Finite Group Databasebeta

← #27157074

Subgroups — #27157074

Order 12416.

266 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 266 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#20261739C211yesmaximal, characteristic, complemented
2C4×#13095521C211yesmaximal, characteristic, complemented
3Q8⋊#11715897C212yesmaximal, complemented
4#20261583C212yesmaximal, complemented
5C4×#13095512C211yesmaximal, characteristic
6C2×#13095521—21no—
7C2×#13095521—21no—
8C4⋊#11715897—21no—
9C4⋊#11715897C2212yes—
10#15690865—21no—
11C4×#11715897C2212yes—
12C2×#13095512C411yescharacteristic
13Q8×Dic97C2211yescharacteristic
14C4×D388—21no—
15C4×D388C411yescharacteristic
16#15690772—21no—
17#15690772C2211yescharacteristic
18#15690768—21no—
19#15690768C411yescharacteristic
20C42×D97—21no—
21C42×D97C2211yescharacteristic
22C4×Q8×C97C411yescharacteristic
23#13095521—24no—
24#13095521—42no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27#13095512—22no—
28#13095512—22no—
29C2×D388—21no—
30C2×D388C2×C411yescharacteristic
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97C2×C411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C4⋊Dic97—21no—
38C4⋊Dic97C2×C411yescharacteristic
39#13095488—21no—
40#13095488—21no—
41#13095488—21no—
42C4×Dic97—21no—
43C4×Dic97—21no—
44C4×Dic97D411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97C2311yescharacteristic
47C2×Q8×C97C2×C411yescharacteristic
48C4⋊C4×C97—21no—
49C4⋊C4×C97C2×C411yescharacteristic
50C42×C97—21no—
51C42×C97C2×C411yescharacteristic
52C22×D97—21no—
53C22×D97C22⋊C411yescharacteristic
54Q8×C97—21no—
55Q8×C97M1612yescomplemented
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22⋊C411yescharacteristic
60C2×C4×C97C22×C411yescharacteristic
61C2×Dic97—21no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65D388—24no—
66D388—21no—
67D388M1612yescomplemented
68C4×D97—24no—
69C4×D97—22no—
70C4×D97—21no—
71C4×D97—21no—
72C4×D97—21no—
73C4×D97—21no—
74C4×D97—42no—
75C4×D97—41no—
76C4×D97—41no—
77#11715897—24no—
78#11715897—42no—
79C22×C97C2×C22⋊C411yescharacteristic
80C2×D97—21no—
81C2×D97—22no—
82C2×D97—21no—
83C2×D97—21no—
84C2×D97—41no—
85C2×D97—41no—
86C4×C97—22no—
87C4×C97—21no—
88C4×C97—41no—
89C4×C97#10412yes—
90C4×C97C2×M1611yescharacteristic
91C4×C97C2×M1611yescharacteristic
92Dic97—21no—
93Dic97—21no—
94Dic97—41no—
95Dic97AGL(1,ℤ/8)11yescharacteristic
96Dic97#13711yescharacteristic
97C2×C97#40711yescharacteristic
98C2×C97#41711yescharacteristic
99C2×C97C2×#10411yescharacteristic
100D97—41no—
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#8760411yescharacteristic
114C4×D4—971no—
115C4×D4—971no—
116C4×D4—1941no—
117C4×D4—1941no—
118#117—971no—
119#117—1941no—
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×Q8C2×#8760411yescharacteristic
129C2×D4—971no—
130C2×D4—1941no—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—971no—
134C22×C4—1941no—
135C22×C4—1941no—
136C22×C4—1941no—
137M16—3882no—
138M16—1944no—
139C2×C8—1942no—
140C2×C8—1942no—
141C4⋊C4—21no—
142C4⋊C4—971no—
143C4⋊C4—1941no—
144C4⋊C4C2×#8760411yescharacteristic
145C22⋊C4—1941no—
146C22⋊C4—1941no—
147C22⋊C4—1941no—
148C42—21no—
149C42—971no—
150C42—971no—
151C42—971no—
152C42—1941no—
153C42—1941no—
154C42C2×#8760411yescharacteristic
155C23—971no—
156C23—1941no—
157Q8—21no—
158Q8#1309552112yescomplemented
159D4—972no—
160D4—1944no—
161D4—1941no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—971no—
165C2×C4—971no—
166C2×C4—971no—
167C2×C4—3881no—
168C2×C4—3881no—
169C2×C4—3882no—
170C2×C4—1941no—
171C2×C4—1941no—
172C2×C4—1944no—
173C2×C4—1941no—
174C2×C4—1942no—
175C2×C4—1941no—
176C2×C4—1941no—
177C2×C4C22⋊#8760411yescharacteristic
178C2×C4C22⋊#8760411yescharacteristic
179C2×C4C22×#8760411yescharacteristic
180C8—3882no—
181C8—1944no—
182C22—3881no—
183C22—3881no—
184C22—1942no—
185C22—1941no—
186C22—1941no—
187C22—1941no—
188C22C2×C22⋊#8760411yescharacteristic
189C4—22no—
190C4—21no—
191C4—41no—
192C4—971no—
193C4—971no—
194C4—3881no—
195C4—1941no—
196C4—1941no—
197C4#1569087512yes—
198C4D97⋊M1611yescharacteristic
199C4C2×#1309552111yescharacteristic
200C2—3882no—
201C2—3881no—
202C2#2026170611yescharacteristic
203C2#2026172611yescharacteristic
204C2C2×#1569087511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#13095512C4
cosocleC4×Dic97C23
hypercenterC4×Q8#87604
hypocenterC97#1489
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20261739C2semidirect
2. C4×#13095521C2semidirect
3. Q8⋊#11715897C2semidirect
4. #20261583C2semidirect
55. Q8×C97M16semidirect
67. D388M16semidirect
103. C97#1489semidirect
158. Q8#13095521semidirect

Maximal subgroups

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

subgroupindex
#202617392
C4×#130955212
Q8⋊#117158972
#202615832
C4×#130955122
#148997