Finite Group Databasebeta

← #46822674

Subgroups — #46822674

Order 34432.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#31771418C211yesmaximal, characteristic, complemented
2#31771414C212yesmaximal, complemented
3#31771410C211yesmaximal, characteristic, complemented
4#31770990C211yesmaximal, characteristic, complemented
5#31770876C211yesmaximal, characteristic, complemented
6#31770871C212yesmaximal, complemented
7#31770781C212yesmaximal, complemented
8#31770780C212yesmaximal, complemented
9#31770721C211yesmaximal, characteristic, complemented
10#31770720C211yesmaximal, characteristic, complemented
11#473×C269C211yesmaximal, characteristic, complemented
12#22575501C2212yescomplemented
13#22575499C2212yes—
14#22575485—22no—
15#22575485—22no—
16#22575484C2212yescomplemented
17#22575481—22no—
18#22575481C2212yes—
19#22575481C2212yescomplemented
20#22575480—22no—
21#22575476—22no—
22#22575476—21no—
23#22575475—21no—
24#22575475C2211yescharacteristic
25#22575473C2211yescharacteristic
26Q16×D269—21no—
27C2×#16864389C2212yescomplemented
28SD16×D269—22no—
29C2×D2152—21no—
30C2×D2152C2211yescharacteristic
31#22575461C2212yescomplemented
32#22575459C2211yescharacteristic, complemented
33#22575442C2212yescomplemented
34#22575388C2211yescharacteristic, complemented
35#22575380C2211yescharacteristic
36#22575379C2211yescharacteristic, complemented
37#22575379C2212yescomplemented
38#22575369C2212yescomplemented
39C8×Dic269C2211yescharacteristic, complemented
40#137×C269C2212yes—
41C2×Q16×C269C2211yescharacteristic, complemented
42#131×C269C2211yescharacteristic
43#108×C269C2211yescharacteristic, complemented
44#101×C269C2212yescomplemented
45#16864403—22no—
46#16864403—22no—
47#16864403—22no—
48#16864402—22no—
49#16864402—42no—
50#16864402C2311yescharacteristic
51#16864402C2312yes—
52Q8×D269—22no—
53Q8×D269—22no—
54C2×D1076—22no—
55C2×D1076C2312yes—
56D4×D269—22no—
57D4×D269—42no—
58Dic1076D411yescharacteristic
59#16864392—22no—
60#16864392D411yescharacteristic
61#16864391—22no—
62#16864391—22no—
63#16864391D412yes—
64#16864390D411yescharacteristic
65#16864389—22no—
66#16864389—22no—
67#16864389—22no—
68#16864389D411yescharacteristic
69D2152—22no—
70D2152—22no—
71D2152—21no—
72D2152—21no—
73D2152D411yescharacteristic
74#16864387D411yescharacteristic
75#16864386C2311yescharacteristic
76#16864385—22no—
77#16864385—21no—
78C2×#14524130C2311yescharacteristic
79C8×D269—22no—
80C8×D269—21no—
81#16864379—42no—
82C4×Dic269C2311yescharacteristic
83#41×C269C2311yescharacteristic
84C2×Q8×C269C2312yes—
85Q16×C269—22no—
86Q16×C269—21no—
87Q16×C269—21no—
88SD16×C269—22no—
89M16×C269—21no—
90M16×C269C2312yes—
91M16×C269C2311yescharacteristic
92C2×C8×C269—21no—
93C2×C8×C269C2311yescharacteristic
94C22×D269—22no—
95C22×D269—42no—
96Dic538—42no—
97Dic538C2×D411yescharacteristic
98Dic538C2×D412yes—
99D1076—22no—
100D1076—22no—
101D1076—42no—
102D1076—42no—
103D1076C2×D411yescharacteristic
104D1076C2×D412yes—
105C22⋊D269—22no—
106C22⋊D269—21no—
107C22⋊D269—42no—
108C2×Dic269—21no—
109C4×D269—22no—
110C4×D269—21no—
111C4×D269—42no—
112C4×D269—42no—
113C4×D269—42no—
114#14524130—21no—
115#14524130C2×D411yescharacteristic
116#14524130C2×D411yescharacteristic
117Q8×C269—22no—
118Q8×C269—22no—
119Q8×C269C2×D411yescharacteristic
120D4×C269—21no—
121D4×C269C2×D411yescharacteristic
122C2×C4×C269—21no—
123C2×C4×C269—22no—
124C2×C4×C269C2411yescharacteristic
125C8×C269—21no—
126C8×C269—22no—
127C8×C269C2×D411yescharacteristic
128C8×C269C2×D411yescharacteristic
129C22×C269—21no—
130C22×C269H(5,𝔽2)11yescharacteristic
131C2×D269—21no—
132C2×D269—22no—
133C2×D269—42no—
134C2×D269—82no—
135C2×D269—162no—
136C4×C269—21no—
137C4×C269—42no—
138C4×C269C22×D411yescharacteristic
139C4×C269C22×D411yescharacteristic
140Dic269—22no—
141Dic269—21no—
142Dic269—41no—
143C2×C269—21no—
144C2×C269—41no—
145C2×C269D4211yescharacteristic
146D269—41no—
147D269—42no—
148D269—82no—
149C269#329011yescharacteristic, complemented, Sylow
150#3290—2691nomaximal, Sylow
151#578—2691no—
152#577—2692no—
153#576—2691no—
154#495—2691no—
155#473D26911yescharacteristic, complemented
156#472—2691no—
157#471—2692no—
158#455—2692no—
159#454—2692no—
160#443—2691no—
161#443—2691no—
1622−1+4—2692no—
163H(5,𝔽2)—2692no—
164#137—2692no—
165#137—5382no—
166#137—5382no—
167#137C2×D26912yes—
168AGL(1,ℤ/8)—2692no—
169AGL(1,ℤ/8)—2692no—
170AGL(1,ℤ/8)—5382no—
171AGL(1,ℤ/8)—5382no—
172#135—2691no—
173#135—2691no—
174#135—5381no—
175#135—5381no—
176#135—5382no—
177C2×Q16—5381no—
178C2×Q16C2×D26911yescharacteristic, complemented
179C2×SD16—2692no—
180C2×SD16—5382no—
181C2×D8—2691no—
182C2×D8—5381no—
183#131—2692no—
184#131—2691no—
185#131C2×D26911yescharacteristic
186#124—2692no—
187#108—2691no—
188#108C2×D26911yescharacteristic, complemented
189#104—2691no—
190#104—2691no—
191#104—2692no—
192#101C2×D26912yescomplemented
193#100—2692no—
194C4×C8—2691no—
195#41—2691no—
196#41—2692no—
197#41—5382no—
198#41—5382no—
199#41—5382no—
200#41—5382no—
201#41—10762no—
202#41C22×D26911yescharacteristic
203C2×Q8—5382no—
204C2×Q8—5382no—
205C2×Q8C22×D26912yes—
206C2×D4—2692no—
207C2×D4—5382no—
208C2×D4—5382no—
209C2×D4—10762no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—2691no—
214Q16—2691no—
215Q16—5382no—
216SD16—22no—
217SD16—2691no—
218SD16—2691no—
219SD16—2692no—
220SD16—5382no—
221SD16—5382no—
222SD16—5382no—
223SD16—5382no—
224SD16—5382no—
225D8—2691no—
226D8—2691no—
227D8—5382no—
228D8—5382no—
229D8—5381no—
230D8—5381no—
231M16—21no—
232M16—2691no—
233M16—5382no—
234M16—5381no—
235M16C22×D26912yes—
236M16C22×D26911yescharacteristic
237C2×C8—21no—
238C2×C8—2691no—
239C2×C8—5382no—
240C2×C8—5381no—
241C2×C8C22×D26911yescharacteristic
242C22⋊C4—10762no—
243C42—2691no—
244C23—5382no—
245C23—10762no—
246Q8—22no—
247Q8—22no—
248Q8—2691no—
249Q8—2692no—
250Q8—10762no—
251Q8D4×D26911yescharacteristic
252D4—21no—
253D4—2691no—
254D4—2692no—
255D4—5382no—
256D4—5381no—
257D4—5382no—
258D4—5382no—
259D4—10762no—
260D4—10762no—
261D4—10762no—
262D4D4×D26911yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—5382no—
266C2×C4—5381no—
267C2×C4—5381no—
268C2×C4—10762no—
269C2×C4—10762no—
270C2×C4—10762no—
271C2×C4C23×D26911yescharacteristic
272C8—21no—
273C8—22no—
274C8—2691no—
275C8—2691no—
276C8—5381no—
277C8D4×D26911yescharacteristic
278C8D4×D26911yescharacteristic
279C22—21no—
280C22—5381no—
281C22—5382no—
282C22—10762no—
283C22—21522no—
284C22—43042no—
285C22#2257549911yescharacteristic
286C4—21no—
287C4—42no—
288C4—5382no—
289C4—5381no—
290C4—10761no—
291C4C2×D4×D26911yescharacteristic
292C4C2×D4×D26911yescharacteristic
293C2—21no—
294C2—41no—
295C2—10762no—
296C2—10761no—
297C2—21522no—
298C2#3177123911yescharacteristic
subgroupG/H
centerC2#31771239
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#473×C269C2
socleC2×C269D42
hypercenter#473D269
hypocenterC269#3290
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #31771418C2semidirect
2. #31771414C2semidirect
3. #31771410C2semidirect
4. #31770990C2semidirect
5. #31770876C2semidirect
6. #31770871C2semidirect
7. #31770781C2semidirect
8. #31770780C2semidirect
9. #31770721C2semidirect
10. #31770720C2semidirect
11. #473×C269C2semidirect
12. #22575501C22semidirect
16. #22575484C22semidirect
19. #22575481C22semidirect
27. C2×#16864389C22semidirect
31. #22575461C22semidirect
32. #22575459C22semidirect
33. #22575442C22semidirect
34. #22575388C22semidirect
36. #22575379C22semidirect
37. #22575379C22semidirect
38. #22575369C22semidirect
39. C8×Dic269C22semidirect
41. C2×Q16×C269C22semidirect
43. #108×C269C22semidirect
44. #101×C269C22semidirect
149. C269#3290semidirect
155. #473D269semidirect
178. C2×Q16C2×D269semidirect
188. #108C2×D269semidirect
192. #101C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714182
#317714142
#317714102
#317709902
#317708762
#317708712
#317707812
#317707802
#317707212
#317707202
#473×C2692
#3290269