Finite Group Databasebeta

← #486×D101 #27181435

Subgroups — #486×D101 #27181435

Order 12928.

528 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C4⋊Q8×D101C211yesmaximal, characteristic, complemented
2#127×D101C211yesmaximal, characteristic
3#20265525C211yesmaximal, characteristic, complemented
4#20265524C211yesmaximal, characteristic, complemented
5#20265523C211yesmaximal, characteristic, complemented
6#20265522C214yesmaximal, complemented
7#102×D101C214yesmaximal, complemented
8C4×C8×D101C211yesmaximal, characteristic, complemented
9#486×C101C211yesmaximal, characteristic, complemented
10C2×Q8×D101—21no—
11C2×D4×D101—21no—
12C2×D4×D101—21no—
13C2×D4×D101C2212yes—
14C2×C8×D101C2212yes—
15#15692255C2211yescharacteristic, complemented
16#15692254—21no—
17#15692254C2212yescomplemented
18#15692253—21no—
19#15692252C2211yescharacteristic
20#15692251—21no—
21#15692251—21no—
22#15692251C2212yes—
23C4⋊C4×D101—21no—
24C4⋊C4×D101C2212yes—
25C42×D101C2211yescharacteristic
26#15692177C2214yescomplemented
27#15692176C2214yescomplemented
28#15692175C2214yescomplemented
29C4×#11717579C2211yescharacteristic, complemented
30C8×Dic101C2212yescomplemented
31C4⋊Q8×C101C2211yescharacteristic, complemented
32#127×C101C2211yescharacteristic
33#102×C101C2214yescomplemented
34C4×C8×C101C2211yescharacteristic, complemented
35C23×D101—22no—
36C2×Dic202—21no—
37C2×Dic202—22no—
38Q8×D101—44no—
39C2×D404—21no—
40C2×D404—21no—
41C2×D404C2312yes—
42C2×C22⋊D101—22no—
43C2×C22⋊D101—22no—
44C2×C22⋊D101—22no—
45D4×D101—28no—
46D4×D101—44no—
47D4×D101—44no—
48C2×C4×D101—22no—
49C2×C4×D101D411yescharacteristic
50C2×C4×D101D411yescharacteristic
51C2×C4×D101C2311yescharacteristic
52C2×#11717579C2312yes—
53C8×D101—24no—
54C4⋊Dic101—21no—
55C4⋊Dic101C2312yes—
56#13108053—22no—
57#13108053—22no—
58C4×Dic101D411yescharacteristic
59C4×Dic101D411yescharacteristic
60C4×Dic101C2311yescharacteristic
61C2×Q8×C101—21no—
62C2×D4×C101—21no—
63C2×D4×C101—21no—
64C2×D4×C101C2312yes—
65C2×C8×C101C2312yes—
66C4⋊C4×C101—21no—
67C4⋊C4×C101C2312yes—
68C42×C101C2311yescharacteristic
69C23×C101—22no—
70C22×D101—22no—
71C22×D101—44no—
72C22×D101—44no—
73C22×D101—44no—
74C22×D101C2×D411yescharacteristic
75Q8×C101—42no—
76D4×C101—24no—
77D4×C101—42no—
78D4×C101—42no—
79C2×C4×C101—22no—
80C2×C4×C101C2×D411yescharacteristic
81C2×C4×C101C2×D411yescharacteristic
82C2×C4×C101C2411yescharacteristic
83C22⋊D101—44no—
84C22⋊D101—44no—
85C22⋊D101—44no—
86C2×Dic101—22no—
87C2×Dic101C2×D411yescharacteristic
88C2×Dic101C2×D411yescharacteristic
89C2×Dic101C2×D411yescharacteristic
90D404—24no—
91D404—42no—
92D404—42no—
93C4×D101—44no—
94C4×D101D814yes—
95C4×D101SD1614yes—
96C4×D101#4114yes—
97Dic202—42no—
98Dic202—44no—
99C8×C101—22no—
100#11717579—22no—
101C22×C101—42no—
102C22×C101—42no—
103C22×C101—42no—
104C22×C101C22×D411yescharacteristic
105C2×D101—42no—
106C2×D101—42no—
107C2×D101—42no—
108C2×D101—88no—
109C2×D101#12412yes—
110C2×D101C2×D812yes—
111C2×D101C2×SD1612yes—
112C4×C101—42no—
113C4×C101C2×D812yes—
114C4×C101C2×SD1612yes—
115C4×C101C2×#4112yes—
116Dic101—42no—
117Dic101#12412yes—
118Dic101C2×D812yes—
119Dic101C2×SD1612yes—
120C2×C101—82no—
121C2×C101C2×#12411yescharacteristic
122C2×C101C22×D811yescharacteristic
123C2×C101C22×SD1611yescharacteristic
124D101—82no—
125D101#48614yesdirect product factor
126C2×#486—1011nomaximal, Sylow
127C101C2×#48611yescharacteristic, complemented, Sylow
128C2×C4⋊Q8—1011no—
129C2×#127—1011no—
130#486—1011no—
131#486—1011no—
132#486—1014no—
133#486—1011no—
134#486D10111yescharacteristic, direct product factor
135C2×#102—1014no—
136C2×C4×C8—1011no—
137C22×Q8—2021no—
138C22×D4—1012no—
139C22×D4—2021no—
140C22×D4—2021no—
141C22×C8—1012no—
142C4⋊Q8—1012no—
143C4⋊Q8—1011no—
144C4⋊Q8—2021no—
145C4⋊Q8—2021no—
146C4⋊Q8C2×D10111yescharacteristic, complemented
147#127—1012no—
148#127—1011no—
149#127—2021no—
150#127—2021no—
151#127C2×D10111yescharacteristic
152C2×C4⋊C4—1012no—
153C2×C4⋊C4—2021no—
154C2×C42—1011no—
155#102—1014no—
156#102—1014no—
157#102—1014no—
158#102C2×D10114yescomplemented
159C4×C8—1012no—
160C4×C8—1011no—
161C4×C8C2×D10111yescharacteristic, complemented
162C24—2022no—
163C2×Q8—21no—
164C2×Q8—4044no—
165C2×Q8—2022no—
166C2×Q8—2021no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—1012no—
170C2×D4—4044no—
171C2×D4—4044no—
172C2×D4—2028no—
173C2×D4—2022no—
174C2×D4—2022no—
175C2×D4—2022no—
176C2×D4—2021no—
177C2×D4—2021no—
178C2×D4C22×D10112yes—
179C22×C4—1011no—
180C22×C4—1011no—
181C22×C4—1011no—
182C22×C4—2022no—
183C2×C8—1012no—
184C2×C8—2024no—
185C2×C8C22×D10112yes—
186C4⋊C4—21no—
187C4⋊C4—1012no—
188C4⋊C4—2022no—
189C4⋊C4—2022no—
190C4⋊C4—2021no—
191C4⋊C4C22×D10112yes—
192C42—1011no—
193C42—1011no—
194C42—1011no—
195C42C22×D10111yescharacteristic
196C23—22no—
197C23—1011no—
198C23—4044no—
199C23—4044no—
200C23—4044no—
201C23—2022no—
202Q8—42no—
203Q8—4044no—
204Q8—4042no—
205D4—24no—
206D4—42no—
207D4—42no—
208D4—4044no—
209D4—4044no—
210D4—4044no—
211D4—4042no—
212D4—4042no—
213D4—2024no—
214C2×C4—22no—
215C2×C4—1011no—
216C2×C4—1011no—
217C2×C4—1011no—
218C2×C4—1014no—
219C2×C4—1014no—
220C2×C4—1014no—
221C2×C4—4044no—
222C2×C4—2022no—
223C2×C4D4×D10111yescharacteristic
224C2×C4D4×D10111yescharacteristic
225C2×C4C23×D10111yescharacteristic
226C8—22no—
227C8—2022no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—8088no—
232C22—1012no—
233C22—1012no—
234C22—1012no—
235C22—4042no—
236C22—4042no—
237C22—4042no—
238C22C2×D4×D10111yescharacteristic
239C4—42no—
240C4—1012no—
241C4—1012no—
242C4—1012no—
243C4—4042no—
244C4D8×D10112yes—
245C4SD16×D10112yes—
246C4#41×D10112yes—
247C2—82no—
248C2—8082no—
249C2—1014no—
250C2#124×D10111yescharacteristic
251C2C2×D8×D10111yescharacteristic
252C2C2×SD16×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#486×C101C2
socleC22×C101C22×D4
prime radical#127×D101C2
hypercenter#486D101
hypocenterC101C2×#486
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D101C2semidirect
3. #20265525C2semidirect
4. #20265524C2semidirect
5. #20265523C2semidirect
6. #20265522C2semidirect
7. #102×D101C2semidirect
8. C4×C8×D101C2semidirect
9. #486×C101C2semidirect
15. #15692255C22semidirect
17. #15692254C22semidirect
26. #15692177C22semidirect
27. #15692176C22semidirect
28. #15692175C22semidirect
29. C4×#11717579C22semidirect
30. C8×Dic101C22semidirect
31. C4⋊Q8×C101C22semidirect
33. #102×C101C22semidirect
34. C4×C8×C101C22semidirect
125. D101#486direct (N × H)
127. C101C2×#486semidirect
134. #486D101direct (N × H)
146. C4⋊Q8C2×D101semidirect
158. #102C2×D101semidirect
161. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1012
#127×D1012
#202655252
#202655242
#202655232
#202655222
#102×D1012
C4×C8×D1012
#486×C1012
C2×#486101