Finite Group Databasebeta

← #27152212

Subgroups — #27152212

Order 12416.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15690843C211yesmaximal, characteristic, complemented
2C2×#15690838C211yesmaximal, characteristic, complemented
3C2×#15690832C211yesmaximal, characteristic, complemented
4C22×#13095500C211yesmaximal, characteristic, complemented
5#20261311C211yesmaximal, characteristic, complemented
6#20261304C211yesmaximal, characteristic, complemented
7#20261129C212yesmaximal, complemented
8#20261124C212yesmaximal, complemented
9#20261119C212yesmaximal, complemented
10#20261108C212yesmaximal, complemented
11#405×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C2×#13095511C2211yescharacteristic
14C22×Dic194C2211yescharacteristic
15C22×D388C2211yescharacteristic
16#15690843—24no—
17#15690838—24no—
18#15690832—24no—
19C2×Dic388C2211yescharacteristic, complemented
20C2×Dic388C2212yescomplemented
21C2×#13095500—24no—
22C2×#13095500C2212yes—
23C2×#13095500C2212yes—
24C2×#13095500C2212yes—
25C2×#13095500C2212yescomplemented
26C2×#13095500C2211yescharacteristic, complemented
27C2×#13095500C2211yescharacteristic, complemented
28C2×D776C2212yes—
29C2×D776C2211yescharacteristic, complemented
30#15690810C2212yescomplemented
31#15690807C2212yes—
32#15690798C2212yes—
33#15690798C2212yescomplemented
34#15690794—22no—
35#15690790—22no—
36#15690785—22no—
37#15690780—22no—
38C2×M16×C97C2211yescharacteristic
39C22×C8×C97C2211yescharacteristic
40#117×C97C2211yescharacteristic
41#97×C97C2212yescomplemented
42C4×C8×C97C2212yescomplemented
43C23×D97—21no—
44#13095511—22no—
45#13095511—22no—
46#13095511—22no—
47#13095511—22no—
48C2×Dic194—22no—
49C2×Dic194—22no—
50C2×Dic194—21no—
51C2×Dic194C2311yescharacteristic
52C2×Dic194C2312yes—
53C2×Dic194C2311yescharacteristic
54C2×D388—22no—
55C2×D388—22no—
56C2×D388—21no—
57C2×D388—24no—
58C2×D388C2312yes—
59C2×D388C2311yescharacteristic
60C2×D388C2311yescharacteristic
61C2×C22⋊D97—21no—
62C2×C22⋊D97—21no—
63C22×Dic97—21no—
64C2×C4×D97—21no—
65C2×C4×D97—21no—
66Dic388—22no—
67Dic388—24no—
68#13095500—24no—
69#13095500—24no—
70#13095500—24no—
71#13095500—22no—
72#13095500—22no—
73#13095500—24no—
74D776—24no—
75D776—22no—
76C4⋊Dic97—24no—
77#13095488—24no—
78#13095488—24no—
79C22×C4×C97C2311yescharacteristic
80M16×C97D414yescomplemented
81C2×C8×C97D414yescomplemented
82C2×C8×C97C2312yes—
83C2×C8×C97C2311yescharacteristic
84C2×C8×C97C2311yescharacteristic
85C4⋊C4×C97D412yes—
86C22⋊C4×C97D412yes—
87C42×C97C2312yes—
88C23×C97C2×D411yescharacteristic
89C22×D97—21no—
90C22×D97—22no—
91C22×D97—21no—
92C22×D97—42no—
93C22×D97—82no—
94C2×C4×C97C2×D414yes—
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2×D412yes—
97C2×C4×C97C2×D412yes—
98C2×C4×C97C2411yescharacteristic
99C22⋊D97—42no—
100C22⋊D97—42no—
101C2×Dic97—21no—
102C2×Dic97—22no—
103C2×Dic97—21no—
104C2×Dic97—42no—
105D388—21no—
106D388—22no—
107D388—21no—
108D388—21no—
109D388—22no—
110D388—21no—
111D388—42no—
112D388—84no—
113C4×D97—42no—
114C4×D97—42no—
115Dic194—22no—
116Dic194—21no—
117Dic194—22no—
118Dic194—21no—
119Dic194—21no—
120Dic194—21no—
121Dic194—42no—
122C8×C97C2×D412yes—
123C8×C97C2×D412yes—
124C8×C97C2×D414yes—
125C22×C97—22no—
126C22×C97#12712yes—
127C22×C97C22×D411yescharacteristic
128C2×D97—41no—
129C2×D97—41no—
130C2×D97—42no—
131C2×D97—42no—
132C2×D97—44no—
133C2×D97—42no—
134C2×D97—84no—
135C4×C97—24no—
136C4×C97#12712yes—
137C4×C97C22×D411yescharacteristic
138C4×C97C22×D411yescharacteristic
139Dic97—42no—
140Dic97—41no—
141Dic97—41no—
142C2×C97—22no—
143C2×C97C2×#12711yescharacteristic
144C2×C97#57712yes—
145D97—81no—
146D97—81no—
147D97—82no—
148#3154—971nomaximal, Sylow
149C97#315411yescharacteristic, complemented, Sylow
150C2×#137—971no—
151C2×AGL(1,ℤ/8)—971no—
152C2×#135—971no—
153C22×SD16—971no—
154#536—971no—
155#535—971no—
156#497—972no—
157#496—972no—
158#495—972no—
159#492—972no—
160#405D9711yescharacteristic, complemented
161C2×#41—971no—
162C2×#41—971no—
163C22×Q8—971no—
164C22×D4—971no—
165#137—1944no—
166AGL(1,ℤ/8)—1944no—
167#135—1944no—
168C2×Q16—971no—
169C2×Q16—972no—
170C2×SD16—972no—
171C2×SD16—972no—
172C2×SD16—971no—
173C2×SD16—971no—
174C2×SD16—972no—
175C2×SD16—972no—
176C2×SD16—1944no—
177C2×D8—971no—
178C2×D8—972no—
179C2×M16C2×D9711yescharacteristic
180C22×C8C2×D9711yescharacteristic
181C4⋊Q8—972no—
182#127—972no—
183#124—972no—
184#124—972no—
185#123—1942no—
186C22⋊Q8—1942no—
187#121—1942no—
188#120—1942no—
189#117C2×D9711yescharacteristic
190#97C2×D9712yescomplemented
191C4×C8C2×D9712yescomplemented
192C24—1941no—
193#41—1942no—
194#41—1942no—
195#41—1942no—
196#41—1942no—
197C2×Q8—971no—
198C2×Q8—972no—
199C2×Q8—971no—
200C2×Q8—1942no—
201C2×Q8—1942no—
202C2×Q8—1941no—
203C2×D4—971no—
204C2×D4—972no—
205C2×D4—971no—
206C2×D4—1941no—
207C2×D4—1941no—
208C2×D4—1944no—
209C2×D4—1941no—
210C2×D4—1942no—
211C2×D4—1942no—
212C22×C4—1941no—
213C22×C4—1941no—
214C22×C4—1941no—
215C22×C4C22×D9711yescharacteristic
216Q16—1942no—
217Q16—1944no—
218SD16—1944no—
219SD16—1944no—
220SD16—1942no—
221SD16—1942no—
222SD16—1944no—
223SD16—1944no—
224D8—1942no—
225D8—1944no—
226M16D38814yescomplemented
227C2×C8D38814yescomplemented
228C2×C8C22×D9711yescharacteristic
229C2×C8C22×D9712yes—
230C2×C8C22×D9711yescharacteristic
231C4⋊C4—1944no—
232C4⋊C4D38812yes—
233C22⋊C4—1944no—
234C22⋊C4—1944no—
235C22⋊C4D38812yes—
236C42C22×D9712yes—
237C23—7762no—
238C23—3882no—
239C23—1941no—
240C23—1942no—
241C23—1941no—
242C23C2×D38811yescharacteristic
243Q8—3882no—
244Q8—1942no—
245Q8—1941no—
246Q8—1942no—
247Q8—1941no—
248Q8—1941no—
249Q8—1941no—
250D4—7764no—
251D4—3882no—
252D4—3882no—
253D4—3882no—
254D4—1941no—
255D4—1942no—
256D4—1941no—
257D4—1941no—
258D4—1942no—
259D4—1941no—
260C2×C4—3882no—
261C2×C4—3882no—
262C2×C4—3882no—
263C2×C4—1941no—
264C2×C4—1942no—
265C2×C4—1941no—
266C2×C4C2×D38814yes—
267C2×C4C2×D38811yescharacteristic
268C2×C4C2×D38812yes—
269C2×C4C2×D38812yes—
270C2×C4C23×D9711yescharacteristic
271C8C2×D38812yes—
272C8C2×D38812yes—
273C8C2×D38814yes—
274C22—22no—
275C22—7764no—
276C22—3881no—
277C22—3881no—
278C22—3882no—
279C22—3882no—
280C22—3884no—
281C22—3882no—
282C22#1569080712yes—
283C22C22×D38811yescharacteristic
284C4—24no—
285C4—3881no—
286C4—3881no—
287C4—3882no—
288C4#1569080712yes—
289C4C22×D38811yescharacteristic
290C4C22×D38811yescharacteristic
291C2—22no—
292C2—7762no—
293C2—7761no—
294C2—7761no—
295C2C2×#1569080711yescharacteristic
296C2#2026154712yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#405×C97C2
socleC22×C97C22×D4
hypercenter#405D97
hypocenterC97#3154
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#15690843C2semidirect
2. C2×#15690838C2semidirect
3. C2×#15690832C2semidirect
4. C22×#13095500C2semidirect
5. #20261311C2semidirect
6. #20261304C2semidirect
7. #20261129C2semidirect
8. #20261124C2semidirect
9. #20261119C2semidirect
10. #20261108C2semidirect
11. #405×C97C2semidirect
19. C2×Dic388C22semidirect
20. C2×Dic388C22semidirect
25. C2×#13095500C22semidirect
26. C2×#13095500C22semidirect
27. C2×#13095500C22semidirect
29. C2×D776C22semidirect
30. #15690810C22semidirect
33. #15690798C22semidirect
41. #97×C97C22semidirect
42. C4×C8×C97C22semidirect
80. M16×C97D4semidirect
81. C2×C8×C97D4semidirect
149. C97#3154semidirect
160. #405D97semidirect
190. #97C2×D97semidirect
191. C4×C8C2×D97semidirect
226. M16D388semidirect
227. C2×C8D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908432
C2×#156908382
C2×#156908322
C22×#130955002
#202613112
#202613042
#202611292
#202611242
#202611192
#202611082
#405×C972
#315497