Finite Group Databasebeta

← #46847959

Subgroups — #46847959

Order 34688.

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

normal subgroups only

This group has 546 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#31774551C211yesmaximal, characteristic, complemented
3#31774532C211yesmaximal, characteristic, complemented
4#31774269C211yesmaximal, characteristic, complemented
5#31774142C212yesmaximal, complemented
6#31774103C211yesmaximal, characteristic, complemented
7#31774049C212yesmaximal, complemented
8#31774039C212yesmaximal, complemented
9C4×#16864639C211yesmaximal, characteristic, complemented
10#31773974C211yesmaximal, characteristic, complemented
11#459×C271C211yesmaximal, characteristic, complemented
12C2×#16864653C2212yescomplemented
13C2×D4×D271—21no—
14C2×D4×D271C2212yes—
15#22576184—28no—
16C2×#16864643C2212yes—
17C2×#16864642C2212yescomplemented
18C2×#16864642C2212yescomplemented
19C2×#16864639C2211yescharacteristic
20C2×#16864637C2212yescomplemented
21#22576160C2211yescharacteristic, complemented
22#22576144—22no—
23#22576140C2212yescomplemented
24#22576134—21no—
25#22576131—21no—
26#22576131C2212yes—
27#22576128—22no—
28C4×D1084C2211yescharacteristic
29C4×D1084C2211yescharacteristic
30C2×C4⋊Dic271—21no—
31#22576093C2211yescharacteristic, complemented
32#22576085C2212yescomplemented
33#22576081C2211yescharacteristic, complemented
34#22576080C2211yescharacteristic
35(C2×D271)⋊C8C2212yescomplemented
36C4×C271⋊C8C2211yescharacteristic, complemented
37C2×D8×C271C2212yescomplemented
38#127×C271C2211yescharacteristic
39C4×D4×C271C2211yescharacteristic, complemented
40C4⋊C8×C271C2211yescharacteristic, complemented
41#102×C271C2212yescomplemented
42C23×D271—22no—
43#16864653—24no—
44#16864653—42no—
45C2×Dic542C2312yes—
46C2×D1084D411yescharacteristic
47C2×D1084C2311yescharacteristic
48C2×C22⋊D271—22no—
49C2×C22⋊D271—22no—
50C2×C22⋊D271—22no—
51D4×D271—24no—
52D4×D271—42no—
53D4×D271—44no—
54C22×Dic271—22no—
55C2×C4×D271—21no—
56C2×C4×D271C2312yes—
57#16864643—24no—
58#16864642—24no—
59#16864642—42no—
60#16864639D414yescomplemented
61#16864637—24no—
62C2×C271⋊C8C2311yescharacteristic
63C2×C271⋊C8C2311yescharacteristic
64C4⋊Dic271—22no—
65C4⋊Dic271—21no—
66C4⋊Dic271—21no—
67C4⋊Dic271D411yescharacteristic
68C4⋊Dic271C2311yescharacteristic
69C22⋊Dic271—22no—
70(C2×D271)⋊C4—22no—
71(C2×D271)⋊C4—21no—
72(C2×D271)⋊C4D412yes—
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—21no—
84C22×D271—44no—
85C22×D271—44no—
86C22×D271—82no—
87C22×D271C2×D412yes—
88Dic542—22no—
89Dic542—22no—
90D1084—22no—
91D1084—21no—
92D1084C2×D412yes—
93C22⋊D271—44no—
94C22⋊D271—44no—
95C22⋊D271—44no—
96C2×Dic271—21no—
97C2×Dic271—22no—
98C2×Dic271—44no—
99C2×Dic271C2×D412yes—
100C4×D271—22no—
101C4×D271—22no—
102C4×D271—42no—
103C271⋊C8—21no—
104C271⋊C8C2×D412yescomplemented
105C23×C271—21no—
106C23×C271—22no—
107D4×C271—22no—
108D4×C271—21no—
109D4×C271—22no—
110D4×C271—42no—
111D4×C271—81no—
112D4×C271C2×D412yescomplemented
113C2×C4×C271—21no—
114C2×C4×C271—42no—
115C2×C4×C271C2×D411yescharacteristic
116C2×C4×C271C2×D411yescharacteristic
117C2×C4×C271C2411yescharacteristic
118C8×C271—22no—
119C22×C271—22no—
120C22×C271—41no—
121C22×C271—41no—
122C22×C271—42no—
123C22×C271—42no—
124C22×C271—42no—
125C22×C271C22×D411yescharacteristic
126C2×D271—22no—
127C2×D271—22no—
128C2×D271—22no—
129C2×D271—22no—
130C2×D271—41no—
131C2×D271—41no—
132C2×D271—88no—
133C4×C271—21no—
134C4×C271—41no—
135C4×C271AGL(1,ℤ/8)12yescomplemented
136C4×C271C22×D411yescharacteristic
137C4×C271H(5,𝔽2)11yescharacteristic
138Dic271—22no—
139Dic271—42no—
140Dic271—41no—
141C2×C271—42no—
142C2×C271—82no—
143C2×C271D4211yescharacteristic
144C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C271#57711yescharacteristic
146D271—42no—
147D271—42no—
148C271#327211yescharacteristic, complemented, Sylow
149#3272—2711nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2712no—
151#547—2711no—
152D42—2711no—
153#492—2711no—
154#468—2712no—
155#461—2711no—
156#459D27111yescharacteristic, complemented
157#450—2712no—
158#449—2712no—
159C4×D8—2711no—
160#436—2711no—
161C2×#41—2712no—
162C22×D4—2712no—
163C22×D4—5421no—
164AGL(1,ℤ/8)—5428no—
165C2×SD16—2712no—
166C2×SD16—2712no—
167C2×SD16—2712no—
168C2×D8—2711no—
169C2×D8C2×D27112yescomplemented
170C2×M16—2712no—
171C4⋊Q8—2711no—
172#127C2×D27111yescharacteristic
173#123—5422no—
174C22⋊Q8—2712no—
175#121—2712no—
176#121—5421no—
177#121—5421no—
178#120—5422no—
179C4×D4—2711no—
180C4×D4—2711no—
181C4×D4C2×D27111yescharacteristic, complemented
182C2×C4⋊C4—5421no—
183#107—2711no—
184C4⋊C8C2×D27111yescharacteristic, complemented
185#103—2712no—
186#102—2711no—
187#102—2711no—
188#102C2×D27112yescomplemented
189C22⋊C8—2712no—
190C4×C8—2711no—
191C24—5422no—
192#41—5424no—
193#41—10842no—
194C2×Q8—2712no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—2711no—
198C2×D4—2711no—
199C2×D4—5422no—
200C2×D4—5424no—
201C2×D4—5422no—
202C2×D4—5422no—
203C2×D4—10842no—
204C2×D4—10844no—
205C2×D4C22×D27112yes—
206C2×D4C22×D27111yescharacteristic
207C22×C4—21no—
208C22×C4—2712no—
209C22×C4—5422no—
210C22×C4—5421no—
211SD16—5424no—
212SD16—5424no—
213SD16—10842no—
214D8—24no—
215D8—2714no—
216M16—5424no—
217C2×C8—2711no—
218C2×C8—2711no—
219C2×C8C22×D27112yes—
220C4⋊C4—2711no—
221C4⋊C4—2711no—
222C4⋊C4—5422no—
223C4⋊C4—5421no—
224C4⋊C4—5421no—
225C4⋊C4C22×D27111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—2712no—
228C22⋊C4—5422no—
229C22⋊C4—5422no—
230C22⋊C4—5421no—
231C42C22×D27111yescharacteristic
232C23—21no—
233C23—22no—
234C23—2712no—
235C23—5421no—
236C23—10844no—
237C23—10844no—
238C23—21682no—
239Q8—5422no—
240Q8—5422no—
241D4—22no—
242D4—21no—
243D4—22no—
244D4—42no—
245D4—81no—
246D4—2712no—
247D4—5422no—
248D4—5421no—
249D4—10844no—
250D4—10844no—
251D4—10844no—
252D4D4×D27112yescomplemented
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—2712no—
256C2×C4—5422no—
257C2×C4—5422no—
258C2×C4—5421no—
259C2×C4—5422no—
260C2×C4—10844no—
261C2×C4—10842no—
262C2×C4D4×D27111yescharacteristic
263C2×C4D4×D27111yescharacteristic
264C2×C4C23×D27111yescharacteristic
265C8—22no—
266C8—2712no—
267C8—5421no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—5422no—
275C22—5422no—
276C22—5422no—
277C22—5422no—
278C22—10841no—
279C22—10841no—
280C22—21688no—
281C22C2×D4×D27111yescharacteristic
282C4—21no—
283C4—41no—
284C4—5422no—
285C4—10842no—
286C4—10841no—
287C4#2257618412yescomplemented
288C4C2×D4×D27111yescharacteristic
289C4#2257620611yescharacteristic
290C2—42no—
291C2—82no—
292C2—10842no—
293C2—10842no—
294C2#3177453111yescharacteristic
295C2C2×#2257618411yescharacteristic
296C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#459×C271C2
socleC22×C271C22×D4
hypercenter#459D271
hypocenterC271#3272
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. #31774551C2semidirect
3. #31774532C2semidirect
4. #31774269C2semidirect
5. #31774142C2semidirect
6. #31774103C2semidirect
7. #31774049C2semidirect
8. #31774039C2semidirect
9. C4×#16864639C2semidirect
10. #31773974C2semidirect
11. #459×C271C2semidirect
12. C2×#16864653C22semidirect
17. C2×#16864642C22semidirect
18. C2×#16864642C22semidirect
20. C2×#16864637C22semidirect
21. #22576160C22semidirect
23. #22576140C22semidirect
31. #22576093C22semidirect
32. #22576085C22semidirect
33. #22576081C22semidirect
35. (C2×D271)⋊C8C22semidirect
36. C4×C271⋊C8C22semidirect
37. C2×D8×C271C22semidirect
39. C4×D4×C271C22semidirect
40. C4⋊C8×C271C22semidirect
41. #102×C271C22semidirect
60. #16864639D4semidirect
104. C271⋊C8C2×D4semidirect
112. D4×C271C2×D4semidirect
135. C4×C271AGL(1,ℤ/8)semidirect
148. C271#3272semidirect
156. #459D271semidirect
169. C2×D8C2×D271semidirect
181. C4×D4C2×D271semidirect
184. C4⋊C8C2×D271semidirect
188. #102C2×D271semidirect
252. D4D4×D271semidirect
287. C4#22576184semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
#317745512
#317745322
#317742692
#317741422
#317741032
#317740492
#317740392
C4×#168646392
#317739742
#459×C2712
#3272271