Finite Group Databasebeta

← #327×D97 #27140211

Subgroups — #327×D97 #27140211

Order 12416.

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

normal subgroups only

This group has 388 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
1#121×D97C211yesmaximal, characteristic
2C22⋊C8×D97C211yesmaximal, characteristic, complemented
3#95×D97C211yesmaximal, 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×D97—21no—
11C22×C4×D97C2211yescharacteristic
12C2×C8×D97—21no—
13#15690785C411yescharacteristic, complemented
14#15690784C2211yescharacteristic
15#15690783—21no—
16#15690782—21no—
17#15690781C411yescharacteristic, complemented
18C4⋊C4×D97C411yescharacteristic, complemented
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, complemented
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—24no—
40C2×C4×D97—21no—
41C2×C4×D97—42no—
42C2×C4×D97—42no—
43C2×C4×D97—42no—
44C2×C4×D97C2×C411yescharacteristic
45C2×C4×D97C2×C411yescharacteristic
46C2×C4×D97D411yescharacteristic
47C2×C4×D97D411yescharacteristic
48C2×#11715895—21no—
49C8×D97—42no—
50C4⋊Dic97C2×C411yescharacteristic, complemented
51#13095490—21no—
52C22⋊Dic97—21no—
53#13095488—21no—
54#13095488—21no—
55C2×D4×C97—21no—
56C2×D4×C97C2×C411yescharacteristic, complemented
57C22×C4×C97—21no—
58C22×C4×C97C2311yescharacteristic
59C2×C8×C97—21no—
60C4⋊C4×C97C2×C411yescharacteristic, complemented
61C22⋊C4×C97—21no—
62C23×C97—21no—
63C23×C97C2×D411yescharacteristic
64C22×D97—22no—
65C22×D97—22no—
66C22×D97—21no—
67C22×D97—42no—
68C22×D97—42no—
69C22×D97—42no—
70C22×D97C22⋊C411yescharacteristic
71C22×D97C22⋊C411yescharacteristic
72C22×D97D812yes—
73C22×D97SD1612yes—
74D4×C97—41no—
75D4×C97—41no—
76D4×C97—41no—
77C2×C4×C97—22no—
78C2×C4×C97—21no—
79C2×C4×C97—41no—
80C2×C4×C97—41no—
81C2×C4×C97—41no—
82C2×C4×C97C22×C411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C22⋊D97—42no—
85C22⋊D97—41no—
86C22⋊D97—41no—
87C22⋊D97—42no—
88C2×Dic97—22no—
89C2×Dic97—21no—
90C2×Dic97—41no—
91C2×Dic97—41no—
92C2×Dic97—41no—
93C2×Dic97C22⋊C411yescharacteristic
94C2×Dic97C22⋊C411yescharacteristic
95D388—41no—
96D388—42no—
97C4×D97—22no—
98C4×D97—22no—
99C4×D97—41no—
100C4×D97—41no—
101C4×D97—48no—
102C4×D97—42no—
103C8×C97—41no—
104#11715895—41no—
105C22×C97—21no—
106C22×C97—21no—
107C22×C97—41no—
108C22×C97—41no—
109C22×C97—41no—
110C22×C97C2×C22⋊C411yescharacteristic
111C22×C97C2×D811yescharacteristic
112C22×C97C2×SD1611yescharacteristic
113C2×D97—21no—
114C2×D97—21no—
115C2×D97—24no—
116C2×D97—24no—
117C2×D97—41no—
118C2×D97—41no—
119C2×D97—41no—
120C2×D97—84no—
121C2×D97#9912yes—
122C2×D97#10212yes—
123C2×D97#10212yes—
124C2×D97#10412yes—
125C4×C97—21no—
126C4×C97—21no—
127C4×C97—42no—
128C4×C97—41no—
129Dic97—21no—
130Dic97—21no—
131Dic97—42no—
132Dic97—41no—
133C2×C97—21no—
134C2×C97—21no—
135C2×C97—81no—
136C2×C97C2×#9911yescharacteristic
137C2×C97C2×#10211yescharacteristic
138C2×C97C2×#10411yescharacteristic
139D97—22no—
140D97—81no—
141D97#32714yesdirect product factor
142C2×#327—971nomaximal, Sylow
143C97C2×#32711yescharacteristic, complemented, Sylow
144C2×#121—971no—
145C2×C22⋊C8—971no—
146C2×#95—971no—
147#327—971no—
148#327—971no—
149#327—971no—
150#327D9711yescharacteristic, direct product factor
151C22×D4—971no—
152C22×D4—1941no—
153C23×C4—971no—
154C23×C4—1941no—
155C22×C8—1941no—
156#121—971no—
157#121—971no—
158#121—971no—
159#121—1941no—
160#121—1941no—
161#121C2×D9711yescharacteristic
162C2×C4⋊C4—971no—
163C2×C22⋊C4—1941no—
164C22⋊C8—971no—
165C22⋊C8—1941no—
166C22⋊C8C2×D9711yescharacteristic, complemented
167#95—971no—
168#95—1941no—
169#95C2×D9711yescharacteristic, complemented
170C24—971no—
171C24—1941no—
172C2×D4—21no—
173C2×D4—971no—
174C2×D4—3882no—
175C2×D4—3882no—
176C2×D4—3882no—
177C2×D4—1941no—
178C2×D4—1941no—
179C2×D4—1941no—
180C2×D4—1941no—
181C2×D4C4×D9711yescharacteristic, complemented
182C22×C4—21no—
183C22×C4—971no—
184C22×C4—971no—
185C22×C4—971no—
186C22×C4—971no—
187C22×C4—971no—
188C22×C4—3882no—
189C22×C4—3882no—
190C22×C4—3882no—
191C22×C4—1941no—
192C22×C4—1944no—
193C22×C4—1941no—
194C22×C4C22×D9711yescharacteristic
195C2×C8—21no—
196C2×C8—3882no—
197C2×C8—1941no—
198C4⋊C4—971no—
199C4⋊C4—1941no—
200C4⋊C4C4×D9711yescharacteristic, complemented
201C22⋊C4—21no—
202C22⋊C4—1941no—
203C22⋊C4—1941no—
204C22⋊C4—1941no—
205C23—21no—
206C23—971no—
207C23—971no—
208C23—972no—
209C23—972no—
210C23—3882no—
211C23—3882no—
212C23—3882no—
213C23—1942no—
214C23—1942no—
215C23—1941no—
216C23D4×D9711yescharacteristic
217D4—41no—
218D4—41no—
219D4—41no—
220D4—3881no—
221D4—3882no—
222D4—3882no—
223D4—3881no—
224D4—3881no—
225D4—3882no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—971no—
232C2×C4—971no—
233C2×C4—3881no—
234C2×C4—3881no—
235C2×C4—3882no—
236C2×C4—3888no—
237C2×C4—3881no—
238C2×C4—3881no—
239C2×C4—3881no—
240C2×C4—1942no—
241C2×C4—1942no—
242C2×C4—1942no—
243C2×C4—1941no—
244C2×C4C2×C4×D9711yescharacteristic
245C2×C4D4×D9711yescharacteristic
246C8—41no—
247C8—3881no—
248C22—21no—
249C22—21no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—7764no—
254C22—972no—
255C22—972no—
256C22—972no—
257C22—972no—
258C22—3881no—
259C22—3881no—
260C22—3881no—
261C22—1944no—
262C22—1944no—
263C22—1941no—
264C22—1941no—
265C22C22⋊C4×D9711yescharacteristic
266C22D8×D9711yescharacteristic
267C22SD16×D9711yescharacteristic
268C4—21no—
269C4—21no—
270C4—42no—
271C4—41no—
272C4—3882no—
273C4—3881no—
274C4—1941no—
275C4—1941no—
276C2—21no—
277C2—21no—
278C2—81no—
279C2—7761no—
280C2—974no—
281C2—1942no—
282C2#99×D9711yescharacteristic
283C2#102×D9711yescharacteristic
284C2#104×D9711yescharacteristic
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
hypocenterC97C2×#327
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. C22⋊C8×D97C2semidirect
3. #95×D97C2semidirect
4. #20260470C2semidirect
5. #20260469C2semidirect
6. #20260468C2semidirect
7. #327×C97C2semidirect
9. C2×D4×D97C4semidirect
13. #15690785C4semidirect
17. #15690781C4semidirect
18. C4⋊C4×D97C4semidirect
20. C22⋊#11715895C22semidirect
23. #15690717C22semidirect
25. C22⋊C8×C97C22semidirect
26. #95×C97C22semidirect
30. C2×D388C2×C4semidirect
50. C4⋊Dic97C2×C4semidirect
56. C2×D4×C97C2×C4semidirect
60. C4⋊C4×C97C2×C4semidirect
141. D97#327direct (N × H)
143. C97C2×#327semidirect
150. #327D97direct (N × H)
166. C22⋊C8C2×D97semidirect
169. #95C2×D97semidirect
181. C2×D4C4×D97semidirect
200. C4⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
#121×D972
C22⋊C8×D972
#95×D972
#202604702
#202604692
#202604682
#327×C972
C2×#32797