Finite Group Databasebeta

← #46847987

Subgroups — #46847987

Order 34688.

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

normal subgroups only

This group has 556 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
1C2×#22576184C212yesmaximal, complemented
2#31774567C211yesmaximal, characteristic, complemented
3D4×D1084C211yesmaximal, characteristic, complemented
4#31774273C211yesmaximal, characteristic, complemented
5#31774149C212yesmaximal, complemented
6#31774098C211yesmaximal, characteristic, complemented
7#31774043C212yesmaximal, complemented
8#31774029C212yesmaximal, complemented
9C4×#16864642C211yesmaximal, characteristic, complemented
10#31773974C211yesmaximal, characteristic, complemented
11#459×C271C211yesmaximal, characteristic, complemented
12C2×#16864653C2212yescomplemented
13C22×D1084—21no—
14C2×D4×D271C2212yes—
15#22576184—28no—
16C2×#16864643C2212yescomplemented
17C2×#16864642C2211yescharacteristic
18C2×#16864639C2212yescomplemented
19C2×#16864639C2212yescomplemented
20C2×#16864637C2212yescomplemented
21#22576157C2211yescharacteristic
22#22576156—21no—
23#22576135C2212yes—
24#22576134—21no—
25#22576132—22no—
26#22576131C2212yescomplemented
27#22576130—22no—
28C4×Dic542C2211yescharacteristic, complemented
29C4×D1084C2211yescharacteristic
30D4×Dic271—21no—
31#22576091C2211yescharacteristic, complemented
32#22576083C2211yescharacteristic, complemented
33#22576082C2212yescomplemented
34#22576081C2211yescharacteristic
35(C2×D271)⋊C8C2212yescomplemented
36C4×C271⋊C8C2211yescharacteristic, complemented
37C2×D8×C271C2212yes—
38#127×C271C2211yescharacteristic, complemented
39C4×D4×C271C2211yescharacteristic
40C4⋊C8×C271C2211yescharacteristic, complemented
41#102×C271C2212yescomplemented
42C23×D271—22no—
43#16864653—24no—
44#16864653—42no—
45C2×Dic542C2311yescharacteristic
46C2×D1084—22no—
47C2×D1084—21no—
48C2×D1084—21no—
49C2×D1084—44no—
50C2×D1084D411yescharacteristic
51C2×D1084C2312yes—
52C2×C22⋊D271—22no—
53C2×C22⋊D271—22no—
54C2×C22⋊D271—22no—
55D4×D271—24no—
56D4×D271—42no—
57C22×Dic271—22no—
58C2×C4×D271C2312yes—
59#16864643—24no—
60#16864642D414yescomplemented
61#16864639—24no—
62#16864639—42no—
63#16864637—24no—
64C2×C271⋊C8C2311yescharacteristic
65C2×C271⋊C8C2311yescharacteristic
66C4⋊Dic271D411yescharacteristic
67C4⋊Dic271C2311yescharacteristic
68Dic271⋊C4—21no—
69C22⋊Dic271—22no—
70(C2×D271)⋊C4—22no—
71(C2×D271)⋊C4D412yes—
72C4×Dic271—21no—
73C2×D4×C271—21no—
74C2×D4×C271—21no—
75C2×D4×C271C2312yes—
76C2×D4×C271C2311yescharacteristic
77C22×C4×C271—21no—
78D8×C271—24no—
79C2×C8×C271C2312yes—
80C4⋊C4×C271C2311yescharacteristic
81C22⋊C4×C271—21no—
82C42×C271C2311yescharacteristic
83C22×D271—22no—
84C22×D271—44no—
85C22×D271—44no—
86C22×D271—82no—
87C22×D271C2×D412yes—
88Dic542—21no—
89Dic542C2×D412yescomplemented
90D1084—22no—
91D1084—22no—
92D1084—22no—
93D1084—44no—
94D1084—42no—
95D1084—81no—
96C22⋊D271—44no—
97C22⋊D271—44no—
98C22⋊D271—82no—
99C2×Dic271—21no—
100C2×Dic271—21no—
101C2×Dic271—44no—
102C2×Dic271C2×D412yes—
103C4×D271—22no—
104C4×D271—22no—
105C271⋊C8—21no—
106C271⋊C8C2×D412yescomplemented
107C23×C271—21no—
108C23×C271—22no—
109D4×C271—22no—
110D4×C271—21no—
111D4×C271—22no—
112D4×C271—42no—
113D4×C271—81no—
114D4×C271C2×D412yes—
115C2×C4×C271—21no—
116C2×C4×C271—42no—
117C2×C4×C271C2×D411yescharacteristic
118C2×C4×C271C2×D411yescharacteristic
119C2×C4×C271C2411yescharacteristic
120C8×C271—22no—
121C22×C271—22no—
122C22×C271—41no—
123C22×C271—41no—
124C22×C271—42no—
125C22×C271—42no—
126C22×C271—42no—
127C22×C271C22×D411yescharacteristic
128C2×D271—22no—
129C2×D271—22no—
130C2×D271—22no—
131C2×D271—42no—
132C2×D271—42no—
133C2×D271—42no—
134C2×D271—88no—
135C4×C271—21no—
136C4×C271—41no—
137C4×C271AGL(1,ℤ/8)12yescomplemented
138C4×C271C22×D411yescharacteristic
139C4×C271H(5,𝔽2)11yescharacteristic
140Dic271—22no—
141Dic271—22no—
142Dic271—41no—
143C2×C271—42no—
144C2×C271—82no—
145C2×C271D4211yescharacteristic
146C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C271#57611yescharacteristic
148D271—42no—
149D271—82no—
150C271#327411yescharacteristic, complemented, Sylow
151#3274—2711nomaximal, Sylow
152C2×AGL(1,ℤ/8)—2712no—
153#550—2711no—
154D42—2711no—
155#493—2711no—
156#469—2712no—
157#460—2711no—
158#459D27111yescharacteristic, complemented
159#449—2712no—
160#447—2712no—
161C4×SD16—2711no—
162#436—2711no—
163C2×#41—2712no—
164C22×D4—2712no—
165C22×D4—5421no—
166AGL(1,ℤ/8)—5428no—
167C2×SD16—2711no—
168C2×SD16—2712no—
169C2×D8—2712no—
170C2×D8—2712no—
171C2×D8C2×D27112yes—
172C2×M16—2712no—
173#127—2711no—
174#127—5421no—
175#127C2×D27111yescharacteristic, complemented
176#121—2712no—
177#121—2712no—
178#121—5422no—
179#121—5421no—
180#120—5422no—
181C4×Q8—2711no—
182C4×D4—2711no—
183C4×D4—5421no—
184C4×D4C2×D27111yescharacteristic
185#106—2711no—
186C4⋊C8C2×D27111yescharacteristic, complemented
187#103—2711no—
188#102—2711no—
189#102—2712no—
190#102C2×D27112yescomplemented
191C22⋊C8—2712no—
192C4×C8—2711no—
193C24—5422no—
194#41—5424no—
195#41—10842no—
196C2×Q8—2711no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—2711no—
200C2×D4—2712no—
201C2×D4—5422no—
202C2×D4—5422no—
203C2×D4—5424no—
204C2×D4—5422no—
205C2×D4—5422no—
206C2×D4—5421no—
207C2×D4—5421no—
208C2×D4—10844no—
209C2×D4—10842no—
210C2×D4C22×D27112yes—
211C2×D4C22×D27111yescharacteristic
212C22×C4—21no—
213C22×C4—2712no—
214C22×C4—5422no—
215SD16—2714no—
216SD16—5424no—
217D8—24no—
218D8—5424no—
219D8—10842no—
220M16—5424no—
221C2×C8—2711no—
222C2×C8—2711no—
223C2×C8C22×D27112yes—
224C4⋊C4—2711no—
225C4⋊C4—2711no—
226C4⋊C4—5421no—
227C4⋊C4C22×D27111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2712no—
230C22⋊C4—5422no—
231C22⋊C4—5422no—
232C42—5421no—
233C42C22×D27111yescharacteristic
234C23—21no—
235C23—22no—
236C23—2712no—
237C23—5422no—
238C23—10844no—
239C23—10844no—
240C23—21682no—
241Q8—2712no—
242Q8—5421no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—5422no—
249D4—5422no—
250D4—5422no—
251D4—10844no—
252D4—10844no—
253D4—10844no—
254D4—10842no—
255D4—21681no—
256D4—21682no—
257D4D4×D27112yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—2712no—
261C2×C4—5421no—
262C2×C4—5422no—
263C2×C4—5422no—
264C2×C4—5421no—
265C2×C4—10844no—
266C2×C4D4×D27111yescharacteristic
267C2×C4D4×D27111yescharacteristic
268C2×C4C23×D27111yescharacteristic
269C8—22no—
270C8—2712no—
271C8—5421no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—5422no—
279C22—5422no—
280C22—5422no—
281C22—10842no—
282C22—10842no—
283C22—10842no—
284C22—21688no—
285C22C2×D4×D27111yescharacteristic
286C4—21no—
287C4—41no—
288C4—5422no—
289C4—5422no—
290C4—10841no—
291C4#2257618412yescomplemented
292C4C2×D4×D27111yescharacteristic
293C4#2257620611yescharacteristic
294C2—42no—
295C2—82no—
296C2—10842no—
297C2—21682no—
298C2#3177453111yescharacteristic
299C2C2×#2257618411yescharacteristic
300C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#459×C271C2
socleC22×C271C22×D4
hypercenter#459D271
hypocenterC271#3274
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. #31774567C2semidirect
3. D4×D1084C2semidirect
4. #31774273C2semidirect
5. #31774149C2semidirect
6. #31774098C2semidirect
7. #31774043C2semidirect
8. #31774029C2semidirect
9. C4×#16864642C2semidirect
10. #31773974C2semidirect
11. #459×C271C2semidirect
12. C2×#16864653C22semidirect
16. C2×#16864643C22semidirect
18. C2×#16864639C22semidirect
19. C2×#16864639C22semidirect
20. C2×#16864637C22semidirect
26. #22576131C22semidirect
28. C4×Dic542C22semidirect
31. #22576091C22semidirect
32. #22576083C22semidirect
33. #22576082C22semidirect
35. (C2×D271)⋊C8C22semidirect
36. C4×C271⋊C8C22semidirect
38. #127×C271C22semidirect
40. C4⋊C8×C271C22semidirect
41. #102×C271C22semidirect
60. #16864642D4semidirect
89. Dic542C2×D4semidirect
106. C271⋊C8C2×D4semidirect
137. C4×C271AGL(1,ℤ/8)semidirect
150. C271#3274semidirect
158. #459D271semidirect
175. #127C2×D271semidirect
186. C4⋊C8C2×D271semidirect
190. #102C2×D271semidirect
291. C4#22576184semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
#317745672
D4×D10842
#317742732
#317741492
#317740982
#317740432
#317740292
C4×#168646422
#317739742
#459×C2712
#3274271