Finite Group Databasebeta

← #27141031

Subgroups — #27141031

Order 12416.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#20261483C211yesmaximal, characteristic, complemented
2#20261124C212yesmaximal, complemented
3#20261101C211yesmaximal, characteristic, complemented
4#20260474C212yesmaximal, complemented
5#330×C97C211yesmaximal, characteristic, complemented
6C2×#13095500—22no—
7C2×D776—22no—
8#15690807C2211yescharacteristic
9#15690798C2212yes—
10#15690795—22no—
11#15690788—22no—
12#15690785—22no—
13#15690730—22no—
14#15690721C2211yescharacteristic
15#125×C97C2211yescharacteristic
16#97×C97C2212yes—
17C2×Dic194D412yes—
18C2×D388—22no—
19C2×D388—21no—
20C2×D388—22no—
21C2×D388D412yes—
22C2×C4×D97—22no—
23#13095500—44no—
24D776—44no—
25C2×#11715895—21no—
26#13095490—22no—
27#13095488—22no—
28#13095488—22no—
29#13095488—22no—
30C2×C8×C97—22no—
31C4⋊C4×C97—22no—
32C4⋊C4×C97D412yes—
33C42×C97C2311yescharacteristic
34C22×D97—22no—
35C22×D97—41no—
36C2×C4×C97—22no—
37C2×C4×C97C2×D412yes—
38C2×C4×C97C2×D411yescharacteristic
39C2×Dic97—22no—
40D388—42no—
41D388—42no—
42D388—42no—
43D388—42no—
44D388—82no—
45C4×D97—82no—
46Dic194—42no—
47C8×C97—24no—
48#11715895—41no—
49C22×C97#12011yescharacteristic
50C2×D97—42no—
51C2×D97—42no—
52C2×D97—42no—
53C2×D97—82no—
54C2×D97—81no—
55C4×C97—22no—
56C4×C97—21no—
57C4×C97—42no—
58Dic97—42no—
59C2×C97#45312yes—
60C2×C97#45411yescharacteristic
61D97—82no—
62D97—161no—
63#1656—971nomaximal, Sylow
64C97#165611yescharacteristic, complemented, Sylow
65#562—971no—
66#496—972no—
67#490—971no—
68#330D9711yescharacteristic, complemented
69#329—972no—
70C2×SD16—1942no—
71C2×D8—1942no—
72#127—971no—
73#125C2×D9711yescharacteristic
74#124—972no—
75#123—1942no—
76C22⋊Q8—1942no—
77#121—1942no—
78#102—1942no—
79#97—971no—
80#97C2×D9712yes—
81C2×Q8—972no—
82C2×D4—972no—
83C2×D4—1942no—
84C2×D4—1942no—
85C2×D4—1941no—
86C22×C4—1942no—
87SD16—3884no—
88D8—3884no—
89C2×C8—22no—
90C2×C8—1941no—
91C4⋊C4—22no—
92C4⋊C4—1942no—
93C4⋊C4D38812yes—
94C22⋊C4—1942no—
95C22⋊C4—1942no—
96C22⋊C4—1942no—
97C42C22×D9711yescharacteristic
98C23—3881no—
99C23—1942no—
100Q8—3882no—
101D4—7762no—
102D4—3882no—
103D4—3882no—
104D4—3882no—
105D4—3882no—
106C2×C4—22no—
107C2×C4—7762no—
108C2×C4—1942no—
109C2×C4D4×D9712yes—
110C2×C4C2×D38811yescharacteristic
111C8—24no—
112C8—3881no—
113C22—7762no—
114C22—7761no—
115C22—3882no—
116C22—3882no—
117C22—3882no—
118C22#1569078011yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—3882no—
123C2—7762no—
124C2—15521no—
125C2#2026090412yes—
126C2#2026091011yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#330×C97C2
socleC22×C97#120
hypercenter#330D97
hypocenterC97#1656
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #20261483C2semidirect
2. #20261124C2semidirect
3. #20261101C2semidirect
4. #20260474C2semidirect
5. #330×C97C2semidirect
64. C97#1656semidirect
68. #330D97semidirect

Maximal subgroups

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

subgroupindex
#202614832
#202611242
#202611012
#202604742
#330×C972
#165697