Finite Group Databasebeta

← #46820124

Subgroups — #46820124

Order 34432.

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×#16864401C211yesmaximal, characteristic, complemented
2C2×#22575483C211yesmaximal, characteristic, complemented
3#31771112C211yesmaximal, characteristic, complemented
4#31770764C214yesmaximal, complemented
5#31770747C214yesmaximal, complemented
6#31770615C211yesmaximal, characteristic, complemented
7#31770610C211yesmaximal, characteristic, complemented
8#31770568C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
10C2×#16864401—24no—
11C2×#16864401—22no—
12C2×#16864401—24no—
13C2×#16864401C2214yes—
14C22×Dic538C2211yescharacteristic
15C22×C22⋊D269—21no—
16C23×Dic269—21no—
17C22×C4×D269C2211yescharacteristic
18#22575483—24no—
19C2×#16864392C2212yescomplemented
20C2×#16864390C2212yescomplemented
21C2×#16864386C2211yescharacteristic
22#22575434—21no—
23#22575429C2212yescomplemented
24#22575424C2212yescomplemented
25#22575421—21no—
26C4×C22⋊D269—22no—
27#22575412C2211yescharacteristic
28#22575378C2214yescomplemented
29#22575374C2214yescomplemented
30#22575365C2214yescomplemented
31C2×#41×C269C2211yescharacteristic
32C22×D4×C269C2211yescharacteristic
33AGL(1,ℤ/8)×C269—24no—
34C2×SD16×C269C2212yescomplemented
35C2×D8×C269C2212yescomplemented
36C2×M16×C269C2211yescharacteristic
37C23×D269D411yescharacteristic
38#16864401—216no—
39#16864401—28no—
40#16864401—28no—
41C2×Dic538—21no—
42C2×Dic538—22no—
43C2×Dic538D414yescomplemented
44C2×Dic538C2312yes—
45C2×C22⋊D269—28no—
46C2×C22⋊D269—24no—
47C2×C22⋊D269—22no—
48C22×Dic269—28no—
49C22×Dic269—24no—
50C22×Dic269—21no—
51C22×Dic269D411yescharacteristic
52C2×C4×D269—22no—
53C2×C4×D269—22no—
54C2×C4×D269D412yes—
55C2×C4×D269C2312yes—
56#16864392—24no—
57#16864390—24no—
58#16864386—22no—
59C2×#14524130C2312yes—
60C4⋊Dic269C2312yes—
61#16864381—22no—
62C22⋊Dic269—22no—
63C22⋊Dic269—21no—
64#16864379—22no—
65C4×Dic269—21no—
66C24×C269—21no—
67#41×C269—22no—
68#41×C269—22no—
69C2×Q8×C269C2311yescharacteristic
70C2×D4×C269—21no—
71C2×D4×C269—21no—
72C2×D4×C269—22no—
73C2×D4×C269D414yescomplemented
74C2×D4×C269C2311yescharacteristic
75C2×D4×C269C2312yes—
76C22×C4×C269—21no—
77C22×C4×C269C2311yescharacteristic
78SD16×C269—24no—
79D8×C269—24no—
80M16×C269—22no—
81C2×C8×C269C2312yes—
82C22×D269—22no—
83C22×D269—42no—
84C22×D269C2×D412yes—
85Dic538—24no—
86Dic538—22no—
87Dic538C2×D414yes—
88C22⋊D269—216no—
89C22⋊D269—44no—
90C2×Dic269—216no—
91C2×Dic269—22no—
92C2×Dic269—24no—
93C2×Dic269—22no—
94C2×Dic269—22no—
95C2×Dic269C2×D412yes—
96C4×D269—24no—
97C4×D269—22no—
98C4×D269—22no—
99#14524130—22no—
100C23×C269—24no—
101C23×C269—22no—
102C23×C269—21no—
103C23×C269—44no—
104C23×C269C2×D411yescharacteristic
105Q8×C269—21no—
106Q8×C269—21no—
107D4×C269—24no—
108D4×C269—22no—
109D4×C269—21no—
110D4×C269—21no—
111D4×C269—42no—
112D4×C269C2×D414yes—
113C2×C4×C269—21no—
114C2×C4×C269—42no—
115C2×C4×C269C2×D411yescharacteristic
116C2×C4×C269C2×D412yes—
117C2×C4×C269C2×D412yes—
118C2×C4×C269C2411yescharacteristic
119C8×C269—22no—
120C22×C269—22no—
121C22×C269—24no—
122C22×C269—48no—
123C22×C269—44no—
124C22×C269—41no—
125C22×C269—42no—
126C22×C269#12012yes—
127C22×C269C22×D411yescharacteristic
128C2×D269—22no—
129C2×D269—22no—
130C2×D269—22no—
131C2×D269—44no—
132C4×C269—41no—
133C4×C269#12012yes—
134C4×C269C22×D411yescharacteristic
135C4×C269C22×D411yescharacteristic
136Dic269—22no—
137Dic269—24no—
138Dic269—42no—
139C2×C269—22no—
140C2×C269—44no—
141C2×C269—81no—
142C2×C269C2×#12011yescharacteristic
143C2×C269#57512yes—
144D269—42no—
145C269#300511yescharacteristic, complemented, Sylow
146#3005—2691nomaximal, Sylow
147C22×#41—2691no—
148C2×#137—2691no—
149C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
150#525—2691no—
151#452—2694no—
152#449—2694no—
153#419—2691no—
154#418—2691no—
155#407—2691no—
156C2×#41—2694no—
157C2×#41—5382no—
158C2×#41—5384no—
159C2×#41—5384no—
160C2×#41C2×D26911yescharacteristic
161C22×Q8—2691no—
162C22×D4—5381no—
163C22×D4C2×D26911yescharacteristic
164C23×C4—2691no—
165C23×C4—5381no—
166#137—5384no—
167AGL(1,ℤ/8)—24no—
168C2×Q16—2692no—
169C2×SD16—2692no—
170C2×SD16C2×D26912yescomplemented
171C2×D8C2×D26912yescomplemented
172C2×M16—2691no—
173C2×M16C2×D26911yescharacteristic
174#123—5381no—
175C22⋊Q8—2692no—
176#121—2692no—
177#120—5381no—
178C4×D4—5382no—
179#117—2691no—
180#103—2694no—
181#102—2694no—
182C22⋊C8—2694no—
183C24—21no—
184C24—2691no—
185#41—22no—
186#41—22no—
187#41—5388no—
188#41—53816no—
189#41—5388no—
190C2×Q8—2694no—
191C2×Q8—2692no—
192C2×Q8—5381no—
193C2×Q8—5382no—
194C2×Q8C22×D26911yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—5384no—
199C2×D4—5388no—
200C2×D4—5382no—
201C2×D4C22⋊D26914yescomplemented
202C2×D4C22×D26911yescharacteristic
203C2×D4C22×D26912yes—
204C22×C4—21no—
205C22×C4—2691no—
206C22×C4—2692no—
207C22×C4—2692no—
208C22×C4—5384no—
209C22×C4—5388no—
210C22×C4—5381no—
211C22×C4—5382no—
212C22×C4—5382no—
213C22×C4C22×D26911yescharacteristic
214Q16—5384no—
215SD16—24no—
216SD16—5384no—
217D8—24no—
218M16—22no—
219M16—5382no—
220C2×C8—2692no—
221C2×C8C22×D26912yes—
222C4⋊C4—2692no—
223C4⋊C4—5382no—
224C22⋊C4—5382no—
225C22⋊C4—5382no—
226C22⋊C4—5381no—
227C42—5381no—
228C23—24no—
229C23—22no—
230C23—21no—
231C23—44no—
232C23—2692no—
233C23—5382no—
234C23—10762no—
235C23D4×D26911yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—2694no—
239Q8—5384no—
240Q8—5382no—
241D4—24no—
242D4—22no—
243D4—21no—
244D4—21no—
245D4—42no—
246D4—53816no—
247D4—10764no—
248D4C2×C22⋊D26914yes—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—2692no—
252C2×C4—53816no—
253C2×C4—5382no—
254C2×C4—5384no—
255C2×C4—5382no—
256C2×C4—5384no—
257C2×C4—5382no—
258C2×C4—5382no—
259C2×C4—5382no—
260C2×C4D4×D26911yescharacteristic
261C2×C4D4×D26912yes—
262C2×C4C2×C22⋊D26912yes—
263C2×C4C23×D26911yescharacteristic
264C8—22no—
265C8—5382no—
266C22—22no—
267C22—24no—
268C22—48no—
269C22—44no—
270C22—42no—
271C22—41no—
272C22—5382no—
273C22—5382no—
274C22—5382no—
275C22—10764no—
276C22#2257542112yes—
277C22C2×D4×D26911yescharacteristic
278C4—41no—
279C4—5382no—
280C4—5384no—
281C4—10762no—
282C4#2257542112yes—
283C4C2×D4×D26911yescharacteristic
284C4C22×C22⋊D26911yescharacteristic
285C2—22no—
286C2—44no—
287C2—81no—
288C2—10762no—
289C2C2×#2257542111yescharacteristic
290C2#3177140112yes—
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#3005
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×#16864401C2semidirect
2. C2×#22575483C2semidirect
3. #31771112C2semidirect
4. #31770764C2semidirect
5. #31770747C2semidirect
6. #31770615C2semidirect
7. #31770610C2semidirect
8. #31770568C2semidirect
9. C2×AGL(1,ℤ/8)×C269C2semidirect
19. C2×#16864392C22semidirect
20. C2×#16864390C22semidirect
23. #22575429C22semidirect
24. #22575424C22semidirect
28. #22575378C22semidirect
29. #22575374C22semidirect
30. #22575365C22semidirect
34. C2×SD16×C269C22semidirect
35. C2×D8×C269C22semidirect
43. C2×Dic538D4semidirect
73. C2×D4×C269D4semidirect
145. C269#3005semidirect
149. C2×AGL(1,ℤ/8)D269semidirect
170. C2×SD16C2×D269semidirect
171. C2×D8C2×D269semidirect
201. C2×D4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C22×#168644012
C2×#225754832
#317711122
#317707642
#317707472
#317706152
#317706102
#317705682
C2×AGL(1,ℤ/8)×C2692
#3005269