Finite Group Databasebeta

← #27140819

Subgroups — #27140819

Order 12416.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261109C211yesmaximal, characteristic, complemented
2#20260930C212yesmaximal, complemented
3#20260484C211yesmaximal, characteristic, complemented
4D4⋊#11715895C212yesmaximal, complemented
5#555×C97C211yesmaximal, characteristic, complemented
6C2×#13095498—22no—
7C2×#13095496—22no—
8#15690807C2211yescharacteristic
9C4⋊#11715895C2212yes—
10#15690730—22no—
11C4×#11715895C2211yescharacteristic
12C4⋊Q8×C97C2211yescharacteristic
13#124×C97—22no—
14#121×C97—22no—
15C4×D4×C97C2212yes—
16#117×C97—22no—
17C2×D388—22no—
18C2×D388—21no—
19#13095498—44no—
20#13095496—44no—
21C2×#11715895—21no—
22C2×#11715895—22no—
23C2×Q8×C97D412yes—
24C2×D4×C97—22no—
25C2×D4×C97D412yes—
26C22×C4×C97—22no—
27C4⋊C4×C97—22no—
28C4⋊C4×C97D412yes—
29C22⋊C4×C97—24no—
30C22⋊C4×C97—22no—
31C42×C97—22no—
32C42×C97C2311yescharacteristic
33C23×C97—22no—
34C22×D97—41no—
35Q8×C97—42no—
36D4×C97—22no—
37D4×C97—22no—
38D4×C97—82no—
39C2×C4×C97—22no—
40C2×C4×C97—22no—
41C2×C4×C97—42no—
42C2×C4×C97—42no—
43C2×C4×C97C2×D411yescharacteristic
44C2×C4×C97C2×D412yes—
45D388—42no—
46D388—42no—
47D388—42no—
48#11715895—22no—
49#11715895—42no—
50C22×C97—42no—
51C22×C97—42no—
52C22×C97—42no—
53C22×C97#12011yescharacteristic
54C2×D97—82no—
55C2×D97—81no—
56C4×C97—21no—
57C4×C97—42no—
58C4×C97—42no—
59C4×C97#13512yes—
60C4×C97AGL(1,ℤ/8)12yes—
61C2×C97—82no—
62C2×C97#44912yes—
63C2×C97#45311yescharacteristic
64D97—161no—
65#1624—971nomaximal, Sylow
66C97#162411yescharacteristic, complemented, Sylow
67#555D9711yescharacteristic, complemented
68#492—971no—
69#459—972no—
70#333—971no—
71D4⋊C8—972no—
72C2×SD16—1942no—
73C2×D8—1942no—
74C4⋊Q8C2×D9711yescharacteristic
75#127—971no—
76#124—22no—
77#121—22no—
78C4×D4C2×D9712yes—
79#117—22no—
80C4⋊C8—972no—
81#102—1942no—
82C4×C8—971no—
83C2×Q8C22⋊D9712yes—
84C2×D4—22no—
85C2×D4—1942no—
86C2×D4—1941no—
87C2×D4C22⋊D9712yes—
88C22×C4—22no—
89SD16—3884no—
90D8—3884no—
91C2×C8—1942no—
92C2×C8—1941no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D9712yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D9711yescharacteristic
99C23—22no—
100C23—3881no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—3882no—
106D4—3882no—
107D4—3882no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D9711yescharacteristic
113C2×C4C2×C22⋊D9712yes—
114C8—3882no—
115C8—1942no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—7762no—
120C22—7761no—
121C22#1569077911yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#1569083012yes—
126C4#1569083512yes—
127C2—82no—
128C2—15521no—
129C2#2026088012yes—
130C2#2026090311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#555×C97C2
socleC22×C97#120
hypercenter#555D97
hypocenterC97#1624
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20261109C2semidirect
2. #20260930C2semidirect
3. #20260484C2semidirect
4. D4⋊#11715895C2semidirect
5. #555×C97C2semidirect
66. C97#1624semidirect
67. #555D97semidirect

Maximal subgroups

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

subgroupindex
#202611092
#202609302
#202604842
D4⋊#117158952
#555×C972
#162497