Finite Group Databasebeta

← #27150973

Subgroups — #27150973

Order 12416.

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

normal subgroups only

This group has 738 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×Dic194C211yesmaximal, characteristic, complemented
2C2×#15690843C212yesmaximal, complemented
3#20261246C211yesmaximal, characteristic
4#20260893C214yesmaximal, complemented
5#20260876C214yesmaximal, complemented
6#20260745C212yesmaximal, complemented
7#407×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9C22×Dic194—28no—
10C22×Dic194—22no—
11C22×Dic194—21no—
12C22×Dic194C2214yes—
13C22×Dic194C2212yes—
14C23×Dic97—21no—
15#15690843—28no—
16C2×Dic388C2214yescomplemented
17C2×#13095500C2214yescomplemented
18#15690793—21no—
19#15690789C2212yes—
20#15690784C2212yes—
21#15690779—21no—
22C4×C22⋊D97—22no—
23#15690769C2211yescharacteristic
24#15690735C2218yescomplemented
25C23×C4×C97C2211yescharacteristic
26C2×M16×C97C2212yes—
27C22⋊C8×C97C2214yes—
28#13095511—24no—
29C2×Dic194—28no—
30C2×Dic194—216no—
31C2×Dic194—24no—
32C2×Dic194—24no—
33C2×Dic194—22no—
34C2×Dic194—216no—
35C2×Dic194D418yescomplemented
36C2×Dic194C2311yescharacteristic
37C2×Dic194C2314yes—
38C2×D388C2311yescharacteristic
39C2×C22⋊D97—22no—
40C2×C22⋊D97—21no—
41C22×Dic97—28no—
42C22×Dic97—24no—
43C22×Dic97—22no—
44C2×C4×D97—21no—
45Dic388—28no—
46#13095500—28no—
47C4⋊Dic97C2312yes—
48#13095490—22no—
49C22⋊Dic97—21no—
50C22⋊Dic97—22no—
51#13095488—22no—
52C4×Dic97—21no—
53C24×C97D411yescharacteristic
54C22×C4×C97—24no—
55C22×C4×C97D412yes—
56C22×C4×C97D411yescharacteristic
57C22×C4×C97C2312yes—
58C22×C4×C97C2311yescharacteristic
59M16×C97—24no—
60C2×C8×C97C2314yes—
61C23×C97—22no—
62C23×C97—24no—
63C23×C97C2×D411yescharacteristic
64C23×C97C2×D412yes—
65C22×D97—21no—
66C2×C4×C97—22no—
67C2×C4×C97—24no—
68C2×C4×C97—24no—
69C2×C4×C97C2×D414yes—
70C2×C4×C97C2×D412yes—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2411yescharacteristic
73C22⋊D97—42no—
74C22⋊D97—44no—
75C2×Dic97—216no—
76C2×Dic97—28no—
77C2×Dic97—24no—
78C2×Dic97—22no—
79C2×Dic97—21no—
80D388—22no—
81C4×D97—42no—
82Dic194—216no—
83Dic194—28no—
84Dic194—24no—
85Dic194—22no—
86Dic194C2×D418yes—
87C8×C97—24no—
88C22×C97—28no—
89C22×C97—22no—
90C22×C97—22no—
91C22×C97—22no—
92C22×C97#12012yes—
93C22×C97#13714yescomplemented
94C22×C97C22×D411yescharacteristic
95C2×D97—41no—
96C2×D97—42no—
97C4×C97—22no—
98C4×C97#12012yes—
99C4×C97C22×D412yes—
100Dic97—28no—
101Dic97—41no—
102Dic97—42no—
103C2×C97—22no—
104C2×C97—24no—
105C2×C97C2×#12011yescharacteristic
106C2×C97C2×#13712yes—
107D97—81no—
108#3004—971nomaximal, Sylow
109C97#300411yescharacteristic, complemented, Sylow
110C23×Q8—971no—
111C2×#137—972no—
112#525—971no—
113#451—974no—
114#448—974no—
115#419—972no—
116#407D9711yescharacteristic, complemented
117C2×#41—971no—
118C22×Q8—972no—
119C22×Q8—974no—
120C22×Q8—1948no—
121C22×Q8—1942no—
122C22×Q8—1941no—
123C23×C4—1941no—
124C23×C4C2×D9711yescharacteristic
125#137—1948no—
126C2×Q16—974no—
127C2×SD16—974no—
128C2×M16C2×D9712yes—
129#123—1941no—
130C22⋊Q8—972no—
131#121—972no—
132#120—1941no—
133C4×D4—1942no—
134#117—971no—
135#103—978no—
136C22⋊C8C2×D9714yes—
137C24D38811yescharacteristic
138#41—1944no—
139C2×Q8—978no—
140C2×Q8—974no—
141C2×Q8—971no—
142C2×Q8—19416no—
143C2×Q8—1948no—
144C2×Q8—19416no—
145C2×Q8—1944no—
146C2×Q8—1944no—
147C2×Q8—1942no—
148C2×D4—971no—
149C2×D4—1942no—
150C2×D4—1941no—
151C22×C4—24no—
152C22×C4—1941no—
153C22×C4—1948no—
154C22×C4—1944no—
155C22×C4—1942no—
156C22×C4D38812yes—
157C22×C4D38811yescharacteristic
158C22×C4C22×D9712yes—
159C22×C4C22×D9711yescharacteristic
160Q16—1948no—
161SD16—1948no—
162M16—24no—
163C2×C8C22×D9714yes—
164C4⋊C4—972no—
165C4⋊C4—1942no—
166C22⋊C4—1941no—
167C22⋊C4—1942no—
168C22⋊C4—1942no—
169C42—1941no—
170C23—22no—
171C23—24no—
172C23—1941no—
173C23C2×D38811yescharacteristic
174C23C2×D38812yes—
175Q8—978no—
176Q8—19416no—
177Q8—1942no—
178Q8—1948no—
179Q8—1944no—
180D4—3882no—
181D4—3884no—
182D4—1942no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—3882no—
187C2×C4—1942no—
188C2×C4—1941no—
189C2×C4—19416no—
190C2×C4—1948no—
191C2×C4—1944no—
192C2×C4D4×D9714yes—
193C2×C4C2×D38812yes—
194C2×C4C2×D38811yescharacteristic
195C2×C4C23×D9711yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—3881no—
202C22—3882no—
203C22#1569078012yes—
204C22#1569084314yescomplemented
205C22C22×D38811yescharacteristic
206C4—22no—
207C4—3881no—
208C4—3882no—
209C4—1948no—
210C4#1569078012yes—
211C4C2×D4×D9712yes—
212C2—22no—
213C2—24no—
214C2—7761no—
215C2C2×#1569078011yescharacteristic
216C2C2×#1569084312yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#407×C97C2
socleC22×C97C22×D4
prime radical#20261246C2
hypercenter#407D97
hypocenterC97#3004
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×Dic194C2semidirect
2. C2×#15690843C2semidirect
4. #20260893C2semidirect
5. #20260876C2semidirect
6. #20260745C2semidirect
7. #407×C97C2semidirect
16. C2×Dic388C22semidirect
17. C2×#13095500C22semidirect
24. #15690735C22semidirect
35. C2×Dic194D4semidirect
93. C22×C97#137semidirect
109. C97#3004semidirect
116. #407D97semidirect
204. C22#15690843semidirect

Maximal subgroups

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

subgroupindex
C23×Dic1942
C2×#156908432
#202612462
#202608932
#202608762
#202607452
#407×C972
#300497