Finite Group Databasebeta

← #27157073

Subgroups — #27157073

Order 12416.

266 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261738C211yesmaximal, characteristic, complemented
2C4×#13095520C211yesmaximal, characteristic, complemented
3#20261585C212yesmaximal, complemented
4#20261583C212yesmaximal, complemented
5#20261203C211yesmaximal, characteristic
6C2×#13095520—21no—
7C2×#13095520—21no—
8#15690877—21no—
9C4⋊#11715897C2212yes—
10#15690865—21no—
11C4×#11715897C2212yes—
12C2×#13095511C411yescharacteristic
13#15690775C411yescharacteristic
14#15690775C2211yescharacteristic
15#15690772C411yescharacteristic
16#15690772C2211yescharacteristic
17#15690770—21no—
18#15690770—21no—
19#15690767—21no—
20C2×C4×Dic97C2211yescharacteristic
21C42×D97—21no—
22#117×C97C411yescharacteristic
23#13095520—24no—
24#13095520—42no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27#13095511—22no—
28#13095511—22no—
29C2×Dic194C2×C411yescharacteristic
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32C22×Dic97D411yescharacteristic
33C22×Dic97D411yescharacteristic
34C2×C4×D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97—21no—
37#13095490—21no—
38#13095490—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C4×Dic97—21no—
43C4×Dic97C2×C411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97C2311yescharacteristic
47C22×C4×C97C2×C411yescharacteristic
48C4⋊C4×C97C2×C411yescharacteristic
49C4⋊C4×C97C2×C411yescharacteristic
50C22⋊C4×C97—21no—
51C42×C97—21no—
52C23×C97C22⋊C411yescharacteristic
53C22×D97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22×C411yescharacteristic
60C22⋊D97—24no—
61C2×Dic97—21no—
62C2×Dic97—22no—
63C2×Dic97—21no—
64C2×Dic97—21no—
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C2×D411yescharacteristic
68C2×Dic97C2×D411yescharacteristic
69D388—21no—
70D388M1612yescomplemented
71C4×D97—24no—
72C4×D97—42no—
73C4×D97—41no—
74C4×D97—41no—
75Dic194—21no—
76Dic194M1612yescomplemented
77#11715897—24no—
78#11715897—42no—
79C22×C97—21no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97C2×C22⋊C411yescharacteristic
83C2×D97—22no—
84C2×D97—41no—
85C2×D97—41no—
86C4×C97—21no—
87C4×C97—41no—
88C4×C97—41no—
89C4×C97C2×M1611yescharacteristic
90C4×C97C2×M1611yescharacteristic
91Dic97—22no—
92Dic97—21no—
93Dic97—21no—
94Dic97#10412yes—
95Dic97AGL(1,ℤ/8)11yescharacteristic
96Dic97#13711yescharacteristic
97C2×C97—41no—
98C2×C97#40711yescharacteristic
99C2×C97#41711yescharacteristic
100C2×C97C2×#10411yescharacteristic
101D97—42no—
102#1489—971nomaximal, Sylow
103C97#148911yescharacteristic, complemented, Sylow
104C4×#41—971no—
105#423—971no—
106C4×M16—971no—
107Q8⋊C8—972no—
108D4⋊C8—972no—
109C2×#41—971no—
110C2×M16—1941no—
111C2×M16—1941no—
112C4×Q8—971no—
113C4×Q8—971no—
114C4×D4—971no—
115C4×D4—971no—
116C4×D4—1941no—
117C4×D4—1941no—
118#117—1941no—
119#117#8760411yescharacteristic
120C2×C42—971no—
121C2×C42—1941no—
122C4⋊C8—972no—
123C4⋊C8—1941no—
124#97—1941no—
125C4×C8—972no—
126#41—1942no—
127#41—1942no—
128C2×Q8—971no—
129C2×D4—971no—
130C2×D4—1941no—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—1941no—
134C22×C4—1941no—
135C22×C4—1941no—
136C22×C4C2×#8760411yescharacteristic
137M16—3882no—
138M16—1944no—
139C2×C8—1942no—
140C2×C8—1942no—
141C4⋊C4—1941no—
142C4⋊C4—1941no—
143C4⋊C4C2×#8760411yescharacteristic
144C4⋊C4C2×#8760411yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—1941no—
147C22⋊C4—1941no—
148C42—21no—
149C42—971no—
150C42—971no—
151C42—971no—
152C42—971no—
153C42—1941no—
154C42—1941no—
155C23—1941no—
156C23C22⋊#8760411yescharacteristic
157Q8—972no—
158Q8—1941no—
159D4—972no—
160D4—1944no—
161D4—1941no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—971no—
167C2×C4—971no—
168C2×C4—971no—
169C2×C4—971no—
170C2×C4—3881no—
171C2×C4—3881no—
172C2×C4—3882no—
173C2×C4—1944no—
174C2×C4—1941no—
175C2×C4—1942no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4C22⋊#8760411yescharacteristic
179C2×C4C22×#8760411yescharacteristic
180C8—3882no—
181C8—1944no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—3881no—
186C22—3881no—
187C22—1942no—
188C22C2×C22⋊#8760411yescharacteristic
189C4—21no—
190C4—41no—
191C4—41no—
192C4—971no—
193C4—971no—
194C4—972no—
195C4—1942no—
196C4—1941no—
197C4—1941no—
198C4D97⋊M1611yescharacteristic
199C4D97⋊M1611yescharacteristic
200C2—41no—
201C2—3882no—
202C2#2026170711yescharacteristic
203C2#2026172611yescharacteristic
204C2D97⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#13095511C4
cosocleC4×Dic97C23
hypercenter#117#87604
hypocenterC97#1489
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20261738C2semidirect
2. C4×#13095520C2semidirect
3. #20261585C2semidirect
4. #20261583C2semidirect
70. D388M16semidirect
76. Dic194M16semidirect
103. C97#1489semidirect

Maximal subgroups

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

subgroupindex
#202617382
C4×#130955202
#202615852
#202615832
#202612032
#148997