Finite Group Databasebeta

← #27157206

Subgroups — #27157206

Order 12416.

256 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261757C211yesmaximal, characteristic, complemented
2#20261694C211yesmaximal, characteristic, complemented
3#20261590C214yesmaximal, complemented
4#20261283C211yesmaximal, characteristic
5C2×#13095519—21no—
6C4⋊#11715897—21no—
7#15690867—21no—
8#15690865—21no—
9#15690865C2212yes—
10#15690865C2212yes—
11C2×#13095510C412yes—
12#15690810C411yescharacteristic
13#15690797C2212yes—
14#15690781—21no—
15#15690781—21no—
16D4×Dic97—21no—
17D4×Dic97—21no—
18C42×D97C2211yescharacteristic
19#127×C97C411yescharacteristic
20#13095519—24no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23#13095510—44no—
24C2×Dic194C2×C412yes—
25C2×C22⋊D97—22no—
26C22×Dic97—22no—
27C2×C4×D97C2×C411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C22⋊Dic97—22no—
33C22⋊Dic97—22no—
34C4×Dic97D411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97C2311yescharacteristic
37C2×D4×C97—21no—
38C2×D4×C97—21no—
39C2×D4×C97C2×C412yes—
40C42×C97C2×C411yescharacteristic
41C23×C97—22no—
42C22×D97C22⋊C411yescharacteristic
43D4×C97—42no—
44D4×C97—42no—
45D4×C97—42no—
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C22×C411yescharacteristic
49C22⋊D97—44no—
50C2×Dic97—22no—
51C2×Dic97—44no—
52C2×Dic97C22⋊C411yescharacteristic
53C2×Dic97C2×D411yescharacteristic
54C2×Dic97C2×D411yescharacteristic
55C4×D97—22no—
56C4×D97—22no—
57C4×D97—41no—
58Dic194—42no—
59#11715897—24no—
60#11715897—42no—
61C22×C97—42no—
62C22×C97—42no—
63C22×C97—42no—
64C22×C97C2×C22⋊C411yescharacteristic
65C2×D97—21no—
66C2×D97—21no—
67C2×D97#10012yes—
68C4×C97—21no—
69C4×C97—21no—
70C4×C97#10412yes—
71Dic97—21no—
72Dic97—21no—
73Dic97—21no—
74Dic97—42no—
75C2×C97—82no—
76C2×C97C2×#10011yescharacteristic
77C2×C97C2×#10411yescharacteristic
78C2×C97#42111yescharacteristic
79D97—22no—
80#1530—971nomaximal, Sylow
81C97#153011yescharacteristic, complemented, Sylow
82#532—971no—
83#432—971no—
84C2×#97—971no—
85#329—974no—
86C2×#41—972no—
87C22×C8—1941no—
88C4⋊Q8—971no—
89#127#8760411yescharacteristic
90#124—972no—
91#121—1941no—
92#121—1941no—
93C4×D4—1941no—
94C4×D4—1941no—
95C2×C42—971no—
96C4⋊C8—1941no—
97C22⋊C8—1941no—
98#97—972no—
99#97—972no—
100#97—1941no—
101#41—3884no—
102C2×Q8—972no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—1942no—
106C2×D4C2×#8760412yes—
107C22×C4—971no—
108C22×C4—971no—
109C22×C4—971no—
110C22×C4—1942no—
111C2×C8—1942no—
112C2×C8—1942no—
113C2×C8—1944no—
114C4⋊C4—1941no—
115C4⋊C4—1941no—
116C22⋊C4—1942no—
117C22⋊C4—1942no—
118C42—971no—
119C42—971no—
120C42—971no—
121C42C2×#8760411yescharacteristic
122C23—22no—
123C23—971no—
124Q8—3882no—
125D4—42no—
126D4—42no—
127D4—42no—
128D4—3884no—
129C2×C4—971no—
130C2×C4—971no—
131C2×C4—971no—
132C2×C4—3881no—
133C2×C4—3884no—
134C2×C4—1942no—
135C2×C4—1942no—
136C2×C4—1942no—
137C2×C4C22⋊#8760411yescharacteristic
138C2×C4C22⋊#8760411yescharacteristic
139C2×C4C22×#8760411yescharacteristic
140C8—3882no—
141C8—1944no—
142C22—42no—
143C22—42no—
144C22—42no—
145C22—972no—
146C22—1941no—
147C22—1941no—
148C22C2×C22⋊#8760411yescharacteristic
149C4—21no—
150C4—21no—
151C4—3882no—
152C4—1941no—
153C4—1941no—
154C4—1941no—
155C4#1569087412yes—
156C2—82no—
157C2—1942no—
158C2D97⋊#10011yescharacteristic
159C2C2×#1569087411yescharacteristic
160C2#2026173311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261283C2
cosocleC4×Dic97C23
hypercenter#127#87604
hypocenterC97#1530
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261757C2semidirect
2. #20261694C2semidirect
3. #20261590C2semidirect
81. C97#1530semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202616942
#202615902
#202612832
#153097