Finite Group Databasebeta

← #27141059

Subgroups — #27141059

Order 12416.

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

normal subgroups only

This group has 196 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#20261105C211yesmaximal, characteristic, complemented
2#20261094C212yesmaximal, complemented
3#20260477C211yesmaximal, characteristic, complemented
4#20260475C212yesmaximal, complemented
5#562×C97C211yesmaximal, characteristic
6#15690810C2211yescharacteristic
7#15690734—22no—
8#15690733—22no—
9#15690731—22no—
10#15690721C2211yescharacteristic
11#15690721C2212yes—
12#125×C97C2211yescharacteristic
13#124×C97C2212yes—
14#123×C97—22no—
15C22⋊Q8×C97—22no—
16#121×C97—22no—
17C2×Dic194—21no—
18C2×#11715895—22no—
19C2×#11715895—21no—
20C4⋊Dic97—22no—
21C2×Q8×C97D412yes—
22C2×D4×C97—22no—
23C2×D4×C97D412yes—
24C22×C4×C97—22no—
25C4⋊C4×C97—22no—
26C4⋊C4×C97—22no—
27C4⋊C4×C97D412yes—
28C22⋊C4×C97—22no—
29C22⋊C4×C97—22no—
30C22⋊C4×C97—22no—
31C42×C97C2311yescharacteristic
32C23×C97—22no—
33Q8×C97—42no—
34D4×C97—42no—
35D4×C97—82no—
36C2×C4×C97—22no—
37C2×C4×C97—22no—
38C2×C4×C97—82no—
39C2×C4×C97C2×D411yescharacteristic
40C2×C4×C97C2×D412yes—
41C2×Dic97—41no—
42Dic194—42no—
43#11715895—41no—
44#11715895—42no—
45C22×C97—42no—
46C22×C97—42no—
47C22×C97—42no—
48C22×C97#12011yescharacteristic
49C4×C97—21no—
50C4×C97—22no—
51C4×C97—42no—
52C4×C97—42no—
53Dic97—81no—
54C2×C97—82no—
55C2×C97#45411yescharacteristic
56C2×C97#45512yes—
57#1662—971nomaximal, Sylow
58C97#166211yescharacteristic, complemented, Sylow
59#562D9711yescharacteristic
60#491—971no—
61#489—972no—
62#330—971no—
63#329—972no—
64C4⋊Q8—971no—
65#125C2×D9711yescharacteristic
66#124C2×D9712yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—1942no—
71#103—1942no—
72#102—1942no—
73#97—971no—
74#97—972no—
75C2×Q8—1941no—
76C2×Q8C22⋊D9712yes—
77C2×D4—22no—
78C2×D4C22⋊D9712yes—
79C22×C4—22no—
80C2×C8—1942no—
81C2×C8—1941no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—1942no—
85C4⋊C4C22⋊D9712yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D9711yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—3882no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—3881no—
99C2×C4C2×C22⋊D9711yescharacteristic
100C2×C4C2×C22⋊D9712yes—
101C8—3881no—
102C8—3882no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#1569077911yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—7761no—
112C2—82no—
113C2#2026090811yescharacteristic
114C2#2026091312yes—
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radical#562×C97C2
socleC22×C97#120
hypercenter#562D97
hypocenterC97#1662
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #20261105C2semidirect
2. #20261094C2semidirect
3. #20260477C2semidirect
4. #20260475C2semidirect
58. C97#1662semidirect

Maximal subgroups

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

subgroupindex
#202611052
#202610942
#202604772
#202604752
#562×C972
#166297