Finite Group Databasebeta

← #27152198

Subgroups — #27152198

Order 12416.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15690840C211yesmaximal, characteristic, complemented
3C2×#15690836C211yesmaximal, characteristic, complemented
4#20261281C211yesmaximal, characteristic, complemented
5C2×#15690797C211yesmaximal, characteristic, complemented
6#20261126C212yesmaximal, complemented
7#20261123C212yesmaximal, complemented
8#20261118C212yesmaximal, complemented
9#20261116C212yesmaximal, complemented
10M16×Dic97C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×Dic194C2211yescharacteristic
14#15690843—24no—
15#15690840—24no—
16#15690836—24no—
17C2×Dic388C2212yescomplemented
18C2×#13095501C2212yescomplemented
19C2×#13095500C2212yescomplemented
20C2×#13095499C2212yes—
21C2×#13095499C2212yescomplemented
22C2×#13095496C2212yes—
23C2×#13095495C2211yescharacteristic
24#15690808C2211yescharacteristic, complemented
25#15690806C2211yescharacteristic, complemented
26#15690799C2211yescharacteristic, complemented
27#15690797—22no—
28#15690797C2212yes—
29#15690797C2212yes—
30#15690797C2211yescharacteristic, complemented
31#15690782—22no—
32C4×C22⋊D97—22no—
33C2×C22⋊Dic97—22no—
34C2×C4×Dic97C2211yescharacteristic
35#15690720C2212yescomplemented
36C8×Dic97C2212yescomplemented
37C2×#41×C97C2211yescharacteristic
38C22×D4×C97C2211yescharacteristic
39AGL(1,ℤ/8)×C97—24no—
40C2×SD16×C97C2212yes—
41C2×D8×C97C2212yes—
42C2×M16×C97C2211yescharacteristic
43#13095511—22no—
44#13095511—22no—
45C2×Dic194—22no—
46C2×Dic194—22no—
47C2×Dic194—21no—
48C2×Dic194C2312yes—
49C2×Dic194C2311yescharacteristic
50C2×D388C2311yescharacteristic
51C2×C22⋊D97—21no—
52C2×C22⋊D97—22no—
53C22×Dic97—21no—
54C22×Dic97D412yes—
55C2×C4×D97—21no—
56Dic388—24no—
57#13095501—24no—
58#13095500—24no—
59#13095499—24no—
60#13095499—24no—
61#13095496—24no—
62#13095495D414yescomplemented
63C2×#11715895C2312yes—
64#13095490—22no—
65C22⋊Dic97—24no—
66C22⋊Dic97—24no—
67C22⋊Dic97—22no—
68#13095488—22no—
69C4×Dic97D412yes—
70C4×Dic97C2311yescharacteristic
71C4×Dic97C2311yescharacteristic
72C24×C97—21no—
73#41×C97—22no—
74#41×C97—22no—
75C2×Q8×C97C2311yescharacteristic
76C2×D4×C97—21no—
77C2×D4×C97—22no—
78C2×D4×C97—22no—
79C2×D4×C97—21no—
80C2×D4×C97C2312yes—
81C2×D4×C97C2311yescharacteristic
82C22×C4×C97—21no—
83C22×C4×C97C2311yescharacteristic
84SD16×C97—24no—
85D8×C97—24no—
86M16×C97D414yescomplemented
87C2×C8×C97C2312yes—
88C23×C97—21no—
89C23×C97—22no—
90C23×C97—44no—
91C23×C97—42no—
92C23×C97C2×D411yescharacteristic
93C22×D97—21no—
94Q8×C97—21no—
95Q8×C97—21no—
96D4×C97—21no—
97D4×C97—22no—
98D4×C97—22no—
99D4×C97—21no—
100D4×C97—42no—
101D4×C97—42no—
102C2×C4×C97—21no—
103C2×C4×C97—42no—
104C2×C4×C97C2×D411yescharacteristic
105C2×C4×C97C2×D412yes—
106C2×C4×C97C2×D412yes—
107C2×C4×C97C2411yescharacteristic
108C22⋊D97—44no—
109C22⋊D97—42no—
110C2×Dic97—22no—
111C2×Dic97—21no—
112C2×Dic97—42no—
113C2×Dic97—42no—
114C2×Dic97C2×D412yes—
115C2×Dic97C2×D412yes—
116D388—21no—
117D388—21no—
118C4×D97—42no—
119Dic194—21no—
120Dic194—22no—
121Dic194—22no—
122Dic194—21no—
123Dic194—42no—
124C8×C97C2×D414yes—
125#11715895C2×D414yes—
126C22×C97—22no—
127C22×C97—41no—
128C22×C97—42no—
129C22×C97—42no—
130C22×C97—44no—
131C22×C97—84no—
132C22×C97#12712yes—
133C22×C97C22×D411yescharacteristic
134C2×D97—41no—
135C2×D97—42no—
136C4×C97—41no—
137C4×C97#12712yes—
138C4×C97C22×D411yescharacteristic
139C4×C97C22×D411yescharacteristic
140Dic97—22no—
141Dic97—22no—
142Dic97—42no—
143Dic97—41no—
144C2×C97—22no—
145C2×C97—81no—
146C2×C97—82no—
147C2×C97C2×#12711yescharacteristic
148C2×C97#57512yes—
149D97—81no—
150#3153—971nomaximal, Sylow
151C97#315311yescharacteristic, complemented, Sylow
152C2×#137—971no—
153C2×#137—971no—
154C2×AGL(1,ℤ/8)—971no—
155C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
156#532—971no—
157C2×#124—971no—
158#497—972no—
159#496—972no—
160#495—972no—
161#495—972no—
162C4×M16—971no—
163C2×#41—971no—
164C2×#41C2×D9711yescharacteristic
165C22×Q8—971no—
166C22×D4C2×D9711yescharacteristic
167#137—1944no—
168#137—1944no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—1944no—
171C2×Q16—972no—
172C2×Q16—972no—
173C2×SD16—972no—
174C2×SD16—972no—
175C2×SD16—972no—
176C2×SD16C2×D9712yes—
177C2×D8—972no—
178C2×D8C2×D9712yes—
179C2×M16—971no—
180C2×M16C2×D9711yescharacteristic
181C4⋊Q8—971no—
182#127—971no—
183#124—971no—
184#124—972no—
185#124—972no—
186#124—971no—
187#124—1942no—
188#121—1942no—
189C4×D4—1942no—
190C2×C22⋊C4—1942no—
191C2×C42—971no—
192#97—972no—
193C4×C8—972no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—1942no—
198#41—1942no—
199C2×Q8—972no—
200C2×Q8—971no—
201C2×Q8—1942no—
202C2×Q8—1942no—
203C2×Q8—1941no—
204C2×Q8C22×D9711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—971no—
210C2×D4—1942no—
211C2×D4—1941no—
212C2×D4C22×D9712yes—
213C2×D4C22×D9711yescharacteristic
214C22×C4—21no—
215C22×C4—972no—
216C22×C4—1941no—
217C22×C4—1941no—
218C22×C4C22×D9711yescharacteristic
219Q16—1944no—
220Q16—1944no—
221SD16—24no—
222SD16—1944no—
223SD16—1944no—
224SD16—1944no—
225D8—24no—
226D8—1944no—
227M16—974no—
228M16C22⋊D9714yescomplemented
229C2×C8—972no—
230C2×C8C22×D9712yes—
231C4⋊C4—1942no—
232C22⋊C4—1942no—
233C22⋊C4—1944no—
234C22⋊C4—1944no—
235C22⋊C4—1942no—
236C42—972no—
237C42—971no—
238C42—971no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—1941no—
244C23D4×D9711yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—3882no—
248Q8—1941no—
249Q8—1942no—
250Q8—1942no—
251Q8—1941no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3882no—
259D4—3884no—
260D4—1941no—
261D4—1941no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—972no—
265C2×C4—972no—
266C2×C4—3882no—
267C2×C4—3882no—
268C2×C4—3882no—
269C2×C4—1942no—
270C2×C4—1941no—
271C2×C4D4×D9711yescharacteristic
272C2×C4D4×D9712yes—
273C2×C4C2×C22⋊D9712yes—
274C2×C4C23×D9711yescharacteristic
275C8—974no—
276C8C2×C22⋊D9714yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—3881no—
284C22—3882no—
285C22#1569080612yes—
286C22C2×D4×D9711yescharacteristic
287C4—41no—
288C4—3881no—
289C4—3882no—
290C4—1942no—
291C4—1942no—
292C4#1569080612yes—
293C4C2×D4×D9711yescharacteristic
294C4C22×C22⋊D9711yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—7761no—
299C2C2×#1569080611yescharacteristic
300C2#2026153112yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×AGL(1,ℤ/8)×C97C2
socleC22×C97C22×D4
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#3153
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15690843C2semidirect
2. C2×#15690840C2semidirect
3. C2×#15690836C2semidirect
4. #20261281C2semidirect
5. C2×#15690797C2semidirect
6. #20261126C2semidirect
7. #20261123C2semidirect
8. #20261118C2semidirect
9. #20261116C2semidirect
10. M16×Dic97C2semidirect
11. C2×AGL(1,ℤ/8)×C97C2semidirect
17. C2×Dic388C22semidirect
18. C2×#13095501C22semidirect
19. C2×#13095500C22semidirect
21. C2×#13095499C22semidirect
24. #15690808C22semidirect
25. #15690806C22semidirect
26. #15690799C22semidirect
30. #15690797C22semidirect
35. #15690720C22semidirect
36. C8×Dic97C22semidirect
62. #13095495D4semidirect
86. M16×C97D4semidirect
151. C97#3153semidirect
155. C2×AGL(1,ℤ/8)D97semidirect
228. M16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908432
C2×#156908402
C2×#156908362
#202612812
C2×#156907972
#202611262
#202611232
#202611182
#202611162
M16×Dic972
C2×AGL(1,ℤ/8)×C972
#315397