Finite Group Databasebeta

← #27184278

Subgroups — #27184278

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#20265991C211yesmaximal, characteristic, complemented
2#20265978C211yesmaximal, characteristic, complemented
3#20265620C211yesmaximal, characteristic, complemented
4#20265616C212yesmaximal, complemented
5#20265611C212yesmaximal, complemented
6#20265607C211yesmaximal, characteristic, complemented
7#20265606C212yesmaximal, complemented
8C2×#15692199C211yesmaximal, characteristic, complemented
9SD32×D101C212yesmaximal, complemented
10#20265595C211yesmaximal, characteristic, complemented
11#509×C101C211yesmaximal, characteristic, complemented
12#15692302—21no—
13#15692300—21no—
14#15692285—21no—
15#15692284—22no—
16#15692281—21no—
17#15692279—22no—
18#15692277C2211yescharacteristic
19#15692274C2212yes—
20#15692274C2212yescomplemented
21C2×Dic404C2211yescharacteristic, complemented
22Q16×D101C2212yescomplemented
23D8×D101C2212yes—
24#15692261C2211yescharacteristic
25Dic808C2212yescomplemented
26#15692201C2211yescharacteristic, complemented
27#15692200C2212yescomplemented
28#15692199C2212yescomplemented
29#15692199C2212yescomplemented
30#15692199C2211yescharacteristic, complemented
31#15692198C2211yescharacteristic, complemented
32#15692196C2211yescharacteristic, complemented
33#15692194C2212yescomplemented
34C2×#13108049C2211yescharacteristic, complemented
35C16×D101C2212yescomplemented
36#135×C101C2211yescharacteristic, complemented
37C2×D8×C101C2211yescharacteristic
38SD32×C101C2212yescomplemented
39D16×C101C2212yescomplemented
40M32×C101C2211yescharacteristic, complemented
41#13108074—21no—
42#13108074—41no—
43#13108074D411yescharacteristic
44#13108073—22no—
45#13108073—22no—
46#13108073—42no—
47#13108073—41no—
48C2×Dic202—21no—
49C2×Dic202—41no—
50Q8×D101—22no—
51C2×C22⋊D101—42no—
52D4×D101—22no—
53D4×D101—41no—
54Dic404—21no—
55Dic404C2312yes—
56Dic404C2311yescharacteristic
57#13108064—22no—
58#13108063—21no—
59#13108062—22no—
60#13108062—22no—
61D808C2311yescharacteristic
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—42no—
79Q8×C101—21no—
80D4×C101—22no—
81D4×C101—21no—
82D4×C101—41no—
83D4×C101—41no—
84C2×C4×C101—41no—
85C2×C4×C101C2×D411yescharacteristic
86C22⋊D101—41no—
87C22⋊D101—42no—
88C22⋊D101—42no—
89C22⋊D101—42no—
90C22⋊D101D812yes—
91C2×Dic101—41no—
92C2×Dic101—42no—
93C2×Dic101—42no—
94D404—21no—
95D404D811yescharacteristic
96C4×D101—42no—
97C4×D101—41no—
98C4×D101C2×D412yes—
99Dic202—21no—
100Dic202—22no—
101Dic202—41no—
102Dic202—42no—
103Dic202D811yescharacteristic
104C8×C101C2×D411yescharacteristic
105C8×C101C2411yescharacteristic
106#11717579C2×D412yes—
107C22×C101—41no—
108C22×C101—42no—
109C22×C101—161no—
110C22×C101C2×D811yescharacteristic
111C2×D101—41no—
112C2×D101—162no—
113C2×D101C2×D812yes—
114C4×C101—41no—
115C4×C101C2×D811yescharacteristic
116C4×C101C22×D411yescharacteristic
117Dic101—42no—
118Dic101—41no—
119Dic101C2×D812yes—
120C2×C101—21no—
121C2×C101—82no—
122C2×C101—81no—
123C2×C101C22×D811yescharacteristic
124D101—22no—
125D101—81no—
126#3416—1011nomaximal, Sylow
127C101#341611yescharacteristic, complemented, Sylow
128#578—1011no—
129#576—1011no—
130#510—1012no—
131#510—1011no—
132#509—1012no—
133#509D10111yescharacteristic, complemented
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, complemented
149C2×Q16—1012no—
150C2×Q16—1011no—
151C2×D8—1012no—
152C2×D8C2×D10111yescharacteristic
153#131—1011no—
154Q32—1011no—
155Q32—1012no—
156SD32—1011no—
157SD32—1012no—
158SD32—1012no—
159SD32—1011no—
160SD32—1012no—
161SD32C2×D10112yescomplemented
162D16—1011no—
163D16C2×D10112yescomplemented
164M32—1012no—
165M32C2×D10111yescharacteristic, complemented
166C2×C16—1011no—
167C2×C16—1012no—
168#41—21no—
169#41—1011no—
170#41—4041no—
171#41—4042no—
172#41—4041no—
173#41—2022no—
174#41—2022no—
175#41—2021no—
176C2×Q8—4041no—
177C2×Q8—2022no—
178C2×Q8—2021no—
179C2×D4—21no—
180C2×D4—4041no—
181C2×D4—4042no—
182C2×D4—2022no—
183Q16—1012no—
184Q16—1011no—
185Q16—2022no—
186Q16—2021no—
187Q16C22×D10111yescharacteristic
188SD16—21no—
189SD16—2022no—
190SD16—2022no—
191SD16—2021no—
192D8—21no—
193D8—1011no—
194D8—2022no—
195D8C22×D10111yescharacteristic
196D8C22×D10112yes—
197M16—1011no—
198M16—1012no—
199C2×C8—1012no—
200C2×C8C22×D10111yescharacteristic
201C16—1011no—
202C16—1011no—
203C16C22×D10112yes—
204C23—41no—
205C23—4042no—
206Q8—21no—
207Q8—1011no—
208Q8—4042no—
209Q8—4041no—
210Q8—2021no—
211Q8—2022no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—1012no—
217D4—1011no—
218D4—4042no—
219D4—4042no—
220D4—4042no—
221D4—4041no—
222D4—2021no—
223C2×C4—41no—
224C2×C4—1012no—
225C2×C4—4042no—
226C2×C4—4042no—
227C2×C4—4042no—
228C2×C4—4041no—
229C2×C4—4041no—
230C2×C4D4×D10111yescharacteristic
231C8—1012no—
232C8D4×D10111yescharacteristic
233C8C23×D10111yescharacteristic
234C22—41no—
235C22—42no—
236C22—161no—
237C22—16162no—
238C22—1012no—
239C22—4041no—
240C22D8×D10111yescharacteristic
241C4—41no—
242C4—1012no—
243C4—4041no—
244C4—4042no—
245C4D8×D10111yescharacteristic
246C4C2×D4×D10111yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—8081no—
251C2—2022no—
252C2C2×D8×D10111yescharacteristic
subgroupG/H
centerC2C2×D8×D101
commutator, cosocleC8×C101C24
FrattiniC8C23×D101
Fitting#509×C101C2
socleC2×C101C22×D8
hypercenter#509D101
hypocenterC101#3416
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. #20265991C2semidirect
2. #20265978C2semidirect
3. #20265620C2semidirect
4. #20265616C2semidirect
5. #20265611C2semidirect
6. #20265607C2semidirect
7. #20265606C2semidirect
8. C2×#15692199C2semidirect
9. SD32×D101C2semidirect
10. #20265595C2semidirect
11. #509×C101C2semidirect
20. #15692274C22semidirect
21. C2×Dic404C22semidirect
22. Q16×D101C22semidirect
25. Dic808C22semidirect
26. #15692201C22semidirect
27. #15692200C22semidirect
28. #15692199C22semidirect
29. #15692199C22semidirect
30. #15692199C22semidirect
31. #15692198C22semidirect
32. #15692196C22semidirect
33. #15692194C22semidirect
34. C2×#13108049C22semidirect
35. C16×D101C22semidirect
36. #135×C101C22semidirect
38. SD32×C101C22semidirect
39. D16×C101C22semidirect
40. M32×C101C22semidirect
127. C101#3416semidirect
133. #509D101semidirect
148. #135C2×D101semidirect
161. SD32C2×D101semidirect
163. D16C2×D101semidirect
165. M32C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659912
#202659782
#202656202
#202656162
#202656112
#202656072
#202656062
C2×#156921992
SD32×D1012
#202655952
#509×C1012
#3416101