Finite Group Databasebeta

← #46820130

Subgroups — #46820130

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×#16864402C211yesmaximal, characteristic, complemented
2C2×#22575486C211yesmaximal, characteristic, complemented
3C2×#22575481C211yesmaximal, characteristic, complemented
4#31771116C211yesmaximal, characteristic, complemented
5#31770769C214yesmaximal, complemented
6#31770752C214yesmaximal, complemented
7#31770615C211yesmaximal, characteristic, complemented
8#31770611C211yesmaximal, characteristic, complemented
9#407×C269C211yesmaximal, characteristic, complemented
10C2×#16864402—24no—
11C2×#16864402—24no—
12C2×#16864402—22no—
13C2×#16864402C2211yescharacteristic
14C2×#16864402C2214yes—
15C22×Dic538C2211yescharacteristic
16C22×D1076C2211yescharacteristic
17C22×C22⋊D269—21no—
18C22×C4×D269—21no—
19#22575486—24no—
20#22575481—24no—
21C2×Dic1076C2212yescomplemented
22C2×#16864391C2212yescomplemented
23C2×#16864391C2212yescomplemented
24C2×D2152C2212yescomplemented
25#22575436—21no—
26#22575432C2212yescomplemented
27#22575427C2212yescomplemented
28#22575422—21no—
29C4×C22⋊D269—22no—
30#22575412C2211yescharacteristic
31#22575378C2214yescomplemented
32#22575375C2214yescomplemented
33C23×C4×C269C2211yescharacteristic
34C2×M16×C269C2211yescharacteristic
35C2×M16×C269C2211yescharacteristic
36C22⋊C8×C269C2214yescomplemented
37C23×D269—21no—
38#16864402—216no—
39#16864402—28no—
40#16864402—28no—
41#16864402—22no—
42#16864402—22no—
43C2×Dic538—22no—
44C2×Dic538—21no—
45C2×Dic538D414yescomplemented
46C2×Dic538C2311yescharacteristic
47C2×Dic538C2312yes—
48C2×D1076—22no—
49C2×D1076—21no—
50C2×D1076D414yescomplemented
51C2×D1076C2311yescharacteristic
52C2×D1076C2312yes—
53C2×C22⋊D269—28no—
54C2×C22⋊D269—24no—
55C2×C22⋊D269—22no—
56C2×C22⋊D269—21no—
57C22×Dic269—21no—
58C2×C4×D269—28no—
59C2×C4×D269—24no—
60C2×C4×D269—21no—
61Dic1076—24no—
62#16864391—24no—
63#16864391—24no—
64D2152—24no—
65C4⋊Dic269C2312yes—
66#16864381—22no—
67C22⋊Dic269—21no—
68C22⋊Dic269—22no—
69#16864379—22no—
70C4×Dic269—21no—
71C24×C269D411yescharacteristic
72C22×C4×C269—22no—
73C22×C4×C269—22no—
74C22×C4×C269D412yes—
75C22×C4×C269D411yescharacteristic
76C22×C4×C269C2312yes—
77C22×C4×C269C2311yescharacteristic
78M16×C269—22no—
79M16×C269—22no—
80C2×C8×C269C2312yes—
81C2×C8×C269C2312yes—
82C22×D269—24no—
83C22×D269—22no—
84C22×D269—21no—
85C22×D269—44no—
86Dic538—24no—
87Dic538—22no—
88Dic538—21no—
89Dic538—21no—
90Dic538C2×D414yes—
91D1076—24no—
92D1076—22no—
93D1076—21no—
94D1076—21no—
95D1076C2×D414yes—
96C22⋊D269—216no—
97C22⋊D269—44no—
98C22⋊D269—42no—
99C2×Dic269—24no—
100C2×Dic269—22no—
101C2×Dic269—22no—
102C2×Dic269—21no—
103C4×D269—216no—
104C4×D269—42no—
105C23×C269—22no—
106C23×C269—42no—
107C23×C269C2×D412yes—
108C23×C269C2×D411yescharacteristic
109C2×C4×C269—22no—
110C2×C4×C269—24no—
111C2×C4×C269—22no—
112C2×C4×C269—22no—
113C2×C4×C269C2×D412yes—
114C2×C4×C269C2×D412yes—
115C2×C4×C269C2×D412yes—
116C2×C4×C269C2×D411yescharacteristic
117C2×C4×C269C2411yescharacteristic
118C8×C269—22no—
119C8×C269—22no—
120C22×C269—22no—
121C22×C269—22no—
122C22×C269—22no—
123C22×C269—22no—
124C22×C269—44no—
125C22×C269#12012yes—
126C22×C269C22×D411yescharacteristic
127C2×D269—24no—
128C2×D269—48no—
129C2×D269—44no—
130C2×D269—41no—
131C2×D269—42no—
132C4×C269—22no—
133C4×C269#12012yes—
134C4×C269C22×D411yescharacteristic
135C4×C269C22×D411yescharacteristic
136Dic269—24no—
137Dic269—41no—
138Dic269—42no—
139C2×C269—22no—
140C2×C269—42no—
141C2×C269C2×#12011yescharacteristic
142C2×C269#57512yes—
143D269—44no—
144D269—81no—
145C269#300511yescharacteristic, complemented, Sylow
146#3005—2691nomaximal, Sylow
147C22×#41—2691no—
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#407D26911yescharacteristic, complemented
156C2×#41—2691no—
157C2×#41—2694no—
158C2×#41—5384no—
159C2×#41—5384no—
160C2×#41—5382no—
161C22×Q8—2691no—
162C22×D4—2691no—
163C22×D4—5381no—
164C23×C4—5381no—
165C23×C4C2×D26911yescharacteristic
166#137—5384no—
167AGL(1,ℤ/8)—5384no—
168C2×Q16—2692no—
169C2×SD16—2692no—
170C2×SD16—2692no—
171C2×D8—2692no—
172C2×M16C2×D26911yescharacteristic
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⋊C8C2×D26914yescomplemented
183C24—5381no—
184C24D107611yescharacteristic
185#41—5388no—
186#41—5388no—
187#41—53816no—
188#41—5382no—
189#41—5382no—
190C2×Q8—2694no—
191C2×Q8—2692no—
192C2×Q8—2691no—
193C2×Q8—5382no—
194C2×Q8—5381no—
195C2×D4—2694no—
196C2×D4—2692no—
197C2×D4—2691no—
198C2×D4—5388no—
199C2×D4—5384no—
200C2×D4—5382no—
201C2×D4—5381no—
202C2×D4—5382no—
203C2×D4—5381no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—5388no—
207C22×C4—5381no—
208C22×C4—5384no—
209C22×C4—5381no—
210C22×C4D107612yes—
211C22×C4D107611yescharacteristic
212C22×C4C22×D26912yes—
213C22×C4C22×D26911yescharacteristic
214Q16—5384no—
215SD16—5384no—
216SD16—5384no—
217D8—5384no—
218M16—22no—
219M16—22no—
220C2×C8C22×D26912yes—
221C2×C8C22×D26912yes—
222C4⋊C4—2692no—
223C4⋊C4—5382no—
224C22⋊C4—5381no—
225C22⋊C4—5382no—
226C22⋊C4—5382no—
227C42—5381no—
228C23—22no—
229C23—42no—
230C23—5384no—
231C23—5382no—
232C23—5381no—
233C23—10764no—
234C23C2×D107612yes—
235C23C2×D107611yescharacteristic
236Q8—2694no—
237Q8—5384no—
238Q8—5382no—
239Q8—5381no—
240Q8—5381no—
241D4—2694no—
242D4—53816no—
243D4—5384no—
244D4—5382no—
245D4—5381no—
246D4—5381no—
247D4—10764no—
248D4—10762no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—53816no—
254C2×C4—5384no—
255C2×C4—5382no—
256C2×C4—5382no—
257C2×C4—5381no—
258C2×C4—10762no—
259C2×C4D4×D26912yes—
260C2×C4D4×D26912yes—
261C2×C4C2×D107612yes—
262C2×C4C2×D107611yescharacteristic
263C2×C4C23×D26911yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—5384no—
272C22—10768no—
273C22—10764no—
274C22—10762no—
275C22—10761no—
276C22#2257542312yes—
277C22C22×D107611yescharacteristic
278C4—22no—
279C4—5384no—
280C4—10762no—
281C4—10761no—
282C4#2257542312yes—
283C4C2×D4×D26911yescharacteristic
284C4C2×D4×D26911yescharacteristic
285C2—22no—
286C2—42no—
287C2—10764no—
288C2—21521no—
289C2C2×#2257542311yescharacteristic
290C2#3177140512yes—
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#407×C269C2
socleC22×C269C22×D4
hypercenter#407D269
hypocenterC269#3005
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#16864402C2semidirect
2. C2×#22575486C2semidirect
3. C2×#22575481C2semidirect
4. #31771116C2semidirect
5. #31770769C2semidirect
6. #31770752C2semidirect
7. #31770615C2semidirect
8. #31770611C2semidirect
9. #407×C269C2semidirect
21. C2×Dic1076C22semidirect
22. C2×#16864391C22semidirect
23. C2×#16864391C22semidirect
24. C2×D2152C22semidirect
26. #22575432C22semidirect
27. #22575427C22semidirect
31. #22575378C22semidirect
32. #22575375C22semidirect
36. C22⋊C8×C269C22semidirect
45. C2×Dic538D4semidirect
50. C2×D1076D4semidirect
145. C269#3005semidirect
155. #407D269semidirect
182. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
C22×#168644022
C2×#225754862
C2×#225754812
#317711162
#317707692
#317707522
#317706152
#317706112
#407×C2692
#3005269