Finite Group Databasebeta

← #27150966

Subgroups — #27150966

Order 12416.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D388C211yesmaximal, characteristic, complemented
2C2×#15690838C212yesmaximal, complemented
3#20261246C211yesmaximal, characteristic, complemented
4#20260888C214yesmaximal, complemented
5#20260871C214yesmaximal, complemented
6#20260741C212yesmaximal, complemented
7#407×C97C211yesmaximal, characteristic, complemented
8C24×D97—21no—
9C2×#13095511C2211yescharacteristic, complemented
10C22×D388—28no—
11C22×D388—22no—
12C22×D388—21no—
13C22×D388C2214yes—
14C22×D388C2212yes—
15#15690838—28no—
16C2×#13095500C2214yescomplemented
17C2×D776C2214yescomplemented
18#15690793—21no—
19#15690789C2212yescomplemented
20#15690784C2212yescomplemented
21#15690779—21no—
22C4×C22⋊D97—22no—
23#15690769C2211yescharacteristic, complemented
24#15690732C2218yescomplemented
25C23×C4×C97C2211yescharacteristic
26C2×M16×C97C2212yescomplemented
27C22⋊C8×C97C2214yescomplemented
28C23×D97—28no—
29C23×D97—24no—
30C23×D97—22no—
31C23×D97—48no—
32#13095511—24no—
33C2×Dic194C2311yescharacteristic, complemented
34C2×D388—28no—
35C2×D388—216no—
36C2×D388—24no—
37C2×D388—24no—
38C2×D388—22no—
39C2×D388—216no—
40C2×D388D418yescomplemented
41C2×D388C2314yes—
42C2×D388C2311yescharacteristic, complemented
43C2×C22⋊D97—22no—
44C2×C22⋊D97—21no—
45C2×C4×D97—21no—
46#13095500—28no—
47D776—28no—
48C4⋊Dic97C2312yescomplemented
49#13095490—22no—
50C22⋊Dic97—21no—
51C22⋊Dic97—22no—
52#13095488—22no—
53C4×Dic97—21no—
54C24×C97D411yescharacteristic
55C22×C4×C97—24no—
56C22×C4×C97D412yes—
57C22×C4×C97D411yescharacteristic
58C22×C4×C97C2312yes—
59C22×C4×C97C2311yescharacteristic
60M16×C97—24no—
61C2×C8×C97C2314yescomplemented
62C23×C97—22no—
63C23×C97—24no—
64C23×C97C2×D411yescharacteristic
65C23×C97C2×D412yes—
66C22×D97—216no—
67C22×D97—28no—
68C22×D97—24no—
69C22×D97—21no—
70C22×D97—432no—
71C22×D97—48no—
72C22×D97—48no—
73C22×D97—48no—
74C2×C4×C97—22no—
75C2×C4×C97—24no—
76C2×C4×C97—24no—
77C2×C4×C97C2×D414yes—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2411yescharacteristic
81C22⋊D97—42no—
82C22⋊D97—44no—
83C2×Dic97—22no—
84C2×Dic97—21no—
85D388—216no—
86D388—28no—
87D388—24no—
88D388—22no—
89D388C2×D418yes—
90C4×D97—42no—
91Dic194—22no—
92C8×C97—24no—
93C22×C97—28no—
94C22×C97—22no—
95C22×C97—22no—
96C22×C97—22no—
97C22×C97#12012yes—
98C22×C97AGL(1,ℤ/8)14yescomplemented
99C22×C97C22×D411yescharacteristic
100C2×D97—28no—
101C2×D97—416no—
102C2×D97—432no—
103C2×D97—48no—
104C2×D97—41no—
105C2×D97—42no—
106C4×C97—22no—
107C4×C97#12012yes—
108C4×C97C22×D412yes—
109Dic97—41no—
110Dic97—42no—
111C2×C97—22no—
112C2×C97—24no—
113C2×C97C2×#12011yescharacteristic
114C2×C97C2×AGL(1,ℤ/8)12yes—
115D97—48no—
116D97—81no—
117#3003—971nomaximal, Sylow
118C97#300311yescharacteristic, complemented, Sylow
119C23×D4—971no—
120C2×AGL(1,ℤ/8)—972no—
121#525—971no—
122#450—974no—
123#447—974no—
124#418—972no—
125#407D9711yescharacteristic, complemented
126C25—1941no—
127C2×#41—971no—
128C22×D4—972no—
129C22×D4—974no—
130C22×D4—1948no—
131C22×D4—1942no—
132C22×D4—1941no—
133C23×C4C2×D9711yescharacteristic
134AGL(1,ℤ/8)—1948no—
135C2×SD16—974no—
136C2×D8—974no—
137C2×M16C2×D9712yescomplemented
138#123—1941no—
139C22⋊Q8—972no—
140#121—972no—
141#120—1941no—
142C4×D4—1942no—
143#117—971no—
144#102—978no—
145C22⋊C8C2×D9714yescomplemented
146C24—3888no—
147C24—1948no—
148C24—1942no—
149C24—1944no—
150C24D38811yescharacteristic
151#41—1944no—
152C2×Q8—971no—
153C2×D4—978no—
154C2×D4—974no—
155C2×D4—971no—
156C2×D4—19416no—
157C2×D4—1948no—
158C2×D4—19416no—
159C2×D4—1942no—
160C2×D4—1941no—
161C2×D4—1944no—
162C2×D4—1944no—
163C2×D4—1942no—
164C22×C4—24no—
165C22×C4—1941no—
166C22×C4D38812yes—
167C22×C4D38811yescharacteristic
168C22×C4C22×D9712yes—
169C22×C4C22×D9711yescharacteristic
170SD16—1948no—
171D8—1948no—
172M16—24no—
173C2×C8C22×D9714yescomplemented
174C4⋊C4—972no—
175C4⋊C4—1942no—
176C22⋊C4—1941no—
177C22⋊C4—1942no—
178C22⋊C4—1942no—
179C42—1941no—
180C23—22no—
181C23—24no—
182C23—38832no—
183C23—3888no—
184C23—3888no—
185C23—3888no—
186C23—19416no—
187C23—1941no—
188C23—1948no—
189C23—1944no—
190C23C2×D38811yescharacteristic
191C23C2×D38812yes—
192Q8—1942no—
193D4—978no—
194D4—3882no—
195D4—3884no—
196D4—19416no—
197D4—1948no—
198D4—1942no—
199D4—1944no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—3882no—
204C2×C4—1942no—
205C2×C4—1941no—
206C2×C4D4×D9714yes—
207C2×C4C2×D38812yes—
208C2×C4C2×D38811yescharacteristic
209C2×C4C23×D9711yescharacteristic
210C8—24no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—38816no—
216C22—38832no—
217C22—3881no—
218C22—3882no—
219C22—3888no—
220C22—1948no—
221C22#1569078012yes—
222C22#1569083814yescomplemented
223C22C22×D38811yescharacteristic
224C4—22no—
225C4—3881no—
226C4—3882no—
227C4#1569078012yes—
228C4C2×D4×D9712yes—
229C2—22no—
230C2—24no—
231C2—7761no—
232C2—3888no—
233C2C2×#1569078011yescharacteristic
234C2C2×#1569083812yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#407×C97C2
socleC22×C97C22×D4
hypercenter#407D97
hypocenterC97#3003
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×D388C2semidirect
2. C2×#15690838C2semidirect
3. #20261246C2semidirect
4. #20260888C2semidirect
5. #20260871C2semidirect
6. #20260741C2semidirect
7. #407×C97C2semidirect
9. C2×#13095511C22semidirect
16. C2×#13095500C22semidirect
17. C2×D776C22semidirect
19. #15690789C22semidirect
20. #15690784C22semidirect
23. #15690769C22semidirect
24. #15690732C22semidirect
26. C2×M16×C97C22semidirect
27. C22⋊C8×C97C22semidirect
33. C2×Dic194C23semidirect
40. C2×D388D4semidirect
42. C2×D388C23semidirect
48. C4⋊Dic97C23semidirect
61. C2×C8×C97C23semidirect
98. C22×C97AGL(1,ℤ/8)semidirect
118. C97#3003semidirect
125. #407D97semidirect
137. C2×M16C2×D97semidirect
145. C22⋊C8C2×D97semidirect
173. C2×C8C22×D97semidirect
222. C22#15690838semidirect

Maximal subgroups

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

subgroupindex
C23×D3882
C2×#156908382
#202612462
#202608882
#202608712
#202607412
#407×C972
#300397