Finite Group Databasebeta

← #46820131

Subgroups — #46820131

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×#16864403C211yesmaximal, characteristic, complemented
2C2×#22575484C211yesmaximal, characteristic, complemented
3C2×#22575481C211yesmaximal, characteristic, complemented
4#31771117C211yesmaximal, characteristic, complemented
5#31770770C214yesmaximal, complemented
6#31770753C214yesmaximal, complemented
7#31770609C211yesmaximal, characteristic, complemented
8#31770568C211yesmaximal, characteristic, complemented
9#419×C269C211yesmaximal, characteristic, complemented
10C2×#16864403—24no—
11C2×#16864403—22no—
12C2×#16864403—24no—
13C2×#16864403C2214yes—
14C2×#16864402C2211yescharacteristic
15C22×D1076—21no—
16C22×D1076C2211yescharacteristic
17C22×C4×D269—21no—
18C22×C4×D269C2211yescharacteristic
19#22575484—24no—
20#22575481—24no—
21C2×#16864392C2212yescomplemented
22C2×#16864391C2212yescomplemented
23C2×#16864389C2212yescomplemented
24C2×D2152C2212yescomplemented
25C2×#16864386C2211yescharacteristic
26#22575437—21no—
27#22575433C2212yescomplemented
28#22575428C2212yescomplemented
29#22575423—21no—
30C4×D1076—22no—
31#22575373C2214yescomplemented
32#22575365C2214yescomplemented
33C22×Q8×C269C2211yescharacteristic
34C2×M16×C269C2211yescharacteristic
35#117×C269C2211yescharacteristic
36#103×C269C2214yescomplemented
37C23×D269—21no—
38C23×D269D411yescharacteristic
39#16864403—216no—
40#16864403—28no—
41#16864403—28no—
42#16864402—22no—
43#16864402—22no—
44C2×Dic538C2311yescharacteristic
45C2×D1076—28no—
46C2×D1076—24no—
47C2×D1076—22no—
48C2×D1076—21no—
49C2×D1076—22no—
50C2×D1076D414yescomplemented
51C2×D1076C2311yescharacteristic
52C2×D1076C2312yes—
53C2×C22⋊D269—21no—
54C22×Dic269D411yescharacteristic
55C2×C4×D269—28no—
56C2×C4×D269—24no—
57C2×C4×D269—22no—
58C2×C4×D269—22no—
59C2×C4×D269—21no—
60C2×C4×D269D412yes—
61C2×C4×D269C2312yes—
62#16864392—24no—
63#16864391—24no—
64#16864389—24no—
65D2152—24no—
66#16864386—22no—
67C2×#14524130C2312yes—
68C4⋊Dic269—22no—
69#16864379—22no—
70#16864379—22no—
71C2×Q8×C269—22no—
72C2×Q8×C269—21no—
73C2×Q8×C269D414yescomplemented
74C2×Q8×C269C2312yes—
75C22×C4×C269—21no—
76C22×C4×C269C2311yescharacteristic
77M16×C269—22no—
78C2×C8×C269C2312yes—
79C4⋊C4×C269C2312yes—
80C22⋊C4×C269—21no—
81C42×C269—21no—
82C22×D269—22no—
83C22×D269—24no—
84C22×D269—22no—
85C22×D269—21no—
86C22×D269—42no—
87C22×D269—44no—
88C22×D269C2×D412yes—
89Dic538—21no—
90Dic538—21no—
91D1076—216no—
92D1076—24no—
93D1076—22no—
94D1076—21no—
95D1076—21no—
96D1076—44no—
97D1076C2×D414yes—
98C22⋊D269—42no—
99C2×Dic269—22no—
100C2×Dic269—21no—
101C2×Dic269C2×D412yes—
102C4×D269—216no—
103C4×D269—24no—
104C4×D269—22no—
105C4×D269—22no—
106C4×D269—42no—
107#14524130—22no—
108C23×C269C2×D411yescharacteristic
109Q8×C269—24no—
110Q8×C269—22no—
111Q8×C269C2×D414yes—
112C2×C4×C269—24no—
113C2×C4×C269—22no—
114C2×C4×C269—22no—
115C2×C4×C269C2×D411yescharacteristic
116C2×C4×C269C2×D412yes—
117C2×C4×C269C2×D412yes—
118C2×C4×C269C2411yescharacteristic
119C8×C269—22no—
120C22×C269—22no—
121C22×C269#12012yes—
122C22×C269C22×D411yescharacteristic
123C2×D269—22no—
124C2×D269—24no—
125C2×D269—22no—
126C2×D269—22no—
127C2×D269—44no—
128C2×D269—48no—
129C2×D269—44no—
130C2×D269—41no—
131C2×D269—42no—
132C4×C269—24no—
133C4×C269—42no—
134C4×C269#12012yes—
135C4×C269C22×D411yescharacteristic
136C4×C269C22×D411yescharacteristic
137Dic269—22no—
138Dic269—41no—
139C2×C269—22no—
140C2×C269C2×#12011yescharacteristic
141C2×C269#57512yes—
142D269—42no—
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#419D26911yescharacteristic, complemented
154#418—2691no—
155#407—2691no—
156C2×#41—2694no—
157C2×#41—2691no—
158C2×#41—5382no—
159C2×#41—5384no—
160C2×#41—5384no—
161C22×Q8C2×D26911yescharacteristic
162C22×D4—2691no—
163C22×D4—5381no—
164C23×C4—2691no—
165C23×C4—5381no—
166#137—5384no—
167AGL(1,ℤ/8)—5384no—
168C2×Q16—2692no—
169C2×SD16—2692no—
170C2×SD16—2692no—
171C2×D8—2692no—
172C2×M16—2691no—
173C2×M16C2×D26911yescharacteristic
174#123—5381no—
175C22⋊Q8—2692no—
176#121—2692no—
177#120—5381no—
178C4×D4—5382no—
179#117C2×D26911yescharacteristic
180#103C2×D26914yescomplemented
181#102—2694no—
182C22⋊C8—2694no—
183C24—2691no—
184C24—5381no—
185#41—5388no—
186#41—53816no—
187#41—5388no—
188#41—5382no—
189#41—5382no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—2691no—
193C2×Q8D107614yescomplemented
194C2×Q8C22×D26912yes—
195C2×D4—2694no—
196C2×D4—2692no—
197C2×D4—2691no—
198C2×D4—5384no—
199C2×D4—5388no—
200C2×D4—5382no—
201C2×D4—5381no—
202C2×D4—5381no—
203C2×D4—5382no—
204C22×C4—21no—
205C22×C4—2691no—
206C22×C4—2692no—
207C22×C4—2692no—
208C22×C4—5384no—
209C22×C4—5388no—
210C22×C4—5382no—
211C22×C4—5382no—
212C22×C4—5381no—
213C22×C4C22×D26911yescharacteristic
214Q16—5384no—
215SD16—5384no—
216SD16—5384no—
217D8—5384no—
218M16—22no—
219M16—5382no—
220C2×C8—2692no—
221C2×C8C22×D26912yes—
222C4⋊C4—5382no—
223C4⋊C4C22×D26912yes—
224C22⋊C4—21no—
225C22⋊C4—5382no—
226C22⋊C4—5382no—
227C42—21no—
228C23—2692no—
229C23—5384no—
230C23—5382no—
231C23—5382no—
232C23—5381no—
233C23—10762no—
234C23—10764no—
235C23D4×D26911yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—5381no—
239Q8—5381no—
240Q8C2×D107614yes—
241D4—2694no—
242D4—53816no—
243D4—5384no—
244D4—5381no—
245D4—5382no—
246D4—5381no—
247D4—10764no—
248D4—10762no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—2692no—
253C2×C4—53816no—
254C2×C4—5382no—
255C2×C4—5384no—
256C2×C4—5382no—
257C2×C4—5382no—
258C2×C4—5381no—
259C2×C4—10762no—
260C2×C4D4×D26911yescharacteristic
261C2×C4D4×D26912yes—
262C2×C4C2×D107612yes—
263C2×C4C23×D26911yescharacteristic
264C8—22no—
265C8—5382no—
266C22—22no—
267C22—5382no—
268C22—5382no—
269C22—5384no—
270C22—5382no—
271C22—10764no—
272C22—10768no—
273C22—10764no—
274C22—10762no—
275C22—10761no—
276C22#2257542312yes—
277C22C2×D4×D26911yescharacteristic
278C4—24no—
279C4—42no—
280C4—5382no—
281C4—10761no—
282C4#2257542312yes—
283C4C2×D4×D26911yescharacteristic
284C4C22×D107611yescharacteristic
285C2—22no—
286C2—10762no—
287C2—10764no—
288C2—21521no—
289C2C2×#2257542311yescharacteristic
290C2#3177140412yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#419×C269C2
socleC22×C269C22×D4
hypercenter#419D269
hypocenterC269#3005
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#16864403C2semidirect
2. C2×#22575484C2semidirect
3. C2×#22575481C2semidirect
4. #31771117C2semidirect
5. #31770770C2semidirect
6. #31770753C2semidirect
7. #31770609C2semidirect
8. #31770568C2semidirect
9. #419×C269C2semidirect
21. C2×#16864392C22semidirect
22. C2×#16864391C22semidirect
23. C2×#16864389C22semidirect
24. C2×D2152C22semidirect
27. #22575433C22semidirect
28. #22575428C22semidirect
31. #22575373C22semidirect
32. #22575365C22semidirect
36. #103×C269C22semidirect
50. C2×D1076D4semidirect
73. C2×Q8×C269D4semidirect
145. C269#3005semidirect
153. #419D269semidirect
180. #103C2×D269semidirect
193. C2×Q8D1076semidirect

Maximal subgroups

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

subgroupindex
C22×#168644032
C2×#225754842
C2×#225754812
#317711172
#317707702
#317707532
#317706092
#317705682
#419×C2692
#3005269