Finite Group Databasebeta

← #27182926

Subgroups — #27182926

Order 12928.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265989C212yesmaximal, complemented
2#20265984C212yesmaximal, complemented
3#20265550C211yesmaximal, characteristic, complemented
4#20265441C212yesmaximal, complemented
5#20265440C212yesmaximal, complemented
6#20265361C212yesmaximal, complemented
7#20265346C212yesmaximal, complemented
8#20265295C211yesmaximal, characteristic, complemented
9#443×C101C211yesmaximal, characteristic, complemented
10#15692302C2212yes—
11#15692301C2212yescomplemented
12#15692288—22no—
13#15692288C2212yescomplemented
14#15692287—22no—
15#15692285C2212yescomplemented
16#15692283—22no—
17#15692283C2212yes—
18#15692280C2212yes—
19#15692279—22no—
20#15692277—22no—
21#15692277C2211yescharacteristic
22#15692276—22no—
23C2×#13108063—22no—
24C2×#13108063C2212yescomplemented
25C2×#13108063C2212yescomplemented
26SD16×D101—22no—
27#15692261C2212yescomplemented
28#15692255C2211yescharacteristic, complemented
29#15692252C2211yescharacteristic
30#15692191C2211yescharacteristic, complemented
31#15692183C2212yescomplemented
32#15692173C2212yescomplemented
33#15692171C2212yescomplemented
34#135×C101C2211yescharacteristic
35#131×C101C2212yescomplemented
36#108×C101C2211yescharacteristic, complemented
37#104×C101C2212yescomplemented
38C4×C8×C101C2211yescharacteristic, complemented
39#13108075—22no—
40#13108074—22no—
41#13108074—22no—
42#13108074C2312yes—
43#13108073—22no—
44#13108073—42no—
45C2×Dic202—22no—
46C2×Dic202C2312yes—
47Q8×D101—22no—
48C2×D404—22no—
49C2×D404—41no—
50C2×D404C2312yes—
51D4×D101—22no—
52D4×D101—42no—
53Dic404—22no—
54Dic404D411yescharacteristic, complemented
55#13108064—22no—
56#13108063—22no—
57#13108063—22no—
58#13108063—22no—
59#13108063—22no—
60#13108063—22no—
61#13108063—22no—
62#13108063D412yescomplemented
63#13108062—22no—
64#13108061—22no—
65D808—22no—
66D808D411yescharacteristic, complemented
67#13108059—22no—
68#13108058C2312yes—
69#13108057—22no—
70C8×D101—22no—
71C4⋊Dic101—41no—
72#41×C101C2312yes—
73Q16×C101D411yescharacteristic, complemented
74SD16×C101D412yescomplemented
75D8×C101D411yescharacteristic, complemented
76M16×C101—22no—
77M16×C101C2312yes—
78C2×C8×C101—22no—
79C2×C8×C101C2311yescharacteristic
80C2×C8×C101C2311yescharacteristic
81C42×C101C2311yescharacteristic
82C22×D101—22no—
83C22×D101—42no—
84Q8×C101C2×D412yes—
85D4×C101—22no—
86D4×C101C2×D412yes—
87C2×C4×C101—21no—
88C2×C4×C101—22no—
89C2×C4×C101C2411yescharacteristic
90C22⋊D101—22no—
91C22⋊D101—42no—
92C22⋊D101—42no—
93C2×Dic101—22no—
94C2×Dic101—42no—
95D404—22no—
96D404—22no—
97D404—42no—
98D404—82no—
99D404C2×D412yes—
100C4×D101—22no—
101C4×D101—42no—
102C4×D101—42no—
103Dic202—22no—
104Dic202—22no—
105Dic202—42no—
106Dic202C2×D412yes—
107C8×C101—22no—
108C8×C101C2×D411yescharacteristic
109C8×C101C2×D411yescharacteristic
110C8×C101C2×D411yescharacteristic
111C8×C101C2×D411yescharacteristic
112#11717579—22no—
113C22×C101—22no—
114C22×C101H(5,𝔽2)11yescharacteristic
115C2×D101—22no—
116C2×D101—42no—
117C2×D101—82no—
118C2×D101—162no—
119C4×C101—22no—
120C4×C101—22no—
121C4×C101C22×D411yescharacteristic
122C4×C101C22×D411yescharacteristic
123Dic101—22no—
124Dic101—42no—
125C2×C101—21no—
126C2×C101—42no—
127C2×C101D4211yescharacteristic
128D101—42no—
129D101—82no—
130#3291—1011nomaximal, Sylow
131C101#329111yescharacteristic, complemented, Sylow
132#577—1012no—
133#577—1012no—
134#492—1011no—
135#471—1012no—
136#471—1012no—
137#456—1012no—
138#453—1012no—
139#443—1011no—
140#443D10111yescharacteristic, complemented
1412−1+4—1012no—
142H(5,𝔽2)—1012no—
143#137—1012no—
144#137—1012no—
145#137—2022no—
146#137—2022no—
147AGL(1,ℤ/8)—1012no—
148AGL(1,ℤ/8)—1012no—
149AGL(1,ℤ/8)—2022no—
150AGL(1,ℤ/8)—2022no—
151#135—1011no—
152#135—2022no—
153#135—2022no—
154#135C2×D10111yescharacteristic
155C2×SD16—1012no—
156C2×SD16—1012no—
157C2×SD16—2022no—
158C2×SD16—2022no—
159#131—1012no—
160#131C2×D10112yescomplemented
161C4⋊Q8—1011no—
162#127—1011no—
163#108—1011no—
164#108C2×D10111yescharacteristic, complemented
165#104—1012no—
166#104C2×D10112yescomplemented
167#101—1012no—
168#100—1012no—
169C4×C8C2×D10111yescharacteristic, complemented
170#41—1012no—
171#41—4042no—
172#41—2022no—
173#41—2022no—
174#41—2022no—
175#41—2022no—
176#41C22×D10112yes—
177C2×Q8—1012no—
178C2×Q8—2022no—
179C2×Q8—2022no—
180C2×D4—1012no—
181C2×D4—4041no—
182C2×D4—4042no—
183C2×D4—2022no—
184C2×D4—2022no—
185Q16—1011no—
186Q16—2022no—
187Q16—2022no—
188Q16D40411yescharacteristic, complemented
189SD16—1012no—
190SD16—2022no—
191SD16—2022no—
192SD16—2022no—
193SD16—2022no—
194SD16—2022no—
195SD16—2022no—
196SD16—2022no—
197SD16—2022no—
198SD16D40412yescomplemented
199D8—1011no—
200D8—2022no—
201D8—2022no—
202D8D40411yescharacteristic, complemented
203M16—22no—
204M16—1012no—
205M16—2022no—
206M16C22×D10112yes—
207C2×C8—22no—
208C2×C8—2022no—
209C2×C8C22×D10111yescharacteristic
210C2×C8C22×D10111yescharacteristic
211C4⋊C4—4041no—
212C42C22×D10111yescharacteristic
213C23—4042no—
214C23—2022no—
215Q8—1012no—
216Q8—4042no—
217Q8—2022no—
218Q8—2022no—
219Q8C2×D40412yes—
220D4—22no—
221D4—8082no—
222D4—1012no—
223D4—4042no—
224D4—4042no—
225D4—4042no—
226D4—2022no—
227D4—2022no—
228D4—2022no—
229D4C2×D40412yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—4042no—
233C2×C4—4042no—
234C2×C4—4042no—
235C2×C4—2022no—
236C2×C4—2022no—
237C2×C4C23×D10111yescharacteristic
238C8—22no—
239C8—2022no—
240C8D4×D10111yescharacteristic
241C8D4×D10111yescharacteristic
242C8C2×D40411yescharacteristic
243C8C2×D40411yescharacteristic
244C22—22no—
245C22—8082no—
246C22—16162no—
247C22—4042no—
248C22—2022no—
249C22#1569230111yescharacteristic
250C4—22no—
251C4—22no—
252C4—4042no—
253C4—2022no—
254C4C2×D4×D10111yescharacteristic
255C4C22×D40411yescharacteristic
256C2—21no—
257C2—42no—
258C2—8082no—
259C2—4042no—
260C2D4×D40411yescharacteristic
subgroupG/H
centerC2D4×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#443×C101C2
socleC2×C101D42
hypercenter#443D101
hypocenterC101#3291
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #20265989C2semidirect
2. #20265984C2semidirect
3. #20265550C2semidirect
4. #20265441C2semidirect
5. #20265440C2semidirect
6. #20265361C2semidirect
7. #20265346C2semidirect
8. #20265295C2semidirect
9. #443×C101C2semidirect
11. #15692301C22semidirect
13. #15692288C22semidirect
15. #15692285C22semidirect
24. C2×#13108063C22semidirect
25. C2×#13108063C22semidirect
27. #15692261C22semidirect
28. #15692255C22semidirect
30. #15692191C22semidirect
31. #15692183C22semidirect
32. #15692173C22semidirect
33. #15692171C22semidirect
35. #131×C101C22semidirect
36. #108×C101C22semidirect
37. #104×C101C22semidirect
38. C4×C8×C101C22semidirect
54. Dic404D4semidirect
62. #13108063D4semidirect
66. D808D4semidirect
73. Q16×C101D4semidirect
74. SD16×C101D4semidirect
75. D8×C101D4semidirect
131. C101#3291semidirect
140. #443D101semidirect
160. #131C2×D101semidirect
164. #108C2×D101semidirect
166. #104C2×D101semidirect
169. C4×C8C2×D101semidirect
188. Q16D404semidirect
198. SD16D404semidirect
202. D8D404semidirect

Maximal subgroups

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

subgroupindex
#202659892
#202659842
#202655502
#202654412
#202654402
#202653612
#202653462
#202652952
#443×C1012
#3291101