Finite Group Databasebeta

← #27182752

Subgroups — #27182752

Order 12928.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15692280C212yesmaximal, complemented
2D4×D404C211yesmaximal, characteristic, complemented
3#20265553C211yesmaximal, characteristic, complemented
4#20265428C212yesmaximal, complemented
5#20265377C211yesmaximal, characteristic, complemented
6#20265369C211yesmaximal, characteristic, complemented
7#20265322C212yesmaximal, complemented
8#20265310C212yesmaximal, complemented
9C4×#13108061C211yesmaximal, characteristic, complemented
10#20265255C211yesmaximal, characteristic, complemented
11#550×C101C211yesmaximal, characteristic, complemented
12C22×D404C2212yes—
13C2×D4×D101—21no—
14#15692280—28no—
15C2×#13108061C2211yescharacteristic
16C2×#13108061C2212yescomplemented
17C2×#13108059C2212yes—
18C2×#13108059C2212yescomplemented
19C2×#13108059C2212yescomplemented
20C2×#13108058C2212yescomplemented
21#15692252C2211yescharacteristic
22#15692230—21no—
23#15692229C2212yes—
24#15692225—22no—
25C4×D404C2211yescharacteristic
26#15692186C2211yescharacteristic, complemented
27C4⋊#11717579C2211yescharacteristic, complemented
28#15692179C2211yescharacteristic, complemented
29#15692176C2212yescomplemented
30#15692175C2211yescharacteristic
31#15692175C2212yescomplemented
32C22⋊#11717579C2212yescomplemented
33C4×#11717579C2211yescharacteristic, complemented
34C2×#41×C101C2212yescomplemented
35#127×C101—21no—
36#127×C101C2211yescharacteristic, complemented
37#121×C101—22no—
38#121×C101C2212yescomplemented
39C4×Q8×C101C2211yescharacteristic, complemented
40C4×D4×C101—21no—
41C4×D4×C101C2211yescharacteristic
42C23×D101—22no—
43C2×D404—24no—
44C2×D404—21no—
45C2×D404—22no—
46C2×D404—21no—
47C2×D404—42no—
48C2×D404C2312yes—
49C2×D404C2311yescharacteristic
50C2×C22⋊D101—22no—
51D4×D101—44no—
52C2×C4×D101—21no—
53#13108061—24no—
54#13108061D414yescomplemented
55#13108059—24no—
56#13108059—24no—
57#13108059—42no—
58#13108058—24no—
59C2×#11717579C2311yescharacteristic
60C2×#11717579C2312yes—
61C2×#11717579C2311yescharacteristic
62C4⋊Dic101C2311yescharacteristic
63#13108051—21no—
64#13108051—22no—
65#41×C101—24no—
66#41×C101—42no—
67C2×Q8×C101C2311yescharacteristic
68C2×D4×C101—22no—
69C2×D4×C101—22no—
70C2×D4×C101—21no—
71C2×D4×C101—21no—
72C2×D4×C101D411yescharacteristic
73C2×D4×C101C2312yes—
74C22×C4×C101—22no—
75C22×C4×C101C2312yes—
76C4⋊C4×C101—21no—
77C4⋊C4×C101D411yescharacteristic
78C4⋊C4×C101C2311yescharacteristic
79C22⋊C4×C101—22no—
80C22⋊C4×C101D412yes—
81C42×C101—21no—
82C42×C101C2311yescharacteristic
83C23×C101—22no—
84C23×C101C2×D412yes—
85C22×D101—21no—
86C22×D101—22no—
87C22×D101—44no—
88C22×D101—44no—
89C22×D101—82no—
90Q8×C101—21no—
91Q8×C101C2×D412yescomplemented
92D4×C101—22no—
93D4×C101—22no—
94D4×C101—22no—
95D4×C101—44no—
96D4×C101—42no—
97D4×C101—81no—
98D4×C101—82no—
99C2×C4×C101—21no—
100C2×C4×C101—22no—
101C2×C4×C101—22no—
102C2×C4×C101—21no—
103C2×C4×C101—44no—
104C2×C4×C101C2×D411yescharacteristic
105C2×C4×C101C2×D412yes—
106C2×C4×C101C2×D411yescharacteristic
107C2×C4×C101C2411yescharacteristic
108C22⋊D101—44no—
109C2×Dic101—21no—
110D404—22no—
111D404—21no—
112D404—22no—
113D404—44no—
114D404—42no—
115D404—81no—
116D404C2×D412yes—
117C4×D101—42no—
118#11717579—22no—
119#11717579—21no—
120#11717579C2×D412yescomplemented
121C22×C101—22no—
122C22×C101—22no—
123C22×C101—22no—
124C22×C101—42no—
125C22×C101—42no—
126C22×C101—42no—
127C22×C101C22×D411yescharacteristic
128C2×D101—22no—
129C2×D101—41no—
130C2×D101—41no—
131C2×D101—42no—
132C2×D101—42no—
133C2×D101—42no—
134C2×D101—88no—
135C4×C101—22no—
136C4×C101—22no—
137C4×C101—21no—
138C4×C101—41no—
139C4×C101AGL(1,ℤ/8)12yescomplemented
140C4×C101C22×D411yescharacteristic
141C4×C101H(5,𝔽2)11yescharacteristic
142Dic101—41no—
143C2×C101—42no—
144C2×C101—82no—
145C2×C101D4211yescharacteristic
146C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C101#57611yescharacteristic
148D101—42no—
149D101—82no—
150#3274—1011nomaximal, Sylow
151C101#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1012no—
153#550D10111yescharacteristic, complemented
154D42—1011no—
155#493—1011no—
156#469—1012no—
157#460—1011no—
158#459—1011no—
159#449—1012no—
160#447—1012no—
161C4×SD16—1011no—
162#436—1011no—
163C2×#41C2×D10112yescomplemented
164C22×D4—1012no—
165C22×D4—2021no—
166AGL(1,ℤ/8)—2028no—
167C2×SD16—1011no—
168C2×SD16—1012no—
169C2×D8—1012no—
170C2×D8—1012no—
171C2×D8—1012no—
172C2×M16—1012no—
173#127—21no—
174#127—1011no—
175#127C2×D10111yescharacteristic, complemented
176#121—22no—
177#121—1012no—
178#121—2021no—
179#121C2×D10112yescomplemented
180#120—2022no—
181C4×Q8C2×D10111yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—1011no—
184C4×D4C2×D10111yescharacteristic
185#106—1011no—
186C4⋊C8—1011no—
187#103—1011no—
188#102—1012no—
189#102—1011no—
190#102—1012no—
191C22⋊C8—1012no—
192C4×C8—1011no—
193C24—2022no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D10111yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—1011no—
202C2×D4—1012no—
203C2×D4—4042no—
204C2×D4—4044no—
205C2×D4—2024no—
206C2×D4—2022no—
207C2×D4—2021no—
208C2×D4—2022no—
209C2×D4—2021no—
210C2×D4C22⋊D10111yescharacteristic
211C2×D4C22×D10112yes—
212C22×C4—22no—
213C22×C4—2021no—
214C22×C4C22×D10112yes—
215SD16—1014no—
216SD16—2024no—
217D8—4042no—
218D8—2024no—
219D8—2024no—
220M16—2024no—
221C2×C8—1011no—
222C2×C8—1011no—
223C2×C8—1012no—
224C4⋊C4—21no—
225C4⋊C4—1011no—
226C4⋊C4C22⋊D10111yescharacteristic
227C4⋊C4C22×D10111yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—2021no—
230C22⋊C4—2022no—
231C22⋊C4C22⋊D10112yes—
232C42—21no—
233C42C22×D10111yescharacteristic
234C23—22no—
235C23—8082no—
236C23—4044no—
237C23—4044no—
238C23—2022no—
239C23—2021no—
240C23C2×C22⋊D10112yes—
241Q8—21no—
242Q8D4×D10112yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—8081no—
251D4—1012no—
252D4—4044no—
253D4—4044no—
254D4—4042no—
255D4—2022no—
256D4—2022no—
257D4—2021no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—4042no—
264C2×C4—2021no—
265C2×C4C2×C22⋊D10111yescharacteristic
266C2×C4C2×C22⋊D10112yes—
267C2×C4C2×C22⋊D10111yescharacteristic
268C2×C4C23×D10111yescharacteristic
269C8—1012no—
270C8—2022no—
271C8—2021no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—8088no—
279C22—4042no—
280C22—4042no—
281C22—4042no—
282C22—4041no—
283C22—4041no—
284C22—2022no—
285C22C22×C22⋊D10111yescharacteristic
286C4—22no—
287C4—22no—
288C4—21no—
289C4—41no—
290C4—4041no—
291C4#1569228012yescomplemented
292C4C2×D4×D10111yescharacteristic
293C4#1569230011yescharacteristic
294C2—42no—
295C2—82no—
296C2—8082no—
297C2—4042no—
298C2D4×C22⋊D10111yescharacteristic
299C2C2×#1569228011yescharacteristic
300C2#2026598011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#550×C101C2
socleC22×C101C22×D4
hypercenter#550D101
hypocenterC101#3274
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. D4×D404C2semidirect
3. #20265553C2semidirect
4. #20265428C2semidirect
5. #20265377C2semidirect
6. #20265369C2semidirect
7. #20265322C2semidirect
8. #20265310C2semidirect
9. C4×#13108061C2semidirect
10. #20265255C2semidirect
11. #550×C101C2semidirect
16. C2×#13108061C22semidirect
18. C2×#13108059C22semidirect
19. C2×#13108059C22semidirect
20. C2×#13108058C22semidirect
26. #15692186C22semidirect
27. C4⋊#11717579C22semidirect
28. #15692179C22semidirect
29. #15692176C22semidirect
31. #15692175C22semidirect
32. C22⋊#11717579C22semidirect
33. C4×#11717579C22semidirect
34. C2×#41×C101C22semidirect
36. #127×C101C22semidirect
38. #121×C101C22semidirect
39. C4×Q8×C101C22semidirect
54. #13108061D4semidirect
91. Q8×C101C2×D4semidirect
120. #11717579C2×D4semidirect
139. C4×C101AGL(1,ℤ/8)semidirect
151. C101#3274semidirect
153. #550D101semidirect
163. C2×#41C2×D101semidirect
175. #127C2×D101semidirect
179. #121C2×D101semidirect
181. C4×Q8C2×D101semidirect
242. Q8D4×D101semidirect
291. C4#15692280semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
D4×D4042
#202655532
#202654282
#202653772
#202653692
#202653222
#202653102
C4×#131080612
#202652552
#550×C1012
#3274101