Finite Group Databasebeta

← #27157900

Subgroups — #27157900

Order 12416.

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

normal subgroups only

This group has 336 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
1C2×#15690874C214yesmaximal, complemented
2C4×#13095521C211yesmaximal, characteristic, complemented
3C4×C4⋊#87604C211yesmaximal, characteristic, complemented
4#20261283C211yesmaximal, characteristic
5C2×#13095521C2212yes—
6C2×C4⋊#87604—21no—
7C2×C4×#87604C2212yes—
8#15690874—28no—
9#15690865—21no—
10C4×#11715897—21no—
11(C2×#87604)⋊C4—21no—
12C2×#13095510C2212yes—
13#15690810C411yescharacteristic, complemented
14#15690797C412yescomplemented
15#15690781—21no—
16#15690781—21no—
17D4×Dic97—21no—
18D4×Dic97—21no—
19C42×D97C2211yescharacteristic
20#127×C97C411yescharacteristic, complemented
21C22×#87604—22no—
22#13095521—22no—
23#13095521D414yescomplemented
24C2×#11715897—22no—
25C4⋊#87604—24no—
26C4×#87604—24no—
27#13095510—24no—
28#13095510—24no—
29C2×Dic194C2×C412yes—
30C2×C22⋊D97—22no—
31C22×Dic97—22no—
32C2×C4×D97D411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C2×C4×D97C2311yescharacteristic
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C22⋊Dic97—22no—
38C22⋊Dic97—22no—
39C4×Dic97C2×C411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C2×D4×C97—21no—
43C2×D4×C97—21no—
44C2×D4×C97C2×C412yes—
45C42×C97C2×C411yescharacteristic
46C23×C97—22no—
47C22×D97C2×D411yescharacteristic
48C2×#87604—24no—
49C2×#87604—42no—
50C2×#87604—42no—
51D4×C97—24no—
52D4×C97—42no—
53D4×C97—42no—
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22×C411yescharacteristic
57C22⋊D97—42no—
58C22⋊D97—42no—
59C2×Dic97—22no—
60C2×Dic97—44no—
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C4×D97—22no—
65C4×D97—41no—
66C4×D97C2×D412yes—
67C4×D97#4112yes—
68Dic194—24no—
69#11715897—24no—
70C22×C97—42no—
71C22×C97—42no—
72C22×C97—42no—
73C22×C97C2×C22⋊C411yescharacteristic
74C2×D97—21no—
75C2×D97—21no—
76C2×D97#12411yescharacteristic
77C2×D97#12711yescharacteristic
78#87604—44no—
79C4×C97—21no—
80C4×C97#10412yescomplemented
81C4×C97C4×D412yes—
82Dic97—21no—
83Dic97—21no—
84Dic97—42no—
85Dic97#12112yes—
86C2×C97—82no—
87C2×C97#39011yescharacteristic
88C2×C97C2×#10411yescharacteristic
89C2×C97#42111yescharacteristic
90D97—21no—
91D97—21no—
92#1980—971nomaximal, Sylow
93C97#198011yescharacteristic, complemented, Sylow
94#532—971no—
95C2×#104—974no—
96C4×M16—971no—
97C4×C4⋊C4—971no—
98C2×#41—972no—
99C2×M16—972no—
100C4⋊Q8—971no—
101#127#8760411yescharacteristic, complemented
102#124—972no—
103#121—1941no—
104#121—1941no—
105C4×D4—1941no—
106C4×D4—1941no—
107C2×C4⋊C4—1941no—
108C2×C42—971no—
109C2×C42—972no—
110#104—1948no—
111#97—1941no—
112C4×C8—1941no—
113#95—1941no—
114#41—1944no—
115#41—1944no—
116C2×Q8—972no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—1942no—
120C2×D4C2×#8760412yes—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—1942no—
125C22×C4—1942no—
126M16—974no—
127M16—1942no—
128C2×C8—1942no—
129C4⋊C4—1944no—
130C4⋊C4—1941no—
131C4⋊C4—1941no—
132C22⋊C4—1942no—
133C22⋊C4—1942no—
134C42—971no—
135C42—971no—
136C42—971no—
137C42—1944no—
138C42C2×#8760411yescharacteristic
139C23—22no—
140C23—971no—
141Q8—1944no—
142D4—24no—
143D4—42no—
144D4—42no—
145D4—3882no—
146D4—3882no—
147C2×C4—971no—
148C2×C4—972no—
149C2×C4—972no—
150C2×C4—971no—
151C2×C4—971no—
152C2×C4—3881no—
153C2×C4—3884no—
154C2×C4—3882no—
155C2×C4—3882no—
156C2×C4—1944no—
157C2×C4—1942no—
158C2×C4—1942no—
159C2×C4C22⋊#8760411yescharacteristic
160C2×C4C22⋊#8760411yescharacteristic
161C2×C4C22×#8760411yescharacteristic
162C8—1944no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—971no—
167C22—971no—
168C22—1941no—
169C22—1941no—
170C22C2×C22⋊#8760411yescharacteristic
171C4—21no—
172C4—972no—
173C4—3884no—
174C4—3882no—
175C4—1941no—
176C4—1941no—
177C4#1569087412yescomplemented
178C4D4×#8760412yes—
179C2—82no—
180C2—1941no—
181C2—1941no—
182C2#2026168711yescharacteristic
183C2C2×#1569087411yescharacteristic
184C2#2026173311yescharacteristic
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#1980
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. C2×#15690874C2semidirect
2. C4×#13095521C2semidirect
3. C4×C4⋊#87604C2semidirect
13. #15690810C4semidirect
14. #15690797C4semidirect
20. #127×C97C4semidirect
23. #13095521D4semidirect
80. C4×C97#104semidirect
93. C97#1980semidirect
101. #127#87604semidirect
177. C4#15690874semidirect

Maximal subgroups

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

subgroupindex
C2×#156908742
C4×#130955212
C4×C4⋊#876042
#202612832
#198097