Finite Group Databasebeta

← #27157912

Subgroups — #27157912

Order 12416.

260 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#13095521C211yesmaximal, characteristic
2C4×C4⋊#87604C211yesmaximal, characteristic
3#20261605C214yesmaximal
4#20261290C211yesmaximal, characteristic
5C2×#13095521C2212yes—
6C2×C4⋊#87604—21no—
7C2×C4×#87604C2212yes—
8#15690865—21no—
9C4×#11715897—21no—
10(C2×#87604)⋊C4—21no—
11#15690810C411yescharacteristic, complemented
12#15690801C412yescomplemented
13#15690790—21no—
14#15690786—21no—
15Q8×Dic97—21no—
16#15690775—21no—
17#15690768C2212yes—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21#13095521—22no—
22#13095521Q814yescomplemented
23C2×#11715897—22no—
24C4⋊#87604—24no—
25C4×#87604—24no—
26C2×Dic194—21no—
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×C4×D97C2311yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97—22no—
32C4⋊Dic97C2×C412yes—
33#13095490—22no—
34#13095488—22no—
35C4×Dic97—22no—
36C4×Dic97C2×C411yescharacteristic
37C4×Dic97D411yescharacteristic
38C4×Dic97D411yescharacteristic
39C2×Q8×C97—21no—
40C4⋊C4×C97—21no—
41C4⋊C4×C97C2×C412yes—
42C42×C97C2×C411yescharacteristic
43C22×D97C2×D411yescharacteristic
44C2×#87604—24no—
45C2×#87604—42no—
46C2×#87604—42no—
47Q8×C97—42no—
48C2×C4×C97—22no—
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22×C411yescharacteristic
52C2×Dic97—22no—
53C2×Dic97C22⋊C411yescharacteristic
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C2×D411yescharacteristic
56C4×D97—22no—
57C4×D97—41no—
58C4×D97C2×Q812yes—
59C4×D97#4112yes—
60Dic194—42no—
61#11715897—24no—
62C22×C97C2×C22⋊C411yescharacteristic
63C2×D97—21no—
64C2×D97—21no—
65C2×D97#12411yescharacteristic
66C2×D97C4⋊Q811yescharacteristic
67#87604—44no—
68C4×C97—21no—
69C4×C97—42no—
70C4×C97#10412yes—
71C4×C97C4×Q812yes—
72Dic97—21no—
73Dic97—21no—
74Dic97—42no—
75Dic97C22⋊Q812yes—
76C2×C97#39111yescharacteristic
77C2×C97C2×#10411yescharacteristic
78C2×C97#42111yescharacteristic
79D97—21no—
80D97—21no—
81#1986—971nomaximal, Sylow
82C97#198611yescharacteristic, complemented, Sylow
83#533—971no—
84C4×M16—971no—
85C4×C4⋊C4—971no—
86#339—974no—
87C2×M16—972no—
88C4⋊Q8—971no—
89C4⋊Q8#8760411yescharacteristic, complemented
90#125—972no—
91C22⋊Q8—1941no—
92C22⋊Q8—1941no—
93C4×Q8—1941no—
94C4×Q8—1941no—
95#117—972no—
96C2×C4⋊C4—1941no—
97C2×C42—971no—
98C2×C42—972no—
99#97—1941no—
100C4×C8—1941no—
101#95—1941no—
102C2×Q8—21no—
103C2×Q8—1941no—
104C22×C4—971no—
105C22×C4—971no—
106C22×C4—971no—
107C22×C4—1942no—
108M16—974no—
109M16—1942no—
110C2×C8—1942no—
111C4⋊C4—21no—
112C4⋊C4—972no—
113C4⋊C4—1942no—
114C4⋊C4—1942no—
115C4⋊C4—1944no—
116C4⋊C4—1941no—
117C4⋊C4C2×#8760412yes—
118C22⋊C4—1942no—
119C42—971no—
120C42—971no—
121C42—971no—
122C42—1942no—
123C42—1944no—
124C42C2×#8760411yescharacteristic
125C23—971no—
126Q8—42no—
127Q8—3882no—
128C2×C4—22no—
129C2×C4—971no—
130C2×C4—972no—
131C2×C4—972no—
132C2×C4—971no—
133C2×C4—971no—
134C2×C4—3881no—
135C2×C4—3882no—
136C2×C4—3882no—
137C2×C4—1944no—
138C2×C4—1942no—
139C2×C4—1942no—
140C2×C4C22⋊#8760411yescharacteristic
141C2×C4C22⋊#8760411yescharacteristic
142C2×C4C22×#8760411yescharacteristic
143C8—1944no—
144C22—971no—
145C22—971no—
146C22—1941no—
147C22—1941no—
148C22C2×C22⋊#8760411yescharacteristic
149C4—21no—
150C4—42no—
151C4—972no—
152C4—3884no—
153C4—3882no—
154C4—1941no—
155C4—1941no—
156C4#1569087412yes—
157C4Q8×#8760412yes—
158C2—1941no—
159C2—1941no—
160C2(C2×#87604)⋊Q811yescharacteristic
161C2C2×#1569087411yescharacteristic
162C2#2026173411yescharacteristic
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#1986
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #15690810C4semidirect
12. #15690801C4semidirect
19. C4⋊Q8×C97C4semidirect
22. #13095521Q8semidirect
82. C97#1986semidirect
89. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
C4×#130955212
C4×C4⋊#876042
#202616052
#202612902
#198697