Finite Group Databasebeta

← #489×D251 #44560691

Subgroups — #489×D251 #44560691

Order 32128.

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×D251C211yesmaximal, characteristic, complemented
2#124×D251C211yesmaximal, characteristic
3#30632049C211yesmaximal, characteristic, complemented
4#30632048C212yesmaximal, complemented
5#30632047C211yesmaximal, characteristic, complemented
6#30632046C211yesmaximal, characteristic, complemented
7#30632045C212yesmaximal, complemented
8#103×D251C212yesmaximal, complemented
9#102×D251C212yesmaximal
10#97×D251C211yesmaximal, characteristic, complemented
11#489×C251C211yesmaximal, characteristic, complemented
12C2×Q8×D251—21no—
13C2×Q8×D251C2211yescharacteristic
14C2×D4×D251C2211yescharacteristic
15C2×C8×D251C2212yes—
16#22548715C2211yescharacteristic, complemented
17#22548714—21no—
18#22548714C2212yescomplemented
19#22548713—21no—
20#22548704C2211yescharacteristic
21#22548703C2211yescharacteristic
22#22548702C2211yescharacteristic
23#22548701—21no—
24#22548701—21no—
25C4⋊C4×D251—21no—
26C4⋊C4×D251C2212yes—
27C22⋊C4×D251—21no—
28C22⋊C4×D251—21no—
29C42×D251C2211yescharacteristic
30#22548640C2212yescomplemented
31#22548639C2212yescomplemented
32#22548638C2212yescomplemented
33#22548637C2212yes—
34#22548636C2212yes—
35#22548635C2212yes—
36#22548626C2211yescharacteristic, complemented
37#22548625C2212yescomplemented
38C4⋊Q8×C251C2211yescharacteristic, complemented
39#124×C251C2211yescharacteristic
40#103×C251C2212yescomplemented
41#102×C251C2212yes—
42#97×C251C2211yescharacteristic, complemented
43C23×D251—21no—
44C2×Dic502—22no—
45C2×Dic502—21no—
46C2×Dic502—21no—
47C2×Dic502C2311yescharacteristic
48Q8×D251—24no—
49Q8×D251—44no—
50C2×D1004C2311yescharacteristic
51C2×C22⋊D251—21no—
52D4×D251—24no—
53C2×C4×D251—22no—
54C2×C4×D251—21no—
55C2×C4×D251D411yescharacteristic
56C2×C4×D251D411yescharacteristic
57C2×C4×D251C2311yescharacteristic
58C2×C251⋊C8C2312yes—
59C8×D251—24no—
60C4⋊Dic251—21no—
61C4⋊Dic251C2312yes—
62Dic251⋊C4—22no—
63Dic251⋊C4—22no—
64C22⋊Dic251—21no—
65C22⋊Dic251—21no—
66(C2×D251)⋊C4—21no—
67(C2×D251)⋊C4—21no—
68(C2×D251)⋊C4—21no—
69(C2×D251)⋊C4—21no—
70C4×Dic251D411yescharacteristic
71C4×Dic251D411yescharacteristic
72C4×Dic251C2311yescharacteristic
73C2×Q8×C251—21no—
74C2×Q8×C251C2311yescharacteristic
75C2×D4×C251C2311yescharacteristic
76C2×C8×C251C2312yes—
77C4⋊C4×C251—21no—
78C4⋊C4×C251C2312yes—
79C22⋊C4×C251—21no—
80C22⋊C4×C251—21no—
81C42×C251C2311yescharacteristic
82C22×D251—21no—
83C22×D251—42no—
84C22×D251—42no—
85C22×D251—42no—
86C22×D251C2×D411yescharacteristic
87Dic502—22no—
88Dic502—44no—
89Dic502—42no—
90Dic502—42no—
91D1004—22no—
92C22⋊D251—42no—
93C2×Dic251—21no—
94C2×Dic251—22no—
95C2×Dic251C2×D411yescharacteristic
96C2×Dic251C2×D411yescharacteristic
97C2×Dic251C2×D411yescharacteristic
98C4×D251—22no—
99C4×D251—22no—
100C4×D251—42no—
101C4×D251—44no—
102C4×D251#4114yes—
103C251⋊C8—22no—
104C23×C251—21no—
105Q8×C251—22no—
106Q8×C251—42no—
107D4×C251—22no—
108C2×C4×C251—21no—
109C2×C4×C251—22no—
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2×D411yescharacteristic
112C2×C4×C251C2411yescharacteristic
113C8×C251—22no—
114C22×C251—41no—
115C22×C251—41no—
116C22×C251—41no—
117C22×C251C22×D411yescharacteristic
118C2×D251—41no—
119C2×D251—41no—
120C2×D251—41no—
121C2×D251—84no—
122C2×D251#12412yes—
123C2×D251AGL(1,ℤ/8)12yes—
124C2×D251#13712yes—
125C4×C251—21no—
126C4×C251—21no—
127C4×C251—41no—
128C4×C251—42no—
129C4×C251C2×#4112yes—
130Dic251—21no—
131Dic251—21no—
132Dic251—41no—
133Dic251—42no—
134Dic251#12412yes—
135C2×C251—81no—
136C2×C251C2×#12411yescharacteristic
137C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C251C2×#13711yescharacteristic
139D251—81no—
140D251#48914yesdirect product factor
141C251C2×#48911yescharacteristic, complemented, Sylow
142C2×#489—2511nomaximal, Sylow
143C2×C4⋊Q8—2511no—
144C2×#124—2511no—
145#489—2511no—
146#489—2511no—
147#489—2512no—
148#489—2512no—
149#489—2511no—
150#489D25111yescharacteristic, direct product factor
151C2×#103—2512no—
152C2×#102—2512no—
153C2×#97—2511no—
154C22×Q8—5021no—
155C22×Q8—2511no—
156C22×D4—2511no—
157C22×C8—2512no—
158C4⋊Q8—5021no—
159C4⋊Q8—5021no—
160C4⋊Q8—2512no—
161C4⋊Q8—2511no—
162C4⋊Q8C2×D25111yescharacteristic, complemented
163#124—5021no—
164#124—5021no—
165#124—2511no—
166#124—2511no—
167#124—2511no—
168#124C2×D25111yescharacteristic
169C2×C4⋊C4—5021no—
170C2×C4⋊C4—2512no—
171C2×C22⋊C4—5021no—
172C2×C22⋊C4—5021no—
173C2×C42—2511no—
174#103—2512no—
175#103—2512no—
176#103—2512no—
177#103C2×D25112yescomplemented
178#102—2512no—
179#102—2512no—
180#102—2512no—
181#102C2×D25112yes—
182#97—2512no—
183#97—2511no—
184#97C2×D25111yescharacteristic, complemented
185C24—5021no—
186C2×Q8—21no—
187C2×Q8—10044no—
188C2×Q8—5021no—
189C2×Q8—5022no—
190C2×Q8—5024no—
191C2×Q8—5021no—
192C2×Q8—2511no—
193C2×Q8C22×D25111yescharacteristic
194C2×D4—5024no—
195C2×D4—5021no—
196C2×D4—2511no—
197C2×D4C22×D25111yescharacteristic
198C22×C4—5021no—
199C22×C4—5022no—
200C22×C4—2511no—
201C22×C4—2511no—
202C22×C4—2511no—
203C2×C8—5024no—
204C2×C8—2512no—
205C2×C8C22×D25112yes—
206C4⋊C4—21no—
207C4⋊C4—5022no—
208C4⋊C4—5022no—
209C4⋊C4—5021no—
210C4⋊C4—2512no—
211C4⋊C4C22×D25112yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—5021no—
215C22⋊C4—5021no—
216C22⋊C4—5021no—
217C22⋊C4—5021no—
218C22⋊C4—5021no—
219C22⋊C4—5021no—
220C42—2511no—
221C42—2511no—
222C42—2511no—
223C42C22×D25111yescharacteristic
224C23—21no—
225C23—10042no—
226C23—10042no—
227C23—10042no—
228C23—5021no—
229C23—2511no—
230Q8—22no—
231Q8—42no—
232Q8—10042no—
233Q8—10044no—
234Q8—10042no—
235Q8—5022no—
236D4—22no—
237D4—10042no—
238D4—5022no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—10044no—
242C2×C4—10042no—
243C2×C4—5022no—
244C2×C4—5022no—
245C2×C4—5021no—
246C2×C4—5022no—
247C2×C4—2511no—
248C2×C4—2511no—
249C2×C4—2511no—
250C2×C4—2514no—
251C2×C4D4×D25111yescharacteristic
252C2×C4D4×D25111yescharacteristic
253C2×C4C23×D25111yescharacteristic
254C8—22no—
255C8—5022no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—20084no—
260C22—10041no—
261C22—10041no—
262C22—10041no—
263C22—2512no—
264C22—2512no—
265C22—2512no—
266C22C2×D4×D25111yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—10041no—
272C4—10042no—
273C4—5021no—
274C4—5021no—
275C4—2512no—
276C4#41×D25112yes—
277C2—81no—
278C2—20081no—
279C2—2514no—
280C2#124×D25111yescharacteristic
281C2AGL(1,ℤ/8)×D25111yescharacteristic
282C2#137×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#489×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
hypercenter#489D251
hypocenterC251C2×#489
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D251C2semidirect
3. #30632049C2semidirect
4. #30632048C2semidirect
5. #30632047C2semidirect
6. #30632046C2semidirect
7. #30632045C2semidirect
8. #103×D251C2semidirect
10. #97×D251C2semidirect
11. #489×C251C2semidirect
16. #22548715C22semidirect
18. #22548714C22semidirect
30. #22548640C22semidirect
31. #22548639C22semidirect
32. #22548638C22semidirect
36. #22548626C22semidirect
37. #22548625C22semidirect
38. C4⋊Q8×C251C22semidirect
40. #103×C251C22semidirect
42. #97×C251C22semidirect
140. D251#489direct (N × H)
141. C251C2×#489semidirect
150. #489D251direct (N × H)
162. C4⋊Q8C2×D251semidirect
177. #103C2×D251semidirect
184. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D2512
#124×D2512
#306320492
#306320482
#306320472
#306320462
#306320452
#103×D2512
#102×D2512
#97×D2512
#489×C2512
C2×#489251