Finite Group Databasebeta

← #27157168

Subgroups — #27157168

Order 12416.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261753C211yesmaximal, characteristic
2#20261679C211yesmaximal, characteristic
3#20261588C214yesmaximal
4#20261290C211yesmaximal, characteristic
5C2×C4⋊#87604—21no—
6#15690877—21no—
7#15690867C2212yes—
8C4×#11715897—21no—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4C2212yes—
11#15690810C411yescharacteristic, complemented
12#15690801—21no—
13#15690790C2212yes—
14#15690786—21no—
15Q8×Dic97—21no—
16#15690775C412yescomplemented
17#15690768—21no—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21C2×#11715897—22no—
22C4⋊#87604—42no—
23C2×Dic194C2×C412yes—
24C2×C4×D97D411yescharacteristic
25C2×C4×D97D411yescharacteristic
26C2×C4×D97C2311yescharacteristic
27C4⋊Dic97—21no—
28C4⋊Dic97—21no—
29C4⋊Dic97—22no—
30#13095490—22no—
31#13095488—22no—
32C4×Dic97—22no—
33C4×Dic97C2×C411yescharacteristic
34C4×Dic97D411yescharacteristic
35C4×Dic97D411yescharacteristic
36C2×Q8×C97—21no—
37C4⋊C4×C97—21no—
38C4⋊C4×C97C2×C412yes—
39C42×C97C2×C411yescharacteristic
40C22×D97C2×D411yescharacteristic
41C2×#87604—42no—
42C2×#87604—42no—
43C2×#87604—42no—
44Q8×C97—42no—
45C2×C4×C97—22no—
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C22×C411yescharacteristic
49C2×Dic97—22no—
50C2×Dic97C22⋊C411yescharacteristic
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C2×D411yescharacteristic
53C4×D97—41no—
54C4×D97—41no—
55C4×D97SD1612yes—
56C4×D97Q1612yes—
57Dic194—42no—
58#11715897—42no—
59C22×C97C2×C22⋊C411yescharacteristic
60C2×D97—21no—
61C2×D97—21no—
62C2×D97C2×SD1611yescharacteristic
63C2×D97C2×Q1611yescharacteristic
64#87604—82no—
65C4×C97—21no—
66C4×C97—21no—
67C4×C97—42no—
68C4×C97#10312yes—
69Dic97—21no—
70Dic97—21no—
71Dic97—42no—
72Dic97#10312yes—
73C2×C97#41011yescharacteristic
74C2×C97C2×#10311yescharacteristic
75C2×C97#42111yescharacteristic
76D97—21no—
77D97—21no—
78#1518—971nomaximal, Sylow
79C97#151811yescharacteristic, complemented, Sylow
80#533—971no—
81#431—971no—
82#382—971no—
83#328—974no—
84C4⋊Q8—971no—
85C4⋊Q8#8760411yescharacteristic, complemented
86#125—1941no—
87C22⋊Q8—972no—
88C22⋊Q8—1941no—
89C4×Q8—972no—
90C4×Q8—1941no—
91#117—1941no—
92C2×C4⋊C4—1941no—
93C2×C42—971no—
94C4⋊C8—1941no—
95C22⋊C8—972no—
96C4×C8—1941no—
97#95—972no—
98#95—1941no—
99C2×Q8—21no—
100C2×Q8—972no—
101C22×C4—971no—
102C22×C4—971no—
103C22×C4—971no—
104C22×C4—1942no—
105C2×C8—1942no—
106C4⋊C4—21no—
107C4⋊C4—3882no—
108C4⋊C4—1942no—
109C4⋊C4—1942no—
110C4⋊C4—1941no—
111C4⋊C4—1941no—
112C4⋊C4C2×#8760412yes—
113C22⋊C4—1942no—
114C42—971no—
115C42—971no—
116C42—971no—
117C42—1942no—
118C42C2×#8760411yescharacteristic
119C23—971no—
120Q8—42no—
121Q8—3882no—
122C2×C4—22no—
123C2×C4—971no—
124C2×C4—972no—
125C2×C4—972no—
126C2×C4—971no—
127C2×C4—971no—
128C2×C4—3881no—
129C2×C4—3881no—
130C2×C4—3882no—
131C2×C4—3882no—
132C2×C4—3882no—
133C2×C4—1942no—
134C2×C4C22⋊#8760411yescharacteristic
135C2×C4C22⋊#8760411yescharacteristic
136C2×C4C22×#8760411yescharacteristic
137C8—3882no—
138C22—971no—
139C22—971no—
140C22—1941no—
141C22—1941no—
142C22C2×C22⋊#8760411yescharacteristic
143C4—21no—
144C4—21no—
145C4—42no—
146C4—7762no—
147C4—972no—
148C4—3882no—
149C4—1941no—
150C4—1941no—
151C4#1569087312yes—
152C2—1941no—
153C2—1941no—
154C2#2026171611yescharacteristic
155C2C2×#1569087311yescharacteristic
156C2#2026173311yescharacteristic
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#1518
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
16. #15690775C4semidirect
19. C4⋊Q8×C97C4semidirect
79. C97#1518semidirect
85. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617532
#202616792
#202615882
#202612902
#151897