Finite Group Databasebeta

← #489×D271 #46846690

Subgroups — #489×D271 #46846690

Order 34688.

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×D271C211yesmaximal, characteristic, complemented
2#124×D271C211yesmaximal, characteristic
3#31774259C211yesmaximal, characteristic, complemented
4#31774258C212yesmaximal, complemented
5#31774257C211yesmaximal, characteristic, complemented
6#31774256C211yesmaximal, characteristic, complemented
7#31774255C212yesmaximal, complemented
8#103×D271C212yesmaximal, complemented
9#102×D271C212yesmaximal
10#97×D271C211yesmaximal, characteristic, complemented
11#489×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271—21no—
13C2×Q8×D271C2211yescharacteristic
14C2×D4×D271C2211yescharacteristic
15C2×C8×D271C2212yes—
16#22576160C2211yescharacteristic, complemented
17#22576159—21no—
18#22576159C2212yescomplemented
19#22576158—21no—
20#22576149C2211yescharacteristic
21#22576148C2211yescharacteristic
22#22576147C2211yescharacteristic
23#22576146—21no—
24#22576146—21no—
25C4⋊C4×D271—21no—
26C4⋊C4×D271C2212yes—
27C22⋊C4×D271—21no—
28C22⋊C4×D271—21no—
29C42×D271C2211yescharacteristic
30#22576085C2212yescomplemented
31#22576084C2212yescomplemented
32#22576083C2212yescomplemented
33#22576082C2212yes—
34#22576081C2212yes—
35#22576080C2212yes—
36#22576071C2211yescharacteristic, complemented
37#22576070C2212yescomplemented
38C4⋊Q8×C271C2211yescharacteristic, complemented
39#124×C271C2211yescharacteristic
40#103×C271C2212yescomplemented
41#102×C271C2212yes—
42#97×C271C2211yescharacteristic, complemented
43C23×D271—21no—
44C2×Dic542—21no—
45C2×Dic542—22no—
46C2×Dic542—21no—
47C2×Dic542C2311yescharacteristic
48Q8×D271—24no—
49Q8×D271—44no—
50C2×D1084C2311yescharacteristic
51C2×C22⋊D271—21no—
52D4×D271—24no—
53C2×C4×D271—22no—
54C2×C4×D271—21no—
55C2×C4×D271D411yescharacteristic
56C2×C4×D271D411yescharacteristic
57C2×C4×D271C2311yescharacteristic
58C2×C271⋊C8C2312yes—
59C8×D271—24no—
60C4⋊Dic271—21no—
61C4⋊Dic271C2312yes—
62Dic271⋊C4—22no—
63Dic271⋊C4—22no—
64C22⋊Dic271—21no—
65C22⋊Dic271—21no—
66(C2×D271)⋊C4—21no—
67(C2×D271)⋊C4—21no—
68(C2×D271)⋊C4—21no—
69(C2×D271)⋊C4—21no—
70C4×Dic271D411yescharacteristic
71C4×Dic271D411yescharacteristic
72C4×Dic271C2311yescharacteristic
73C2×Q8×C271—21no—
74C2×Q8×C271C2311yescharacteristic
75C2×D4×C271C2311yescharacteristic
76C2×C8×C271C2312yes—
77C4⋊C4×C271—21no—
78C4⋊C4×C271C2312yes—
79C22⋊C4×C271—21no—
80C22⋊C4×C271—21no—
81C42×C271C2311yescharacteristic
82C22×D271—21no—
83C22×D271—42no—
84C22×D271—42no—
85C22×D271—42no—
86C22×D271C2×D411yescharacteristic
87Dic542—22no—
88Dic542—44no—
89Dic542—42no—
90Dic542—42no—
91D1084—22no—
92C22⋊D271—42no—
93C2×Dic271—21no—
94C2×Dic271—22no—
95C2×Dic271C2×D411yescharacteristic
96C2×Dic271C2×D411yescharacteristic
97C2×Dic271C2×D411yescharacteristic
98C4×D271—22no—
99C4×D271—22no—
100C4×D271—42no—
101C4×D271—44no—
102C4×D271#4114yes—
103C271⋊C8—22no—
104C23×C271—21no—
105Q8×C271—22no—
106Q8×C271—42no—
107D4×C271—22no—
108C2×C4×C271—21no—
109C2×C4×C271—22no—
110C2×C4×C271C2×D411yescharacteristic
111C2×C4×C271C2×D411yescharacteristic
112C2×C4×C271C2411yescharacteristic
113C8×C271—22no—
114C22×C271—41no—
115C22×C271—41no—
116C22×C271—41no—
117C22×C271C22×D411yescharacteristic
118C2×D271—41no—
119C2×D271—41no—
120C2×D271—41no—
121C2×D271—84no—
122C2×D271#12412yes—
123C2×D271AGL(1,ℤ/8)12yes—
124C2×D271#13712yes—
125C4×C271—21no—
126C4×C271—21no—
127C4×C271—41no—
128C4×C271—42no—
129C4×C271C2×#4112yes—
130Dic271—21no—
131Dic271—21no—
132Dic271—41no—
133Dic271—42no—
134Dic271#12412yes—
135C2×C271—81no—
136C2×C271C2×#12411yescharacteristic
137C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C271C2×#13711yescharacteristic
139D271—81no—
140D271#48914yesdirect product factor
141C271C2×#48911yescharacteristic, complemented, Sylow
142C2×#489—2711nomaximal, Sylow
143C2×C4⋊Q8—2711no—
144C2×#124—2711no—
145#489—2711no—
146#489—2711no—
147#489—2712no—
148#489—2712no—
149#489—2711no—
150#489D27111yescharacteristic, direct product factor
151C2×#103—2712no—
152C2×#102—2712no—
153C2×#97—2711no—
154C22×Q8—2711no—
155C22×Q8—5421no—
156C22×D4—2711no—
157C22×C8—2712no—
158C4⋊Q8—2712no—
159C4⋊Q8—2711no—
160C4⋊Q8—5421no—
161C4⋊Q8—5421no—
162C4⋊Q8C2×D27111yescharacteristic, complemented
163#124—2711no—
164#124—2711no—
165#124—2711no—
166#124—5421no—
167#124—5421no—
168#124C2×D27111yescharacteristic
169C2×C4⋊C4—2712no—
170C2×C4⋊C4—5421no—
171C2×C22⋊C4—5421no—
172C2×C22⋊C4—5421no—
173C2×C42—2711no—
174#103—2712no—
175#103—2712no—
176#103—2712no—
177#103C2×D27112yescomplemented
178#102—2712no—
179#102—2712no—
180#102—2712no—
181#102C2×D27112yes—
182#97—2712no—
183#97—2711no—
184#97C2×D27111yescharacteristic, complemented
185C24—5421no—
186C2×Q8—21no—
187C2×Q8—2711no—
188C2×Q8—5421no—
189C2×Q8—5422no—
190C2×Q8—5424no—
191C2×Q8—5421no—
192C2×Q8—10844no—
193C2×Q8C22×D27111yescharacteristic
194C2×D4—2711no—
195C2×D4—5424no—
196C2×D4—5421no—
197C2×D4C22×D27111yescharacteristic
198C22×C4—2711no—
199C22×C4—2711no—
200C22×C4—2711no—
201C22×C4—5421no—
202C22×C4—5422no—
203C2×C8—2712no—
204C2×C8—5424no—
205C2×C8C22×D27112yes—
206C4⋊C4—21no—
207C4⋊C4—2712no—
208C4⋊C4—5422no—
209C4⋊C4—5422no—
210C4⋊C4—5421no—
211C4⋊C4C22×D27112yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—5421no—
215C22⋊C4—5421no—
216C22⋊C4—5421no—
217C22⋊C4—5421no—
218C22⋊C4—5421no—
219C22⋊C4—5421no—
220C42—2711no—
221C42—2711no—
222C42—2711no—
223C42C22×D27111yescharacteristic
224C23—21no—
225C23—2711no—
226C23—5421no—
227C23—10842no—
228C23—10842no—
229C23—10842no—
230Q8—22no—
231Q8—42no—
232Q8—5422no—
233Q8—10842no—
234Q8—10842no—
235Q8—10844no—
236D4—22no—
237D4—5422no—
238D4—10842no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—2711no—
242C2×C4—2714no—
243C2×C4—2711no—
244C2×C4—2711no—
245C2×C4—5422no—
246C2×C4—5422no—
247C2×C4—5421no—
248C2×C4—5422no—
249C2×C4—10842no—
250C2×C4—10844no—
251C2×C4D4×D27111yescharacteristic
252C2×C4D4×D27111yescharacteristic
253C2×C4C23×D27111yescharacteristic
254C8—22no—
255C8—5422no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—2712no—
260C22—2712no—
261C22—2712no—
262C22—10841no—
263C22—10841no—
264C22—10841no—
265C22—21684no—
266C22C2×D4×D27111yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—2712no—
272C4—5421no—
273C4—5421no—
274C4—10841no—
275C4—10842no—
276C4#41×D27112yes—
277C2—81no—
278C2—2714no—
279C2—21681no—
280C2#124×D27111yescharacteristic
281C2AGL(1,ℤ/8)×D27111yescharacteristic
282C2#137×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#489×C271C2
socleC22×C271C22×D4
prime radicalC2×D4×D271C22
hypercenter#489D271
hypocenterC271C2×#489
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D271C2semidirect
3. #31774259C2semidirect
4. #31774258C2semidirect
5. #31774257C2semidirect
6. #31774256C2semidirect
7. #31774255C2semidirect
8. #103×D271C2semidirect
10. #97×D271C2semidirect
11. #489×C271C2semidirect
16. #22576160C22semidirect
18. #22576159C22semidirect
30. #22576085C22semidirect
31. #22576084C22semidirect
32. #22576083C22semidirect
36. #22576071C22semidirect
37. #22576070C22semidirect
38. C4⋊Q8×C271C22semidirect
40. #103×C271C22semidirect
42. #97×C271C22semidirect
140. D271#489direct (N × H)
141. C271C2×#489semidirect
150. #489D271direct (N × H)
162. C4⋊Q8C2×D271semidirect
177. #103C2×D271semidirect
184. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D2712
#124×D2712
#317742592
#317742582
#317742572
#317742562
#317742552
#103×D2712
#102×D2712
#97×D2712
#489×C2712
C2×#489271