Finite Group Databasebeta

← #27157874

Subgroups — #27157874

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×#15690874C214yesmaximal, complemented
3#20261671C211yesmaximal, characteristic, complemented
4#20261283C211yesmaximal, characteristic
5C2×#13095521C2212yes—
6C2×C4×#87604C2212yes—
7#15690877—22no—
8#15690874—28no—
9(C2×#87604)⋊C4—22no—
10C2×#13095510C2212yes—
11#15690810C411yescharacteristic, complemented
12#15690797C412yescomplemented
13#15690781—22no—
14D4×Dic97—22no—
15C42×D97C2211yescharacteristic
16#127×C97C411yescharacteristic, complemented
17C22×#87604—22no—
18#13095521—24no—
19C2×#11715897—22no—
20C4×#87604—22no—
21C4×#87604D414yescomplemented
22#13095510—24no—
23#13095510—24no—
24C2×Dic194C2×C412yes—
25C2×C22⋊D97—22no—
26C22×Dic97—22no—
27C2×C4×D97D412yes—
28C2×C4×D97C2311yescharacteristic
29C4⋊Dic97—22no—
30C22⋊Dic97—24no—
31C4×Dic97C2×C411yescharacteristic
32C4×Dic97D412yes—
33C2×D4×C97—22no—
34C2×D4×C97C2×C412yes—
35C42×C97C2×C411yescharacteristic
36C23×C97—22no—
37C22×D97C2×D411yescharacteristic
38C2×#87604—24no—
39C2×#87604—44no—
40D4×C97—24no—
41D4×C97—44no—
42C2×C4×C97C22⋊C412yes—
43C2×C4×C97C22×C411yescharacteristic
44C22⋊D97—44no—
45C2×Dic97—22no—
46C2×Dic97—42no—
47C2×Dic97—42no—
48C2×Dic97C22⋊C412yes—
49C2×Dic97C2×D411yescharacteristic
50C4×D97—42no—
51C4×D97C2×D412yes—
52C4×D97#4112yes—
53Dic194—24no—
54#11715897—42no—
55C22×C97—42no—
56C22×C97—44no—
57C22×C97C2×C22⋊C411yescharacteristic
58C2×D97—22no—
59C2×D97#12112yes—
60#87604—44no—
61C4×C97—22no—
62C4×C97C4×D412yes—
63Dic97—22no—
64Dic97—42no—
65Dic97#12411yescharacteristic
66Dic97#12711yescharacteristic
67C2×C97—82no—
68C2×C97#39011yescharacteristic
69C2×C97#42112yes—
70D97—22no—
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—971no—
80#127#8760411yescharacteristic, complemented
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×Q8—972no—
92C2×D4—22no—
93C2×D4—1942no—
94C2×D4C2×#8760412yes—
95C22×C4—971no—
96C22×C4—972no—
97C22×C4—1942no—
98C22×C4—1942no—
99M16—1944no—
100C2×C8—1942no—
101C4⋊C4—1942no—
102C22⋊C4—1944no—
103C42—972no—
104C42—971no—
105C42—974no—
106C42—1942no—
107C42C2×#8760411yescharacteristic
108C23—22no—
109C23—971no—
110Q8—1944no—
111D4—24no—
112D4—44no—
113D4—3884no—
114C2×C4—971no—
115C2×C4—972no—
116C2×C4—972no—
117C2×C4—972no—
118C2×C4—3882no—
119C2×C4—3882no—
120C2×C4—3882no—
121C2×C4—3884no—
122C2×C4—1944no—
123C2×C4—1942no—
124C2×C4C22⋊#8760412yes—
125C2×C4C22×#8760411yescharacteristic
126C8—3882no—
127C22—42no—
128C22—44no—
129C22—972no—
130C22—1942no—
131C22C2×C22⋊#8760411yescharacteristic
132C4—22no—
133C4—971no—
134C4—971no—
135C4—3884no—
136C4—3882no—
137C4—1942no—
138C4D4×#8760412yes—
139C2—82no—
140C2—1942no—
141C2#2026168711yescharacteristic
142C2#2026173312yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261283C2
cosocleC2×C4×D97C23
hypercenter#127#87604
hypocenterC97#1963
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261740C2semidirect
2. C2×#15690874C2semidirect
3. #20261671C2semidirect
11. #15690810C4semidirect
12. #15690797C4semidirect
16. #127×C97C4semidirect
21. C4×#87604D4semidirect
72. C97#1963semidirect
80. #127#87604semidirect

Maximal subgroups

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

subgroupindex
#202617402
C2×#156908742
#202616712
#202612832
#196397