Finite Group Databasebeta

← #46845604

Subgroups — #46845604

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×#16864654C211yesmaximal, characteristic, complemented
2C2×#22576190C211yesmaximal, characteristic, complemented
3C2×#22576185C211yesmaximal, characteristic, complemented
4#31774057C214yesmaximal, complemented
5#31774040C214yesmaximal, complemented
6#31773905C211yesmaximal, characteristic, complemented
7#31773901C211yesmaximal, characteristic, complemented
8#31773861C211yesmaximal, characteristic, complemented
9#525×C271C211yesmaximal, characteristic, complemented
10C2×#16864654—24no—
11C2×#16864654—24no—
12C2×#16864654—22no—
13C2×#16864654C2214yes—
14C22×Dic542C2211yescharacteristic
15C22×D1084C2211yescharacteristic
16C22×C22⋊D271—21no—
17C22×C4×D271—21no—
18#22576190—24no—
19#22576185—24no—
20C2×#16864644C2212yescomplemented
21C2×#16864642C2212yescomplemented
22C2×#16864641C2212yescomplemented
23C2×#16864639C2212yescomplemented
24C2×#16864638C2211yescharacteristic
25C2×#16864638C2211yescharacteristic
26#22576083C2214yescomplemented
27#22576080C2214yescomplemented
28C22⋊(C271⋊C8)C2214yescomplemented
29C2×#41×C271C2211yescharacteristic
30C23×C4×C271C2211yescharacteristic
31#123×C271—21no—
32C22⋊Q8×C271C2212yescomplemented
33#121×C271C2212yescomplemented
34#120×C271—21no—
35C4×D4×C271—22no—
36#117×C271C2211yescharacteristic
37C23×D271—21no—
38#16864654—216no—
39#16864654—28no—
40#16864654—28no—
41C2×Dic542—22no—
42C2×Dic542—21no—
43C2×Dic542D414yescomplemented
44C2×Dic542C2312yes—
45C2×D1084—22no—
46C2×D1084—21no—
47C2×D1084D414yescomplemented
48C2×D1084C2312yes—
49C2×C22⋊D271—28no—
50C2×C22⋊D271—24no—
51C22×Dic271—21no—
52C2×C4×D271—28no—
53C2×C4×D271—24no—
54#16864644—24no—
55#16864642—24no—
56#16864641—24no—
57#16864639—24no—
58#16864638—22no—
59#16864638—22no—
60C2×C271⋊C8C2312yes—
61C2×C271⋊C8C2312yes—
62C24×C271D411yescharacteristic
63#41×C271—22no—
64#41×C271—22no—
65C2×Q8×C271C2311yescharacteristic
66C2×D4×C271—22no—
67C2×D4×C271—21no—
68C2×D4×C271C2311yescharacteristic
69C22×C4×C271—22no—
70C22×C4×C271—22no—
71C22×C4×C271—21no—
72C22×C4×C271D411yescharacteristic
73C22×C4×C271D412yes—
74C22×C4×C271C2312yes—
75C22×C4×C271C2311yescharacteristic
76C4⋊C4×C271—22no—
77C4⋊C4×C271C2312yes—
78C22⋊C4×C271—21no—
79C22⋊C4×C271—22no—
80C22⋊C4×C271—22no—
81C42×C271—21no—
82C22×D271—24no—
83C22×D271—22no—
84C22×D271—44no—
85Dic542—24no—
86Dic542—22no—
87Dic542C2×D414yes—
88D1084—24no—
89D1084—22no—
90D1084C2×D414yes—
91C22⋊D271—216no—
92C2×Dic271—24no—
93C2×Dic271—22no—
94C4×D271—216no—
95C271⋊C8—22no—
96C271⋊C8—22no—
97C23×C271—22no—
98C23×C271—21no—
99C23×C271—42no—
100C23×C271C2×D412yes—
101C23×C271C2×D411yescharacteristic
102Q8×C271—21no—
103Q8×C271—21no—
104D4×C271—21no—
105D4×C271—21no—
106D4×C271—44no—
107D4×C271—42no—
108C2×C4×C271—22no—
109C2×C4×C271—24no—
110C2×C4×C271—22no—
111C2×C4×C271—22no—
112C2×C4×C271—22no—
113C2×C4×C271—21no—
114C2×C4×C271—42no—
115C2×C4×C271C2×D412yes—
116C2×C4×C271C2×D412yes—
117C2×C4×C271C2×D412yes—
118C2×C4×C271C2×D411yescharacteristic
119C2×C4×C271C2411yescharacteristic
120C22×C271—22no—
121C22×C271—22no—
122C22×C271—22no—
123C22×C271—22no—
124C22×C271—44no—
125C22×C271—41no—
126C22×C271—42no—
127C22×C271#12012yes—
128C22×C271C22×D411yescharacteristic
129C2×D271—24no—
130C2×D271—48no—
131C2×D271—44no—
132C4×C271—22no—
133C4×C271—41no—
134C4×C271—42no—
135C4×C271#12012yes—
136C4×C271C22×D411yescharacteristic
137C4×C271C22×D411yescharacteristic
138Dic271—24no—
139C2×C271—22no—
140C2×C271—42no—
141C2×C271—81no—
142C2×C271C2×#12011yescharacteristic
143C2×C271#57512yes—
144D271—44no—
145C271#300511yescharacteristic, complemented, Sylow
146#3005—2711nomaximal, Sylow
147C22×#41—2711no—
148C2×#137—2711no—
149C2×AGL(1,ℤ/8)—2711no—
150#525D27111yescharacteristic, complemented
151#452—2714no—
152#449—2714no—
153#419—2711no—
154#418—2711no—
155#407—2711no—
156C2×#41—2714no—
157C2×#41—5424no—
158C2×#41—5424no—
159C2×#41—5422no—
160C2×#41C2×D27111yescharacteristic
161C22×Q8—2711no—
162C22×D4—2711no—
163C22×D4—5421no—
164C23×C4—5421no—
165C23×C4C2×D27111yescharacteristic
166#137—5424no—
167AGL(1,ℤ/8)—5424no—
168C2×Q16—2712no—
169C2×SD16—2712no—
170C2×SD16—2712no—
171C2×D8—2712no—
172C2×M16—2711no—
173C2×M16—2711no—
174#123—21no—
175C22⋊Q8C2×D27112yescomplemented
176#121C2×D27112yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D27111yescharacteristic
180#103—2714no—
181#102—2714no—
182C22⋊C8—2714no—
183C24—5421no—
184C24C22⋊D27111yescharacteristic
185#41—22no—
186#41—22no—
187#41—5428no—
188#41—5428no—
189#41—54216no—
190C2×Q8—2714no—
191C2×Q8—2712no—
192C2×Q8—5422no—
193C2×Q8—5421no—
194C2×Q8C22×D27111yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—2714no—
198C2×D4—2712no—
199C2×D4—5428no—
200C2×D4—5424no—
201C2×D4—5422no—
202C2×D4—5421no—
203C2×D4C22×D27111yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—5428no—
208C22×C4—5421no—
209C22×C4—5424no—
210C22×C4C22⋊D27112yes—
211C22×C4C22⋊D27111yescharacteristic
212C22×C4C22×D27112yes—
213C22×C4C22×D27111yescharacteristic
214Q16—5424no—
215SD16—5424no—
216SD16—5424no—
217D8—5424no—
218M16—5422no—
219M16—5422no—
220C2×C8—2712no—
221C2×C8—2712no—
222C4⋊C4—22no—
223C4⋊C4C22×D27112yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—5424no—
232C23—5422no—
233C23—10844no—
234C23C2×C22⋊D27112yes—
235C23C2×C22⋊D27111yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—2714no—
239Q8—5424no—
240Q8—5422no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—2714no—
246D4—54216no—
247D4—5424no—
248D4—5422no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—54216no—
257C2×C4—5424no—
258C2×C4—5422no—
259C2×C4D4×D27112yes—
260C2×C4D4×D27112yes—
261C2×C4C2×C22⋊D27112yes—
262C2×C4C2×C22⋊D27111yescharacteristic
263C2×C4C23×D27111yescharacteristic
264C8—5422no—
265C8—5422no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—5424no—
274C22—10848no—
275C22—10844no—
276C22#2257612812yes—
277C22C22×C22⋊D27111yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—5424no—
282C4#2257612812yes—
283C4C2×D4×D27111yescharacteristic
284C4C2×D4×D27111yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—10844no—
289C2C2×#2257612811yescharacteristic
290C2#3177469512yes—
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#525×C271C2
socleC22×C271C22×D4
hypercenter#525D271
hypocenterC271#3005
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#16864654C2semidirect
2. C2×#22576190C2semidirect
3. C2×#22576185C2semidirect
4. #31774057C2semidirect
5. #31774040C2semidirect
6. #31773905C2semidirect
7. #31773901C2semidirect
8. #31773861C2semidirect
9. #525×C271C2semidirect
20. C2×#16864644C22semidirect
21. C2×#16864642C22semidirect
22. C2×#16864641C22semidirect
23. C2×#16864639C22semidirect
26. #22576083C22semidirect
27. #22576080C22semidirect
28. C22⋊(C271⋊C8)C22semidirect
32. C22⋊Q8×C271C22semidirect
33. #121×C271C22semidirect
43. C2×Dic542D4semidirect
47. C2×D1084D4semidirect
145. C271#3005semidirect
150. #525D271semidirect
175. C22⋊Q8C2×D271semidirect
176. #121C2×D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646542
C2×#225761902
C2×#225761852
#317740572
#317740402
#317739052
#317739012
#317738612
#525×C2712
#3005271