Finite Group Databasebeta

← #46822670

Subgroups — #46822670

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#31771411C212yesmaximal, complemented
3#31771406C211yesmaximal, characteristic, complemented
4#31770877C211yesmaximal, characteristic, complemented
5#31770872C211yesmaximal, characteristic, complemented
6#31770865C212yesmaximal, complemented
7#31770782C212yesmaximal, complemented
8#31770777C212yesmaximal, complemented
9#31770723C211yesmaximal, characteristic, complemented
10#31770723C211yesmaximal, characteristic, complemented
11#495×C269C211yesmaximal, characteristic, complemented
12#22575501C2212yescomplemented
13#22575498C2212yes—
14#22575485—22no—
15#22575484C2212yes—
16#22575484C2212yescomplemented
17#22575482—22no—
18#22575480—22no—
19#22575479C2212yes—
20#22575479C2212yescomplemented
21#22575477—22no—
22#22575476—21no—
23#22575475C2211yescharacteristic
24#22575475C2211yescharacteristic
25#22575474—22no—
26#22575472—21no—
27Q16×D269—21no—
28SD16×D269—22no—
29D8×D269—21no—
30#22575459C2211yescharacteristic
31#22575459C2212yescomplemented
32#22575459C2211yescharacteristic, complemented
33#22575389C2211yescharacteristic, complemented
34#22575389C2211yescharacteristic, complemented
35#22575381C2211yescharacteristic
36#22575381C2211yescharacteristic, complemented
37#22575381C2212yescomplemented
38#22575372C2212yescomplemented
39#22575370C2212yescomplemented
40C2×Q16×C269C2211yescharacteristic, complemented
41C2×SD16×C269C2212yescomplemented
42C2×D8×C269C2211yescharacteristic
43#124×C269C2212yescomplemented
44C4×C8×C269C2211yescharacteristic, complemented
45#16864403—22no—
46#16864403—22no—
47#16864402—42no—
48#16864402C2311yescharacteristic
49#16864402C2312yes—
50#16864402C2311yescharacteristic
51#16864401—22no—
52#16864401—22no—
53Q8×D269—22no—
54Q8×D269—22no—
55C2×C22⋊D269—42no—
56D4×D269—22no—
57D4×D269—22no—
58Dic1076D411yescharacteristic
59Dic1076D411yescharacteristic
60#16864392—22no—
61#16864392—22no—
62#16864391D411yescharacteristic
63#16864391D411yescharacteristic
64#16864391D412yes—
65#16864390—22no—
66#16864390—22no—
67#16864389—22no—
68#16864389—22no—
69D2152D411yescharacteristic
70D2152D411yescharacteristic
71#16864387—22no—
72#16864387—22no—
73#16864386C2311yescharacteristic
74#16864386C2312yes—
75#16864386C2311yescharacteristic
76#16864385—21no—
77#16864385—22no—
78#16864385—21no—
79C8×D269—21no—
80C8×D269—22no—
81C8×D269—21no—
82C2×Q8×C269C2312yes—
83C2×D4×C269C2312yes—
84Q16×C269—21no—
85Q16×C269—21no—
86SD16×C269—22no—
87SD16×C269—22no—
88D8×C269—21no—
89D8×C269—21no—
90C2×C8×C269C2311yescharacteristic
91C2×C8×C269C2311yescharacteristic
92C22⋊C4×C269—42no—
93C42×C269C2311yescharacteristic
94C22×D269—42no—
95Dic538—42no—
96Dic538C2×D411yescharacteristic
97Dic538C2×D412yes—
98Dic538C2×D411yescharacteristic
99D1076—42no—
100D1076C2×D411yescharacteristic
101D1076C2×D412yes—
102D1076C2×D411yescharacteristic
103C22⋊D269—21no—
104C22⋊D269—22no—
105C22⋊D269—21no—
106C22⋊D269—42no—
107C22⋊D269—42no—
108C2×Dic269—42no—
109C4×D269—21no—
110C4×D269—22no—
111C4×D269—21no—
112C4×D269—42no—
113C4×D269—42no—
114#14524130—21no—
115#14524130—22no—
116#14524130—21no—
117C23×C269—22no—
118Q8×C269—22no—
119Q8×C269—22no—
120D4×C269—22no—
121D4×C269—22no—
122C2×C4×C269—21no—
123C2×C4×C269—22no—
124C2×C4×C269C2411yescharacteristic
125C8×C269C2×D411yescharacteristic
126C8×C269C2×D411yescharacteristic
127C8×C269C2×D411yescharacteristic
128C8×C269C2×D411yescharacteristic
129C22×C269—42no—
130C22×C269—82no—
131C22×C269H(5,𝔽2)11yescharacteristic
132C2×D269—21no—
133C2×D269—22no—
134C2×D269—21no—
135C2×D269—162no—
136C4×C269—41no—
137C4×C269—42no—
138C4×C269C22×D411yescharacteristic
139C4×C269C22×D411yescharacteristic
140Dic269—21no—
141Dic269—22no—
142Dic269—21no—
143C2×C269—21no—
144C2×C269—82no—
145C2×C269D4211yescharacteristic
146D269—41no—
147D269—42no—
148D269—41no—
149C269#329011yescharacteristic, complemented, Sylow
150#3290—2691nomaximal, Sylow
151#578—2691no—
152#577—2692no—
153#576—2691no—
154#495D26911yescharacteristic, complemented
155#473—2691no—
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—2692no—
166#137—5382no—
167#137—5382no—
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—5382no—
180C2×SD16C2×D26912yescomplemented
181C2×D8—5381no—
182C2×D8C2×D26911yescharacteristic
183#131—2691no—
184#131—2692no—
185#131—2691no—
186#124C2×D26912yescomplemented
187#108—2691no—
188#108—2691no—
189#104—2691no—
190#104—2691no—
191#104—2692no—
192#101—2692no—
193#100—2692no—
194C4×C8C2×D26911yescharacteristic, complemented
195#41—2691no—
196#41—2692no—
197#41—2691no—
198#41—5382no—
199#41—5382no—
200#41—5382no—
201#41—5382no—
202#41—10762no—
203C2×Q8—5382no—
204C2×Q8—5382no—
205C2×Q8C22×D26912yes—
206C2×D4—5382no—
207C2×D4—5382no—
208C2×D4—10762no—
209C2×D4C22×D26912yes—
210Q16—21no—
211Q16—21no—
212Q16—2691no—
213Q16—2691no—
214Q16—5382no—
215Q16—5382no—
216SD16—22no—
217SD16—22no—
218SD16—2691no—
219SD16—2691no—
220SD16—2692no—
221SD16—5382no—
222SD16—5382no—
223SD16—5382no—
224SD16—5382no—
225D8—21no—
226D8—21no—
227D8—2691no—
228D8—2691no—
229D8—5382no—
230D8—5382no—
231M16—2691no—
232M16—2692no—
233M16—2691no—
234M16—5381no—
235M16—5382no—
236M16—5381no—
237C2×C8—5381no—
238C2×C8—5382no—
239C2×C8—5381no—
240C2×C8C22×D26911yescharacteristic
241C2×C8C22×D26911yescharacteristic
242C22⋊C4—42no—
243C42C22×D26911yescharacteristic
244C23—22no—
245C23—10762no—
246Q8—22no—
247Q8—22no—
248Q8—2691no—
249Q8—2692no—
250Q8—2691no—
251Q8—10762no—
252D4—22no—
253D4—22no—
254D4—2691no—
255D4—2692no—
256D4—2691no—
257D4—5381no—
258D4—5382no—
259D4—5381no—
260D4—10762no—
261D4—10762no—
262D4—10762no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—5381no—
266C2×C4—5382no—
267C2×C4—5381no—
268C2×C4—10762no—
269C2×C4—10762no—
270C2×C4—10762no—
271C2×C4C23×D26911yescharacteristic
272C8—5381no—
273C8—5382no—
274C8—5381no—
275C8D4×D26911yescharacteristic
276C8D4×D26911yescharacteristic
277C8D4×D26911yescharacteristic
278C8D4×D26911yescharacteristic
279C22—42no—
280C22—82no—
281C22—5381no—
282C22—5382no—
283C22—5381no—
284C22—43042no—
285C22#2257549811yescharacteristic
286C4—41no—
287C4—42no—
288C4—5381no—
289C4—5382no—
290C4—5381no—
291C4C2×D4×D26911yescharacteristic
292C4C2×D4×D26911yescharacteristic
293C2—21no—
294C2—82no—
295C2—10761no—
296C2—10762no—
297C2—10761no—
298C2#3177124011yescharacteristic
subgroupG/H
centerC2#31771240
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#495×C269C2
socleC2×C269D42
hypercenter#495D269
hypocenterC269#3290
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #31771418C2semidirect
2. #31771411C2semidirect
3. #31771406C2semidirect
4. #31770877C2semidirect
5. #31770872C2semidirect
6. #31770865C2semidirect
7. #31770782C2semidirect
8. #31770777C2semidirect
9. #31770723C2semidirect
10. #31770723C2semidirect
11. #495×C269C2semidirect
12. #22575501C22semidirect
16. #22575484C22semidirect
20. #22575479C22semidirect
31. #22575459C22semidirect
32. #22575459C22semidirect
33. #22575389C22semidirect
34. #22575389C22semidirect
36. #22575381C22semidirect
37. #22575381C22semidirect
38. #22575372C22semidirect
39. #22575370C22semidirect
40. C2×Q16×C269C22semidirect
41. C2×SD16×C269C22semidirect
43. #124×C269C22semidirect
44. C4×C8×C269C22semidirect
149. C269#3290semidirect
154. #495D269semidirect
178. C2×Q16C2×D269semidirect
180. C2×SD16C2×D269semidirect
186. #124C2×D269semidirect
194. C4×C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714182
#317714112
#317714062
#317708772
#317708722
#317708652
#317707822
#317707772
#317707232
#495×C2692
#3290269