Finite Group Databasebeta

← #483×D271 #46846238

Subgroups — #483×D271 #46846238

Order 34688.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22⋊Q8×D271C211yesmaximal, characteristic
2C2×C4⋊C4×D271C211yesmaximal, characteristic
3#31774229C212yesmaximal, complemented
4#31774228C211yesmaximal, characteristic, complemented
5#31774227C211yesmaximal, characteristic, complemented
6#31774226C212yesmaximal, complemented
7#31774225C211yesmaximal, characteristic, complemented
8#106×D271C212yesmaximal, complemented
9#103×D271C212yesmaximal, complemented
10C22⋊C8×D271C211yesmaximal, characteristic
11#483×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271C2211yescharacteristic
13C22×C4×D271—21no—
14C22×C4×D271C2211yescharacteristic
15C2×C8×D271C2212yes—
16#22576140C2211yescharacteristic
17#22576139C2211yescharacteristic
18#22576138—21no—
19#22576137—21no—
20#22576136C2211yescharacteristic
21C2×C4⋊Dic271C2211yescharacteristic
22C2×Dic271⋊C4—21no—
23C4⋊C4×D271—21no—
24C4⋊C4×D271—21no—
25C4⋊C4×D271—21no—
26C4⋊C4×D271C2212yes—
27C4⋊C4×D271C2212yes—
28C4⋊C4×D271C2211yescharacteristic
29C22⋊C4×D271—21no—
30#22576092C2212yescomplemented
31#22576091C2212yescomplemented
32#22576091C2212yescomplemented
33#22576085C2212yescomplemented
34#22576084C2212yescomplemented
35#22576083C2212yescomplemented
36C22⋊(C271⋊C8)C2211yescharacteristic
37(C2×D271)⋊C8C2212yes—
38C22⋊Q8×C271C2211yescharacteristic
39C2×C4⋊C4×C271C2211yescharacteristic
40#106×C271C2212yescomplemented
41#103×C271C2212yescomplemented
42C22⋊C8×C271C2211yescharacteristic
43C23×D271D411yescharacteristic
44C2×Dic542—21no—
45C2×Dic542C2311yescharacteristic
46Q8×D271—24no—
47C22×Dic271—21no—
48C22×Dic271D411yescharacteristic
49C2×C4×D271—24no—
50C2×C4×D271—22no—
51C2×C4×D271—22no—
52C2×C4×D271—21no—
53C2×C4×D271—21no—
54C2×C4×D271—44no—
55C2×C4×D271D411yescharacteristic
56C2×C4×D271D411yescharacteristic
57C2×C4×D271C2311yescharacteristic
58C2×C4×D271C2311yescharacteristic
59C2×C271⋊C8C2312yes—
60C8×D271—24no—
61C4⋊Dic271—21no—
62C4⋊Dic271—21no—
63C4⋊Dic271C2311yescharacteristic
64C4⋊Dic271C2312yes—
65Dic271⋊C4—21no—
66Dic271⋊C4—21no—
67Dic271⋊C4—22no—
68Dic271⋊C4—22no—
69Dic271⋊C4—21no—
70C22⋊Dic271—21no—
71(C2×D271)⋊C4—21no—
72(C2×D271)⋊C4—21no—
73C2×Q8×C271C2311yescharacteristic
74C22×C4×C271—21no—
75C22×C4×C271C2311yescharacteristic
76C2×C8×C271C2312yes—
77C4⋊C4×C271—21no—
78C4⋊C4×C271—21no—
79C4⋊C4×C271C2311yescharacteristic
80C4⋊C4×C271C2312yes—
81C22⋊C4×C271—21no—
82C22×D271—22no—
83C22×D271—22no—
84C22×D271SD1614yes—
85C22×D271C2×D411yescharacteristic
86C22×D271C2×D411yescharacteristic
87Dic542—22no—
88Dic542—42no—
89C2×Dic271—22no—
90C2×Dic271—21no—
91C2×Dic271—21no—
92C2×Dic271—22no—
93C2×Dic271—42no—
94C2×Dic271C2×D411yescharacteristic
95C2×Dic271C2×D411yescharacteristic
96C4×D271—22no—
97C4×D271—22no—
98C4×D271—22no—
99C4×D271—42no—
100C4×D271—44no—
101C4×D271—44no—
102C4×D271—42no—
103C4×D271#4114yes—
104C271⋊C8—22no—
105C23×C271C2×D411yescharacteristic
106Q8×C271—22no—
107C2×C4×C271—22no—
108C2×C4×C271—21no—
109C2×C4×C271—21no—
110C2×C4×C271—22no—
111C2×C4×C271—42no—
112C2×C4×C271C2×D411yescharacteristic
113C2×C4×C271C2411yescharacteristic
114C8×C271—22no—
115C22×C271—21no—
116C22×C271—21no—
117C22×C271C2×SD1612yes—
118C22×C271C22×D411yescharacteristic
119C2×D271—21no—
120C2×D271—21no—
121C2×D271—28no—
122C2×D271#12312yes—
123C2×D271C2×SD1612yes—
124C2×D271C2×SD1612yes—
125C2×D271#13712yes—
126C4×C271—21no—
127C4×C271—41no—
128C4×C271—42no—
129C4×C271—41no—
130C4×C271C2×#4112yes—
131Dic271—21no—
132Dic271—41no—
133Dic271—42no—
134Dic271—41no—
135Dic271#12312yes—
136C2×C271—22no—
137C2×C271C2×#12311yescharacteristic
138C2×C271C22×SD1611yescharacteristic
139C2×C271C2×#13711yescharacteristic
140D271—22no—
141D271#48314yesdirect product factor
142C271C2×#48311yescharacteristic, complemented, Sylow
143C2×#483—2711nomaximal, Sylow
144C2×C22⋊Q8—2711no—
145C22×C4⋊C4—2711no—
146#483—2711no—
147#483—2711no—
148#483—2712no—
149#483—2712no—
150#483—2711no—
151#483D27111yescharacteristic, direct product factor
152C2×#106—2712no—
153C2×#103—2712no—
154C2×C22⋊C8—2711no—
155C22×Q8—2711no—
156C23×C4—2711no—
157C23×C4—5421no—
158C22×C8—2712no—
159C22⋊Q8—2711no—
160C22⋊Q8—2711no—
161C22⋊Q8—2711no—
162C22⋊Q8—5421no—
163C22⋊Q8—5421no—
164C22⋊Q8C2×D27111yescharacteristic
165C2×C4⋊C4—2711no—
166C2×C4⋊C4—2712no—
167C2×C4⋊C4—2712no—
168C2×C4⋊C4—2711no—
169C2×C4⋊C4—5421no—
170C2×C4⋊C4—5421no—
171C2×C4⋊C4—5421no—
172C2×C4⋊C4—5421no—
173C2×C4⋊C4C2×D27111yescharacteristic
174C2×C22⋊C4—5421no—
175#106—2712no—
176#106—2712no—
177#106—2712no—
178#106C2×D27112yescomplemented
179#103—2712no—
180#103—2712no—
181#103—2712no—
182#103C2×D27112yescomplemented
183C22⋊C8—2711no—
184C22⋊C8—2712no—
185C22⋊C8C2×D27111yescharacteristic
186C24—2711no—
187C2×Q8—2711no—
188C2×Q8—5421no—
189C2×Q8—5424no—
190C2×Q8C22×D27111yescharacteristic
191C22×C4—21no—
192C22×C4—2711no—
193C22×C4—2711no—
194C22×C4—2711no—
195C22×C4—2711no—
196C22×C4—2711no—
197C22×C4—5424no—
198C22×C4—5421no—
199C22×C4—5421no—
200C22×C4—5422no—
201C22×C4—5422no—
202C22×C4—5421no—
203C22×C4—10844no—
204C22×C4C22×D27111yescharacteristic
205C2×C8—2712no—
206C2×C8—5424no—
207C2×C8C22×D27112yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—2712no—
211C4⋊C4—2711no—
212C4⋊C4—5422no—
213C4⋊C4—5422no—
214C4⋊C4—5421no—
215C4⋊C4—5421no—
216C4⋊C4—5421no—
217C4⋊C4—5421no—
218C4⋊C4—5421no—
219C4⋊C4C22×D27112yes—
220C4⋊C4C22×D27111yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—5421no—
223C22⋊C4—5421no—
224C22⋊C4—5421no—
225C23—2711no—
226C23—2714no—
227C23—2711no—
228C23—5422no—
229C23—5422no—
230C23D4×D27111yescharacteristic
231Q8—22no—
232Q8—5422no—
233Q8—10842no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—2711no—
240C2×C4—2714no—
241C2×C4—2711no—
242C2×C4—5422no—
243C2×C4—5422no—
244C2×C4—5422no—
245C2×C4—5422no—
246C2×C4—5421no—
247C2×C4—5421no—
248C2×C4—5422no—
249C2×C4—10844no—
250C2×C4—10844no—
251C2×C4—10842no—
252C2×C4—10842no—
253C2×C4—10842no—
254C2×C4D4×D27111yescharacteristic
255C2×C4C23×D27111yescharacteristic
256C8—22no—
257C8—5422no—
258C22—21no—
259C22—21no—
260C22—2712no—
261C22—2712no—
262C22—2712no—
263C22—2712no—
264C22—5428no—
265C22—5421no—
266C22—5421no—
267C22SD16×D27112yes—
268C22C2×D4×D27111yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—2712no—
274C4—5421no—
275C4—10841no—
276C4—10842no—
277C4—10841no—
278C4#41×D27112yes—
279C2—22no—
280C2—2714no—
281C2—5422no—
282C2#123×D27111yescharacteristic
283C2C2×SD16×D27111yescharacteristic
284C2#137×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#483×C271C2
socleC22×C271C22×D4
prime radicalC23×D271D4
hypercenter#483D271
hypocenterC271C2×#483
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #31774229C2semidirect
4. #31774228C2semidirect
5. #31774227C2semidirect
6. #31774226C2semidirect
7. #31774225C2semidirect
8. #106×D271C2semidirect
9. #103×D271C2semidirect
11. #483×C271C2semidirect
30. #22576092C22semidirect
31. #22576091C22semidirect
32. #22576091C22semidirect
33. #22576085C22semidirect
34. #22576084C22semidirect
35. #22576083C22semidirect
40. #106×C271C22semidirect
41. #103×C271C22semidirect
141. D271#483direct (N × H)
142. C271C2×#483semidirect
151. #483D271direct (N × H)
178. #106C2×D271semidirect
182. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2712
C2×C4⋊C4×D2712
#317742292
#317742282
#317742272
#317742262
#317742252
#106×D2712
#103×D2712
C22⋊C8×D2712
#483×C2712
C2×#483271