Finite Group Databasebeta

← #27157120

Subgroups — #27157120

Order 12416.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261754C211yesmaximal, characteristic
2C4×C4⋊#87604C211yesmaximal, characteristic
3#20261589C214yesmaximal
4#20261290C211yesmaximal, characteristic
5C2×C4⋊#87604—21no—
6C2×C4×#87604—21no—
7C4⋊#11715897—21no—
8#15690867C2212yes—
9C4×#11715897—21no—
10(C2×#87604)⋊C4C2212yes—
11#15690810C411yescharacteristic, complemented
12#15690801—21no—
13#15690790—21no—
14#15690786C2212yes—
15Q8×Dic97C412yescomplemented
16#15690775—21no—
17#15690768—21no—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21C2×#11715897—22no—
22C4⋊#87604—24no—
23C4×#87604—42no—
24C2×Dic194—21no—
25C2×C4×D97D411yescharacteristic
26C2×C4×D97D411yescharacteristic
27C2×C4×D97C2311yescharacteristic
28C4⋊Dic97—21no—
29C4⋊Dic97—22no—
30C4⋊Dic97C2×C412yes—
31#13095490—22no—
32#13095488—22no—
33C4×Dic97—22no—
34C4×Dic97C2×C411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97D411yescharacteristic
37C2×Q8×C97C2×C412yes—
38C4⋊C4×C97—21no—
39C4⋊C4×C97—21no—
40C42×C97C2×C411yescharacteristic
41C22×D97C2×D411yescharacteristic
42C2×#87604—24no—
43C2×#87604—42no—
44C2×#87604—42no—
45Q8×C97—42no—
46C2×C4×C97—22no—
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C22⋊C411yescharacteristic
49C2×C4×C97C22×C411yescharacteristic
50C2×Dic97—22no—
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C22⋊C411yescharacteristic
53C2×Dic97C2×D411yescharacteristic
54C4×D97—22no—
55C4×D97—41no—
56C4×D97SD1612yes—
57C4×D97Q1612yes—
58Dic194—42no—
59#11715897—42no—
60C22×C97C2×C22⋊C411yescharacteristic
61C2×D97—21no—
62C2×D97—21no—
63C2×D97C2×SD1611yescharacteristic
64C2×D97C2×Q1611yescharacteristic
65#87604—44no—
66C4×C97—21no—
67C4×C97—42no—
68C4×C97#10312yes—
69C4×C97#10412yes—
70Dic97—21no—
71Dic97—21no—
72Dic97—42no—
73Dic97#10312yes—
74C2×C97#41011yescharacteristic
75C2×C97C2×#10311yescharacteristic
76C2×C97C2×#10411yescharacteristic
77D97—21no—
78D97—21no—
79#1503—971nomaximal, Sylow
80C97#150311yescharacteristic, complemented, Sylow
81#533—971no—
82#431—971no—
83C4×C4⋊C4—971no—
84#328—974no—
85C4⋊Q8—971no—
86C4⋊Q8#8760411yescharacteristic, complemented
87#125—1941no—
88C22⋊Q8—972no—
89C22⋊Q8—1941no—
90C4×Q8—972no—
91C4×Q8—1941no—
92#117—1941no—
93C2×C4⋊C4—1941no—
94C2×C42—971no—
95C2×C42—1941no—
96C4⋊C8—1941no—
97C22⋊C8—972no—
98C4×C8—1941no—
99#95—972no—
100C2×Q8—1941no—
101C2×Q8C2×#8760412yes—
102C22×C4—971no—
103C22×C4—971no—
104C22×C4—971no—
105C22×C4—1942no—
106C2×C8—1942no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—972no—
110C4⋊C4—1942no—
111C4⋊C4—1942no—
112C4⋊C4—1944no—
113C4⋊C4—1941no—
114C22⋊C4—1942no—
115C42—971no—
116C42—971no—
117C42—971no—
118C42—3882no—
119C42—1942no—
120C42C2×#8760411yescharacteristic
121C23—971no—
122Q8—42no—
123Q8—3882no—
124C2×C4—22no—
125C2×C4—971no—
126C2×C4—972no—
127C2×C4—972no—
128C2×C4—971no—
129C2×C4—971no—
130C2×C4—3881no—
131C2×C4—3882no—
132C2×C4—3882no—
133C2×C4—1944no—
134C2×C4—1942no—
135C2×C4—1942no—
136C2×C4C22⋊#8760411yescharacteristic
137C2×C4C22⋊#8760411yescharacteristic
138C2×C4C22×#8760411yescharacteristic
139C8—3882no—
140C22—971no—
141C22—971no—
142C22—1941no—
143C22—1941no—
144C22C2×C22⋊#8760411yescharacteristic
145C4—21no—
146C4—42no—
147C4—972no—
148C4—3884no—
149C4—3882no—
150C4—1941no—
151C4—1941no—
152C4#1569087312yes—
153C4#1569087512yes—
154C2—1941no—
155C2—1941no—
156C2#2026171511yescharacteristic
157C2C2×#1569087311yescharacteristic
158C2C2×#1569087511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1503
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #15690810C4semidirect
15. Q8×Dic97C4semidirect
19. C4⋊Q8×C97C4semidirect
80. C97#1503semidirect
86. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617542
C4×C4⋊#876042
#202615892
#202612902
#150397