Finite Group Databasebeta

← #27182898

Subgroups — #27182898

Order 12928.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265983C212yesmaximal, complemented
2#20265978C212yesmaximal, complemented
3#20265444C212yesmaximal, complemented
4#20265437C212yesmaximal, complemented
5#20265354C211yesmaximal, characteristic, complemented
6#20265349C211yesmaximal, characteristic, complemented
7#20265344C212yesmaximal, complemented
8#20265300C212yesmaximal, complemented
9#496×C101C211yesmaximal, characteristic, complemented
10#15692303C2211yescharacteristic, complemented
11#15692300C2211yescharacteristic
12#15692300C2212yescomplemented
13#15692286C2212yescomplemented
14#15692284—22no—
15#15692282—22no—
16#15692281C2212yes—
17#15692281C2212yescomplemented
18#15692281C2212yescomplemented
19#15692279—22no—
20#15692279—22no—
21#15692277C2212yes—
22#15692276—22no—
23#15692274—22no—
24SD16×D101—22no—
25D8×D101—22no—
26#15692261C2212yescomplemented
27#15692261C2212yescomplemented
28#15692191C2212yescomplemented
29#15692183C2212yescomplemented
30#15692183C2212yescomplemented
31#15692174C2211yescharacteristic, complemented
32#15692172C2212yescomplemented
33#15692172C2211yescharacteristic, complemented
34C2×SD16×C101C2212yescomplemented
35C2×D8×C101C2212yescomplemented
36#127×C101C2211yescharacteristic, complemented
37#124×C101C2211yescharacteristic, complemented
38#97×C101C2211yescharacteristic, complemented
39#13108075—22no—
40#13108074—41no—
41#13108074C2312yes—
42#13108074C2312yes—
43#13108073—22no—
44#13108073—22no—
45#13108073—22no—
46Q8×D101—22no—
47C2×C22⋊D101—41no—
48C2×C22⋊D101—42no—
49D4×D101—22no—
50D4×D101—22no—
51D4×D101—22no—
52Dic404D412yescomplemented
53#13108064—22no—
54#13108063D412yescomplemented
55#13108063D412yescomplemented
56#13108062—22no—
57#13108062—22no—
58#13108062—22no—
59#13108061—22no—
60D808D412yescomplemented
61#13108059—22no—
62#13108059—22no—
63#13108059—22no—
64#13108058C2312yes—
65#13108058C2312yes—
66#13108057—22no—
67#13108057—22no—
68C8×D101—22no—
69C8×D101—22no—
70C2×Q8×C101C2311yescharacteristic
71C2×D4×C101—41no—
72C2×D4×C101C2311yescharacteristic
73C2×D4×C101C2312yes—
74SD16×C101—22no—
75SD16×C101—22no—
76D8×C101—22no—
77D8×C101—22no—
78C2×C8×C101C2312yes—
79C22⋊C4×C101—41no—
80C42×C101C2311yescharacteristic
81C23×C101—21no—
82C23×C101—22no—
83C22×D101—42no—
84C22×D101—41no—
85Q8×C101—22no—
86D4×C101—22no—
87D4×C101—22no—
88D4×C101—22no—
89D4×C101—82no—
90C2×C4×C101—21no—
91C2×C4×C101—21no—
92C2×C4×C101C2411yescharacteristic
93C22⋊D101—22no—
94C22⋊D101—22no—
95C22⋊D101—42no—
96C22⋊D101—42no—
97C22⋊D101—42no—
98C2×Dic101—42no—
99C2×Dic101—41no—
100D404—41no—
101D404C2×D412yes—
102D404C2×D412yes—
103C4×D101—22no—
104C4×D101—22no—
105C4×D101—42no—
106Dic202—41no—
107Dic202C2×D412yes—
108Dic202C2×D412yes—
109C8×C101C2×D412yes—
110C8×C101C2×D412yes—
111#11717579—22no—
112#11717579—22no—
113C22×C101—41no—
114C22×C101—42no—
115C22×C101—82no—
116C22×C101—81no—
117C22×C101H(5,𝔽2)11yescharacteristic
118C2×D101—22no—
119C2×D101—22no—
120C2×D101—162no—
121C2×D101—161no—
122C4×C101—41no—
123C4×C101—41no—
124C4×C101C22×D412yes—
125Dic101—22no—
126Dic101—22no—
127C2×C101—21no—
128C2×C101—82no—
129C2×C101—81no—
130C2×C101D4211yescharacteristic
131D101—42no—
132D101—42no—
133#3288—1011nomaximal, Sylow
134C101#328811yescharacteristic, complemented, Sylow
135#577—1012no—
136#576—1012no—
137#496D10111yescharacteristic, complemented
138#472—1012no—
139#471—1012no—
140#455—1011no—
141#454—1011no—
142#453—1012no—
143#444—1012no—
1442−1+4—1011no—
145H(5,𝔽2)—1011no—
146H(5,𝔽2)—1012no—
147#137—1012no—
148#137—2022no—
149AGL(1,ℤ/8)—1012no—
150AGL(1,ℤ/8)—1012no—
151AGL(1,ℤ/8)—1012no—
152AGL(1,ℤ/8)—2022no—
153AGL(1,ℤ/8)—2022no—
154AGL(1,ℤ/8)—2022no—
155#135—1012no—
156#135—2022no—
157#135—2022no—
158C2×SD16—2022no—
159C2×SD16C2×D10112yescomplemented
160C2×D8—2022no—
161C2×D8C2×D10112yescomplemented
162#131—1012no—
163#131—1012no—
164#127C2×D10111yescharacteristic, complemented
165#124C2×D10111yescharacteristic, complemented
166#108—1012no—
167#104—1012no—
168#104—1012no—
169#101—1011no—
170#100—1012no—
171#100—1011no—
172#97C2×D10111yescharacteristic, complemented
173#41—1012no—
174#41—1012no—
175#41—4041no—
176#41—2022no—
177#41—2022no—
178#41—2022no—
179#41—2022no—
180C2×Q8—2022no—
181C2×Q8C22×D10111yescharacteristic
182C2×D4—41no—
183C2×D4—4041no—
184C2×D4—4042no—
185C2×D4—2022no—
186C2×D4—2022no—
187C2×D4—2022no—
188C2×D4C22×D10111yescharacteristic
189C2×D4C22×D10112yes—
190Q16—1012no—
191Q16—2022no—
192SD16—22no—
193SD16—22no—
194SD16—1012no—
195SD16—1012no—
196SD16—2022no—
197SD16—2022no—
198SD16—2022no—
199SD16—2022no—
200D8—22no—
201D8—22no—
202D8—1012no—
203D8—2022no—
204D8—2022no—
205D8—2022no—
206M16—1012no—
207M16—1012no—
208M16—2022no—
209M16—2022no—
210C2×C8—2022no—
211C2×C8—2022no—
212C2×C8C22×D10112yes—
213C22⋊C4—41no—
214C42C22×D10111yescharacteristic
215C23—21no—
216C23—22no—
217C23—4042no—
218C23—4041no—
219Q8—22no—
220Q8—1012no—
221Q8—1012no—
222Q8—4041no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—1012no—
228D4—1012no—
229D4—4041no—
230D4—4042no—
231D4—4042no—
232D4—4042no—
233D4—2022no—
234D4—2022no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—4042no—
238C2×C4—4041no—
239C2×C4—4042no—
240C2×C4—2022no—
241C2×C4—2022no—
242C2×C4C23×D10111yescharacteristic
243C8—2022no—
244C8—2022no—
245C8D4×D10112yes—
246C8D4×D10112yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—16161no—
252C22—16162no—
253C22—2022no—
254C22—2022no—
255C22#1569230011yescharacteristic
256C4—41no—
257C4—41no—
258C4—2022no—
259C4—2022no—
260C4C2×D4×D10112yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—4042no—
265C2—4042no—
266C2#2026581211yescharacteristic
subgroupG/H
centerC2#20265812
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#496×C101C2
socleC2×C101D42
hypercenter#496D101
hypocenterC101#3288
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20265983C2semidirect
2. #20265978C2semidirect
3. #20265444C2semidirect
4. #20265437C2semidirect
5. #20265354C2semidirect
6. #20265349C2semidirect
7. #20265344C2semidirect
8. #20265300C2semidirect
9. #496×C101C2semidirect
10. #15692303C22semidirect
12. #15692300C22semidirect
13. #15692286C22semidirect
17. #15692281C22semidirect
18. #15692281C22semidirect
26. #15692261C22semidirect
27. #15692261C22semidirect
28. #15692191C22semidirect
29. #15692183C22semidirect
30. #15692183C22semidirect
31. #15692174C22semidirect
32. #15692172C22semidirect
33. #15692172C22semidirect
34. C2×SD16×C101C22semidirect
35. C2×D8×C101C22semidirect
36. #127×C101C22semidirect
37. #124×C101C22semidirect
38. #97×C101C22semidirect
52. Dic404D4semidirect
54. #13108063D4semidirect
55. #13108063D4semidirect
60. D808D4semidirect
134. C101#3288semidirect
137. #496D101semidirect
159. C2×SD16C2×D101semidirect
161. C2×D8C2×D101semidirect
164. #127C2×D101semidirect
165. #124C2×D101semidirect
172. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659832
#202659782
#202654442
#202654372
#202653542
#202653492
#202653442
#202653002
#496×C1012
#3288101