Finite Group Databasebeta

← #27184277

Subgroups — #27184277

Order 12928.

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

normal subgroups only

This group has 360 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#20265990C211yesmaximal, characteristic, complemented
2#20265979C211yesmaximal, characteristic, complemented
3#20265619C212yesmaximal, complemented
4#20265615C211yesmaximal, characteristic, complemented
5#20265614C212yesmaximal, complemented
6#20265610C212yesmaximal, complemented
7#20265607C211yesmaximal, characteristic, complemented
8C2×#15692198C211yesmaximal, characteristic, complemented
9SD32×D101C212yesmaximal, complemented
10#20265595C211yesmaximal, characteristic, complemented
11#510×C101C211yesmaximal, characteristic, complemented
12#15692303—21no—
13#15692301—21no—
14#15692287—22no—
15#15692286—21no—
16#15692282—22no—
17#15692280—21no—
18#15692278C2212yes—
19#15692278C2212yescomplemented
20#15692277C2211yescharacteristic, complemented
21Q16×D101C2212yescomplemented
22C2×D808C2211yescharacteristic
23D8×D101C2212yes—
24#15692261C2211yescharacteristic
25#15692201C2211yescharacteristic, complemented
26#15692200C2212yescomplemented
27#15692199C2211yescharacteristic, complemented
28#15692198C2212yescomplemented
29#15692198C2212yescomplemented
30#15692198C2211yescharacteristic, complemented
31D1616C2212yescomplemented
32#15692196C2211yescharacteristic, complemented
33#15692194C2212yescomplemented
34C2×#13108049C2211yescharacteristic, complemented
35C16×D101C2212yescomplemented
36#135×C101C2211yescharacteristic
37C2×Q16×C101C2211yescharacteristic, complemented
38Q32×C101C2212yescomplemented
39SD32×C101C2212yescomplemented
40M32×C101C2211yescharacteristic, complemented
41#13108075—22no—
42#13108075—22no—
43#13108075—41no—
44#13108074—21no—
45#13108074—42no—
46#13108074—41no—
47#13108074D411yescharacteristic
48Q8×D101—22no—
49Q8×D101—41no—
50C2×D404—21no—
51C2×D404—41no—
52D4×D101—22no—
53D4×D101—42no—
54Dic404C2311yescharacteristic
55#13108064—22no—
56#13108063—21no—
57#13108061—22no—
58#13108061—22no—
59D808—21no—
60D808C2311yescharacteristic
61D808C2312yes—
62#13108059—22no—
63#13108058D411yescharacteristic
64#13108057D412yes—
65C8×D101C2312yes—
66#13108049C2311yescharacteristic
67#13108049C2311yescharacteristic
68#41×C101—21no—
69C2×Q8×C101—21no—
70Q16×C101—21no—
71Q16×C101C2312yes—
72Q16×C101C2311yescharacteristic
73SD16×C101—21no—
74D8×C101C2311yescharacteristic
75C2×C8×C101C2311yescharacteristic
76C16×C101C2312yes—
77C22×D101—41no—
78C22×D101—42no—
79Q8×C101—21no—
80Q8×C101—22no—
81Q8×C101—41no—
82D4×C101—21no—
83D4×C101—41no—
84C2×C4×C101—41no—
85C2×C4×C101—41no—
86C2×C4×C101C2×D411yescharacteristic
87C22⋊D101—41no—
88C22⋊D101—42no—
89C22⋊D101D812yes—
90D404—22no—
91D404—21no—
92D404—42no—
93D404—42no—
94D404—41no—
95D404D811yescharacteristic
96C4×D101—42no—
97C4×D101—41no—
98C4×D101—42no—
99C4×D101—42no—
100C4×D101C2×D412yes—
101Dic202—21no—
102Dic202—42no—
103Dic202D811yescharacteristic
104C8×C101C2×D411yescharacteristic
105C8×C101C2411yescharacteristic
106#11717579C2×D412yes—
107C22×C101—41no—
108C22×C101C2×D811yescharacteristic
109C2×D101—41no—
110C2×D101—42no—
111C2×D101—162no—
112C2×D101—161no—
113C2×D101C2×D812yes—
114C4×C101—42no—
115C4×C101—41no—
116C4×C101C2×D811yescharacteristic
117C4×C101C22×D411yescharacteristic
118Dic101—41no—
119Dic101C2×D812yes—
120C2×C101—21no—
121C2×C101—81no—
122C2×C101C22×D811yescharacteristic
123D101—22no—
124D101—81no—
125D101—82no—
126#3416—1011nomaximal, Sylow
127C101#341611yescharacteristic, complemented, Sylow
128#578—1011no—
129#576—1011no—
130#510—1012no—
131#510D10111yescharacteristic, complemented
132#509—1012no—
133#509—1011no—
134#508—1011no—
135#508—1012no—
136C2×SD32—1011no—
137C2×SD32—1012no—
138#504—1011no—
1392−1+4—2021no—
140H(5,𝔽2)—2021no—
141#137—2022no—
142#137—2021no—
143AGL(1,ℤ/8)—2022no—
144AGL(1,ℤ/8)—2021no—
145#135—1012no—
146#135—1012no—
147#135—1011no—
148#135C2×D10111yescharacteristic
149C2×Q16—1012no—
150C2×Q16C2×D10111yescharacteristic, complemented
151C2×D8—1012no—
152C2×D8—1011no—
153#131—1011no—
154Q32—1011no—
155Q32C2×D10112yescomplemented
156SD32—1012no—
157SD32—1012no—
158SD32—1011no—
159SD32—1011no—
160SD32—1012no—
161SD32C2×D10112yescomplemented
162D16—1011no—
163D16—1012no—
164M32—1012no—
165M32C2×D10111yescharacteristic, complemented
166C2×C16—1011no—
167C2×C16—1012no—
168#41—21no—
169#41—1011no—
170#41—4042no—
171#41—4041no—
172#41—4041no—
173#41—2022no—
174#41—2022no—
175#41—2021no—
176C2×Q8—21no—
177C2×Q8—4041no—
178C2×Q8—2022no—
179C2×D4—4041no—
180C2×D4—4042no—
181C2×D4—2022no—
182C2×D4—2021no—
183Q16—21no—
184Q16—1011no—
185Q16—2022no—
186Q16C22×D10112yes—
187Q16C22×D10111yescharacteristic
188SD16—21no—
189SD16—2022no—
190SD16—2022no—
191SD16—2021no—
192D8—1011no—
193D8—1012no—
194D8—2022no—
195D8—2021no—
196D8C22×D10111yescharacteristic
197M16—1011no—
198M16—1012no—
199C2×C8—1012no—
200C2×C8C22×D10111yescharacteristic
201C16—1011no—
202C16—1011no—
203C16C22×D10112yes—
204C23—4042no—
205C23—4041no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—1011no—
210Q8—4042no—
211Q8—2021no—
212D4—21no—
213D4—41no—
214D4—1012no—
215D4—1011no—
216D4—4042no—
217D4—4042no—
218D4—4042no—
219D4—4041no—
220D4—4041no—
221D4—2022no—
222D4—2021no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1012no—
226C2×C4—4042no—
227C2×C4—4042no—
228C2×C4—4042no—
229C2×C4—4041no—
230C2×C4D4×D10111yescharacteristic
231C8—1012no—
232C8D4×D10111yescharacteristic
233C8C23×D10111yescharacteristic
234C22—41no—
235C22—16161no—
236C22—16162no—
237C22—1012no—
238C22—4041no—
239C22—4042no—
240C22D8×D10111yescharacteristic
241C4—41no—
242C4—42no—
243C4—1012no—
244C4—4041no—
245C4D8×D10111yescharacteristic
246C4C2×D4×D10111yescharacteristic
247C2—21no—
248C2—81no—
249C2—8082no—
250C2—8081no—
251C2—2022no—
252C2C2×D8×D10111yescharacteristic
subgroupG/H
centerC2C2×D8×D101
commutator, cosocleC8×C101C24
FrattiniC8C23×D101
Fitting#510×C101C2
socleC2×C101C22×D8
hypercenter#510D101
hypocenterC101#3416
LCSG ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #20265990C2semidirect
2. #20265979C2semidirect
3. #20265619C2semidirect
4. #20265615C2semidirect
5. #20265614C2semidirect
6. #20265610C2semidirect
7. #20265607C2semidirect
8. C2×#15692198C2semidirect
9. SD32×D101C2semidirect
10. #20265595C2semidirect
11. #510×C101C2semidirect
19. #15692278C22semidirect
20. #15692277C22semidirect
21. Q16×D101C22semidirect
25. #15692201C22semidirect
26. #15692200C22semidirect
27. #15692199C22semidirect
28. #15692198C22semidirect
29. #15692198C22semidirect
30. #15692198C22semidirect
31. D1616C22semidirect
32. #15692196C22semidirect
33. #15692194C22semidirect
34. C2×#13108049C22semidirect
35. C16×D101C22semidirect
37. C2×Q16×C101C22semidirect
38. Q32×C101C22semidirect
39. SD32×C101C22semidirect
40. M32×C101C22semidirect
127. C101#3416semidirect
131. #510D101semidirect
150. C2×Q16C2×D101semidirect
155. Q32C2×D101semidirect
161. SD32C2×D101semidirect
165. M32C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659902
#202659792
#202656192
#202656152
#202656142
#202656102
#202656072
C2×#156921982
SD32×D1012
#202655952
#510×C1012
#3416101