Finite Group Databasebeta

← #27152210

Subgroups — #27152210

Order 12416.

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

normal subgroups only

This group has 564 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×#15690830C211yesmaximal, characteristic, complemented
3C22×#13095499C211yesmaximal, characteristic, complemented
4#20261306C211yesmaximal, characteristic, complemented
5#20261302C211yesmaximal, characteristic, complemented
6#20261126C212yesmaximal, complemented
7#20261123C212yesmaximal, complemented
8#20261118C212yesmaximal, complemented
9#20261106C212yesmaximal, complemented
10#20260693C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×Dic194C2211yescharacteristic
14#15690843—24no—
15#15690830—24no—
16C2×Dic388C2212yescomplemented
17C2×#13095501C2211yescharacteristic, complemented
18C2×#13095500C2212yescomplemented
19C2×#13095499—24no—
20C2×#13095499C2212yes—
21C2×#13095499C2212yes—
22C2×#13095499C2211yescharacteristic, complemented
23C2×#13095498C2211yescharacteristic, complemented
24C2×#13095496C2211yescharacteristic, complemented
25C22×#11715895C2211yescharacteristic
26#15690808C2212yescomplemented
27#15690806C2212yes—
28#15690797C2212yes—
29#15690797C2212yescomplemented
30#15690791—22no—
31#15690789—22no—
32#15690781—22no—
33#15690779—22no—
34#15690769C2211yescharacteristic
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×Dic194C2311yescharacteristic
49C2×Dic194C2312yes—
50C2×D388C2311yescharacteristic
51C2×C22⋊D97—21no—
52C2×C22⋊D97—24no—
53C22×Dic97—21no—
54C2×C4×D97—21no—
55Dic388—24no—
56#13095501—22no—
57#13095500—24no—
58#13095499—24no—
59#13095499—24no—
60#13095499—22no—
61#13095498—22no—
62#13095496—22no—
63C2×#11715895D414yescomplemented
64C2×#11715895C2311yescharacteristic
65C2×#11715895C2311yescharacteristic
66C4⋊Dic97D412yes—
67#13095490—24no—
68C22⋊Dic97—24no—
69C22⋊Dic97—24no—
70C22⋊Dic97D412yes—
71C4×Dic97C2312yes—
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—42no—
91C23×C97—82no—
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—42no—
109C22⋊D97—84no—
110C2×Dic97—21no—
111C2×Dic97—22no—
112C2×Dic97—42no—
113C2×Dic97C2×D414yes—
114D388—21no—
115D388—21no—
116C4×D97—42no—
117Dic194—22no—
118Dic194—21no—
119Dic194—22no—
120Dic194—21no—
121Dic194—42no—
122C8×C97C2×D414yes—
123#11715895C2×D412yes—
124#11715895C2×D412yes—
125C22×C97—22no—
126C22×C97—41no—
127C22×C97—42no—
128C22×C97—44no—
129C22×C97—42no—
130C22×C97—84no—
131C22×C97#12712yes—
132C22×C97C22×D411yescharacteristic
133C2×D97—41no—
134C2×D97—42no—
135C4×C97—41no—
136C4×C97#12712yes—
137C4×C97C22×D411yescharacteristic
138C4×C97C22×D411yescharacteristic
139Dic97—24no—
140Dic97—42no—
141Dic97—41no—
142C2×C97—22no—
143C2×C97—81no—
144C2×C97—82no—
145C2×C97C2×#12711yescharacteristic
146C2×C97#57712yes—
147D97—81no—
148#3154—971nomaximal, Sylow
149C97#315411yescharacteristic, complemented, Sylow
150C2×#137—971no—
151C2×AGL(1,ℤ/8)D9711yescharacteristic, complemented
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#405—971no—
161C2×#41—971no—
162C2×#41C2×D9711yescharacteristic
163C22×Q8—971no—
164C22×D4C2×D9711yescharacteristic
165#137—1944no—
166AGL(1,ℤ/8)—24no—
167#135—1944no—
168C2×Q16—971no—
169C2×Q16—972no—
170C2×SD16—971no—
171C2×SD16—972no—
172C2×SD16—972no—
173C2×SD16—971no—
174C2×SD16—972no—
175C2×SD16—1944no—
176C2×SD16C2×D9712yes—
177C2×D8—971no—
178C2×D8C2×D9712yes—
179C2×M16C2×D9711yescharacteristic
180C22×C8—971no—
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#117—971no—
190#97—972no—
191C4×C8—972no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—1942no—
196#41—1942no—
197C2×Q8—971no—
198C2×Q8—972no—
199C2×Q8—1942no—
200C2×Q8—1942no—
201C2×Q8—1941no—
202C2×Q8C22×D9711yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—971no—
208C2×D4—1944no—
209C2×D4—1941no—
210C2×D4C22×D9711yescharacteristic
211C2×D4C22×D9712yes—
212C22×C4—21no—
213C22×C4—1941no—
214C22×C4—1941no—
215C22×C4C22×D9711yescharacteristic
216Q16—1942no—
217Q16—1944no—
218SD16—24no—
219SD16—1944no—
220SD16—1944no—
221SD16—1942no—
222SD16—1942no—
223SD16—1944no—
224D8—24no—
225D8—1942no—
226M16C22⋊D9714yescomplemented
227C2×C8—974no—
228C2×C8—971no—
229C2×C8—971no—
230C2×C8C22×D9712yes—
231C4⋊C4—972no—
232C4⋊C4—1944no—
233C22⋊C4—972no—
234C22⋊C4—1944no—
235C22⋊C4—1944no—
236C42—972no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—1941no—
242C23D4×D9711yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—3882no—
246Q8—1942no—
247Q8—1941no—
248Q8—1942no—
249Q8—1941no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—7764no—
257D4—3882no—
258D4—1941no—
259D4—1941no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—974no—
263C2×C4—3882no—
264C2×C4—3882no—
265C2×C4—1941no—
266C2×C4—1942no—
267C2×C4D4×D9711yescharacteristic
268C2×C4D4×D9712yes—
269C2×C4C2×C22⋊D9712yes—
270C2×C4C23×D9711yescharacteristic
271C8—972no—
272C8—972no—
273C8C2×C22⋊D9714yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—3881no—
281C22—3882no—
282C22#1569080612yes—
283C22C2×D4×D9711yescharacteristic
284C4—41no—
285C4—3881no—
286C4—3882no—
287C4—1944no—
288C4#1569080612yes—
289C4C2×D4×D9711yescharacteristic
290C4C22×C22⋊D9711yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—7761no—
295C2C2×#1569080611yescharacteristic
296C2#2026154512yes—
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#3154
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×#15690830C2semidirect
3. C22×#13095499C2semidirect
4. #20261306C2semidirect
5. #20261302C2semidirect
6. #20261126C2semidirect
7. #20261123C2semidirect
8. #20261118C2semidirect
9. #20261106C2semidirect
10. #20260693C2semidirect
11. C2×AGL(1,ℤ/8)×C97C2semidirect
16. C2×Dic388C22semidirect
17. C2×#13095501C22semidirect
18. C2×#13095500C22semidirect
22. C2×#13095499C22semidirect
23. C2×#13095498C22semidirect
24. C2×#13095496C22semidirect
26. #15690808C22semidirect
29. #15690797C22semidirect
35. #15690720C22semidirect
36. C8×Dic97C22semidirect
63. C2×#11715895D4semidirect
86. M16×C97D4semidirect
149. C97#3154semidirect
151. C2×AGL(1,ℤ/8)D97semidirect
226. 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×#156908302
C22×#130954992
#202613062
#202613022
#202611262
#202611232
#202611182
#202611062
#202606932
C2×AGL(1,ℤ/8)×C972
#315497