Finite Group Databasebeta

← #27150304

Subgroups — #27150304

Order 12416.

484 conjugacy classes of subgroups; only the 297 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 297 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
1C2×#15690817C212yesmaximal
2#20260762C216yesmaximal, complemented
3#20260759C213yesmaximal, complemented
4C2×C4⋊#11715895C213yesmaximal, complemented
5C4×#41×C97C211yesmaximal, characteristic
6#15690817—28no—
7C2×#13095495C2216yes—
8C22×#11715895C2216yes—
9C4⋊#11715895C2211yescharacteristic
10C4⋊#11715895C2213yes—
11C4⋊#11715895C2216yes—
12C4⋊#11715895C2216yes—
13C2×#41×C97C2211yescharacteristic
14C4×Q8×C97C411yescharacteristic
15C4×Q8×C97C411yescharacteristic
16C4×D4×C97C413yes—
17C4×D4×C97C413yes—
18#117×C97C2213yes—
19C2×C42×C97C2213yes—
20#13095495—212no—
21C2×#11715895—212no—
22C2×#11715895C2312yes—
23C2×#11715895C2316yes—
24#41×C97D414yes—
25#41×C97Q814yes—
26C2×Q8×C97C2×C411yescharacteristic
27C2×D4×C97C2×C413yes—
28C22×C4×C97C2×C413yes—
29C22×C4×C97C2×C413yes—
30C22×C4×C97C2313yes—
31C4⋊C4×C97C2×C413yes—
32C4⋊C4×C97C2×C413yes—
33C22⋊C4×C97C2×C416yes—
34C42×C97C2×C413yes—
35C42×C97C2×C413yes—
36C42×C97C2311yescharacteristic
37C42×C97C2313yes—
38C23×C97C22×C413yes—
39Q8×C97C4⋊C414yes—
40D4×C97C4⋊C4112yes—
41C2×C4×C97—26no—
42C2×C4×C97—43no—
43C2×C4×C97C22×C411yescharacteristic
44C2×C4×C97C22×C413yes—
45C2×C4×C97C22×C413yes—
46C2×C4×C97C22×C413yes—
47C2×C4×C97C22×C411yescharacteristic
48C2×C4×C97C2×D416yes—
49C2×C4×C97C2×Q816yes—
50C2×C4×C97C2411yescharacteristic
51#11715895—22no—
52#11715895—26no—
53C22×C97—23no—
54C22×C97—23no—
55C22×C97C2×C4⋊C416yes—
56C22×C97C23×C411yescharacteristic
57C4×C97—23no—
58C4×C97—21no—
59C4×C97—23no—
60C4×C97C2×C4⋊C416yes—
61C4×C97#13112yescomplemented
62C4×C97C22×D411yescharacteristic
63C4×C97C22×Q811yescharacteristic
64C2×C97—26no—
65C2×C97C22×C4⋊C411yescharacteristic
66C2×C97C2×#13111yescharacteristic
67C2×C97#56811yescharacteristic
68#2906—971nomaximal, Sylow
69C97#290611yescharacteristic, complemented, Sylow
70C2×#131—972no—
71C4×#41D9711yescharacteristic
72#424—976no—
73#423—973no—
74C2×C4⋊C8—973no—
75C2×#41C2×D9711yescharacteristic
76#131—1948no—
77C2×M16—976no—
78C22×C8—976no—
79C4×Q8Dic9711yescharacteristic
80C4×Q8Dic9711yescharacteristic
81C4×D4Dic9713yes—
82C4×D4Dic9713yes—
83#117C2×D9713yes—
84C2×C42C2×D9713yes—
85C4⋊C8—971no—
86C4⋊C8—973no—
87C4⋊C8—976no—
88C4⋊C8—976no—
89#41Dic19414yes—
90#41D38814yes—
91C2×Q8C2×Dic9711yescharacteristic
92C2×D4C2×Dic9713yes—
93C22×C4C2×Dic9713yes—
94C22×C4C2×Dic9713yes—
95C22×C4C22×D9713yes—
96M16—19412no—
97C2×C8—972no—
98C2×C8—976no—
99C2×C8—19412no—
100C4⋊C4C2×Dic9713yes—
101C4⋊C4C2×Dic9713yes—
102C22⋊C4C2×Dic9716yes—
103C42C2×Dic9713yes—
104C42C2×Dic9713yes—
105C42C22×D9711yescharacteristic
106C42C22×D9713yes—
107C23C22×Dic9713yes—
108Q8C4⋊Dic9714yes—
109D4C4⋊Dic97112yes—
110C2×C4—26no—
111C2×C4—43no—
112C2×C4C22×Dic9711yescharacteristic
113C2×C4C22×Dic9713yes—
114C2×C4C22×Dic9713yes—
115C2×C4C22×Dic9713yes—
116C2×C4C22×Dic9711yescharacteristic
117C2×C4C2×D38816yes—
118C2×C4C2×Dic19416yes—
119C2×C4C23×D9711yescharacteristic
120C8—1942no—
121C8—1946no—
122C22—23no—
123C22—23no—
124C22C2×C4⋊Dic9716yes—
125C22C23×Dic9711yescharacteristic
126C4—23no—
127C4—21no—
128C4—23no—
129C4C2×C4⋊Dic9716yes—
130C4#1569081712yescomplemented
131C4C22×D38811yescharacteristic
132C4C22×Dic19411yescharacteristic
133C2—26no—
134C2C22×C4⋊Dic9711yescharacteristic
135C2C2×#1569081711yescharacteristic
136C2#2026150211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
FittingC4×#41×C97C2
prime radicalC2×#41×C97C22
cosocleC2×C4×C97C24
hypercenterC4×#41D97
hypocenterC97#2906
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #20260762C2semidirect
3. #20260759C2semidirect
4. C2×C4⋊#11715895C2semidirect
61. C4×C97#131semidirect
69. C97#2906semidirect
130. C4#15690817semidirect

Maximal subgroups

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

subgroupindex
C2×#156908172
#202607622
#202607592
C2×C4⋊#117158952
C4×#41×C972
#290697