Finite Group Databasebeta

← #27142278

Subgroups — #27142278

Order 12416.

380 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261294C211yesmaximal, characteristic
2C2×#15690725C211yesmaximal, characteristic, complemented
3C2×#15690724C212yesmaximal, complemented
4#20260501C212yesmaximal, complemented
5#518×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97C2211yescharacteristic
7#15690793—21no—
8#15690791C412yescomplemented
9#15690782—22no—
10#15690779—21no—
11#15690778C412yescomplemented
12C2×C22⋊Dic97C2212yes—
13C2×C22⋊Dic97C2211yescharacteristic
14#15690725—22no—
15#15690724—24no—
16C22×D4×C97C2211yescharacteristic
17C4×D4×C97—24no—
18#117×C97—21no—
19C2×C22⋊C4×C97—21no—
20C2×C22⋊C4×C97C2212yes—
21C23×D97C2×C411yescharacteristic, complemented
22C2×C22⋊D97—21no—
23C2×C22⋊D97—21no—
24C2×C22⋊D97—22no—
25C2×C22⋊D97—22no—
26C2×C22⋊D97—22no—
27C22×Dic97—21no—
28C22×Dic97—22no—
29C22×Dic97C2×C411yescharacteristic, complemented
30#13095490—22no—
31C22⋊Dic97—22no—
32C22⋊Dic97—21no—
33C22⋊Dic97—22no—
34C22⋊Dic97—22no—
35C22⋊Dic97C2×C412yescomplemented
36#13095488—22no—
37C24×C97D411yescharacteristic
38C24×C97C2311yescharacteristic
39C2×D4×C97—22no—
40C2×D4×C97—22no—
41C2×D4×C97C2×C412yes—
42C2×D4×C97D412yes—
43C22×C4×C97—22no—
44C22×C4×C97—22no—
45C22×C4×C97D411yescharacteristic
46C4⋊C4×C97—22no—
47C22⋊C4×C97—22no—
48C22⋊C4×C97—22no—
49C22⋊C4×C97D414yescomplemented
50C42×C97—22no—
51C23×C97—21no—
52C23×C97—21no—
53C23×C97—81no—
54C23×C97—81no—
55C23×C97C22⋊C412yes—
56C23×C97C22×C411yescharacteristic
57C23×C97C2×D411yescharacteristic
58C23×C97C2×D412yes—
59C23×C97C2×D411yescharacteristic
60C23×C97#4112yes—
61C22×D97—21no—
62C22×D97—22no—
63C22×D97—81no—
64D4×C97—24no—
65D4×C97—24no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—24no—
69C2×C4×C97—82no—
70C2×C4×C97—82no—
71C2×C4×C97C22⋊C412yes—
72C22⋊D97—41no—
73C22⋊D97—42no—
74C22⋊D97—41no—
75C22⋊D97—82no—
76C2×Dic97—22no—
77C2×Dic97—21no—
78C2×Dic97—22no—
79C2×Dic97—42no—
80C2×Dic97—81no—
81C2×Dic97—82no—
82C22×C97—22no—
83C22×C97—41no—
84C22×C97—42no—
85C22×C97—42no—
86C22×C97—41no—
87C22×C97—41no—
88C22×C97—42no—
89C22×C97—42no—
90C22×C97—81no—
91C22×C97C2×C22⋊C411yescharacteristic
92C22×C97C4×D412yes—
93C22×C97#12011yescharacteristic
94C22×C97#12112yes—
95C22×C97#12311yescharacteristic
96C2×D97—42no—
97C2×D97—41no—
98C2×D97—82no—
99C4×C97—22no—
100C4×C97—44no—
101Dic97—42no—
102Dic97—41no—
103Dic97—82no—
104C2×C97—21no—
105C2×C97—22no—
106C2×C97—21no—
107C2×C97—22no—
108C2×C97—21no—
109C2×C97—42no—
110C2×C97#38611yescharacteristic
111D97—81no—
112#1913—971nomaximal, Sylow
113C97#191311yescharacteristic, complemented, Sylow
114#534—971no—
115#518D9711yescharacteristic, complemented
116C2×#99—971no—
117C2×#99—972no—
118#342—972no—
119C22×D4—971no—
120C22×D4C2×D9711yescharacteristic
121#123—972no—
122#123—1941no—
123#121—1942no—
124#120—972no—
125#120—1941no—
126C4×D4—24no—
127#117—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—971no—
130C2×C22⋊C4—972no—
131C2×C22⋊C4C2×D9712yes—
132#99—1942no—
133#99—1944no—
134C24—971no—
135C24C22⋊D9711yescharacteristic
136C24C22×D9711yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—1942no—
140C2×D4—1942no—
141C2×D4—1941no—
142C2×D4—1941no—
143C2×D4—1942no—
144C2×D4C4×D9712yes—
145C2×D4D38812yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—971no—
149C22×C4—1942no—
150C22×C4—1941no—
151C22×C4C22⋊D9711yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—1942no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—972no—
157C22⋊C4—1942no—
158C22⋊C4—1942no—
159C22⋊C4—1942no—
160C22⋊C4—1941no—
161C22⋊C4—1942no—
162C22⋊C4C22⋊D9714yescomplemented
163C42—22no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—7761no—
169C23—1942no—
170C23—1941no—
171C23#1309548812yes—
172C23C2×C4×D9711yescharacteristic
173C23C2×C22⋊D9711yescharacteristic
174C23C2×C22⋊D9712yes—
175C23C2×D38811yescharacteristic
176C23#1309551112yes—
177D4—24no—
178D4—24no—
179D4—7762no—
180D4—3881no—
181D4—3882no—
182D4—3881no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—7762no—
189C2×C4—7761no—
190C2×C4—3882no—
191C2×C4—1942no—
192C2×C4—1941no—
193C2×C4—1942no—
194C2×C4#1309548812yes—
195C22—22no—
196C22—42no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—7762no—
205C22—3882no—
206C22—3881no—
207C22C2×#1309548811yescharacteristic
208C22C4×C22⋊D9712yes—
209C22#1569077911yescharacteristic
210C22#1569078412yes—
211C22#1569079311yescharacteristic
212C4—22no—
213C4—44no—
214C4—7762no—
215C4—3882no—
216C4—3881no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—7761no—
224C2#2026061411yescharacteristic
subgroupG/H
centerC2#20260614
commutatorC23×C97C22×C4
FrattiniC24C22×D97
Fitting#518×C97C2
socleC2×C97#386
prime radical#20261294C2
cosocleC24×C97C23
hypercenter#518D97
hypocenterC97#1913
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#15690725C2semidirect
3. C2×#15690724C2semidirect
4. #20260501C2semidirect
5. #518×C97C2semidirect
8. #15690791C4semidirect
11. #15690778C4semidirect
21. C23×D97C2×C4semidirect
29. C22×Dic97C2×C4semidirect
35. C22⋊Dic97C2×C4semidirect
49. C22⋊C4×C97D4semidirect
113. C97#1913semidirect
115. #518D97semidirect
162. C22⋊C4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612942
C2×#156907252
C2×#156907242
#202605012
#518×C972
#191397