Finite Group Databasebeta

← #46848052

Subgroups — #46848052

Order 34688.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#22576182C211yesmaximal, characteristic, complemented
2C2×D8×D271C211yesmaximal, characteristic, complemented
3#31774544C212yesmaximal, complemented
4#31774280C211yesmaximal, characteristic, complemented
5#31774139C211yesmaximal, characteristic, complemented
6#31774111C212yesmaximal, complemented
7#31774049C212yesmaximal, complemented
8#31774040C212yesmaximal, complemented
9#31773979C211yesmaximal, characteristic, complemented
10C8×C22⋊D271C211yesmaximal, characteristic, complemented
11#466×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2212yescomplemented
13C2×D4×D271C2212yes—
14#22576182—24no—
15C2×Dic1084C2211yescharacteristic, complemented
16C2×#16864643C2212yescomplemented
17C2×#16864642C2212yescomplemented
18C2×D2168C2211yescharacteristic
19C2×#16864639C2212yes—
20D8×D271—24no—
21C2×C8×D271C2211yescharacteristic
22#22576147C2212yescomplemented
23#22576142—22no—
24#22576140C2212yes—
25#22576131—22no—
26#22576128—22no—
27#22576122C2212yescomplemented
28C4×C22⋊D271C2211yescharacteristic
29C22⋊C4×D271—22no—
30Dic271⋊C8C2211yescharacteristic, complemented
31#22576083C2212yescomplemented
32#22576080C2212yescomplemented
33C22⋊(C271⋊C8)C2211yescharacteristic, complemented
34(C2×D271)⋊C8C2211yescharacteristic
35C8×Dic271C2211yescharacteristic, complemented
36C2×D8×C271C2211yescharacteristic
37C22×C8×C271C2211yescharacteristic, complemented
38#121×C271C2212yescomplemented
39#107×C271C2211yescharacteristic, complemented
40#102×C271C2212yescomplemented
41C23×D271—22no—
42#16864654—24no—
43#16864654—42no—
44C2×Dic542C2312yes—
45C2×D1084C2312yes—
46C2×C22⋊D271—22no—
47C2×C22⋊D271—22no—
48C2×C22⋊D271D411yescharacteristic
49D4×D271—24no—
50D4×D271—42no—
51C2×C4×D271—22no—
52C2×C4×D271—22no—
53C2×C4×D271C2311yescharacteristic
54Dic1084—22no—
55#16864643—24no—
56#16864642—42no—
57D2168D414yescomplemented
58#16864639—24no—
59C2×C271⋊C8C2311yescharacteristic
60C2×C271⋊C8C2311yescharacteristic
61C8×D271—22no—
62C4⋊Dic271—22no—
63Dic271⋊C4D411yescharacteristic
64C22⋊Dic271—22no—
65C22⋊Dic271—22no—
66C22⋊Dic271D411yescharacteristic
67(C2×D271)⋊C4—22no—
68(C2×D271)⋊C4—22no—
69(C2×D271)⋊C4D411yescharacteristic
70C4×Dic271C2311yescharacteristic
71C2×D4×C271—22no—
72C2×D4×C271C2312yes—
73C22×C4×C271C2311yescharacteristic
74D8×C271—22no—
75C2×C8×C271—22no—
76C2×C8×C271C2311yescharacteristic
77C2×C8×C271C2311yescharacteristic
78C4⋊C4×C271C2312yes—
79C22⋊C4×C271—22no—
80C22×D271—22no—
81C22×D271—44no—
82C22×D271—44no—
83C22×D271—82no—
84C22×D271C2×D411yescharacteristic
85Dic542—22no—
86Dic542—22no—
87D1084—22no—
88D1084C2×D414yes—
89C22⋊D271—22no—
90C22⋊D271—44no—
91C22⋊D271—44no—
92C2×Dic271—22no—
93C2×Dic271C2×D411yescharacteristic
94C2×Dic271C2×D411yescharacteristic
95C2×Dic271C2×D411yescharacteristic
96C4×D271—21no—
97C4×D271—21no—
98C4×D271—44no—
99C4×D271—82no—
100C271⋊C8—21no—
101C271⋊C8—21no—
102C23×C271—22no—
103C23×C271C2×D411yescharacteristic
104D4×C271—22no—
105D4×C271—22no—
106D4×C271—82no—
107C2×C4×C271—21no—
108C2×C4×C271—21no—
109C2×C4×C271—22no—
110C2×C4×C271C2×D411yescharacteristic
111C2×C4×C271C2411yescharacteristic
112C8×C271—21no—
113C8×C271C2×D412yes—
114C22×C271—21no—
115C22×C271—21no—
116C22×C271—21no—
117C22×C271—42no—
118C22×C271—42no—
119C22×C271—42no—
120C22×C271C22×D411yescharacteristic
121C2×D271—21no—
122C2×D271—21no—
123C2×D271—24no—
124C2×D271—42no—
125C2×D271—42no—
126C2×D271—88no—
127C2×D271#13512yescomplemented
128C4×C271—21no—
129C4×C271—42no—
130C4×C271C22×D411yescharacteristic
131C4×C271H(5,𝔽2)11yescharacteristic
132Dic271—21no—
133Dic271—21no—
134Dic271—21no—
135Dic271—42no—
136C2×C271—41no—
137C2×C271—82no—
138C2×C271D4211yescharacteristic
139C2×C271C2×#13511yescharacteristic
140C2×C271#57611yescharacteristic
141D271—22no—
142D271—44no—
143C271#328011yescharacteristic, complemented, Sylow
144#3280—2711nomaximal, Sylow
145C2×#135—2711no—
146C22×D8—2711no—
147#546—2712no—
148#495—2711no—
149#467—2711no—
150#466D27111yescharacteristic, complemented
151#463—2712no—
152#450—2712no—
153#449—2712no—
154C4×D8—2711no—
155C8×D4—2711no—
156C2×#41—2712no—
157C22×D4—2712no—
158#135—5424no—
159C2×Q16—2711no—
160C2×SD16—2712no—
161C2×SD16—2712no—
162C2×D8—2712no—
163C2×D8—2711no—
164C2×D8—5424no—
165C2×D8C2×D27111yescharacteristic
166C22×C8—2711no—
167C22×C8C2×D27111yescharacteristic, complemented
168#124—2712no—
169#123—5422no—
170C22⋊Q8—2712no—
171#121—5422no—
172#121C2×D27112yescomplemented
173#120—5422no—
174C4×D4—2711no—
175C4×D4—2712no—
176C2×C22⋊C4—5422no—
177#107C2×D27111yescharacteristic, complemented
178C4⋊C8—2711no—
179#103—2712no—
180#102—2712no—
181#102C2×D27112yescomplemented
182C22⋊C8—2711no—
183C22⋊C8—2711no—
184C4×C8—2711no—
185C24—5422no—
186#41—5424no—
187#41—10842no—
188C2×Q8—2712no—
189C2×D4—22no—
190C2×D4—2711no—
191C2×D4—2712no—
192C2×D4—5422no—
193C2×D4—5424no—
194C2×D4—5422no—
195C2×D4—10842no—
196C2×D4C22×D27112yes—
197C22×C4—2711no—
198C22×C4—5422no—
199C22×C4—5422no—
200C22×C4C22×D27111yescharacteristic
201Q16—5422no—
202SD16—5424no—
203SD16—10842no—
204D8—22no—
205D8—2714no—
206D8—5424no—
207C2×C8—22no—
208C2×C8—2711no—
209C2×C8—2711no—
210C2×C8—5422no—
211C2×C8C22×D27111yescharacteristic
212C2×C8C22×D27111yescharacteristic
213C4⋊C4—2711no—
214C4⋊C4—5422no—
215C4⋊C4C22×D27112yes—
216C22⋊C4—22no—
217C22⋊C4—2711no—
218C22⋊C4—2711no—
219C22⋊C4—5422no—
220C22⋊C4—5422no—
221C22⋊C4—5422no—
222C22⋊C4—5422no—
223C42—2711no—
224C23—22no—
225C23—2711no—
226C23—5422no—
227C23—10844no—
228C23—10844no—
229C23—21682no—
230C23D4×D27111yescharacteristic
231Q8—5422no—
232Q8—5422no—
233D4—22no—
234D4—22no—
235D4—82no—
236D4—2714no—
237D4—5422no—
238D4—5422no—
239D4—10844no—
240D4—10844no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—2711no—
245C2×C4—2711no—
246C2×C4—2711no—
247C2×C4—5421no—
248C2×C4—5421no—
249C2×C4—5422no—
250C2×C4—10844no—
251C2×C4—21682no—
252C2×C4D4×D27111yescharacteristic
253C2×C4C23×D27111yescharacteristic
254C8—21no—
255C8—5421no—
256C8—5421no—
257C8D4×D27112yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—2712no—
265C22—5421no—
266C22—5421no—
267C22—5424no—
268C22—10842no—
269C22—10842no—
270C22—21688no—
271C22C2×D4×D27111yescharacteristic
272C4—21no—
273C4—42no—
274C4—5421no—
275C4—5421no—
276C4—5421no—
277C4—10842no—
278C4C2×D4×D27111yescharacteristic
279C4#2257620611yescharacteristic
280C2—41no—
281C2—82no—
282C2—5422no—
283C2—10844no—
284C2#3177453111yescharacteristic
285C2#135×D27111yescharacteristic
286C2#3177469811yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#466×C271C2
socleC22×C271C22×D4
hypercenter#466D271
hypocenterC271#3280
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#22576182C2semidirect
2. C2×D8×D271C2semidirect
3. #31774544C2semidirect
4. #31774280C2semidirect
5. #31774139C2semidirect
6. #31774111C2semidirect
7. #31774049C2semidirect
8. #31774040C2semidirect
9. #31773979C2semidirect
10. C8×C22⋊D271C2semidirect
11. #466×C271C2semidirect
12. C2×#16864654C22semidirect
15. C2×Dic1084C22semidirect
16. C2×#16864643C22semidirect
17. C2×#16864642C22semidirect
22. #22576147C22semidirect
27. #22576122C22semidirect
30. Dic271⋊C8C22semidirect
31. #22576083C22semidirect
32. #22576080C22semidirect
33. C22⋊(C271⋊C8)C22semidirect
35. C8×Dic271C22semidirect
37. C22×C8×C271C22semidirect
38. #121×C271C22semidirect
39. #107×C271C22semidirect
40. #102×C271C22semidirect
57. D2168D4semidirect
127. C2×D271#135semidirect
143. C271#3280semidirect
150. #466D271semidirect
167. C22×C8C2×D271semidirect
172. #121C2×D271semidirect
177. #107C2×D271semidirect
181. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761822
C2×D8×D2712
#317745442
#317742802
#317741392
#317741112
#317740492
#317740402
#317739792
C8×C22⋊D2712
#466×C2712
#3280271