Finite Group Databasebeta

← #480×D101 #27181000

Subgroups — #480×D101 #27181000

Order 12928.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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#121×D101C211yesmaximal, characteristic
2C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
3#20265492C212yesmaximal, complemented
4#20265491C211yesmaximal, characteristic, complemented
5#20265490C211yesmaximal, characteristic, complemented
6#20265489C212yesmaximal, complemented
7#20265488C211yesmaximal, characteristic, complemented
8#107×D101C212yesmaximal, complemented
9#102×D101C212yesmaximal, complemented
10C22⋊C8×D101C211yesmaximal, characteristic, complemented
11#480×C101C211yesmaximal, characteristic, complemented
12C2×D4×D101—21no—
13C2×D4×D101C2211yescharacteristic
14C22×C4×D101—21no—
15C22×C4×D101C2211yescharacteristic
16C2×C8×D101C2212yes—
17#15692230C2211yescharacteristic
18#15692229C2211yescharacteristic
19#15692228—21no—
20#15692227—21no—
21#15692226C2211yescharacteristic
22C2×C4⋊Dic101C2211yescharacteristic, complemented
23C2×#13108053—21no—
24C4⋊C4×D101—21no—
25C4⋊C4×D101—21no—
26C4⋊C4×D101C2212yes—
27C4⋊C4×D101C2211yescharacteristic
28C4⋊C4×D101C2212yescomplemented
29C22⋊C4×D101—21no—
30#15692189C2212yescomplemented
31#15692188C2212yescomplemented
32#15692188C2212yescomplemented
33#15692177C2212yescomplemented
34#15692176C2212yescomplemented
35#15692175C2212yescomplemented
36C22⋊#11717579C2211yescharacteristic, complemented
37#15692167C2212yescomplemented
38#121×C101C2211yescharacteristic
39C2×C4⋊C4×C101C2211yescharacteristic, complemented
40#107×C101C2212yescomplemented
41#102×C101C2212yescomplemented
42C22⋊C8×C101C2211yescharacteristic, complemented
43C23×D101—21no—
44C23×D101D411yescharacteristic
45C2×D404—21no—
46C2×D404C2311yescharacteristic
47C2×C22⋊D101—21no—
48C2×C22⋊D101—21no—
49C2×C22⋊D101—21no—
50D4×D101—24no—
51D4×D101—44no—
52C22×Dic101—21no—
53C22×Dic101D411yescharacteristic
54C2×C4×D101—24no—
55C2×C4×D101—22no—
56C2×C4×D101—22no—
57C2×C4×D101—21no—
58C2×C4×D101—44no—
59C2×C4×D101D411yescharacteristic
60C2×C4×D101D411yescharacteristic
61C2×C4×D101C2311yescharacteristic
62C2×C4×D101C2311yescharacteristic
63C2×#11717579C2312yes—
64C8×D101—24no—
65C4⋊Dic101—21no—
66C4⋊Dic101C2312yes—
67C4⋊Dic101C2311yescharacteristic
68#13108053—22no—
69#13108053—22no—
70#13108053—21no—
71C22⋊Dic101—21no—
72#13108051—21no—
73#13108051—21no—
74C2×D4×C101—21no—
75C2×D4×C101C2311yescharacteristic
76C22×C4×C101—21no—
77C22×C4×C101C2311yescharacteristic
78C2×C8×C101C2312yes—
79C4⋊C4×C101—21no—
80C4⋊C4×C101C2312yes—
81C4⋊C4×C101C2311yescharacteristic
82C22⋊C4×C101—21no—
83C23×C101—21no—
84C23×C101C2×D411yescharacteristic
85C22×D101—22no—
86C22×D101—22no—
87C22×D101—21no—
88C22×D101—42no—
89C22×D101—42no—
90C22×D101—42no—
91C22×D101D814yes—
92C22×D101C2×D411yescharacteristic
93C22×D101C2×D411yescharacteristic
94D4×C101—22no—
95D4×C101—42no—
96C2×C4×C101—22no—
97C2×C4×C101—21no—
98C2×C4×C101—22no—
99C2×C4×C101—42no—
100C2×C4×C101C2×D411yescharacteristic
101C2×C4×C101C2411yescharacteristic
102C22⋊D101—42no—
103C22⋊D101—42no—
104C22⋊D101—42no—
105C2×Dic101—22no—
106C2×Dic101—21no—
107C2×Dic101—22no—
108C2×Dic101—42no—
109C2×Dic101C2×D411yescharacteristic
110C2×Dic101C2×D411yescharacteristic
111D404—22no—
112D404—42no—
113C4×D101—22no—
114C4×D101—22no—
115C4×D101—22no—
116C4×D101—44no—
117C4×D101—44no—
118C4×D101—42no—
119C4×D101#4114yes—
120C8×C101—22no—
121#11717579—22no—
122C22×C101—21no—
123C22×C101—21no—
124C22×C101—41no—
125C22×C101—41no—
126C22×C101—41no—
127C22×C101C2×D812yes—
128C22×C101C22×D411yescharacteristic
129C2×D101—21no—
130C2×D101—21no—
131C2×D101—28no—
132C2×D101—41no—
133C2×D101—41no—
134C2×D101—41no—
135C2×D101—84no—
136C2×D101#12312yes—
137C2×D101C2×D812yes—
138C2×D101C2×D812yes—
139C2×D101#13712yes—
140C4×C101—21no—
141C4×C101—42no—
142C4×C101—41no—
143C4×C101C2×#4112yes—
144Dic101—21no—
145Dic101—42no—
146Dic101—41no—
147Dic101#12312yes—
148C2×C101—22no—
149C2×C101—81no—
150C2×C101C2×#12311yescharacteristic
151C2×C101C22×D811yescharacteristic
152C2×C101C2×#13711yescharacteristic
153D101—22no—
154D101—81no—
155D101#48014yesdirect product factor
156C2×#480—1011nomaximal, Sylow
157C101C2×#48011yescharacteristic, complemented, Sylow
158C2×#121—1011no—
159C22×C4⋊C4—1011no—
160#480—1011no—
161#480—1011no—
162#480—1012no—
163#480—1012no—
164#480—1011no—
165#480D10111yescharacteristic, direct product factor
166C2×#107—1012no—
167C2×#102—1012no—
168C2×C22⋊C8—1011no—
169C22×D4—1011no—
170C22×D4—2021no—
171C23×C4—1011no—
172C23×C4—2021no—
173C22×C8—1012no—
174#121—1011no—
175#121—1011no—
176#121—1011no—
177#121—2021no—
178#121—2021no—
179#121C2×D10111yescharacteristic
180C2×C4⋊C4—1011no—
181C2×C4⋊C4—1012no—
182C2×C4⋊C4—1012no—
183C2×C4⋊C4—1011no—
184C2×C4⋊C4—2021no—
185C2×C4⋊C4—2021no—
186C2×C4⋊C4—2021no—
187C2×C4⋊C4C2×D10111yescharacteristic, complemented
188C2×C22⋊C4—2021no—
189#107—1012no—
190#107—1012no—
191#107—1012no—
192#107C2×D10112yescomplemented
193#102—1012no—
194#102—1012no—
195#102—1012no—
196#102C2×D10112yescomplemented
197C22⋊C8—1011no—
198C22⋊C8—1012no—
199C22⋊C8C2×D10111yescharacteristic, complemented
200C24—1011no—
201C24—2021no—
202C2×D4—21no—
203C2×D4—1011no—
204C2×D4—4044no—
205C2×D4—2021no—
206C2×D4—2024no—
207C2×D4—2021no—
208C2×D4—2021no—
209C2×D4—2021no—
210C2×D4C22×D10111yescharacteristic
211C22×C4—21no—
212C22×C4—1011no—
213C22×C4—1011no—
214C22×C4—1011no—
215C22×C4—1011no—
216C22×C4—1011no—
217C22×C4—4044no—
218C22×C4—2024no—
219C22×C4—2021no—
220C22×C4—2022no—
221C22×C4—2022no—
222C22×C4—2021no—
223C22×C4C22×D10111yescharacteristic
224C2×C8—1012no—
225C2×C8—2024no—
226C2×C8C22×D10112yes—
227C4⋊C4—21no—
228C4⋊C4—1012no—
229C4⋊C4—1011no—
230C4⋊C4—2022no—
231C4⋊C4—2022no—
232C4⋊C4—2021no—
233C4⋊C4—2021no—
234C4⋊C4C22×D10112yes—
235C4⋊C4C22×D10111yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—2021no—
238C22⋊C4—2021no—
239C22⋊C4—2021no—
240C23—21no—
241C23—1011no—
242C23—1011no—
243C23—1014no—
244C23—4042no—
245C23—4042no—
246C23—4042no—
247C23—2022no—
248C23—2022no—
249C23—2021no—
250C23D4×D10111yescharacteristic
251D4—22no—
252D4—42no—
253D4—4042no—
254D4—4042no—
255D4—4042no—
256D4—4042no—
257D4—2022no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—1011no—
263C2×C4—1014no—
264C2×C4—1011no—
265C2×C4—4044no—
266C2×C4—4044no—
267C2×C4—4042no—
268C2×C4—4042no—
269C2×C4—2022no—
270C2×C4—2022no—
271C2×C4—2022no—
272C2×C4—2021no—
273C2×C4—2022no—
274C2×C4—2022no—
275C2×C4D4×D10111yescharacteristic
276C2×C4C23×D10111yescharacteristic
277C8—22no—
278C8—2022no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—8084no—
285C22—1012no—
286C22—1012no—
287C22—1012no—
288C22—1012no—
289C22—4041no—
290C22—4041no—
291C22—4041no—
292C22—2028no—
293C22—2021no—
294C22—2021no—
295C22D8×D10112yes—
296C22C2×D4×D10111yescharacteristic
297C4—21no—
298C4—42no—
299C4—41no—
300C4—1012no—
301C4—4042no—
302C4—4041no—
303C4—2021no—
304C4#41×D10112yes—
305C2—22no—
306C2—81no—
307C2—8081no—
308C2—1014no—
309C2—2022no—
310C2#123×D10111yescharacteristic
311C2C2×D8×D10111yescharacteristic
312C2#137×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#480×C101C2
socleC22×C101C22×D4
prime radical#121×D101C2
hypercenter#480D101
hypocenterC101C2×#480
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D101C2semidirect
3. #20265492C2semidirect
4. #20265491C2semidirect
5. #20265490C2semidirect
6. #20265489C2semidirect
7. #20265488C2semidirect
8. #107×D101C2semidirect
9. #102×D101C2semidirect
10. C22⋊C8×D101C2semidirect
11. #480×C101C2semidirect
22. C2×C4⋊Dic101C22semidirect
28. C4⋊C4×D101C22semidirect
30. #15692189C22semidirect
31. #15692188C22semidirect
32. #15692188C22semidirect
33. #15692177C22semidirect
34. #15692176C22semidirect
35. #15692175C22semidirect
36. C22⋊#11717579C22semidirect
37. #15692167C22semidirect
39. C2×C4⋊C4×C101C22semidirect
40. #107×C101C22semidirect
41. #102×C101C22semidirect
42. C22⋊C8×C101C22semidirect
155. D101#480direct (N × H)
157. C101C2×#480semidirect
165. #480D101direct (N × H)
187. C2×C4⋊C4C2×D101semidirect
192. #107C2×D101semidirect
196. #102C2×D101semidirect
199. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#121×D1012
C2×C4⋊C4×D1012
#202654922
#202654912
#202654902
#202654892
#202654882
#107×D1012
#102×D1012
C22⋊C8×D1012
#480×C1012
C2×#480101