Finite Group Databasebeta

← #46845605

Subgroups — #46845605

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×#22576185C211yesmaximal, characteristic, complemented
3#31774404C211yesmaximal, characteristic, complemented
4#31774058C214yesmaximal, complemented
5#31774041C214yesmaximal, complemented
6#31773906C211yesmaximal, characteristic, complemented
7#31773903C211yesmaximal, characteristic, complemented
8#31773860C211yesmaximal, characteristic, complemented
9C2×#137×C271C211yesmaximal, characteristic, complemented
10C2×#16864655—24no—
11C2×#16864655—22no—
12C2×#16864655—24no—
13C2×#16864655C2214yes—
14C22×D1084—21no—
15C22×D1084C2211yescharacteristic
16C22×C4×D271—21no—
17C22×C4×D271C2211yescharacteristic
18#22576185—24no—
19C2×#16864641C2212yescomplemented
20C2×#16864639C2212yescomplemented
21C2×#16864638C2211yescharacteristic
22#22576141—21no—
23#22576136C2212yescomplemented
24#22576131C2212yescomplemented
25#22576128—21no—
26C4×C22⋊D271—22no—
27#22576119C2211yescharacteristic
28#22576084C2214yescomplemented
29#22576082C2214yescomplemented
30(C2×D271)⋊C8C2214yescomplemented
31C2×#41×C271C2211yescharacteristic
32C22×Q8×C271C2211yescharacteristic
33#137×C271—24no—
34C2×Q16×C271C2212yescomplemented
35C2×SD16×C271C2212yescomplemented
36C2×M16×C271C2211yescharacteristic
37C23×D271—21no—
38C23×D271D411yescharacteristic
39#16864655—216no—
40#16864655—28no—
41#16864655—28no—
42C2×D1084—28no—
43C2×D1084—24no—
44C2×D1084—21no—
45C2×D1084—22no—
46C2×D1084D414yescomplemented
47C2×D1084C2312yes—
48C2×C22⋊D271—22no—
49C22×Dic271D411yescharacteristic
50C2×C4×D271—28no—
51C2×C4×D271—24no—
52C2×C4×D271—22no—
53C2×C4×D271—22no—
54C2×C4×D271D412yes—
55C2×C4×D271C2312yes—
56#16864641—24no—
57#16864639—24no—
58#16864638—22no—
59C2×C271⋊C8C2312yes—
60C4⋊Dic271C2312yes—
61Dic271⋊C4—22no—
62C22⋊Dic271—22no—
63C22⋊Dic271—21no—
64(C2×D271)⋊C4—22no—
65C4×Dic271—21no—
66#41×C271—22no—
67#41×C271—22no—
68C2×Q8×C271—22no—
69C2×Q8×C271—21no—
70C2×Q8×C271D414yescomplemented
71C2×Q8×C271C2311yescharacteristic
72C2×Q8×C271C2312yes—
73C2×D4×C271—21no—
74C2×D4×C271C2311yescharacteristic
75C22×C4×C271—21no—
76C22×C4×C271—21no—
77C22×C4×C271C2311yescharacteristic
78Q16×C271—24no—
79SD16×C271—24no—
80M16×C271—22no—
81C2×C8×C271C2312yes—
82C22×D271—22no—
83C22×D271—24no—
84C22×D271—22no—
85C22×D271—42no—
86C22×D271—44no—
87C22×D271C2×D412yes—
88D1084—216no—
89D1084—24no—
90D1084—22no—
91D1084C2×D414yes—
92C22⋊D271—44no—
93C2×Dic271—22no—
94C2×Dic271—22no—
95C2×Dic271C2×D412yes—
96C4×D271—216no—
97C4×D271—24no—
98C4×D271—22no—
99C4×D271—22no—
100C271⋊C8—22no—
101C23×C271—21no—
102C23×C271C2×D411yescharacteristic
103Q8×C271—24no—
104Q8×C271—22no—
105Q8×C271—21no—
106Q8×C271—21no—
107Q8×C271C2×D414yes—
108D4×C271—21no—
109D4×C271—21no—
110D4×C271—42no—
111C2×C4×C271—24no—
112C2×C4×C271—22no—
113C2×C4×C271—21no—
114C2×C4×C271—42no—
115C2×C4×C271C2×D411yescharacteristic
116C2×C4×C271C2×D412yes—
117C2×C4×C271C2×D412yes—
118C2×C4×C271C2411yescharacteristic
119C8×C271—22no—
120C22×C271—22no—
121C22×C271—41no—
122C22×C271—42no—
123C22×C271#12012yes—
124C22×C271C22×D411yescharacteristic
125C2×D271—22no—
126C2×D271—24no—
127C2×D271—22no—
128C2×D271—22no—
129C2×D271—44no—
130C2×D271—48no—
131C2×D271—44no—
132C4×C271—24no—
133C4×C271—41no—
134C4×C271#12012yes—
135C4×C271C22×D411yescharacteristic
136C4×C271C22×D411yescharacteristic
137Dic271—22no—
138Dic271—42no—
139C2×C271—22no—
140C2×C271—81no—
141C2×C271C2×#12011yescharacteristic
142C2×C271#57512yes—
143D271—42no—
144D271—44no—
145C271#300511yescharacteristic, complemented, Sylow
146#3005—2711nomaximal, Sylow
147C22×#41—2711no—
148C2×#137D27111yescharacteristic, complemented
149C2×AGL(1,ℤ/8)—2711no—
150#525—2711no—
151#452—2714no—
152#449—2714no—
153#419—2711no—
154#418—2711no—
155#407—2711no—
156C2×#41—2714no—
157C2×#41—5422no—
158C2×#41—5424no—
159C2×#41—5424no—
160C2×#41C2×D27111yescharacteristic
161C22×Q8C2×D27111yescharacteristic
162C22×D4—2711no—
163C22×D4—5421no—
164C23×C4—2711no—
165C23×C4—5421no—
166#137—24no—
167AGL(1,ℤ/8)—5424no—
168C2×Q16C2×D27112yescomplemented
169C2×SD16—2712no—
170C2×SD16C2×D27112yescomplemented
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#117—2711no—
180#103—2714no—
181#102—2714no—
182C22⋊C8—2714no—
183C24—2711no—
184C24—5421no—
185#41—22no—
186#41—22no—
187#41—5428no—
188#41—54216no—
189#41—5428no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8C22⋊D27114yescomplemented
193C2×Q8C22×D27111yescharacteristic
194C2×Q8C22×D27112yes—
195C2×D4—21no—
196C2×D4—2714no—
197C2×D4—2712no—
198C2×D4—5424no—
199C2×D4—5428no—
200C2×D4—5422no—
201C2×D4—5421no—
202C2×D4—5422no—
203C2×D4C22×D27111yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—2711no—
207C22×C4—2712no—
208C22×C4—2712no—
209C22×C4—5424no—
210C22×C4—5428no—
211C22×C4—5422no—
212C22×C4—5422no—
213C22×C4C22×D27111yescharacteristic
214Q16—24no—
215SD16—24no—
216SD16—5424no—
217D8—5424no—
218M16—22no—
219M16—5422no—
220C2×C8—2712no—
221C2×C8C22×D27112yes—
222C4⋊C4—2712no—
223C4⋊C4—5422no—
224C22⋊C4—5422no—
225C22⋊C4—5422no—
226C22⋊C4—5421no—
227C42—5421no—
228C23—21no—
229C23—2712no—
230C23—5424no—
231C23—5422no—
232C23—5422no—
233C23—10842no—
234C23—10844no—
235C23D4×D27111yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—21no—
239Q8—21no—
240Q8C2×C22⋊D27114yes—
241D4—21no—
242D4—21no—
243D4—42no—
244D4—2714no—
245D4—54216no—
246D4—5424no—
247D4—5422no—
248D4—10844no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—2712no—
254C2×C4—54216no—
255C2×C4—5422no—
256C2×C4—5424no—
257C2×C4—5422no—
258C2×C4—5422no—
259C2×C4—5422no—
260C2×C4D4×D27111yescharacteristic
261C2×C4D4×D27112yes—
262C2×C4C2×C22⋊D27112yes—
263C2×C4C23×D27111yescharacteristic
264C8—22no—
265C8—5422no—
266C22—22no—
267C22—41no—
268C22—42no—
269C22—5424no—
270C22—5422no—
271C22—5422no—
272C22—5422no—
273C22—10844no—
274C22—10848no—
275C22—10844no—
276C22#2257612812yes—
277C22C2×D4×D27111yescharacteristic
278C4—24no—
279C4—41no—
280C4—5422no—
281C4—10842no—
282C4#2257612812yes—
283C4C2×D4×D27111yescharacteristic
284C4C22×C22⋊D27111yescharacteristic
285C2—22no—
286C2—81no—
287C2—10842no—
288C2—10844no—
289C2C2×#2257612811yescharacteristic
290C2#3177469612yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#137×C271C2
socleC22×C271C22×D4
hypercenterC2×#137D271
hypocenterC271#3005
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#16864655C2semidirect
2. C2×#22576185C2semidirect
3. #31774404C2semidirect
4. #31774058C2semidirect
5. #31774041C2semidirect
6. #31773906C2semidirect
7. #31773903C2semidirect
8. #31773860C2semidirect
9. C2×#137×C271C2semidirect
19. C2×#16864641C22semidirect
20. C2×#16864639C22semidirect
23. #22576136C22semidirect
24. #22576131C22semidirect
28. #22576084C22semidirect
29. #22576082C22semidirect
30. (C2×D271)⋊C8C22semidirect
34. C2×Q16×C271C22semidirect
35. C2×SD16×C271C22semidirect
46. C2×D1084D4semidirect
70. C2×Q8×C271D4semidirect
145. C271#3005semidirect
148. C2×#137D271semidirect
168. C2×Q16C2×D271semidirect
170. C2×SD16C2×D271semidirect
192. C2×Q8C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646552
C2×#225761852
#317744042
#317740582
#317740412
#317739062
#317739032
#317738602
C2×#137×C2712
#3005271