62−1+4×C97C411yescharacteristic
7#137×C97—21no—
8C4⋊Q8×C97C2211yescharacteristic
9#104×C97—21no—
10#101×C97C411yescharacteristic
11#13095495—41no—
12C2×#11715895—41no—
13#41×C97—21no—
14#41×C97—41no—
15#41×C97—41no—
16C2×Q8×C97—41no—
17C2×Q8×C97C2×C411yescharacteristic
18C2×Q8×C97D411yescharacteristic
19Q16×C97—41no—
20SD16×C97—41no—
21M16×C97—21no—
22C4⋊C4×C97—41no—
23C42×C97D411yescharacteristic
24Q8×C97—21no—
25Q8×C97—41no—
26Q8×C97—41no—
27Q8×C97—41no—
28D4×C97—21no—
29D4×C97—41no—
30D4×C97—41no—
31C2×C4×C97—21no—
32C2×C4×C97—21no—
33C2×C4×C97—21no—
34C2×C4×C97—41no—
35C2×C4×C97—42no—
36C2×C4×C97C22⋊C411yescharacteristic
37C8×C97—41no—
38#11715895—81no—
39#11715895—81no—
40C22×C97—41no—
41C22×C97#9911yescharacteristic
42C4×C97—21no—
43C4×C97—21no—
44C4×C97—21no—
45C4×C97—41no—
46C4×C97—41no—
47C4×C97—41no—
48C2×C97—21no—
49C2×C97—81no—
50C2×C97AΣL(1,𝔽16)11yescharacteristic
51#1405—971nomaximal, Sylow
52C97#140511yescharacteristic, complemented, Sylow
53#456D9711yescharacteristic
54#356—971no—
55#333—971no—
562−1+4Dic9711yescharacteristic
57#137—21no—
58C4⋊Q8C2×D9711yescharacteristic
59C4⋊C8—1941no—
60#104—21no—
61#101—1941no—
62#101Dic9711yescharacteristic
63#41—21no—
64#41—41no—
65#41—41no—
66C2×Q8—41no—
67C2×Q8C2×Dic9711yescharacteristic
68C2×Q8C22⋊D9711yescharacteristic
69Q16—41no—
70SD16—41no—
71M16—21no—
72M16—3881no—
73C2×C8—3881no—
74C4⋊C4—41no—
75C42C22⋊D9711yescharacteristic
76Q8—21no—
77Q8—41no—
78Q8—41no—
79Q8—41no—
80D4—21no—
81D4—41no—
82D4—41no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—41no—
87C2×C4—42no—
88C2×C4C22⋊Dic9711yescharacteristic
89C8—41no—
90C8—7761no—
91C8—7761no—
92C22—41no—
93C22#1569072511yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—41no—
100C2—21no—
101C2—81no—
102C2#2026052211yescharacteristic
subgroupG/H
centerC2#20260522
commutatorC2×Q8×C97C2×C4
FrattiniC4⋊Q8C2×D97
Fitting#456×C97C2
socleC2×C97AΣL(1,𝔽16)
prime radical2−1+4×C97C4
cosocleC4⋊Q8×C97C22
hypercenter#456D97
hypocenterC97#1405
LCSG ▷ C2×Q8×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #20260534C2semidirect
2. #20260484C2semidirect
52. C97#1405semidirect

Maximal subgroups

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

subgroupindex
#202605342
#202604842
#456×C972
#140597