Finite Group Databasebeta

← #489×D101 #27181458

Subgroups — #489×D101 #27181458

Order 12928.

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

normal subgroups only

This group has 436 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
1C4⋊Q8×D101C211yesmaximal, characteristic, complemented
2#124×D101C211yesmaximal, characteristic
3#20265539C211yesmaximal, characteristic, complemented
4#20265538C212yesmaximal, complemented
5#20265537C211yesmaximal, characteristic, complemented
6#20265536C211yesmaximal, characteristic, complemented
7#20265535C212yesmaximal, complemented
8#103×D101C212yesmaximal, complemented
9#102×D101C212yesmaximal
10#97×D101C211yesmaximal, characteristic, complemented
11#489×C101C211yesmaximal, characteristic, complemented
12C2×Q8×D101—21no—
13C2×Q8×D101C2211yescharacteristic
14C2×D4×D101C2211yescharacteristic
15C2×C8×D101C2212yes—
16#15692255C2211yescharacteristic, complemented
17#15692254—21no—
18#15692254C2212yescomplemented
19#15692253—21no—
20#15692244C2211yescharacteristic
21#15692243C2211yescharacteristic
22#15692242C2211yescharacteristic
23#15692241—21no—
24#15692241—21no—
25C4⋊C4×D101—21no—
26C4⋊C4×D101C2212yes—
27C22⋊C4×D101—21no—
28C22⋊C4×D101—21no—
29C42×D101C2211yescharacteristic
30#15692180C2212yescomplemented
31#15692179C2212yescomplemented
32#15692178C2212yescomplemented
33#15692177C2212yes—
34#15692176C2212yes—
35#15692175C2212yes—
36#15692166C2211yescharacteristic, complemented
37#15692165C2212yescomplemented
38C4⋊Q8×C101C2211yescharacteristic, complemented
39#124×C101C2211yescharacteristic
40#103×C101C2212yescomplemented
41#102×C101C2212yes—
42#97×C101C2211yescharacteristic, complemented
43C23×D101—21no—
44C2×Dic202—21no—
45C2×Dic202—22no—
46C2×Dic202—21no—
47C2×Dic202C2311yescharacteristic
48Q8×D101—24no—
49Q8×D101—44no—
50C2×D404C2311yescharacteristic
51C2×C22⋊D101—21no—
52D4×D101—24no—
53C2×C4×D101—22no—
54C2×C4×D101—21no—
55C2×C4×D101D411yescharacteristic
56C2×C4×D101D411yescharacteristic
57C2×C4×D101C2311yescharacteristic
58C2×#11717579C2312yes—
59C8×D101—24no—
60C4⋊Dic101—21no—
61C4⋊Dic101C2312yes—
62#13108053—22no—
63#13108053—22no—
64C22⋊Dic101—21no—
65C22⋊Dic101—21no—
66#13108051—21no—
67#13108051—21no—
68#13108051—21no—
69#13108051—21no—
70C4×Dic101D411yescharacteristic
71C4×Dic101D411yescharacteristic
72C4×Dic101C2311yescharacteristic
73C2×Q8×C101—21no—
74C2×Q8×C101C2311yescharacteristic
75C2×D4×C101C2311yescharacteristic
76C2×C8×C101C2312yes—
77C4⋊C4×C101—21no—
78C4⋊C4×C101C2312yes—
79C22⋊C4×C101—21no—
80C22⋊C4×C101—21no—
81C42×C101C2311yescharacteristic
82C23×C101—21no—
83C22×D101—21no—
84C22×D101—42no—
85C22×D101—42no—
86C22×D101—42no—
87C22×D101C2×D411yescharacteristic
88Q8×C101—22no—
89Q8×C101—42no—
90D4×C101—22no—
91C2×C4×C101—21no—
92C2×C4×C101—22no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2411yescharacteristic
96C22⋊D101—42no—
97C2×Dic101—21no—
98C2×Dic101—22no—
99C2×Dic101C2×D411yescharacteristic
100C2×Dic101C2×D411yescharacteristic
101C2×Dic101C2×D411yescharacteristic
102D404—22no—
103C4×D101—22no—
104C4×D101—22no—
105C4×D101—42no—
106C4×D101—44no—
107C4×D101#4114yes—
108Dic202—22no—
109Dic202—44no—
110Dic202—42no—
111Dic202—42no—
112C8×C101—22no—
113#11717579—22no—
114C22×C101—41no—
115C22×C101—41no—
116C22×C101—41no—
117C22×C101C22×D411yescharacteristic
118C2×D101—41no—
119C2×D101—41no—
120C2×D101—41no—
121C2×D101—84no—
122C2×D101#12412yes—
123C2×D101AGL(1,ℤ/8)12yes—
124C2×D101#13712yes—
125C4×C101—21no—
126C4×C101—21no—
127C4×C101—41no—
128C4×C101—42no—
129C4×C101C2×#4112yes—
130Dic101—21no—
131Dic101—21no—
132Dic101—41no—
133Dic101—42no—
134Dic101#12412yes—
135C2×C101—81no—
136C2×C101C2×#12411yescharacteristic
137C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C101C2×#13711yescharacteristic
139D101—81no—
140D101#48914yesdirect product factor
141C2×#489—1011nomaximal, Sylow
142C101C2×#48911yescharacteristic, complemented, Sylow
143C2×C4⋊Q8—1011no—
144C2×#124—1011no—
145#489—1011no—
146#489—1011no—
147#489—1012no—
148#489—1012no—
149#489—1011no—
150#489D10111yescharacteristic, direct product factor
151C2×#103—1012no—
152C2×#102—1012no—
153C2×#97—1011no—
154C22×Q8—1011no—
155C22×Q8—2021no—
156C22×D4—1011no—
157C22×C8—1012no—
158C4⋊Q8—1012no—
159C4⋊Q8—1011no—
160C4⋊Q8—2021no—
161C4⋊Q8—2021no—
162C4⋊Q8C2×D10111yescharacteristic, complemented
163#124—1011no—
164#124—1011no—
165#124—1011no—
166#124—2021no—
167#124—2021no—
168#124C2×D10111yescharacteristic
169C2×C4⋊C4—1012no—
170C2×C4⋊C4—2021no—
171C2×C22⋊C4—2021no—
172C2×C22⋊C4—2021no—
173C2×C42—1011no—
174#103—1012no—
175#103—1012no—
176#103—1012no—
177#103C2×D10112yescomplemented
178#102—1012no—
179#102—1012no—
180#102—1012no—
181#102C2×D10112yes—
182#97—1012no—
183#97—1011no—
184#97C2×D10111yescharacteristic, complemented
185C24—2021no—
186C2×Q8—21no—
187C2×Q8—1011no—
188C2×Q8—4044no—
189C2×Q8—2021no—
190C2×Q8—2022no—
191C2×Q8—2024no—
192C2×Q8—2021no—
193C2×Q8C22×D10111yescharacteristic
194C2×D4—1011no—
195C2×D4—2024no—
196C2×D4—2021no—
197C2×D4C22×D10111yescharacteristic
198C22×C4—1011no—
199C22×C4—1011no—
200C22×C4—1011no—
201C22×C4—2021no—
202C22×C4—2022no—
203C2×C8—1012no—
204C2×C8—2024no—
205C2×C8C22×D10112yes—
206C4⋊C4—21no—
207C4⋊C4—1012no—
208C4⋊C4—2022no—
209C4⋊C4—2022no—
210C4⋊C4—2021no—
211C4⋊C4C22×D10112yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—2021no—
215C22⋊C4—2021no—
216C22⋊C4—2021no—
217C22⋊C4—2021no—
218C22⋊C4—2021no—
219C22⋊C4—2021no—
220C42—1011no—
221C42—1011no—
222C42—1011no—
223C42C22×D10111yescharacteristic
224C23—21no—
225C23—1011no—
226C23—4042no—
227C23—4042no—
228C23—4042no—
229C23—2021no—
230Q8—22no—
231Q8—42no—
232Q8—4042no—
233Q8—4042no—
234Q8—4044no—
235Q8—2022no—
236D4—22no—
237D4—4042no—
238D4—2022no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—1011no—
242C2×C4—1014no—
243C2×C4—1011no—
244C2×C4—1011no—
245C2×C4—4044no—
246C2×C4—4042no—
247C2×C4—2022no—
248C2×C4—2022no—
249C2×C4—2021no—
250C2×C4—2022no—
251C2×C4D4×D10111yescharacteristic
252C2×C4D4×D10111yescharacteristic
253C2×C4C23×D10111yescharacteristic
254C8—22no—
255C8—2022no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—8084no—
260C22—1012no—
261C22—1012no—
262C22—1012no—
263C22—4041no—
264C22—4041no—
265C22—4041no—
266C22C2×D4×D10111yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—1012no—
272C4—4041no—
273C4—4042no—
274C4—2021no—
275C4—2021no—
276C4#41×D10112yes—
277C2—81no—
278C2—8081no—
279C2—1014no—
280C2#124×D10111yescharacteristic
281C2AGL(1,ℤ/8)×D10111yescharacteristic
282C2#137×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#489×C101C2
socleC22×C101C22×D4
prime radicalC2×D4×D101C22
hypercenter#489D101
hypocenterC101C2×#489
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D101C2semidirect
3. #20265539C2semidirect
4. #20265538C2semidirect
5. #20265537C2semidirect
6. #20265536C2semidirect
7. #20265535C2semidirect
8. #103×D101C2semidirect
10. #97×D101C2semidirect
11. #489×C101C2semidirect
16. #15692255C22semidirect
18. #15692254C22semidirect
30. #15692180C22semidirect
31. #15692179C22semidirect
32. #15692178C22semidirect
36. #15692166C22semidirect
37. #15692165C22semidirect
38. C4⋊Q8×C101C22semidirect
40. #103×C101C22semidirect
42. #97×C101C22semidirect
140. D101#489direct (N × H)
142. C101C2×#489semidirect
150. #489D101direct (N × H)
162. C4⋊Q8C2×D101semidirect
177. #103C2×D101semidirect
184. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1012
#124×D1012
#202655392
#202655382
#202655372
#202655362
#202655352
#103×D1012
#102×D1012
#97×D1012
#489×C1012
C2×#489101