Finite Group Databasebeta

← #27157872

Subgroups — #27157872

Order 12416.

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

normal subgroups only

This group has 320 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#20261740C211yesmaximal, characteristic, complemented
2C2×#15690875C214yesmaximal, complemented
3#20261671C211yesmaximal, characteristic, complemented
4#20261286C211yesmaximal, characteristic
5C2×#13095521C2212yes—
6C2×C4×#87604C2212yes—
7#15690877—22no—
8#15690875—28no—
9(C2×#87604)⋊C4—22no—
10C2×#13095512C2212yes—
11#15690807C411yescharacteristic, complemented
12#15690799C412yescomplemented
13#15690785—22no—
14#15690772—22no—
15C42×D97C2211yescharacteristic
16C4⋊Q8×C97C411yescharacteristic, complemented
17C22×#87604—22no—
18#13095521—24no—
19C2×#11715897—22no—
20C4×#87604—22no—
21C4×#87604D414yescomplemented
22#13095512—24no—
23#13095512—24no—
24C2×D388—22no—
25C2×D388—22no—
26C2×D388C2×C412yes—
27C2×C4×D97—22no—
28C2×C4×D97D412yes—
29C2×C4×D97C2311yescharacteristic
30#13095488—24no—
31C4×Dic97C2×C411yescharacteristic
32C4×Dic97D412yes—
33C2×Q8×C97C2×C412yes—
34C4⋊C4×C97—22no—
35C42×C97C2×C411yescharacteristic
36C22×D97—22no—
37C22×D97C2×D411yescharacteristic
38C2×#87604—24no—
39C2×#87604—44no—
40Q8×C97—24no—
41C2×C4×C97—22no—
42C2×C4×C97C22⋊C412yes—
43C2×C4×C97C22×C411yescharacteristic
44C2×Dic97C22⋊C412yes—
45C2×Dic97C2×D411yescharacteristic
46D388—24no—
47D388—44no—
48D388—44no—
49C4×D97—42no—
50C4×D97—42no—
51C4×D97—42no—
52C4×D97C2×D412yes—
53C4×D97#4112yes—
54#11715897—42no—
55C22×C97C2×C22⋊C411yescharacteristic
56C2×D97—22no—
57C2×D97—42no—
58C2×D97—44no—
59C2×D97#12112yes—
60#87604—44no—
61C4×C97—22no—
62C4×C97—42no—
63C4×C97C4×D412yes—
64Dic97—22no—
65Dic97#12411yescharacteristic
66Dic97#12711yescharacteristic
67C2×C97#39011yescharacteristic
68C2×C97#42112yes—
69D97—22no—
70D97—82no—
71#1963—971nomaximal, Sylow
72C97#196311yescharacteristic, complemented, Sylow
73#532—971no—
74#423—971no—
75C2×#104—974no—
76#376—971no—
77C2×#41—972no—
78C2×M16—972no—
79C4⋊Q8#8760411yescharacteristic, complemented
80#127—971no—
81#124—972no—
82#121—1942no—
83C4×D4—1942no—
84C2×C42—971no—
85C2×C42—972no—
86C4⋊C8—1942no—
87#104—1948no—
88#95—1942no—
89#41—1944no—
90#41—1944no—
91C2×Q8C2×#8760412yes—
92C2×D4—972no—
93C2×D4—1942no—
94C2×D4—1942no—
95C22×C4—971no—
96C22×C4—972no—
97C22×C4—1942no—
98C22×C4—1942no—
99M16—1944no—
100C2×C8—1942no—
101C4⋊C4—22no—
102C22⋊C4—1944no—
103C42—972no—
104C42—971no—
105C42—974no—
106C42—1942no—
107C42C2×#8760411yescharacteristic
108C23—971no—
109C23—1942no—
110Q8—24no—
111D4—3884no—
112D4—3884no—
113D4—1944no—
114C2×C4—22no—
115C2×C4—971no—
116C2×C4—972no—
117C2×C4—972no—
118C2×C4—972no—
119C2×C4—3882no—
120C2×C4—3882no—
121C2×C4—3882no—
122C2×C4—3884no—
123C2×C4—1944no—
124C2×C4C22⋊#8760412yes—
125C2×C4C22×#8760411yescharacteristic
126C8—3882no—
127C22—972no—
128C22—3882no—
129C22—3884no—
130C22—1942no—
131C22C2×C22⋊#8760411yescharacteristic
132C4—22no—
133C4—42no—
134C4—971no—
135C4—971no—
136C4—3884no—
137C4—1942no—
138C4D4×#8760412yes—
139C2—7762no—
140C2—1942no—
141C2#2026168711yescharacteristic
142C2#2026173412yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261286C2
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1963
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
1. #20261740C2semidirect
2. C2×#15690875C2semidirect
3. #20261671C2semidirect
11. #15690807C4semidirect
12. #15690799C4semidirect
16. C4⋊Q8×C97C4semidirect
21. C4×#87604D4semidirect
72. C97#1963semidirect
79. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617402
C2×#156908752
#202616712
#202612862
#196397