Finite Group Databasebeta

← #46820110

Subgroups — #46820110

Order 34432.

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×D269C211yesmaximal, characteristic, complemented
2C2×#22575478C211yesmaximal, characteristic, complemented
3#31771112C211yesmaximal, characteristic, complemented
4#31770754C214yesmaximal, complemented
5#31770737C214yesmaximal, complemented
6#31770611C211yesmaximal, characteristic, complemented
7#31770610C211yesmaximal, characteristic, complemented
8#31770568C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
10C24×D269—21no—
11C22×D1076C2211yescharacteristic
12C22×C22⋊D269—21no—
13C2×D4×D269—24no—
14C2×D4×D269—22no—
15C2×D4×D269—24no—
16C2×D4×D269C2214yes—
17C22×C4×D269C2211yescharacteristic
18#22575478—24no—
19C2×#16864389C2212yescomplemented
20C2×#16864387C2212yescomplemented
21C2×#16864386C2211yescharacteristic, complemented
22#22575434—21no—
23#22575429C2212yescomplemented
24#22575424C2212yescomplemented
25#22575421—21no—
26C4×C22⋊D269—22no—
27#22575412C2211yescharacteristic, complemented
28#22575375C2214yescomplemented
29#22575374C2214yescomplemented
30#22575365C2214yescomplemented
31C2×#41×C269C2211yescharacteristic, complemented
32C22×D4×C269C2211yescharacteristic
33AGL(1,ℤ/8)×C269—24no—
34C2×SD16×C269C2212yescomplemented
35C2×D8×C269C2212yescomplemented
36C2×M16×C269C2211yescharacteristic, complemented
37C23×D269—28no—
38C23×D269—24no—
39C23×D269—21no—
40C23×D269—48no—
41C23×D269D411yescharacteristic
42C2×D1076—21no—
43C2×D1076—22no—
44C2×D1076D414yescomplemented
45C2×D1076C2312yes—
46C2×C22⋊D269—28no—
47C2×C22⋊D269—22no—
48C2×C22⋊D269—24no—
49D4×D269—216no—
50D4×D269—28no—
51D4×D269—28no—
52C22×Dic269D411yescharacteristic
53C2×C4×D269—22no—
54C2×C4×D269—22no—
55C2×C4×D269D412yes—
56C2×C4×D269C2312yes—
57#16864389—24no—
58#16864387—24no—
59#16864386—22no—
60C2×#14524130C2312yescomplemented
61C4⋊Dic269C2312yescomplemented
62#16864381—22no—
63C22⋊Dic269—22no—
64C22⋊Dic269—21no—
65#16864379—22no—
66C4×Dic269—21no—
67C24×C269—21no—
68#41×C269—22no—
69#41×C269—22no—
70C2×Q8×C269C2311yescharacteristic, complemented
71C2×D4×C269—21no—
72C2×D4×C269—21no—
73C2×D4×C269—22no—
74C2×D4×C269D414yescomplemented
75C2×D4×C269C2312yes—
76C2×D4×C269C2311yescharacteristic, complemented
77C22×C4×C269—21no—
78C22×C4×C269C2311yescharacteristic
79SD16×C269—24no—
80D8×C269—24no—
81M16×C269—22no—
82C2×C8×C269C2312yescomplemented
83C22×D269—22no—
84C22×D269—24no—
85C22×D269—216no—
86C22×D269—24no—
87C22×D269—22no—
88C22×D269—432no—
89C22×D269—48no—
90C22×D269—44no—
91C22×D269—48no—
92C22×D269C2×D412yes—
93D1076—24no—
94D1076—22no—
95D1076C2×D414yes—
96C22⋊D269—216no—
97C22⋊D269—44no—
98C2×Dic269—22no—
99C2×Dic269—22no—
100C2×Dic269C2×D412yes—
101C4×D269—24no—
102C4×D269—22no—
103C4×D269—22no—
104#14524130—22no—
105C23×C269—24no—
106C23×C269—22no—
107C23×C269—21no—
108C23×C269—44no—
109C23×C269C2×D411yescharacteristic
110Q8×C269—21no—
111Q8×C269—21no—
112D4×C269—24no—
113D4×C269—22no—
114D4×C269—21no—
115D4×C269—21no—
116D4×C269—42no—
117D4×C269C2×D414yes—
118C2×C4×C269—21no—
119C2×C4×C269—42no—
120C2×C4×C269C2×D411yescharacteristic
121C2×C4×C269C2×D412yes—
122C2×C4×C269C2×D412yes—
123C2×C4×C269C2411yescharacteristic
124C8×C269—22no—
125C22×C269—22no—
126C22×C269—24no—
127C22×C269—48no—
128C22×C269—44no—
129C22×C269—41no—
130C22×C269—42no—
131C22×C269#12012yes—
132C22×C269C22×D411yescharacteristic
133C2×D269—28no—
134C2×D269—22no—
135C2×D269—22no—
136C2×D269—24no—
137C2×D269—432no—
138C2×D269—48no—
139C2×D269—44no—
140C2×D269AGL(1,ℤ/8)14yescomplemented
141C4×C269—41no—
142C4×C269#12012yes—
143C4×C269C22×D411yescharacteristic
144C4×C269C22×D411yescharacteristic
145Dic269—22no—
146Dic269—42no—
147C2×C269—22no—
148C2×C269—44no—
149C2×C269—81no—
150C2×C269C2×#12011yescharacteristic
151C2×C269C2×AGL(1,ℤ/8)12yes—
152D269—24no—
153D269—44no—
154C269#300311yescharacteristic, complemented, Sylow
155#3003—2691nomaximal, Sylow
156C23×D4—2691no—
157C2×AGL(1,ℤ/8)—2691no—
158C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
159#525—2691no—
160#450—2694no—
161#447—2694no—
162#418—2691no—
163#418—2691no—
164#407—2691no—
165C25—5381no—
166C2×#41C2×D26911yescharacteristic, complemented
167C22×D4—2694no—
168C22×D4—2691no—
169C22×D4—5382no—
170C22×D4—5381no—
171C22×D4—5384no—
172C22×D4—5384no—
173C22×D4C2×D26911yescharacteristic
174C23×C4—2691no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—5384no—
177C2×SD16—2692no—
178C2×SD16C2×D26912yescomplemented
179C2×D8—2692no—
180C2×D8C2×D26912yescomplemented
181C2×M16—2691no—
182C2×M16C2×D26911yescharacteristic, complemented
183#123—5381no—
184C22⋊Q8—2692no—
185#121—2692no—
186#120—5381no—
187C4×D4—5382no—
188#117—2691no—
189#102—2694no—
190#102—2694no—
191C22⋊C8—2694no—
192C24—21no—
193C24—2691no—
194C24—5384no—
195C24—5388no—
196C24—5381no—
197C24—10768no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D26911yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—2694no—
205C2×D4—2692no—
206C2×D4—5384no—
207C2×D4—5388no—
208C2×D4—53816no—
209C2×D4—5388no—
210C2×D4—5388no—
211C2×D4—5382no—
212C2×D4—5381no—
213C2×D4—5382no—
214C2×D4C22⋊D26914yescomplemented
215C2×D4C22×D26912yes—
216C2×D4C22×D26911yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—2691no—
219C22×C4—2692no—
220C22×C4—2692no—
221C22×C4—5382no—
222C22×C4—5382no—
223C22×C4C22×D26911yescharacteristic
224SD16—24no—
225SD16—5384no—
226D8—24no—
227D8—5384no—
228M16—22no—
229M16—5382no—
230C2×C8—2692no—
231C2×C8C22×D26912yescomplemented
232C4⋊C4—2692no—
233C4⋊C4—5382no—
234C22⋊C4—5382no—
235C22⋊C4—5382no—
236C22⋊C4—5381no—
237C42—5381no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—2692no—
243C23—5384no—
244C23—53816no—
245C23—5382no—
246C23—5384no—
247C23—5382no—
248C23—10764no—
249C23—10768no—
250C23—107632no—
251C23—10768no—
252C23D4×D26911yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—2694no—
261D4—53816no—
262D4—5384no—
263D4—5382no—
264D4—10764no—
265D4C2×C22⋊D26914yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—2692no—
269C2×C4—5382no—
270C2×C4—5384no—
271C2×C4—5382no—
272C2×C4—5382no—
273C2×C4—5382no—
274C2×C4D4×D26911yescharacteristic
275C2×C4D4×D26912yes—
276C2×C4C2×C22⋊D26912yes—
277C2×C4C23×D26911yescharacteristic
278C8—22no—
279C8—5382no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—2694no—
287C22—5388no—
288C22—5382no—
289C22—5382no—
290C22—5384no—
291C22—10768no—
292C22—107632no—
293C22—10764no—
294C22#2257542112yes—
295C22C2×D4×D26911yescharacteristic
296C4—41no—
297C4—5382no—
298C4—10762no—
299C4#2257542112yes—
300C4C2×D4×D26911yescharacteristic
301C4C22×C22⋊D26911yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—5384no—
306C2—10764no—
307C2C2×#2257542111yescharacteristic
308C2AGL(1,ℤ/8)×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×AGL(1,ℤ/8)×C269C2
socleC22×C269C22×D4
hypercenterC2×AGL(1,ℤ/8)D269
hypocenterC269#3003
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
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×D269C2semidirect
2. C2×#22575478C2semidirect
3. #31771112C2semidirect
4. #31770754C2semidirect
5. #31770737C2semidirect
6. #31770611C2semidirect
7. #31770610C2semidirect
8. #31770568C2semidirect
9. C2×AGL(1,ℤ/8)×C269C2semidirect
19. C2×#16864389C22semidirect
20. C2×#16864387C22semidirect
21. C2×#16864386C22semidirect
23. #22575429C22semidirect
24. #22575424C22semidirect
27. #22575412C22semidirect
28. #22575375C22semidirect
29. #22575374C22semidirect
30. #22575365C22semidirect
31. C2×#41×C269C22semidirect
34. C2×SD16×C269C22semidirect
35. C2×D8×C269C22semidirect
36. C2×M16×C269C22semidirect
44. C2×D1076D4semidirect
60. C2×#14524130C23semidirect
61. C4⋊Dic269C23semidirect
70. C2×Q8×C269C23semidirect
74. C2×D4×C269D4semidirect
76. C2×D4×C269C23semidirect
82. C2×C8×C269C23semidirect
140. C2×D269AGL(1,ℤ/8)semidirect
154. C269#3003semidirect
158. C2×AGL(1,ℤ/8)D269semidirect
166. C2×#41C2×D269semidirect
178. C2×SD16C2×D269semidirect
180. C2×D8C2×D269semidirect
182. C2×M16C2×D269semidirect
200. C2×Q8C22×D269semidirect
214. C2×D4C22⋊D269semidirect
216. C2×D4C22×D269semidirect
231. C2×C8C22×D269semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2692
C2×#225754782
#317711122
#317707542
#317707372
#317706112
#317706102
#317705682
C2×AGL(1,ℤ/8)×C2692
#3003269