Finite Group Databasebeta

← #27140072

Subgroups — #27140072

Order 12416.

316 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C2×C4×C22⋊D97C211yesmaximal, characteristic
2#20260699C211yesmaximal, characteristic, complemented
3#20260460C212yesmaximal, complemented
4#20260458C212yesmaximal, complemented
5#407×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97C411yescharacteristic
7C22×C4×D97C411yescharacteristic
8C2×#13095495—21no—
9C4×C22⋊D97—21no—
10C4×C22⋊D97—21no—
11C4×C22⋊D97—21no—
12C4×C22⋊D97—21no—
13C2×#13095490C2211yescharacteristic
14C2×C22⋊Dic97C411yescharacteristic
15C2×#13095488C2211yescharacteristic
16C2×C4×Dic97C411yescharacteristic
17C22⋊#11715895—21no—
18C22⋊#11715895C2212yes—
19C23×C4×C97C2211yescharacteristic
20C2×M16×C97—21no—
21C22⋊C8×C97—21no—
22C22⋊C8×C97C2212yes—
23C23×D97C2×C411yescharacteristic
24C2×C22⋊D97—21no—
25C2×C22⋊D97—21no—
26C2×C22⋊D97—22no—
27C2×C22⋊D97—22no—
28C22×Dic97C2×C411yescharacteristic
29C22×Dic97C2×C411yescharacteristic
30C22×Dic97C2×C411yescharacteristic
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97—42no—
34#13095495—42no—
35C2×#11715895—22no—
36#13095490—21no—
37#13095490—21no—
38C22⋊Dic97—21no—
39C22⋊Dic97—21no—
40#13095488—21no—
41#13095488—21no—
42C4×Dic97—21no—
43C4×Dic97—21no—
44C24×C97C2×C411yescharacteristic
45C22×C4×C97—42no—
46C22×C4×C97C2×C411yescharacteristic
47C22×C4×C97D411yescharacteristic
48C22×C4×C97D411yescharacteristic
49C22×C4×C97D411yescharacteristic
50C22×C4×C97D411yescharacteristic
51C22×C4×C97C2311yescharacteristic
52M16×C97—42no—
53C2×C8×C97—22no—
54C23×C97—21no—
55C23×C97—21no—
56C23×C97—41no—
57C23×C97—41no—
58C23×C97C22⋊C411yescharacteristic
59C23×C97C22⋊C411yescharacteristic
60C23×C97C22×C411yescharacteristic
61C22×D97—21no—
62C22×D97—21no—
63C22×D97—41no—
64C22×D97—41no—
65C22×D97M1612yescomplemented
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97—41no—
70C2×C4×C97—41no—
71C2×C4×C97—41no—
72C2×C4×C97C22⋊C411yescharacteristic
73C2×C4×C97C22⋊C411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C22⋊D97—24no—
77C22⋊D97—24no—
78C2×Dic97—21no—
79C2×Dic97—21no—
80C2×Dic97—21no—
81C2×Dic97—21no—
82C2×Dic97—41no—
83C2×Dic97—41no—
84C2×Dic97M1612yescomplemented
85C4×D97—42no—
86C4×D97—42no—
87C8×C97—42no—
88#11715895—42no—
89C22×C97—21no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97—21no—
96C22×C97—21no—
97C22×C97—42no—
98C22×C97—42no—
99C22×C97C2×C22⋊C411yescharacteristic
100C22×C97C2×M1611yescharacteristic
101C22×C97C2×M1611yescharacteristic
102C2×D97—22no—
103C2×D97—42no—
104C2×D97—41no—
105C2×D97—41no—
106C2×D97—42no—
107C4×C97—21no—
108C4×C97—21no—
109C4×C97—21no—
110C4×C97#9912yes—
111Dic97—22no—
112Dic97—41no—
113Dic97—41no—
114C2×C97—21no—
115C2×C97—21no—
116C2×C97—41no—
117C2×C97—41no—
118C2×C97#40711yescharacteristic
119C2×C97C2×#9911yescharacteristic
120C2×C97#41311yescharacteristic
121D97—42no—
122#1465—971nomaximal, Sylow
123C97#146511yescharacteristic, complemented, Sylow
124C2×C4×D4—971no—
125#407—971no—
126#407D9711yescharacteristic, complemented
127#324—972no—
128#323—972no—
129C22×D4—971no—
130C23×C4—971no—
131C23×C4C2×D9711yescharacteristic
132C2×M16—21no—
133C2×M16—1941no—
134C4×D4—1941no—
135C4×D4—1941no—
136C4×D4—1941no—
137C4×D4—1941no—
138C2×C4⋊C4—971no—
139C2×C22⋊C4—971no—
140C2×C22⋊C4—971no—
141C2×C42—971no—
142C22⋊C8—21no—
143C22⋊C8—972no—
144C22⋊C8—1941no—
145C22⋊C8C2×D9712yes—
146C24—971no—
147C24C4×D9711yescharacteristic
148C2×D4—1942no—
149C2×D4—1942no—
150C2×D4—1941no—
151C2×D4—1941no—
152C22×C4—42no—
153C22×C4—971no—
154C22×C4—971no—
155C22×C4—971no—
156C22×C4—3882no—
157C22×C4—1941no—
158C22×C4—1941no—
159C22×C4C4×D9711yescharacteristic
160C22×C4D38811yescharacteristic
161C22×C4D38811yescharacteristic
162C22×C4C22⋊D9711yescharacteristic
163C22×C4C22⋊D9711yescharacteristic
164C22×C4C22×D9711yescharacteristic
165M16—42no—
166M16—3882no—
167C2×C8—22no—
168C2×C8—1942no—
169C4⋊C4—1941no—
170C4⋊C4—1941no—
171C22⋊C4—1941no—
172C22⋊C4—1941no—
173C22⋊C4—1941no—
174C22⋊C4—1941no—
175C42—1941no—
176C42—1941no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—972no—
182C23—3881no—
183C23—3881no—
184C23—1941no—
185C23—1941no—
186C23#1309548811yescharacteristic
187C23#1309548811yescharacteristic
188C23C2×C4×D9711yescharacteristic
189D4—1944no—
190D4—1944no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—972no—
198C2×C4—3882no—
199C2×C4—3882no—
200C2×C4—3881no—
201C2×C4—3881no—
202C2×C4—1941no—
203C2×C4—1941no—
204C2×C4—1941no—
205C2×C4—1941no—
206C2×C4#1309548811yescharacteristic
207C2×C4#1309548811yescharacteristic
208C2×C4C2×C22⋊D9711yescharacteristic
209C2×C4C2×D38811yescharacteristic
210C8—42no—
211C8—3882no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—42no—
222C22—3882no—
223C22—3882no—
224C22—3881no—
225C22—3881no—
226C22—1942no—
227C22C2×#1309548811yescharacteristic
228C22M16×D9711yescharacteristic
229C22M16×D9711yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—3881no—
234C4—3881no—
235C4—1942no—
236C4#1569072412yes—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2—3882no—
242C2#2026069811yescharacteristic
243C2C2×#1569072411yescharacteristic
244C2#2026071711yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#407×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C4
cosocleC22×C4×C97C23
hypercenter#407D97
hypocenterC97#1465
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #20260699C2semidirect
3. #20260460C2semidirect
4. #20260458C2semidirect
5. #407×C97C2semidirect
65. C22×D97M16semidirect
84. C2×Dic97M16semidirect
123. C97#1465semidirect
126. #407D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D972
#202606992
#202604602
#202604582
#407×C972
#146597