Finite Group Databasebeta

← #27180607

Subgroups — #27180607

Order 12928.

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

normal subgroups only

This group has 584 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×#15692279C212yesmaximal, complemented
2#127×D101C211yesmaximal, characteristic, complemented
3C4×#13108073C211yesmaximal, characteristic, complemented
4#20265378C211yesmaximal, characteristic, complemented
5#20265376C212yesmaximal, complemented
6#20265374C211yesmaximal, characteristic, complemented
7#20265372C211yesmaximal, characteristic, complemented
8#20265370C212yesmaximal, complemented
9#20265199C211yesmaximal, characteristic, complemented
10#20265180C212yesmaximal, complemented
11#459×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2211yescharacteristic
13C2×D4×D101—21no—
14C2×D4×D101—21no—
15C2×D4×D101C2212yes—
16#15692279—28no—
17C2×#13108063C2212yescomplemented
18C2×#13108062C2212yescomplemented
19C2×#13108059C2212yescomplemented
20C2×#13108057C2212yes—
21#15692252C2211yescharacteristic
22#15692251—21no—
23#15692251—21no—
24#15692251C2212yes—
25C4×Dic202C2211yescharacteristic, complemented
26#15692220C2211yescharacteristic, complemented
27D4×Dic101C2211yescharacteristic, complemented
28C4×C22⋊D101—21no—
29#15692215—21no—
30#15692213C2211yescharacteristic
31#15692211—21no—
32C2×C4×Dic101—21no—
33C42×D101C2211yescharacteristic
34C4⋊#11717579C2211yescharacteristic, complemented
35#15692184C2212yescomplemented
36#15692177C2212yescomplemented
37#15692176C2212yescomplemented
38#15692175C2212yescomplemented
39C2×D8×C101C2212yescomplemented
40#127×C101C2211yescharacteristic
41C4×D4×C101C2211yescharacteristic, complemented
42C4⋊C8×C101C2211yescharacteristic, complemented
43#102×C101C2212yescomplemented
44C23×D101—22no—
45#13108073—22no—
46#13108073D414yescomplemented
47C2×Dic202C2311yescharacteristic
48C2×D404—21no—
49C2×D404—21no—
50C2×D404C2312yes—
51C2×C22⋊D101—21no—
52C2×C22⋊D101—22no—
53C2×C22⋊D101—22no—
54C2×C22⋊D101—22no—
55D4×D101—24no—
56D4×D101—24no—
57D4×D101—44no—
58D4×D101—44no—
59C22×Dic101—21no—
60C22×Dic101—21no—
61C2×C4×D101D411yescharacteristic
62C2×C4×D101D411yescharacteristic
63C2×C4×D101C2311yescharacteristic
64#13108063—24no—
65#13108062—24no—
66#13108059—24no—
67#13108057—24no—
68C2×#11717579C2312yes—
69C4⋊Dic101C2311yescharacteristic
70#13108053—21no—
71#13108053—21no—
72C22⋊Dic101—21no—
73#13108051—21no—
74C4×Dic101—21no—
75C4×Dic101—21no—
76C4×Dic101—21no—
77C4×Dic101D411yescharacteristic
78C4×Dic101D411yescharacteristic
79C4×Dic101C2311yescharacteristic
80C2×D4×C101—21no—
81C2×D4×C101—21no—
82C2×D4×C101C2312yes—
83C2×D4×C101C2311yescharacteristic
84C22×C4×C101—21no—
85D8×C101—24no—
86C2×C8×C101C2312yes—
87C4⋊C4×C101C2311yescharacteristic
88C22⋊C4×C101—21no—
89C42×C101C2311yescharacteristic
90C23×C101—21no—
91C23×C101—22no—
92C22×D101—22no—
93C22×D101—44no—
94C22×D101—44no—
95C22×D101—44no—
96C22×D101C2×D411yescharacteristic
97D4×C101—22no—
98D4×C101—21no—
99D4×C101—22no—
100D4×C101—42no—
101D4×C101—42no—
102D4×C101C2×D412yescomplemented
103C2×C4×C101—21no—
104C2×C4×C101—42no—
105C2×C4×C101C2×D411yescharacteristic
106C2×C4×C101C2×D411yescharacteristic
107C2×C4×C101C2411yescharacteristic
108C22⋊D101—24no—
109C22⋊D101—44no—
110C22⋊D101—44no—
111C22⋊D101—44no—
112C2×Dic101—21no—
113C2×Dic101—24no—
114C2×Dic101—21no—
115C2×Dic101—42no—
116C2×Dic101C2×D411yescharacteristic
117C2×Dic101C2×D411yescharacteristic
118C2×Dic101C2×D411yescharacteristic
119D404—22no—
120D404—22no—
121D404—42no—
122D404—42no—
123C4×D101—22no—
124C4×D101—22no—
125C4×D101C2×D412yes—
126C4×D101#4112yes—
127Dic202—21no—
128Dic202C2×D412yescomplemented
129C8×C101—22no—
130#11717579—22no—
131C22×C101—22no—
132C22×C101—41no—
133C22×C101—41no—
134C22×C101—42no—
135C22×C101—42no—
136C22×C101—42no—
137C22×C101C22×D411yescharacteristic
138C2×D101—21no—
139C2×D101—21no—
140C2×D101—42no—
141C2×D101—42no—
142C2×D101—42no—
143C2×D101—88no—
144C2×D101#12112yes—
145C4×C101—21no—
146C4×C101—41no—
147C4×C101AGL(1,ℤ/8)12yescomplemented
148C4×C101C22×D411yescharacteristic
149C4×C101C2×#4111yescharacteristic
150Dic101—22no—
151Dic101—21no—
152Dic101—41no—
153Dic101#12112yes—
154Dic101AGL(1,ℤ/8)12yescomplemented
155C2×C101—42no—
156C2×C101—82no—
157C2×C101C2×#12111yescharacteristic
158C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
160D101—22no—
161D101—82no—
162#3038—1011nomaximal, Sylow
163C101#303811yescharacteristic, complemented, Sylow
164C2×AGL(1,ℤ/8)—1012no—
165C2×#127—1011no—
166C4×#41—1011no—
167#460—1011no—
168#460—1011no—
169#460—1012no—
170#459—1012no—
171#459—1011no—
172#459D10111yescharacteristic, complemented
173#423—1011no—
174#418—1012no—
175C2×#41—1011no—
176C22×D4—1012no—
177C22×D4—2021no—
178C22×D4—2021no—
179AGL(1,ℤ/8)—2028no—
180C2×SD16—1012no—
181C2×SD16—1012no—
182C2×D8—1012no—
183C2×D8C2×D10112yescomplemented
184C2×M16—1012no—
185#127—1012no—
186#127—1011no—
187#127—2021no—
188#127—2021no—
189#127C2×D10111yescharacteristic
190C4×Q8—1011no—
191C4×Q8—1011no—
192C4×D4—1011no—
193C4×D4—2021no—
194C4×D4—2021no—
195C4×D4C2×D10111yescharacteristic, complemented
196#117—1011no—
197#117—2021no—
198C2×C42—1011no—
199C2×C42—2021no—
200C4⋊C8—1012no—
201C4⋊C8—1011no—
202C4⋊C8C2×D10111yescharacteristic, complemented
203#102—1012no—
204#102—1012no—
205#102—1012no—
206#102C2×D10112yescomplemented
207C24—2022no—
208#41—1014no—
209#41—2022no—
210C2×Q8—1011no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—1012no—
214C2×D4—4044no—
215C2×D4—4044no—
216C2×D4—2024no—
217C2×D4—2024no—
218C2×D4—2022no—
219C2×D4—2022no—
220C2×D4—2022no—
221C2×D4—2021no—
222C2×D4—2021no—
223C2×D4—2021no—
224C2×D4C22×D10112yes—
225C2×D4C22×D10111yescharacteristic
226C22×C4—21no—
227C22×C4—1011no—
228C22×C4—1011no—
229C22×C4—1011no—
230C22×C4—2021no—
231C22×C4—2021no—
232SD16—2024no—
233SD16—2024no—
234D8—24no—
235D8—2024no—
236M16—2024no—
237C2×C8—1012no—
238C2×C8C22×D10112yes—
239C4⋊C4—1011no—
240C4⋊C4—2021no—
241C4⋊C4—2021no—
242C4⋊C4C22×D10111yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—2021no—
245C22⋊C4—2021no—
246C42—1011no—
247C42—1011no—
248C42—1011no—
249C42—2021no—
250C42—2021no—
251C42—2021no—
252C42C22×D10111yescharacteristic
253C23—21no—
254C23—22no—
255C23—1011no—
256C23—4044no—
257C23—4044no—
258C23—4044no—
259C23—2022no—
260Q8—1012no—
261Q8—2021no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—4042no—
268D4—4044no—
269D4—4044no—
270D4—4044no—
271D4—4042no—
272D4—2024no—
273D4—2022no—
274D4—2022no—
275D4D4×D10112yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—1011no—
279C2×C4—1011no—
280C2×C4—1012no—
281C2×C4—1012no—
282C2×C4—1011no—
283C2×C4—4042no—
284C2×C4—2022no—
285C2×C4—2024no—
286C2×C4—2021no—
287C2×C4—2021no—
288C2×C4—2022no—
289C2×C4D4×D10111yescharacteristic
290C2×C4D4×D10111yescharacteristic
291C2×C4C23×D10111yescharacteristic
292C8—22no—
293C8—2022no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—8088no—
301C22—1012no—
302C22—4042no—
303C22—4042no—
304C22—4042no—
305C22—2021no—
306C22—2021no—
307C22C2×D4×D10111yescharacteristic
308C4—21no—
309C4—41no—
310C4—1012no—
311C4—1012no—
312C4—4041no—
313C4—2022no—
314C4—2021no—
315C4#1569227912yescomplemented
316C4C2×D4×D10111yescharacteristic
317C4#41×D10111yescharacteristic
318C2—42no—
319C2—82no—
320C2—8082no—
321C2—2022no—
322C2#121×D10111yescharacteristic
323C2AGL(1,ℤ/8)×D10111yescharacteristic
324C2C2×#1569227911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#459×C101C2
socleC22×C101C22×D4
hypercenter#459D101
hypocenterC101#3038
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. #127×D101C2semidirect
3. C4×#13108073C2semidirect
4. #20265378C2semidirect
5. #20265376C2semidirect
6. #20265374C2semidirect
7. #20265372C2semidirect
8. #20265370C2semidirect
9. #20265199C2semidirect
10. #20265180C2semidirect
11. #459×C101C2semidirect
17. C2×#13108063C22semidirect
18. C2×#13108062C22semidirect
19. C2×#13108059C22semidirect
25. C4×Dic202C22semidirect
26. #15692220C22semidirect
27. D4×Dic101C22semidirect
34. C4⋊#11717579C22semidirect
35. #15692184C22semidirect
36. #15692177C22semidirect
37. #15692176C22semidirect
38. #15692175C22semidirect
39. C2×D8×C101C22semidirect
41. C4×D4×C101C22semidirect
42. C4⋊C8×C101C22semidirect
43. #102×C101C22semidirect
46. #13108073D4semidirect
102. D4×C101C2×D4semidirect
128. Dic202C2×D4semidirect
147. C4×C101AGL(1,ℤ/8)semidirect
154. Dic101AGL(1,ℤ/8)semidirect
163. C101#3038semidirect
172. #459D101semidirect
183. C2×D8C2×D101semidirect
195. C4×D4C2×D101semidirect
202. C4⋊C8C2×D101semidirect
206. #102C2×D101semidirect
275. D4D4×D101semidirect
315. C4#15692279semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
#127×D1012
C4×#131080732
#202653782
#202653762
#202653742
#202653722
#202653702
#202651992
#202651802
#459×C1012
#3038101