Finite Group Databasebeta

← #46845610

Subgroups — #46845610

Order 34688.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16864655C211yesmaximal, characteristic, complemented
2C2×#22576191C211yesmaximal, characteristic, complemented
3C2×#22576188C211yesmaximal, characteristic, complemented
4#31774409C211yesmaximal, characteristic, complemented
5#31774062C214yesmaximal, complemented
6#31774045C214yesmaximal, complemented
7#31773901C211yesmaximal, characteristic, complemented
8#31773860C211yesmaximal, characteristic, complemented
9#419×C271C211yesmaximal, characteristic, complemented
10C2×#16864655—24no—
11C2×#16864655—22no—
12C2×#16864655—24no—
13C2×#16864655C2214yes—
14C2×#16864654C2211yescharacteristic
15C22×D1084—21no—
16C22×D1084C2211yescharacteristic
17C22×C4×D271—21no—
18C22×C4×D271C2211yescharacteristic
19#22576191—24no—
20#22576188—24no—
21C2×#16864644C2212yescomplemented
22C2×#16864643C2212yescomplemented
23C2×#16864641C2212yescomplemented
24C2×D2168C2212yescomplemented
25C2×#16864638C2211yescharacteristic
26#22576144—21no—
27#22576140C2212yescomplemented
28#22576135C2212yescomplemented
29#22576130—21no—
30C4×D1084—22no—
31#22576080C2214yescomplemented
32(C2×D271)⋊C8C2214yescomplemented
33C22×Q8×C271C2211yescharacteristic
34C2×M16×C271C2211yescharacteristic
35#117×C271C2211yescharacteristic
36#103×C271C2214yescomplemented
37C23×D271—21no—
38C23×D271D411yescharacteristic
39#16864655—216no—
40#16864655—28no—
41#16864655—28no—
42#16864654—22no—
43#16864654—22no—
44C2×Dic542C2311yescharacteristic
45C2×D1084—28no—
46C2×D1084—24no—
47C2×D1084—22no—
48C2×D1084—21no—
49C2×D1084—22no—
50C2×D1084D414yescomplemented
51C2×D1084C2311yescharacteristic
52C2×D1084C2312yes—
53C2×C22⋊D271—21no—
54C22×Dic271D411yescharacteristic
55C2×C4×D271—28no—
56C2×C4×D271—24no—
57C2×C4×D271—22no—
58C2×C4×D271—22no—
59C2×C4×D271—21no—
60C2×C4×D271D412yes—
61C2×C4×D271C2312yes—
62#16864644—24no—
63#16864643—24no—
64#16864641—24no—
65D2168—24no—
66#16864638—22no—
67C2×C271⋊C8C2312yes—
68C4⋊Dic271—22no—
69(C2×D271)⋊C4—22no—
70(C2×D271)⋊C4—22no—
71C2×Q8×C271—22no—
72C2×Q8×C271—21no—
73C2×Q8×C271D414yescomplemented
74C2×Q8×C271C2312yes—
75C22×C4×C271—21no—
76C22×C4×C271C2311yescharacteristic
77M16×C271—22no—
78C2×C8×C271C2312yes—
79C4⋊C4×C271C2312yes—
80C22⋊C4×C271—21no—
81C42×C271—21no—
82C22×D271—22no—
83C22×D271—24no—
84C22×D271—22no—
85C22×D271—21no—
86C22×D271—42no—
87C22×D271—44no—
88C22×D271C2×D412yes—
89Dic542—21no—
90Dic542—21no—
91D1084—216no—
92D1084—24no—
93D1084—22no—
94D1084—21no—
95D1084—21no—
96D1084—44no—
97D1084C2×D414yes—
98C22⋊D271—42no—
99C2×Dic271—22no—
100C2×Dic271—21no—
101C2×Dic271C2×D412yes—
102C4×D271—216no—
103C4×D271—24no—
104C4×D271—22no—
105C4×D271—22no—
106C4×D271—42no—
107C271⋊C8—22no—
108C23×C271C2×D411yescharacteristic
109Q8×C271—24no—
110Q8×C271—22no—
111Q8×C271C2×D414yes—
112C2×C4×C271—24no—
113C2×C4×C271—22no—
114C2×C4×C271—22no—
115C2×C4×C271C2×D411yescharacteristic
116C2×C4×C271C2×D412yes—
117C2×C4×C271C2×D412yes—
118C2×C4×C271C2411yescharacteristic
119C8×C271—22no—
120C22×C271—22no—
121C22×C271#12012yes—
122C22×C271C22×D411yescharacteristic
123C2×D271—22no—
124C2×D271—24no—
125C2×D271—22no—
126C2×D271—22no—
127C2×D271—44no—
128C2×D271—48no—
129C2×D271—44no—
130C2×D271—41no—
131C2×D271—42no—
132C4×C271—24no—
133C4×C271—42no—
134C4×C271#12012yes—
135C4×C271C22×D411yescharacteristic
136C4×C271C22×D411yescharacteristic
137Dic271—22no—
138Dic271—41no—
139C2×C271—22no—
140C2×C271C2×#12011yescharacteristic
141C2×C271#57512yes—
142D271—42no—
143D271—44no—
144D271—81no—
145C271#300511yescharacteristic, complemented, Sylow
146#3005—2711nomaximal, Sylow
147C22×#41—2711no—
148C2×#137—2711no—
149C2×AGL(1,ℤ/8)—2711no—
150#525—2711no—
151#452—2714no—
152#449—2714no—
153#419D27111yescharacteristic, complemented
154#418—2711no—
155#407—2711no—
156C2×#41—2714no—
157C2×#41—2711no—
158C2×#41—5422no—
159C2×#41—5424no—
160C2×#41—5424no—
161C22×Q8C2×D27111yescharacteristic
162C22×D4—2711no—
163C22×D4—5421no—
164C23×C4—2711no—
165C23×C4—5421no—
166#137—5424no—
167AGL(1,ℤ/8)—5424no—
168C2×Q16—2712no—
169C2×SD16—2712no—
170C2×SD16—2712no—
171C2×D8—2712no—
172C2×M16—2711no—
173C2×M16C2×D27111yescharacteristic
174#123—5421no—
175C22⋊Q8—2712no—
176#121—2712no—
177#120—5421no—
178C4×D4—5422no—
179#117C2×D27111yescharacteristic
180#103C2×D27114yescomplemented
181#102—2714no—
182C22⋊C8—2714no—
183C24—2711no—
184C24—5421no—
185#41—5428no—
186#41—54216no—
187#41—5428no—
188#41—5422no—
189#41—5422no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—2711no—
193C2×Q8D108414yescomplemented
194C2×Q8C22×D27112yes—
195C2×D4—2714no—
196C2×D4—2712no—
197C2×D4—2711no—
198C2×D4—5424no—
199C2×D4—5428no—
200C2×D4—5422no—
201C2×D4—5421no—
202C2×D4—5421no—
203C2×D4—5422no—
204C22×C4—21no—
205C22×C4—2711no—
206C22×C4—2712no—
207C22×C4—2712no—
208C22×C4—5424no—
209C22×C4—5428no—
210C22×C4—5422no—
211C22×C4—5422no—
212C22×C4—5421no—
213C22×C4C22×D27111yescharacteristic
214Q16—5424no—
215SD16—5424no—
216SD16—5424no—
217D8—5424no—
218M16—22no—
219M16—5422no—
220C2×C8—2712no—
221C2×C8C22×D27112yes—
222C4⋊C4—5422no—
223C4⋊C4C22×D27112yes—
224C22⋊C4—21no—
225C22⋊C4—5422no—
226C22⋊C4—5422no—
227C42—21no—
228C23—2712no—
229C23—5424no—
230C23—5422no—
231C23—5422no—
232C23—5421no—
233C23—10842no—
234C23—10844no—
235C23D4×D27111yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—5421no—
239Q8—5421no—
240Q8C2×D108414yes—
241D4—2714no—
242D4—54216no—
243D4—5424no—
244D4—5421no—
245D4—5422no—
246D4—5421no—
247D4—10844no—
248D4—10842no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—2712no—
253C2×C4—54216no—
254C2×C4—5422no—
255C2×C4—5424no—
256C2×C4—5422no—
257C2×C4—5422no—
258C2×C4—5421no—
259C2×C4—10842no—
260C2×C4D4×D27111yescharacteristic
261C2×C4D4×D27112yes—
262C2×C4C2×D108412yes—
263C2×C4C23×D27111yescharacteristic
264C8—22no—
265C8—5422no—
266C22—22no—
267C22—5424no—
268C22—5422no—
269C22—5422no—
270C22—5422no—
271C22—10844no—
272C22—10848no—
273C22—10844no—
274C22—10841no—
275C22—10842no—
276C22#2257613012yes—
277C22C2×D4×D27111yescharacteristic
278C4—24no—
279C4—42no—
280C4—5422no—
281C4—10841no—
282C4#2257613012yes—
283C4C2×D4×D27111yescharacteristic
284C4C22×D108411yescharacteristic
285C2—22no—
286C2—10842no—
287C2—10844no—
288C2—21681no—
289C2C2×#2257613011yescharacteristic
290C2#3177469612yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#419×C271C2
socleC22×C271C22×D4
hypercenter#419D271
hypocenterC271#3005
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#16864655C2semidirect
2. C2×#22576191C2semidirect
3. C2×#22576188C2semidirect
4. #31774409C2semidirect
5. #31774062C2semidirect
6. #31774045C2semidirect
7. #31773901C2semidirect
8. #31773860C2semidirect
9. #419×C271C2semidirect
21. C2×#16864644C22semidirect
22. C2×#16864643C22semidirect
23. C2×#16864641C22semidirect
24. C2×D2168C22semidirect
27. #22576140C22semidirect
28. #22576135C22semidirect
31. #22576080C22semidirect
32. (C2×D271)⋊C8C22semidirect
36. #103×C271C22semidirect
50. C2×D1084D4semidirect
73. C2×Q8×C271D4semidirect
145. C271#3005semidirect
153. #419D271semidirect
180. #103C2×D271semidirect
193. C2×Q8D1084semidirect

Maximal subgroups

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

subgroupindex
C22×#168646552
C2×#225761912
C2×#225761882
#317744092
#317740622
#317740452
#317739012
#317738602
#419×C2712
#3005271