Finite Group Databasebeta

← #27184272

Subgroups — #27184272

Order 12928.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#20265979C211yesmaximal, characteristic, complemented
2#20265978C211yesmaximal, characteristic, complemented
3#20265615C211yesmaximal, characteristic, complemented
4#20265614C212yesmaximal, complemented
5#20265611C212yesmaximal, complemented
6#20265608C212yesmaximal, complemented
7#20265607C211yesmaximal, characteristic, complemented
8C2×#15692196C211yesmaximal, characteristic, complemented
9D16×D101C212yesmaximal, complemented
10#20265595C211yesmaximal, characteristic, complemented
11#509×C101C211yesmaximal, characteristic, complemented
12#15692301—21no—
13#15692300—21no—
14#15692282—22no—
15#15692281—21no—
16#15692280—21no—
17#15692279—22no—
18#15692278C2212yescomplemented
19#15692277C2211yescharacteristic, complemented
20#15692274C2212yescomplemented
21C2×D808C2211yescharacteristic, complemented
22D8×D101C2212yescomplemented
23D8×D101C2212yescomplemented
24#15692261C2211yescharacteristic
25#15692201C2211yescharacteristic, complemented
26#15692200C2212yescomplemented
27#15692199C2211yescharacteristic, complemented
28#15692198C2211yescharacteristic, complemented
29D1616C2212yescomplemented
30#15692196C2211yescharacteristic, complemented
31#15692196C2212yescomplemented
32#15692196C2212yescomplemented
33#15692194C2212yescomplemented
34C2×#13108049C2211yescharacteristic, complemented
35C16×D101C2212yescomplemented
36#135×C101C2211yescharacteristic, complemented
37C2×D8×C101C2211yescharacteristic, complemented
38SD32×C101C2212yescomplemented
39D16×C101C2212yescomplemented
40M32×C101C2211yescharacteristic, complemented
41#13108075—22no—
42#13108074—21no—
43#13108074—41no—
44#13108074D411yescharacteristic
45#13108073—22no—
46#13108073—41no—
47C2×D404—21no—
48C2×D404—41no—
49C2×C22⋊D101—42no—
50D4×D101—22no—
51D4×D101—22no—
52D4×D101—42no—
53D4×D101—41no—
54Dic404C2311yescharacteristic
55#13108063—21no—
56#13108062—22no—
57#13108061—22no—
58D808—21no—
59D808C2312yes—
60D808C2311yescharacteristic
61#13108059—22no—
62#13108059—22no—
63#13108058D411yescharacteristic
64#13108057D412yes—
65C8×D101C2312yes—
66#13108049C2311yescharacteristic
67#13108049C2311yescharacteristic
68#41×C101—21no—
69C2×D4×C101—21no—
70Q16×C101C2311yescharacteristic
71SD16×C101—21no—
72D8×C101—21no—
73D8×C101C2311yescharacteristic
74D8×C101C2312yes—
75C2×C8×C101C2311yescharacteristic
76C16×C101C2312yes—
77C23×C101—41no—
78C22×D101—41no—
79C22×D101—42no—
80C22×D101—42no—
81Q8×C101—21no—
82D4×C101—22no—
83D4×C101—21no—
84D4×C101—41no—
85D4×C101—41no—
86C2×C4×C101—41no—
87C2×C4×C101C2×D411yescharacteristic
88C22⋊D101—41no—
89C22⋊D101—42no—
90C22⋊D101—42no—
91C22⋊D101—42no—
92C22⋊D101D812yes—
93C2×Dic101—42no—
94D404—21no—
95D404—22no—
96D404—42no—
97D404—41no—
98D404D811yescharacteristic
99C4×D101—41no—
100C4×D101—42no—
101C4×D101C2×D412yes—
102Dic202—21no—
103Dic202D811yescharacteristic
104C8×C101C2×D411yescharacteristic
105C8×C101C2411yescharacteristic
106#11717579C2×D412yes—
107C22×C101—42no—
108C22×C101—41no—
109C22×C101—161no—
110C22×C101C2×D811yescharacteristic
111C2×D101—42no—
112C2×D101—41no—
113C2×D101—162no—
114C2×D101—162no—
115C2×D101—161no—
116C2×D101C2×D812yes—
117C4×C101—41no—
118C4×C101C2×D811yescharacteristic
119C4×C101C22×D411yescharacteristic
120Dic101—41no—
121Dic101C2×D812yes—
122C2×C101—21no—
123C2×C101—81no—
124C2×C101—82no—
125C2×C101C22×D811yescharacteristic
126D101—22no—
127D101—82no—
128D101—81no—
129#3415—1011nomaximal, Sylow
130C101#341511yescharacteristic, complemented, Sylow
131#576—1011no—
132#576—1011no—
133#509—1012no—
134#509—1012no—
135#509—1011no—
136#509D10111yescharacteristic, complemented
137#508—1011no—
138#508—1012no—
139C2×D16—1011no—
140C2×D16—1012no—
141#504—1011no—
142H(5,𝔽2)—2021no—
143H(5,𝔽2)—2021no—
144AGL(1,ℤ/8)—2022no—
145AGL(1,ℤ/8)—2022no—
146AGL(1,ℤ/8)—2021no—
147AGL(1,ℤ/8)—2021no—
148#135—1012no—
149#135—1012no—
150#135—1011no—
151#135C2×D10111yescharacteristic, complemented
152C2×D8—1012no—
153C2×D8—1012no—
154C2×D8—1011no—
155C2×D8C2×D10111yescharacteristic, complemented
156#131—1011no—
157Q32—1011no—
158SD32—1011no—
159SD32—1011no—
160SD32—1012no—
161SD32C2×D10112yescomplemented
162D16—1012no—
163D16—1011no—
164D16—1012no—
165D16—1012no—
166D16C2×D10112yescomplemented
167M32—1012no—
168M32C2×D10111yescharacteristic, complemented
169C2×C16—1011no—
170C2×C16—1012no—
171#41—21no—
172#41—1011no—
173#41—4041no—
174#41—4041no—
175#41—2022no—
176#41—2022no—
177#41—2021no—
178C2×D4—21no—
179C2×D4—4041no—
180C2×D4—4042no—
181C2×D4—4041no—
182C2×D4—4042no—
183C2×D4—2022no—
184C2×D4—2022no—
185C2×D4—2021no—
186Q16—1011no—
187Q16C22×D10111yescharacteristic
188SD16—21no—
189SD16—2022no—
190SD16—2022no—
191SD16—2021no—
192D8—21no—
193D8—1012no—
194D8—1011no—
195D8—2022no—
196D8—2022no—
197D8—2021no—
198D8C22×D10112yes—
199D8C22×D10111yescharacteristic
200M16—1011no—
201M16—1012no—
202C2×C8—1012no—
203C2×C8C22×D10111yescharacteristic
204C16—1011no—
205C16—1011no—
206C16C22×D10112yes—
207C23—41no—
208C23—4042no—
209C23—4042no—
210C23—4041no—
211Q8—21no—
212Q8—1011no—
213Q8—2021no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—1012no—
219D4—1011no—
220D4—4042no—
221D4—4042no—
222D4—4042no—
223D4—4042no—
224D4—4041no—
225D4—4041no—
226D4—2021no—
227D4—2022no—
228C2×C4—41no—
229C2×C4—1012no—
230C2×C4—4042no—
231C2×C4—4042no—
232C2×C4—4041no—
233C2×C4D4×D10111yescharacteristic
234C8—1012no—
235C8D4×D10111yescharacteristic
236C8C23×D10111yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—16161no—
241C22—16162no—
242C22—16162no—
243C22—1012no—
244C22—4042no—
245C22—4041no—
246C22D8×D10111yescharacteristic
247C4—41no—
248C4—1012no—
249C4—4041no—
250C4D8×D10111yescharacteristic
251C4C2×D4×D10111yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—8082no—
256C2—8081no—
257C2—2022no—
258C2C2×D8×D10111yescharacteristic
subgroupG/H
centerC2C2×D8×D101
commutator, cosocleC8×C101C24
FrattiniC8C23×D101
Fitting#509×C101C2
socleC2×C101C22×D8
hypercenter#509D101
hypocenterC101#3415
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20265979C2semidirect
2. #20265978C2semidirect
3. #20265615C2semidirect
4. #20265614C2semidirect
5. #20265611C2semidirect
6. #20265608C2semidirect
7. #20265607C2semidirect
8. C2×#15692196C2semidirect
9. D16×D101C2semidirect
10. #20265595C2semidirect
11. #509×C101C2semidirect
18. #15692278C22semidirect
19. #15692277C22semidirect
20. #15692274C22semidirect
21. C2×D808C22semidirect
22. D8×D101C22semidirect
23. D8×D101C22semidirect
25. #15692201C22semidirect
26. #15692200C22semidirect
27. #15692199C22semidirect
28. #15692198C22semidirect
29. D1616C22semidirect
30. #15692196C22semidirect
31. #15692196C22semidirect
32. #15692196C22semidirect
33. #15692194C22semidirect
34. C2×#13108049C22semidirect
35. C16×D101C22semidirect
36. #135×C101C22semidirect
37. C2×D8×C101C22semidirect
38. SD32×C101C22semidirect
39. D16×C101C22semidirect
40. M32×C101C22semidirect
130. C101#3415semidirect
136. #509D101semidirect
151. #135C2×D101semidirect
155. C2×D8C2×D101semidirect
161. SD32C2×D101semidirect
166. D16C2×D101semidirect
168. M32C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659792
#202659782
#202656152
#202656142
#202656112
#202656082
#202656072
C2×#156921962
D16×D1012
#202655952
#509×C1012
#3415101