Finite Group Databasebeta

← #27182751

Subgroups — #27182751

Order 12928.

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

normal subgroups only

This group has 556 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
1C2×#15692279C212yesmaximal, complemented
2#20265847C211yesmaximal, characteristic, complemented
3#20265812C211yesmaximal, characteristic, complemented
4#20265427C212yesmaximal, complemented
5#20265374C211yesmaximal, characteristic, complemented
6#20265369C211yesmaximal, characteristic, complemented
7#20265319C212yesmaximal, complemented
8#20265309C212yesmaximal, complemented
9C4×#13108063C211yesmaximal, characteristic, complemented
10#20265256C211yesmaximal, characteristic, complemented
11#493×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2212yescomplemented
13C2×D4×D101—21no—
14C2×D4×D101C2212yes—
15#15692279—28no—
16C2×#13108063C2211yescharacteristic
17C2×#13108062C2212yescomplemented
18C2×#13108059C2212yes—
19C2×#13108057C2212yescomplemented
20#15692251—21no—
21#15692227—22no—
22#15692226—21no—
23#15692226C2212yes—
24#15692226C2212yescomplemented
25#15692223—22no—
26C4×Dic202C2211yescharacteristic, complemented
27C4×D404C2211yescharacteristic
28C4×D404C2211yescharacteristic
29D4×Dic101—21no—
30#15692187C2211yescharacteristic, complemented
31C4⋊#11717579C2211yescharacteristic, complemented
32#15692180C2211yescharacteristic, complemented
33#15692177C2211yescharacteristic
34#15692176C2212yescomplemented
35#15692176C2212yescomplemented
36#15692167C2212yescomplemented
37C2×D8×C101C2212yescomplemented
38C2×D8×C101C2212yescomplemented
39#127×C101C2211yescharacteristic
40#127×C101C2211yescharacteristic, complemented
41C4×C8×C101C2211yescharacteristic, complemented
42C23×D101—22no—
43#13108073—24no—
44#13108073—42no—
45C2×Dic202C2311yescharacteristic
46C2×D404D411yescharacteristic
47C2×D404C2311yescharacteristic
48C2×C22⋊D101—22no—
49C2×C22⋊D101—22no—
50C2×C22⋊D101—22no—
51C2×C22⋊D101—22no—
52D4×D101—24no—
53D4×D101—42no—
54D4×D101—44no—
55C22×Dic101—22no—
56C2×C4×D101—21no—
57C2×C4×D101C2312yes—
58#13108063D414yescomplemented
59#13108062—24no—
60#13108059—24no—
61#13108057—24no—
62C2×#11717579C2312yes—
63C4⋊Dic101D411yescharacteristic
64C4⋊Dic101C2311yescharacteristic
65C4⋊Dic101C2311yescharacteristic
66#13108053—21no—
67C22⋊Dic101—22no—
68C22⋊Dic101—22no—
69#13108051—21no—
70#13108051D412yes—
71C4×Dic101—21no—
72C2×D4×C101—21no—
73C2×D4×C101—21no—
74C2×D4×C101—21no—
75C2×D4×C101—21no—
76C2×D4×C101C2312yes—
77C2×D4×C101C2312yes—
78D8×C101—24no—
79D8×C101—42no—
80C2×C8×C101C2311yescharacteristic
81C2×C8×C101C2311yescharacteristic
82C42×C101C2311yescharacteristic
83C23×C101—22no—
84C23×C101—22no—
85C22×D101—21no—
86C22×D101—44no—
87C22×D101—44no—
88C22×D101—82no—
89C22×D101C2×D412yes—
90D4×C101—22no—
91D4×C101—22no—
92D4×C101—22no—
93D4×C101—22no—
94D4×C101—42no—
95D4×C101—42no—
96D4×C101—81no—
97D4×C101—81no—
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101C2411yescharacteristic
101C22⋊D101—44no—
102C22⋊D101—44no—
103C22⋊D101—44no—
104C22⋊D101—82no—
105C2×Dic101—21no—
106C2×Dic101—21no—
107C2×Dic101—21no—
108C2×Dic101—44no—
109C2×Dic101C2×D412yes—
110D404—22no—
111D404—21no—
112D404C2×D412yes—
113C4×D101—22no—
114C4×D101—22no—
115C4×D101—42no—
116Dic202—21no—
117Dic202C2×D412yescomplemented
118C8×C101—21no—
119C8×C101C2×D412yescomplemented
120#11717579—22no—
121C22×C101—42no—
122C22×C101—42no—
123C22×C101—42no—
124C22×C101—42no—
125C22×C101—42no—
126C22×C101—42no—
127C22×C101C22×D411yescharacteristic
128C2×D101—22no—
129C2×D101—22no—
130C2×D101—22no—
131C2×D101—22no—
132C2×D101—41no—
133C2×D101—41no—
134C2×D101—88no—
135C4×C101—21no—
136C4×C101AGL(1,ℤ/8)12yescomplemented
137C4×C101C22×D411yescharacteristic
138C4×C101H(5,𝔽2)11yescharacteristic
139Dic101—22no—
140Dic101—22no—
141Dic101—41no—
142Dic101—41no—
143C2×C101—82no—
144C2×C101—82no—
145C2×C101D4211yescharacteristic
146C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C101#57611yescharacteristic
148D101—42no—
149D101—42no—
150#3274—1011nomaximal, Sylow
151C101#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1012no—
153#550—1011no—
154D42—1011no—
155#493D10111yescharacteristic, complemented
156#469—1012no—
157#460—1011no—
158#459—1011no—
159#449—1012no—
160#447—1012no—
161C4×SD16—1011no—
162#436—1011no—
163C2×#41—1012no—
164C22×D4—1012no—
165C22×D4—2021no—
166AGL(1,ℤ/8)—2028no—
167C2×SD16—1012no—
168C2×SD16—1011no—
169C2×D8—1012no—
170C2×D8C2×D10112yescomplemented
171C2×D8C2×D10112yescomplemented
172C2×M16—1012no—
173#127—2021no—
174#127C2×D10111yescharacteristic
175#127C2×D10111yescharacteristic, complemented
176#121—1012no—
177#121—1012no—
178#121—2022no—
179#121—2021no—
180#120—2022no—
181C4×Q8—1011no—
182C4×D4—1011no—
183C4×D4—1011no—
184C4×D4—2021no—
185#106—1011no—
186C4⋊C8—1011no—
187#103—1011no—
188#102—1012no—
189#102—1012no—
190#102—1011no—
191C22⋊C8—1012no—
192C4×C8C2×D10111yescharacteristic, complemented
193C24—2022no—
194#41—4042no—
195#41—2024no—
196C2×Q8—1011no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—1011no—
202C2×D4—1011no—
203C2×D4—4044no—
204C2×D4—4042no—
205C2×D4—2022no—
206C2×D4—2024no—
207C2×D4—2022no—
208C2×D4—2022no—
209C2×D4—2022no—
210C2×D4C22×D10112yes—
211C2×D4C22×D10112yes—
212C22×C4—1012no—
213C22×C4—2022no—
214C22×C4—2021no—
215SD16—1014no—
216SD16—2024no—
217D8—24no—
218D8—42no—
219D8—2024no—
220M16—2024no—
221C2×C8—1012no—
222C2×C8C22×D10111yescharacteristic
223C2×C8C22×D10111yescharacteristic
224C4⋊C4—1011no—
225C4⋊C4—1011no—
226C4⋊C4—1011no—
227C4⋊C4—2021no—
228C22⋊C4—1012no—
229C22⋊C4—2022no—
230C22⋊C4—2022no—
231C22⋊C4—2021no—
232C42—2021no—
233C42C22×D10111yescharacteristic
234C23—22no—
235C23—22no—
236C23—8082no—
237C23—1012no—
238C23—4044no—
239C23—4044no—
240C23—2021no—
241Q8—1012no—
242Q8—2021no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—8082no—
252D4—1012no—
253D4—4044no—
254D4—4044no—
255D4—4044no—
256D4—2021no—
257D4—2022no—
258C2×C4—1012no—
259C2×C4—4042no—
260C2×C4—4044no—
261C2×C4—2021no—
262C2×C4—2021no—
263C2×C4—2021no—
264C2×C4—2022no—
265C2×C4—2022no—
266C2×C4D4×D10111yescharacteristic
267C2×C4D4×D10111yescharacteristic
268C2×C4C23×D10111yescharacteristic
269C8—21no—
270C8—2022no—
271C8D4×D10112yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—8088no—
279C22—4041no—
280C22—4041no—
281C22—2022no—
282C22—2022no—
283C22—2022no—
284C22—2022no—
285C22C2×D4×D10111yescharacteristic
286C4—21no—
287C4—4041no—
288C4—4041no—
289C4—2022no—
290C4—2022no—
291C4#1569227912yescomplemented
292C4C2×D4×D10111yescharacteristic
293C4#1569230011yescharacteristic
294C2—82no—
295C2—82no—
296C2—4042no—
297C2—4042no—
298C2#2026581211yescharacteristic
299C2C2×#1569227911yescharacteristic
300C2#2026597811yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#493×C101C2
socleC22×C101C22×D4
hypercenter#493D101
hypocenterC101#3274
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. #20265847C2semidirect
3. #20265812C2semidirect
4. #20265427C2semidirect
5. #20265374C2semidirect
6. #20265369C2semidirect
7. #20265319C2semidirect
8. #20265309C2semidirect
9. C4×#13108063C2semidirect
10. #20265256C2semidirect
11. #493×C101C2semidirect
12. C2×#13108073C22semidirect
17. C2×#13108062C22semidirect
19. C2×#13108057C22semidirect
24. #15692226C22semidirect
26. C4×Dic202C22semidirect
30. #15692187C22semidirect
31. C4⋊#11717579C22semidirect
32. #15692180C22semidirect
34. #15692176C22semidirect
35. #15692176C22semidirect
36. #15692167C22semidirect
37. C2×D8×C101C22semidirect
38. C2×D8×C101C22semidirect
40. #127×C101C22semidirect
41. C4×C8×C101C22semidirect
58. #13108063D4semidirect
117. Dic202C2×D4semidirect
119. C8×C101C2×D4semidirect
136. C4×C101AGL(1,ℤ/8)semidirect
151. C101#3274semidirect
155. #493D101semidirect
170. C2×D8C2×D101semidirect
171. C2×D8C2×D101semidirect
175. #127C2×D101semidirect
192. C4×C8C2×D101semidirect
271. C8D4×D101semidirect
291. C4#15692279semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
#202658472
#202658122
#202654272
#202653742
#202653692
#202653192
#202653092
C4×#131080632
#202652562
#493×C1012
#3274101