Finite Group Databasebeta

← #46820127

Subgroups — #46820127

Order 34432.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#22575484C211yesmaximal, characteristic, complemented
2C2×#22575479C211yesmaximal, characteristic, complemented
3#31771116C211yesmaximal, characteristic, complemented
4#31770767C214yesmaximal, complemented
5#31770750C214yesmaximal, complemented
6#31770614C211yesmaximal, characteristic, complemented
7#31770610C211yesmaximal, characteristic, complemented
8#31770569C211yesmaximal, characteristic, complemented
9C22×#41×C269C211yesmaximal, characteristic, complemented
10C2×#16864402C2211yescharacteristic
11#22575484—24no—
12#22575479—24no—
13C2×#16864392C2212yescomplemented
14C2×#16864390C2212yescomplemented
15C2×#16864389C2212yescomplemented
16C2×#16864387C2212yescomplemented
17C2×#16864386C2211yescharacteristic
18C2×#16864386C2211yescharacteristic
19#22575436—21no—
20#22575432C2212yescomplemented
21#22575427C2212yescomplemented
22#22575422—21no—
23C4×C22⋊D269—22no—
24#22575412C2211yescharacteristic
25#22575377C2214yescomplemented
26#22575374C2214yescomplemented
27C22⋊#14524130C2214yescomplemented
28C2×#41×C269—24no—
29C2×#41×C269—24no—
30C2×#41×C269—22no—
31C2×#41×C269C2214yes—
32C22×Q8×C269C2211yescharacteristic
33C22×D4×C269—21no—
34C22×D4×C269C2211yescharacteristic
35C23×C4×C269—21no—
36C23×C4×C269C2211yescharacteristic
37#16864402—22no—
38#16864402—22no—
39C2×Dic538C2311yescharacteristic
40C2×D1076C2311yescharacteristic
41C2×C22⋊D269—22no—
42C2×C22⋊D269—21no—
43C2×C4×D269—21no—
44#16864392—24no—
45#16864390—24no—
46#16864389—24no—
47#16864387—24no—
48#16864386—22no—
49#16864386—22no—
50C2×#14524130C2312yes—
51C2×#14524130C2312yes—
52C4⋊Dic269C2312yes—
53#16864381—22no—
54C22⋊Dic269—21no—
55C22⋊Dic269—22no—
56#16864379—22no—
57C4×Dic269—21no—
58C24×C269—21no—
59C24×C269D411yescharacteristic
60#41×C269—216no—
61#41×C269—28no—
62#41×C269—28no—
63C2×Q8×C269—22no—
64C2×Q8×C269—21no—
65C2×Q8×C269D414yescomplemented
66C2×Q8×C269C2312yes—
67C2×D4×C269—28no—
68C2×D4×C269—24no—
69C2×D4×C269—21no—
70C2×D4×C269—22no—
71C2×D4×C269D414yescomplemented
72C2×D4×C269C2312yes—
73C22×C4×C269—28no—
74C22×C4×C269—21no—
75C22×C4×C269—24no—
76C22×C4×C269—22no—
77C22×C4×C269—22no—
78C22×C4×C269D412yes—
79C22×C4×C269D411yescharacteristic
80C22×C4×C269C2312yes—
81C22×C4×C269C2311yescharacteristic
82C22×D269—21no—
83Dic538—21no—
84Dic538—21no—
85D1076—21no—
86D1076—21no—
87C22⋊D269—44no—
88C22⋊D269—42no—
89C2×Dic269—22no—
90C2×Dic269—21no—
91C4×D269—42no—
92#14524130—22no—
93#14524130—22no—
94C23×C269—22no—
95C23×C269—24no—
96C23×C269—22no—
97C23×C269—42no—
98C23×C269—44no—
99C23×C269C2×D412yes—
100C23×C269C2×D411yescharacteristic
101Q8×C269—24no—
102Q8×C269—22no—
103Q8×C269C2×D414yes—
104D4×C269—216no—
105D4×C269—24no—
106D4×C269—22no—
107D4×C269C2×D414yes—
108C2×C4×C269—216no—
109C2×C4×C269—22no—
110C2×C4×C269—24no—
111C2×C4×C269—22no—
112C2×C4×C269—24no—
113C2×C4×C269—22no—
114C2×C4×C269—22no—
115C2×C4×C269C2×D412yes—
116C2×C4×C269C2×D411yescharacteristic
117C2×C4×C269C2×D412yes—
118C2×C4×C269C2×D412yes—
119C2×C4×C269C2411yescharacteristic
120C22×C269—22no—
121C22×C269—22no—
122C22×C269—24no—
123C22×C269—22no—
124C22×C269—22no—
125C22×C269—44no—
126C22×C269—48no—
127C22×C269—44no—
128C22×C269#12012yes—
129C22×C269C22×D411yescharacteristic
130C2×D269—41no—
131C2×D269—42no—
132C4×C269—22no—
133C4×C269—24no—
134C4×C269#12012yes—
135C4×C269C22×D411yescharacteristic
136C4×C269C22×D411yescharacteristic
137Dic269—41no—
138Dic269—42no—
139C2×C269—22no—
140C2×C269—42no—
141C2×C269—44no—
142C2×C269C2×#12011yescharacteristic
143C2×C269#57512yes—
144D269—81no—
145C269#300511yescharacteristic, complemented, Sylow
146#3005—2691nomaximal, Sylow
147C22×#41D26911yescharacteristic, complemented
148C2×#137—2691no—
149C2×AGL(1,ℤ/8)—2691no—
150#525—2691no—
151#452—2694no—
152#449—2694no—
153#419—2691no—
154#418—2691no—
155#407—2691no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—2691no—
160C2×#41C2×D26914yes—
161C22×Q8C2×D26911yescharacteristic
162C22×D4—21no—
163C22×D4C2×D26911yescharacteristic
164C23×C4—21no—
165C23×C4C2×D26911yescharacteristic
166#137—5384no—
167AGL(1,ℤ/8)—5384no—
168C2×Q16—2692no—
169C2×SD16—2692no—
170C2×SD16—2692no—
171C2×D8—2692no—
172C2×M16—2691no—
173C2×M16—2691no—
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—
184C24C22⋊D26911yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—5382no—
189#41—5382no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—2691no—
193C2×Q8C22⋊D26914yescomplemented
194C2×Q8C22×D26912yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—2691no—
200C2×D4—5382no—
201C2×D4—5381no—
202C2×D4C22⋊D26914yescomplemented
203C2×D4C22×D26912yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—5381no—
210C22×C4C22⋊D26912yes—
211C22×C4C22⋊D26911yescharacteristic
212C22×C4C22×D26912yes—
213C22×C4C22×D26911yescharacteristic
214Q16—5384no—
215SD16—5384no—
216SD16—5384no—
217D8—5384no—
218M16—5382no—
219M16—5382no—
220C2×C8—2692no—
221C2×C8—2692no—
222C4⋊C4—2692no—
223C4⋊C4—5382no—
224C22⋊C4—5381no—
225C22⋊C4—5382no—
226C22⋊C4—5382no—
227C42—5381no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—42no—
232C23—44no—
233C23—5381no—
234C23C2×C22⋊D26912yes—
235C23C2×C22⋊D26911yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—5381no—
239Q8—5381no—
240Q8C2×C22⋊D26914yes—
241D4—216no—
242D4—24no—
243D4—22no—
244D4—5381no—
245D4—5381no—
246D4—10764no—
247D4—10762no—
248D4C2×C22⋊D26914yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—5382no—
257C2×C4—5381no—
258C2×C4—10762no—
259C2×C4C2×C22⋊D26912yes—
260C2×C4C2×C22⋊D26911yescharacteristic
261C2×C4C2×C22⋊D26912yes—
262C2×C4C2×C22⋊D26912yes—
263C2×C4C23×D26911yescharacteristic
264C8—5382no—
265C8—5382no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—10762no—
275C22—10761no—
276C22#2257542212yes—
277C22C22×C22⋊D26911yescharacteristic
278C4—22no—
279C4—24no—
280C4—10762no—
281C4—10761no—
282C4#2257542212yes—
283C4C22×C22⋊D26911yescharacteristic
284C4C22×C22⋊D26911yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—21521no—
289C2C2×#2257542211yescharacteristic
290C2#3177140312yes—
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC22×#41×C269C2
socleC22×C269C22×D4
hypercenterC22×#41D269
hypocenterC269#3005
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#22575484C2semidirect
2. C2×#22575479C2semidirect
3. #31771116C2semidirect
4. #31770767C2semidirect
5. #31770750C2semidirect
6. #31770614C2semidirect
7. #31770610C2semidirect
8. #31770569C2semidirect
9. C22×#41×C269C2semidirect
13. C2×#16864392C22semidirect
14. C2×#16864390C22semidirect
15. C2×#16864389C22semidirect
16. C2×#16864387C22semidirect
20. #22575432C22semidirect
21. #22575427C22semidirect
25. #22575377C22semidirect
26. #22575374C22semidirect
27. C22⋊#14524130C22semidirect
65. C2×Q8×C269D4semidirect
71. C2×D4×C269D4semidirect
145. C269#3005semidirect
147. C22×#41D269semidirect
193. C2×Q8C22⋊D269semidirect
202. C2×D4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754842
C2×#225754792
#317711162
#317707672
#317707502
#317706142
#317706102
#317705692
C22×#41×C2692
#3005269