Finite Group Databasebeta

← #27182727

Subgroups — #27182727

Order 12928.

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

normal subgroups only

This group has 546 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#20265831C211yesmaximal, characteristic, complemented
3#20265812C211yesmaximal, characteristic, complemented
4#20265549C211yesmaximal, characteristic, complemented
5#20265422C212yesmaximal, complemented
6#20265383C211yesmaximal, characteristic, complemented
7#20265329C212yesmaximal, complemented
8#20265319C212yesmaximal, complemented
9C4×#13108059C211yesmaximal, characteristic, complemented
10#20265254C211yesmaximal, characteristic, complemented
11#459×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2212yescomplemented
13C2×D4×D101—21no—
14C2×D4×D101C2212yes—
15#15692279—28no—
16C2×#13108063C2212yes—
17C2×#13108062C2212yescomplemented
18C2×#13108062C2212yescomplemented
19C2×#13108059C2211yescharacteristic
20C2×#13108057C2212yescomplemented
21#15692255C2211yescharacteristic, complemented
22#15692239—22no—
23#15692235C2212yescomplemented
24#15692229—21no—
25#15692226—21no—
26#15692226C2212yes—
27#15692223—22no—
28C4×D404C2211yescharacteristic
29C4×D404C2211yescharacteristic
30C2×C4⋊Dic101—21no—
31#15692188C2211yescharacteristic, complemented
32#15692180C2212yescomplemented
33#15692176C2211yescharacteristic, complemented
34#15692175C2211yescharacteristic
35#15692167C2212yescomplemented
36C4×#11717579C2211yescharacteristic, complemented
37C2×D8×C101C2212yescomplemented
38#127×C101C2211yescharacteristic
39C4×D4×C101C2211yescharacteristic, complemented
40C4⋊C8×C101C2211yescharacteristic, complemented
41#102×C101C2212yescomplemented
42C23×D101—22no—
43#13108073—24no—
44#13108073—42no—
45C2×Dic202C2312yes—
46C2×D404D411yescharacteristic
47C2×D404C2311yescharacteristic
48C2×C22⋊D101—22no—
49C2×C22⋊D101—22no—
50C2×C22⋊D101—22no—
51D4×D101—24no—
52D4×D101—42no—
53D4×D101—44no—
54C22×Dic101—22no—
55C2×C4×D101—21no—
56C2×C4×D101C2312yes—
57#13108063—24no—
58#13108062—24no—
59#13108062—42no—
60#13108059D414yescomplemented
61#13108057—24no—
62C2×#11717579C2311yescharacteristic
63C2×#11717579C2311yescharacteristic
64C4⋊Dic101—21no—
65C4⋊Dic101—21no—
66C4⋊Dic101—22no—
67C4⋊Dic101D411yescharacteristic
68C4⋊Dic101C2311yescharacteristic
69C22⋊Dic101—22no—
70#13108051—22no—
71#13108051—21no—
72#13108051D412yes—
73C2×D4×C101—21no—
74C2×D4×C101—21no—
75C2×D4×C101C2312yes—
76C2×D4×C101C2311yescharacteristic
77C22×C4×C101—21no—
78D8×C101—24no—
79C2×C8×C101C2312yes—
80C4⋊C4×C101C2311yescharacteristic
81C22⋊C4×C101—21no—
82C42×C101C2311yescharacteristic
83C23×C101—21no—
84C23×C101—22no—
85C22×D101—21no—
86C22×D101—44no—
87C22×D101—44no—
88C22×D101—82no—
89C22×D101C2×D412yes—
90D4×C101—22no—
91D4×C101—21no—
92D4×C101—22no—
93D4×C101—42no—
94D4×C101—81no—
95D4×C101C2×D412yescomplemented
96C2×C4×C101—21no—
97C2×C4×C101—42no—
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101C2411yescharacteristic
101C22⋊D101—44no—
102C22⋊D101—44no—
103C22⋊D101—44no—
104C2×Dic101—21no—
105C2×Dic101—22no—
106C2×Dic101—44no—
107C2×Dic101C2×D412yes—
108D404—22no—
109D404—21no—
110D404C2×D412yes—
111C4×D101—22no—
112C4×D101—22no—
113C4×D101—42no—
114Dic202—22no—
115Dic202—22no—
116C8×C101—22no—
117#11717579—21no—
118#11717579C2×D412yescomplemented
119C22×C101—22no—
120C22×C101—41no—
121C22×C101—41no—
122C22×C101—42no—
123C22×C101—42no—
124C22×C101—42no—
125C22×C101C22×D411yescharacteristic
126C2×D101—22no—
127C2×D101—22no—
128C2×D101—22no—
129C2×D101—22no—
130C2×D101—41no—
131C2×D101—41no—
132C2×D101—88no—
133C4×C101—21no—
134C4×C101—41no—
135C4×C101AGL(1,ℤ/8)12yescomplemented
136C4×C101C22×D411yescharacteristic
137C4×C101H(5,𝔽2)11yescharacteristic
138Dic101—22no—
139Dic101—42no—
140Dic101—41no—
141C2×C101—42no—
142C2×C101—82no—
143C2×C101D4211yescharacteristic
144C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C101#57711yescharacteristic
146D101—42no—
147D101—42no—
148#3272—1011nomaximal, Sylow
149C101#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1012no—
151#547—1011no—
152D42—1011no—
153#492—1011no—
154#468—1012no—
155#461—1011no—
156#459D10111yescharacteristic, complemented
157#450—1012no—
158#449—1012no—
159C4×D8—1011no—
160#436—1011no—
161C2×#41—1012no—
162C22×D4—1012no—
163C22×D4—2021no—
164AGL(1,ℤ/8)—2028no—
165C2×SD16—1012no—
166C2×SD16—1012no—
167C2×SD16—1012no—
168C2×D8—1011no—
169C2×D8C2×D10112yescomplemented
170C2×M16—1012no—
171C4⋊Q8—1011no—
172#127C2×D10111yescharacteristic
173#123—2022no—
174C22⋊Q8—1012no—
175#121—1012no—
176#121—2021no—
177#121—2021no—
178#120—2022no—
179C4×D4—1011no—
180C4×D4—1011no—
181C4×D4C2×D10111yescharacteristic, complemented
182C2×C4⋊C4—2021no—
183#107—1011no—
184C4⋊C8C2×D10111yescharacteristic, complemented
185#103—1012no—
186#102—1011no—
187#102—1011no—
188#102C2×D10112yescomplemented
189C22⋊C8—1012no—
190C4×C8—1011no—
191C24—2022no—
192#41—4042no—
193#41—2024no—
194C2×Q8—1012no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—1011no—
198C2×D4—1011no—
199C2×D4—4044no—
200C2×D4—4042no—
201C2×D4—2022no—
202C2×D4—2024no—
203C2×D4—2022no—
204C2×D4—2022no—
205C2×D4C22×D10112yes—
206C2×D4C22×D10111yescharacteristic
207C22×C4—21no—
208C22×C4—1012no—
209C22×C4—2022no—
210C22×C4—2021no—
211SD16—4042no—
212SD16—2024no—
213SD16—2024no—
214D8—24no—
215D8—1014no—
216M16—2024no—
217C2×C8—1011no—
218C2×C8—1011no—
219C2×C8C22×D10112yes—
220C4⋊C4—1011no—
221C4⋊C4—1011no—
222C4⋊C4—2022no—
223C4⋊C4—2021no—
224C4⋊C4—2021no—
225C4⋊C4C22×D10111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1012no—
228C22⋊C4—2022no—
229C22⋊C4—2022no—
230C22⋊C4—2021no—
231C42C22×D10111yescharacteristic
232C23—21no—
233C23—22no—
234C23—8082no—
235C23—1012no—
236C23—4044no—
237C23—4044no—
238C23—2021no—
239Q8—2022no—
240Q8—2022no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—42no—
245D4—81no—
246D4—1012no—
247D4—4044no—
248D4—4044no—
249D4—4044no—
250D4—2021no—
251D4—2022no—
252D4D4×D10112yescomplemented
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—1012no—
256C2×C4—4044no—
257C2×C4—4042no—
258C2×C4—2021no—
259C2×C4—2022no—
260C2×C4—2022no—
261C2×C4—2022no—
262C2×C4D4×D10111yescharacteristic
263C2×C4D4×D10111yescharacteristic
264C2×C4C23×D10111yescharacteristic
265C8—22no—
266C8—1012no—
267C8—2021no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—8088no—
275C22—4041no—
276C22—4041no—
277C22—2022no—
278C22—2022no—
279C22—2022no—
280C22—2022no—
281C22C2×D4×D10111yescharacteristic
282C4—21no—
283C4—41no—
284C4—4042no—
285C4—4041no—
286C4—2022no—
287C4#1569227912yescomplemented
288C4C2×D4×D10111yescharacteristic
289C4#1569230111yescharacteristic
290C2—42no—
291C2—82no—
292C2—4042no—
293C2—4042no—
294C2#2026581111yescharacteristic
295C2C2×#1569227911yescharacteristic
296C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#459×C101C2
socleC22×C101C22×D4
hypercenter#459D101
hypocenterC101#3272
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. #20265831C2semidirect
3. #20265812C2semidirect
4. #20265549C2semidirect
5. #20265422C2semidirect
6. #20265383C2semidirect
7. #20265329C2semidirect
8. #20265319C2semidirect
9. C4×#13108059C2semidirect
10. #20265254C2semidirect
11. #459×C101C2semidirect
12. C2×#13108073C22semidirect
17. C2×#13108062C22semidirect
18. C2×#13108062C22semidirect
20. C2×#13108057C22semidirect
21. #15692255C22semidirect
23. #15692235C22semidirect
31. #15692188C22semidirect
32. #15692180C22semidirect
33. #15692176C22semidirect
35. #15692167C22semidirect
36. C4×#11717579C22semidirect
37. C2×D8×C101C22semidirect
39. C4×D4×C101C22semidirect
40. C4⋊C8×C101C22semidirect
41. #102×C101C22semidirect
60. #13108059D4semidirect
95. D4×C101C2×D4semidirect
118. #11717579C2×D4semidirect
135. C4×C101AGL(1,ℤ/8)semidirect
149. C101#3272semidirect
156. #459D101semidirect
169. C2×D8C2×D101semidirect
181. C4×D4C2×D101semidirect
184. C4⋊C8C2×D101semidirect
188. #102C2×D101semidirect
252. D4D4×D101semidirect
287. C4#15692279semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
#202658312
#202658122
#202655492
#202654222
#202653832
#202653292
#202653192
C4×#131080592
#202652542
#459×C1012
#3272101