Finite Group Databasebeta

← #27152214

Subgroups — #27152214

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×#15690841C211yesmaximal, characteristic, complemented
2C2×#15690835C211yesmaximal, characteristic, complemented
3C2×#15690830C211yesmaximal, characteristic, complemented
4C22×#13095498C211yesmaximal, characteristic, complemented
5#20261304C211yesmaximal, characteristic, complemented
6#20261127C212yesmaximal, complemented
7#20261125C212yesmaximal, complemented
8#20261117C212yesmaximal, complemented
9#20261109C212yesmaximal, complemented
10#20260691C211yesmaximal, characteristic, complemented
11#536×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×D388C2211yescharacteristic
14#15690841—24no—
15#15690835—24no—
16#15690830—24no—
17C2×#13095501C2211yescharacteristic, complemented
18C2×#13095501C2212yescomplemented
19C2×#13095499C2211yescharacteristic, complemented
20C2×#13095498—24no—
21C2×#13095498C2212yes—
22C2×#13095498C2212yes—
23C2×#13095498C2212yes—
24C2×#13095498C2212yescomplemented
25C2×#13095498C2211yescharacteristic, complemented
26C2×#13095496C2212yes—
27C2×#13095496C2211yescharacteristic, complemented
28C2×#13095495C2211yescharacteristic
29C22×#11715895C2211yescharacteristic
30#15690807C2212yes—
31#15690798C2212yes—
32#15690785—22no—
33#15690780—22no—
34#15690721C2212yescomplemented
35C4×#11715895C2212yescomplemented
36C2×#41×C97C2211yescharacteristic
37C22×Q8×C97C2211yescharacteristic
38C4⋊Q8×C97C2212yescomplemented
39#124×C97C2212yescomplemented
40#123×C97—22no—
41C22⋊Q8×C97—22no—
42#117×C97C2211yescharacteristic
43C23×D97—21no—
44#13095511—22no—
45#13095511—22no—
46C2×Dic194C2311yescharacteristic
47C2×D388—22no—
48C2×D388—22no—
49C2×D388—21no—
50C2×D388—24no—
51C2×D388C2312yes—
52C2×D388C2311yescharacteristic
53C2×C22⋊D97—21no—
54C2×C4×D97—21no—
55#13095501—22no—
56#13095501—24no—
57#13095499—22no—
58#13095498—24no—
59#13095498—24no—
60#13095498—24no—
61#13095498—22no—
62#13095498—24no—
63#13095496—24no—
64#13095496—22no—
65#13095495D414yescomplemented
66C2×#11715895D414yescomplemented
67C2×#11715895C2311yescharacteristic
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2312yes—
70#13095488—24no—
71#41×C97—22no—
72#41×C97—22no—
73C2×Q8×C97—21no—
74C2×Q8×C97—22no—
75C2×Q8×C97—22no—
76C2×Q8×C97C2312yes—
77C2×Q8×C97C2311yescharacteristic
78C2×D4×C97—21no—
79C2×D4×C97C2311yescharacteristic
80C22×C4×C97—21no—
81C22×C4×C97—21no—
82C22×C4×C97C2311yescharacteristic
83C4⋊C4×C97—24no—
84C4⋊C4×C97D412yes—
85C22⋊C4×C97—24no—
86C22⋊C4×C97D412yes—
87C42×C97C2312yes—
88C23×C97—21no—
89C23×C97C2×D411yescharacteristic
90C22×D97—22no—
91C22×D97—21no—
92C22×D97—42no—
93C22×D97—82no—
94Q8×C97—22no—
95Q8×C97—22no—
96Q8×C97—21no—
97Q8×C97—21no—
98Q8×C97—42no—
99D4×C97—21no—
100D4×C97—21no—
101D4×C97—42no—
102C2×C4×C97—22no—
103C2×C4×C97—21no—
104C2×C4×C97—42no—
105C2×C4×C97—42no—
106C2×C4×C97C2×D412yes—
107C2×C4×C97C2×D412yes—
108C2×C4×C97C2×D414yes—
109C2×C4×C97C2×D411yescharacteristic
110C2×C4×C97C2411yescharacteristic
111C22⋊D97—42no—
112C2×Dic97—21no—
113D388—22no—
114D388—21no—
115D388—22no—
116D388—21no—
117D388—42no—
118D388—84no—
119C4×D97—42no—
120Dic194—21no—
121Dic194—21no—
122#11715895C2×D412yes—
123#11715895C2×D412yes—
124#11715895C2×D414yes—
125C22×C97—22no—
126C22×C97—41no—
127C22×C97—42no—
128C22×C97#12712yes—
129C22×C97C22×D411yescharacteristic
130C2×D97—41no—
131C2×D97—42no—
132C2×D97—42no—
133C2×D97—44no—
134C2×D97—84no—
135C4×C97—24no—
136C4×C97—41no—
137C4×C97—42no—
138C4×C97#12712yes—
139C4×C97C22×D411yescharacteristic
140C4×C97C22×D411yescharacteristic
141Dic97—41no—
142C2×C97—22no—
143C2×C97—81no—
144C2×C97C2×#12711yescharacteristic
145C2×C97#57712yes—
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#536D9711yescharacteristic, complemented
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×Q8C2×D9711yescharacteristic
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—972no—
174C2×SD16—972no—
175C2×SD16—971no—
176C2×SD16—1944no—
177C2×D8—971no—
178C2×D8—972no—
179C2×M16—971no—
180C22×C8—971no—
181C4⋊Q8C2×D9712yescomplemented
182#127—972no—
183#124—972no—
184#124C2×D9712yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—1942no—
188#120—1942no—
189#117C2×D9711yescharacteristic
190#97—972no—
191C4×C8—972no—
192C24—1941no—
193#41—22no—
194#41—22no—
195#41—1942no—
196#41—1942no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—971no—
201C2×Q8C22×D9712yes—
202C2×Q8C22×D9711yescharacteristic
203C2×D4—21no—
204C2×D4—972no—
205C2×D4—971no—
206C2×D4—1941no—
207C2×D4—1942no—
208C2×D4—1942no—
209C2×D4—1944no—
210C2×D4—1941no—
211C2×D4C22×D9711yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—1941no—
215C22×C4C22×D9711yescharacteristic
216Q16—1942no—
217Q16—1944no—
218SD16—1942no—
219SD16—1944no—
220SD16—1944no—
221SD16—1942no—
222SD16—1944no—
223SD16—1944no—
224D8—1942no—
225D8—1944no—
226M16—974no—
227C2×C8—974no—
228C2×C8—971no—
229C2×C8—972no—
230C2×C8—971no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D9712yes—
233C22⋊C4—24no—
234C22⋊C4—1944no—
235C22⋊C4C22⋊D9712yes—
236C42C22×D9712yes—
237C23—21no—
238C23—7762no—
239C23—3882no—
240C23—1942no—
241C23—1941no—
242C23C2×C22⋊D9711yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—1941no—
249Q8—1941no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—7764no—
254D4—3882no—
255D4—3882no—
256D4—1942no—
257D4—1941no—
258D4—1942no—
259D4—1941no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—3882no—
265C2×C4—1941no—
266C2×C4D4×D9712yes—
267C2×C4D4×D9712yes—
268C2×C4C2×C22⋊D9714yes—
269C2×C4C2×C22⋊D9711yescharacteristic
270C2×C4C23×D9711yescharacteristic
271C8—972no—
272C8—972no—
273C8—974no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—7764no—
278C22—3881no—
279C22—3882no—
280C22—3882no—
281C22—3884no—
282C22#1569080612yes—
283C22C22×C22⋊D9711yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—3881no—
288C4#1569080612yes—
289C4C2×D4×D9711yescharacteristic
290C4C2×D4×D9711yescharacteristic
291C2—22no—
292C2—81no—
293C2—7762no—
294C2—7761no—
295C2C2×#1569080611yescharacteristic
296C2#2026154312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#536×C97C2
socleC22×C97C22×D4
hypercenter#536D97
hypocenterC97#3154
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
2. C2×#15690835C2semidirect
3. C2×#15690830C2semidirect
4. C22×#13095498C2semidirect
5. #20261304C2semidirect
6. #20261127C2semidirect
7. #20261125C2semidirect
8. #20261117C2semidirect
9. #20261109C2semidirect
10. #20260691C2semidirect
11. #536×C97C2semidirect
17. C2×#13095501C22semidirect
18. C2×#13095501C22semidirect
19. C2×#13095499C22semidirect
24. C2×#13095498C22semidirect
25. C2×#13095498C22semidirect
27. C2×#13095496C22semidirect
34. #15690721C22semidirect
35. C4×#11715895C22semidirect
38. C4⋊Q8×C97C22semidirect
39. #124×C97C22semidirect
65. #13095495D4semidirect
66. C2×#11715895D4semidirect
149. C97#3154semidirect
154. #536D97semidirect
181. C4⋊Q8C2×D97semidirect
184. #124C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C2×#156908352
C2×#156908302
C22×#130954982
#202613042
#202611272
#202611252
#202611172
#202611092
#202606912
#536×C972
#315497