Finite Group Databasebeta

← #27140263

Subgroups — #27140263

Order 12416.

326 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 63 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#121×D97C211yesmaximal, characteristic
2#20260697C211yesmaximal, characteristic, complemented
3#20260597C211yesmaximal, characteristic, complemented
4#20260470C211yesmaximal, characteristic, complemented
5#20260469C211yesmaximal, characteristic, complemented
6#20260468C211yesmaximal, characteristic, complemented
7#327×C97C211yesmaximal, characteristic, complemented
8C2×D4×D97—21no—
9C2×D4×D97C411yescharacteristic, complemented
10C22×C4×D97C2211yescharacteristic
11C2×#13095494—21no—
12#15690785C411yescharacteristic, complemented
13#15690784C2211yescharacteristic
14#15690783—21no—
15#15690782—21no—
16#15690781C411yescharacteristic, complemented
17C4⋊C4×D97C411yescharacteristic, complemented
18C2×#13095488—21no—
19C22⋊C4×D97—21no—
20C22⋊#11715895C2211yescharacteristic, complemented
21#15690722—21no—
22#15690717—21no—
23#15690717C2211yescharacteristic, complemented
24#121×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic, complemented
26#95×C97C2211yescharacteristic, complemented
27C23×D97—21no—
28C23×D97D411yescharacteristic
29C2×D388—21no—
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34D4×D97—42no—
35D4×D97—42no—
36D4×D97—42no—
37C22×Dic97—21no—
38C22×Dic97D411yescharacteristic
39C2×C4×D97—21no—
40C2×C4×D97—42no—
41C2×C4×D97C2×C411yescharacteristic
42C2×C4×D97C2×C411yescharacteristic
43C2×C4×D97D411yescharacteristic
44C2×C4×D97D411yescharacteristic
45#13095494—42no—
46C2×#11715895—21no—
47C4⋊Dic97C2×C411yescharacteristic
48#13095490—21no—
49C22⋊Dic97—21no—
50#13095488—21no—
51#13095488—21no—
52#13095488—42no—
53C2×D4×C97—21no—
54C2×D4×C97C2×C411yescharacteristic
55C22×C4×C97—21no—
56C22×C4×C97C2311yescharacteristic
57C2×C8×C97—21no—
58C4⋊C4×C97C2×C411yescharacteristic
59C22⋊C4×C97—21no—
60C23×C97—21no—
61C23×C97C2×D411yescharacteristic
62C22×D97—21no—
63C22×D97—21no—
64C22×D97—22no—
65C22×D97—21no—
66C22×D97—41no—
67C22×D97—42no—
68C22×D97—42no—
69C22×D97—42no—
70C22×D97C22⋊C411yescharacteristic
71C22×D97C22⋊C411yescharacteristic
72D4×C97—41no—
73D4×C97—41no—
74D4×C97—41no—
75C2×C4×C97—21no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97—41no—
79C2×C4×C97—41no—
80C2×C4×C97C22×C411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C22⋊D97—42no—
83C22⋊D97—42no—
84C22⋊D97—41no—
85C22⋊D97—41no—
86C2×Dic97—21no—
87C2×Dic97—41no—
88C2×Dic97—41no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91C2×Dic97C22⋊C411yescharacteristic
92C2×Dic97C22⋊C411yescharacteristic
93D388—41no—
94D388—42no—
95C4×D97—22no—
96C4×D97—22no—
97C4×D97—41no—
98C4×D97—41no—
99C4×D97—42no—
100C8×C97—41no—
101#11715895—41no—
102C22×C97—21no—
103C22×C97—21no—
104C22×C97—41no—
105C22×C97—41no—
106C22×C97—41no—
107C22×C97C2×C22⋊C411yescharacteristic
108C22×C97AGL(1,ℤ/8)11yescharacteristic
109C22×C97AGL(1,ℤ/8)11yescharacteristic
110C2×D97—21no—
111C2×D97—21no—
112C2×D97—21no—
113C2×D97—21no—
114C2×D97—21no—
115C2×D97—42no—
116C2×D97—42no—
117C2×D97—41no—
118C2×D97—41no—
119C2×D97—41no—
120C2×D97—84no—
121C2×D97#9912yes—
122C4×C97—21no—
123C4×C97—21no—
124C4×C97—41no—
125C4×C97—81no—
126Dic97—21no—
127Dic97—21no—
128Dic97—41no—
129Dic97—81no—
130C2×C97—21no—
131C2×C97—21no—
132C2×C97—81no—
133C2×C97C2×#9911yescharacteristic
134C2×C97#41811yescharacteristic
135C2×C97#42111yescharacteristic
136D97—22no—
137D97—41no—
138D97—81no—
139#1510—971nomaximal, Sylow
140C97#151011yescharacteristic, complemented, Sylow
141C2×#121—971no—
142#407—971no—
143#381—971no—
144#327—971no—
145#327—971no—
146#327—971no—
147#327D9711yescharacteristic, complemented
148C22×D4—971no—
149C22×D4—1941no—
150C23×C4—971no—
151C2×M16—1941no—
152#121—971no—
153#121—971no—
154#121—971no—
155#121—1941no—
156#121—1941no—
157#121C2×D9711yescharacteristic
158C2×C4⋊C4—971no—
159C2×C22⋊C4—1941no—
160C2×C22⋊C4—1941no—
161C22⋊C8—971no—
162C22⋊C8—1941no—
163C22⋊C8C2×D9711yescharacteristic, complemented
164#95—971no—
165#95—1941no—
166#95C2×D9711yescharacteristic, complemented
167C24—971no—
168C24—1941no—
169C2×D4—21no—
170C2×D4—971no—
171C2×D4—3882no—
172C2×D4—3882no—
173C2×D4—3882no—
174C2×D4—1941no—
175C2×D4—1941no—
176C2×D4—1941no—
177C2×D4—1941no—
178C2×D4C4×D9711yescharacteristic
179C22×C4—21no—
180C22×C4—971no—
181C22×C4—971no—
182C22×C4—971no—
183C22×C4—971no—
184C22×C4—971no—
185C22×C4—3882no—
186C22×C4—1941no—
187C22×C4—1941no—
188C22×C4C22×D9711yescharacteristic
189M16—3882no—
190C2×C8—21no—
191C2×C8—1941no—
192C4⋊C4—971no—
193C4⋊C4—1941no—
194C4⋊C4C4×D9711yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—3882no—
197C22⋊C4—1941no—
198C22⋊C4—1941no—
199C22⋊C4—1941no—
200C23—21no—
201C23—971no—
202C23—971no—
203C23—3881no—
204C23—3882no—
205C23—3882no—
206C23—3882no—
207C23—1942no—
208C23—1941no—
209C23—1941no—
210C23—1941no—
211C23D4×D9711yescharacteristic
212D4—41no—
213D4—41no—
214D4—41no—
215D4—3881no—
216D4—3882no—
217D4—3882no—
218D4—3881no—
219D4—3881no—
220D4—3882no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—971no—
227C2×C4—971no—
228C2×C4—3881no—
229C2×C4—3881no—
230C2×C4—3882no—
231C2×C4—3881no—
232C2×C4—3881no—
233C2×C4—3881no—
234C2×C4—3881no—
235C2×C4—1942no—
236C2×C4—1942no—
237C2×C4—1941no—
238C2×C4C2×C4×D9711yescharacteristic
239C2×C4D4×D9711yescharacteristic
240C8—41no—
241C8—3881no—
242C22—21no—
243C22—21no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—7764no—
248C22—972no—
249C22—3882no—
250C22—3882no—
251C22—3881no—
252C22—3881no—
253C22—3881no—
254C22—1941no—
255C22—1941no—
256C22—1941no—
257C22—1941no—
258C22—1941no—
259C22C22⋊C4×D9711yescharacteristic
260C22#1569083411yescharacteristic
261C22#1569083711yescharacteristic
262C4—21no—
263C4—21no—
264C4—41no—
265C4—81no—
266C4—7761no—
267C4—3881no—
268C4—1941no—
269C4—1941no—
270C2—21no—
271C2—21no—
272C2—81no—
273C2—7761no—
274C2—3881no—
275C2—1942no—
276C2#99×D9711yescharacteristic
277C2#2026073811yescharacteristic
278C2#2026075011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#327×C97C2
socleC22×C97C2×C22⋊C4
prime radical#121×D97C2
cosocleC22×C4×C97C23
hypercenter#327D97
hypocenterC97#1510
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20260697C2semidirect
3. #20260597C2semidirect
4. #20260470C2semidirect
5. #20260469C2semidirect
6. #20260468C2semidirect
7. #327×C97C2semidirect
9. C2×D4×D97C4semidirect
12. #15690785C4semidirect
16. #15690781C4semidirect
17. C4⋊C4×D97C4semidirect
20. C22⋊#11715895C22semidirect
23. #15690717C22semidirect
25. C22⋊C8×C97C22semidirect
26. #95×C97C22semidirect
140. C97#1510semidirect
147. #327D97semidirect
163. C22⋊C8C2×D97semidirect
166. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
#121×D972
#202606972
#202605972
#202604702
#202604692
#202604682
#327×C972
#151097