Finite Group Databasebeta

← #46845589

Subgroups — #46845589

Order 34688.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×D4×D271C211yesmaximal, characteristic, complemented
2C2×#22576185C211yesmaximal, characteristic, complemented
3#31774404C211yesmaximal, characteristic, complemented
4#31774046C214yesmaximal, complemented
5#31774029C214yesmaximal, complemented
6#31773903C211yesmaximal, characteristic, complemented
7#31773902C211yesmaximal, characteristic, complemented
8#31773860C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
10C24×D271—21no—
11C22×D1084C2211yescharacteristic
12C22×C22⋊D271—21no—
13C2×D4×D271—24no—
14C2×D4×D271—22no—
15C2×D4×D271—24no—
16C2×D4×D271C2214yes—
17C22×C4×D271C2211yescharacteristic
18#22576185—24no—
19C2×#16864641C2212yescomplemented
20C2×#16864639C2212yescomplemented
21C2×#16864638C2211yescharacteristic, complemented
22#22576141—21no—
23#22576136C2212yescomplemented
24#22576131C2212yescomplemented
25#22576128—21no—
26C4×C22⋊D271—22no—
27#22576119C2211yescharacteristic, complemented
28#22576082C2214yescomplemented
29#22576081C2214yescomplemented
30(C2×D271)⋊C8C2214yescomplemented
31C2×#41×C271C2211yescharacteristic, complemented
32C22×D4×C271C2211yescharacteristic
33AGL(1,ℤ/8)×C271—24no—
34C2×SD16×C271C2212yescomplemented
35C2×D8×C271C2212yescomplemented
36C2×M16×C271C2211yescharacteristic, complemented
37C23×D271—28no—
38C23×D271—24no—
39C23×D271—21no—
40C23×D271—48no—
41C23×D271D411yescharacteristic
42C2×D1084—21no—
43C2×D1084—22no—
44C2×D1084D414yescomplemented
45C2×D1084C2312yes—
46C2×C22⋊D271—28no—
47C2×C22⋊D271—22no—
48C2×C22⋊D271—24no—
49D4×D271—216no—
50D4×D271—28no—
51D4×D271—28no—
52C22×Dic271D411yescharacteristic
53C2×C4×D271—22no—
54C2×C4×D271—22no—
55C2×C4×D271D412yes—
56C2×C4×D271C2312yes—
57#16864641—24no—
58#16864639—24no—
59#16864638—22no—
60C2×C271⋊C8C2312yescomplemented
61C4⋊Dic271C2312yescomplemented
62Dic271⋊C4—22no—
63C22⋊Dic271—22no—
64C22⋊Dic271—21no—
65(C2×D271)⋊C4—22no—
66C4×Dic271—21no—
67C24×C271—21no—
68#41×C271—22no—
69#41×C271—22no—
70C2×Q8×C271C2311yescharacteristic, complemented
71C2×D4×C271—21no—
72C2×D4×C271—21no—
73C2×D4×C271—22no—
74C2×D4×C271D414yescomplemented
75C2×D4×C271C2312yes—
76C2×D4×C271C2311yescharacteristic, complemented
77C22×C4×C271—21no—
78C22×C4×C271C2311yescharacteristic
79SD16×C271—24no—
80D8×C271—24no—
81M16×C271—22no—
82C2×C8×C271C2312yescomplemented
83C22×D271—22no—
84C22×D271—24no—
85C22×D271—216no—
86C22×D271—24no—
87C22×D271—22no—
88C22×D271—432no—
89C22×D271—48no—
90C22×D271—44no—
91C22×D271—48no—
92C22×D271C2×D412yes—
93D1084—24no—
94D1084—22no—
95D1084C2×D414yes—
96C22⋊D271—216no—
97C22⋊D271—44no—
98C2×Dic271—22no—
99C2×Dic271—22no—
100C2×Dic271C2×D412yes—
101C4×D271—24no—
102C4×D271—22no—
103C4×D271—22no—
104C271⋊C8—22no—
105C23×C271—24no—
106C23×C271—22no—
107C23×C271—21no—
108C23×C271—44no—
109C23×C271C2×D411yescharacteristic
110Q8×C271—21no—
111Q8×C271—21no—
112D4×C271—24no—
113D4×C271—22no—
114D4×C271—21no—
115D4×C271—21no—
116D4×C271—42no—
117D4×C271C2×D414yes—
118C2×C4×C271—21no—
119C2×C4×C271—42no—
120C2×C4×C271C2×D411yescharacteristic
121C2×C4×C271C2×D412yes—
122C2×C4×C271C2×D412yes—
123C2×C4×C271C2411yescharacteristic
124C8×C271—22no—
125C22×C271—22no—
126C22×C271—24no—
127C22×C271—48no—
128C22×C271—44no—
129C22×C271—41no—
130C22×C271—42no—
131C22×C271#12012yes—
132C22×C271C22×D411yescharacteristic
133C2×D271—28no—
134C2×D271—22no—
135C2×D271—22no—
136C2×D271—24no—
137C2×D271—432no—
138C2×D271—48no—
139C2×D271—44no—
140C2×D271AGL(1,ℤ/8)14yescomplemented
141C4×C271—41no—
142C4×C271#12012yes—
143C4×C271C22×D411yescharacteristic
144C4×C271C22×D411yescharacteristic
145Dic271—22no—
146Dic271—42no—
147C2×C271—22no—
148C2×C271—44no—
149C2×C271—81no—
150C2×C271C2×#12011yescharacteristic
151C2×C271C2×AGL(1,ℤ/8)12yes—
152D271—24no—
153D271—44no—
154C271#300311yescharacteristic, complemented, Sylow
155#3003—2711nomaximal, Sylow
156C23×D4—2711no—
157C2×AGL(1,ℤ/8)—2711no—
158C2×AGL(1,ℤ/8)D27111yescharacteristic, complemented
159#525—2711no—
160#450—2714no—
161#447—2714no—
162#418—2711no—
163#418—2711no—
164#407—2711no—
165C25—5421no—
166C2×#41C2×D27111yescharacteristic, complemented
167C22×D4—2714no—
168C22×D4—2711no—
169C22×D4—5422no—
170C22×D4—5421no—
171C22×D4—5424no—
172C22×D4—5424no—
173C22×D4C2×D27111yescharacteristic
174C23×C4—2711no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—5424no—
177C2×SD16—2712no—
178C2×SD16C2×D27112yescomplemented
179C2×D8—2712no—
180C2×D8C2×D27112yescomplemented
181C2×M16—2711no—
182C2×M16C2×D27111yescharacteristic, complemented
183#123—5421no—
184C22⋊Q8—2712no—
185#121—2712no—
186#120—5421no—
187C4×D4—5422no—
188#117—2711no—
189#102—2714no—
190#102—2714no—
191C22⋊C8—2714no—
192C24—21no—
193C24—2711no—
194C24—5424no—
195C24—5428no—
196C24—5421no—
197C24—10848no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D27111yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—2714no—
205C2×D4—2712no—
206C2×D4—5424no—
207C2×D4—5428no—
208C2×D4—54216no—
209C2×D4—5428no—
210C2×D4—5428no—
211C2×D4—5422no—
212C2×D4—5421no—
213C2×D4—5422no—
214C2×D4C22⋊D27114yescomplemented
215C2×D4C22×D27112yes—
216C2×D4C22×D27111yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—2711no—
219C22×C4—2712no—
220C22×C4—2712no—
221C22×C4—5422no—
222C22×C4—5422no—
223C22×C4C22×D27111yescharacteristic
224SD16—24no—
225SD16—5424no—
226D8—24no—
227D8—5424no—
228M16—22no—
229M16—5422no—
230C2×C8—2712no—
231C2×C8C22×D27112yescomplemented
232C4⋊C4—2712no—
233C4⋊C4—5422no—
234C22⋊C4—5422no—
235C22⋊C4—5422no—
236C22⋊C4—5421no—
237C42—5421no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—2712no—
243C23—5424no—
244C23—54216no—
245C23—5422no—
246C23—5424no—
247C23—5422no—
248C23—10844no—
249C23—10848no—
250C23—108432no—
251C23—10848no—
252C23D4×D27111yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—2714no—
261D4—54216no—
262D4—5424no—
263D4—5422no—
264D4—10844no—
265D4C2×C22⋊D27114yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—2712no—
269C2×C4—5422no—
270C2×C4—5424no—
271C2×C4—5422no—
272C2×C4—5422no—
273C2×C4—5422no—
274C2×C4D4×D27111yescharacteristic
275C2×C4D4×D27112yes—
276C2×C4C2×C22⋊D27112yes—
277C2×C4C23×D27111yescharacteristic
278C8—22no—
279C8—5422no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—2714no—
287C22—5428no—
288C22—5422no—
289C22—5422no—
290C22—5424no—
291C22—10848no—
292C22—108432no—
293C22—10844no—
294C22#2257612812yes—
295C22C2×D4×D27111yescharacteristic
296C4—41no—
297C4—5422no—
298C4—10842no—
299C4#2257612812yes—
300C4C2×D4×D27111yescharacteristic
301C4C22×C22⋊D27111yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—5424no—
306C2—10844no—
307C2C2×#2257612811yescharacteristic
308C2AGL(1,ℤ/8)×D27112yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×AGL(1,ℤ/8)×C271C2
socleC22×C271C22×D4
hypercenterC2×AGL(1,ℤ/8)D271
hypocenterC271#3003
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D271C2semidirect
2. C2×#22576185C2semidirect
3. #31774404C2semidirect
4. #31774046C2semidirect
5. #31774029C2semidirect
6. #31773903C2semidirect
7. #31773902C2semidirect
8. #31773860C2semidirect
9. C2×AGL(1,ℤ/8)×C271C2semidirect
19. C2×#16864641C22semidirect
20. C2×#16864639C22semidirect
21. C2×#16864638C22semidirect
23. #22576136C22semidirect
24. #22576131C22semidirect
27. #22576119C22semidirect
28. #22576082C22semidirect
29. #22576081C22semidirect
30. (C2×D271)⋊C8C22semidirect
31. C2×#41×C271C22semidirect
34. C2×SD16×C271C22semidirect
35. C2×D8×C271C22semidirect
36. C2×M16×C271C22semidirect
44. C2×D1084D4semidirect
60. C2×C271⋊C8C23semidirect
61. C4⋊Dic271C23semidirect
70. C2×Q8×C271C23semidirect
74. C2×D4×C271D4semidirect
76. C2×D4×C271C23semidirect
82. C2×C8×C271C23semidirect
140. C2×D271AGL(1,ℤ/8)semidirect
154. C271#3003semidirect
158. C2×AGL(1,ℤ/8)D271semidirect
166. C2×#41C2×D271semidirect
178. C2×SD16C2×D271semidirect
180. C2×D8C2×D271semidirect
182. C2×M16C2×D271semidirect
200. C2×Q8C22×D271semidirect
214. C2×D4C22⋊D271semidirect
216. C2×D4C22×D271semidirect
231. C2×C8C22×D271semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2712
C2×#225761852
#317744042
#317740462
#317740292
#317739032
#317739022
#317738602
C2×AGL(1,ℤ/8)×C2712
#3003271