Finite Group Databasebeta

← #49134401

Subgroups — #49134401

Order 36224.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#31827267C211yesmaximal, characteristic, complemented
2#31827266C211yesmaximal, characteristic, complemented
3#31826903C211yesmaximal, characteristic, complemented
4#31826902C212yesmaximal, complemented
5#31826899C212yesmaximal, complemented
6#31826896C212yesmaximal, complemented
7#31826895C211yesmaximal, characteristic, complemented
8C2×#22583551C211yesmaximal, characteristic, complemented
9D16×D283C212yesmaximal, complemented
10#31826883C211yesmaximal, characteristic, complemented
11#509×C283C211yesmaximal, characteristic, complemented
12#22583656—21no—
13#22583655—21no—
14#22583637—22no—
15#22583636—21no—
16#22583635—21no—
17#22583634—22no—
18#22583633C2212yescomplemented
19#22583632C2211yescharacteristic, complemented
20#22583629C2212yescomplemented
21C2×D2264C2211yescharacteristic, complemented
22D8×D283C2212yescomplemented
23D8×D283C2212yescomplemented
24#22583616C2211yescharacteristic
25#22583556C2211yescharacteristic, complemented
26#22583555C2212yescomplemented
27#22583554C2211yescharacteristic, complemented
28#22583553C2211yescharacteristic, complemented
29D4528C2212yescomplemented
30#22583551C2211yescharacteristic, complemented
31#22583551C2212yescomplemented
32#22583551C2212yescomplemented
33#22583549C2212yescomplemented
34C2×C283⋊C16C2211yescharacteristic, complemented
35C16×D283C2212yescomplemented
36#135×C283C2211yescharacteristic, complemented
37C2×D8×C283C2211yescharacteristic, complemented
38SD32×C283C2212yescomplemented
39D16×C283C2212yescomplemented
40M32×C283C2211yescharacteristic, complemented
41#17985393—22no—
42#17985392—21no—
43#17985392—41no—
44#17985392D411yescharacteristic
45#17985391—22no—
46#17985391—41no—
47C2×D1132—21no—
48C2×D1132—41no—
49C2×C22⋊D283—42no—
50D4×D283—22no—
51D4×D283—22no—
52D4×D283—41no—
53D4×D283—42no—
54Dic1132C2311yescharacteristic
55#17985381—21no—
56#17985380—22no—
57#17985379—22no—
58D2264—21no—
59D2264C2311yescharacteristic
60D2264C2312yes—
61#17985377—22no—
62#17985377—22no—
63#17985376D411yescharacteristic
64#17985375D412yes—
65C8×D283C2312yes—
66C283⋊C16C2311yescharacteristic
67C283⋊C16C2311yescharacteristic
68#41×C283—21no—
69C2×D4×C283—21no—
70Q16×C283C2311yescharacteristic
71SD16×C283—21no—
72D8×C283—21no—
73D8×C283C2311yescharacteristic
74D8×C283C2312yes—
75C2×C8×C283C2311yescharacteristic
76C16×C283C2312yes—
77C22×D283—41no—
78C22×D283—42no—
79C22×D283—42no—
80Dic566—21no—
81Dic566D811yescharacteristic
82D1132—21no—
83D1132—22no—
84D1132—42no—
85D1132—41no—
86D1132D811yescharacteristic
87C22⋊D283—41no—
88C22⋊D283—42no—
89C22⋊D283—42no—
90C22⋊D283—42no—
91C22⋊D283D812yes—
92C2×Dic283—42no—
93C4×D283—41no—
94C4×D283—42no—
95C4×D283C2×D412yes—
96C283⋊C8C2×D412yes—
97C23×C283—41no—
98Q8×C283—21no—
99D4×C283—22no—
100D4×C283—21no—
101D4×C283—41no—
102D4×C283—41no—
103C2×C4×C283—41no—
104C2×C4×C283C2×D411yescharacteristic
105C8×C283C2×D411yescharacteristic
106C8×C283C2411yescharacteristic
107C22×C283—41no—
108C22×C283—42no—
109C22×C283—161no—
110C22×C283C2×D811yescharacteristic
111C2×D283—41no—
112C2×D283—42no—
113C2×D283—162no—
114C2×D283—162no—
115C2×D283—161no—
116C2×D283C2×D812yes—
117C4×C283—41no—
118C4×C283C2×D811yescharacteristic
119C4×C283C22×D411yescharacteristic
120Dic283—41no—
121Dic283C2×D812yes—
122C2×C283—21no—
123C2×C283—81no—
124C2×C283—82no—
125C2×C283C22×D811yescharacteristic
126D283—22no—
127D283—81no—
128D283—82no—
129C283#341511yescharacteristic, complemented, Sylow
130#3415—2831nomaximal, Sylow
131#576—2831no—
132#576—2831no—
133#509—2832no—
134#509—2832no—
135#509—2831no—
136#509D28311yescharacteristic, complemented
137#508—2831no—
138#508—2832no—
139C2×D16—2831no—
140C2×D16—2832no—
141#504—2831no—
142H(5,𝔽2)—5661no—
143H(5,𝔽2)—5661no—
144AGL(1,ℤ/8)—5662no—
145AGL(1,ℤ/8)—5662no—
146AGL(1,ℤ/8)—5661no—
147AGL(1,ℤ/8)—5661no—
148#135—2832no—
149#135—2832no—
150#135—2831no—
151#135C2×D28311yescharacteristic, complemented
152C2×D8—2832no—
153C2×D8—2832no—
154C2×D8—2831no—
155C2×D8C2×D28311yescharacteristic, complemented
156#131—2831no—
157Q32—2831no—
158SD32—2831no—
159SD32—2831no—
160SD32—2832no—
161SD32C2×D28312yescomplemented
162D16—2832no—
163D16—2831no—
164D16—2832no—
165D16—2832no—
166D16C2×D28312yescomplemented
167M32—2832no—
168M32C2×D28311yescharacteristic, complemented
169C2×C16—2831no—
170C2×C16—2832no—
171#41—21no—
172#41—2831no—
173#41—5662no—
174#41—5662no—
175#41—5661no—
176#41—11321no—
177#41—11321no—
178C2×D4—21no—
179C2×D4—5662no—
180C2×D4—5662no—
181C2×D4—5661no—
182C2×D4—11321no—
183C2×D4—11322no—
184C2×D4—11321no—
185C2×D4—11322no—
186Q16—2831no—
187Q16C22×D28311yescharacteristic
188SD16—21no—
189SD16—5662no—
190SD16—5662no—
191SD16—5661no—
192D8—21no—
193D8—2831no—
194D8—2832no—
195D8—5662no—
196D8—5662no—
197D8—5661no—
198D8C22×D28312yes—
199D8C22×D28311yescharacteristic
200M16—2831no—
201M16—2832no—
202C2×C8—2832no—
203C2×C8C22×D28311yescharacteristic
204C16—2831no—
205C16—2831no—
206C16C22×D28312yes—
207C23—41no—
208C23—11322no—
209C23—11322no—
210C23—11321no—
211Q8—21no—
212Q8—2831no—
213Q8—5661no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—2832no—
219D4—2831no—
220D4—5661no—
221D4—5662no—
222D4—11322no—
223D4—11322no—
224D4—11322no—
225D4—11322no—
226D4—11321no—
227D4—11321no—
228C2×C4—41no—
229C2×C4—2832no—
230C2×C4—11322no—
231C2×C4—11322no—
232C2×C4—11321no—
233C2×C4D4×D28311yescharacteristic
234C8—2832no—
235C8D4×D28311yescharacteristic
236C8C23×D28311yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—2832no—
241C22—11322no—
242C22—11321no—
243C22—45281no—
244C22—45282no—
245C22—45282no—
246C22D8×D28311yescharacteristic
247C4—41no—
248C4—2832no—
249C4—11321no—
250C4D8×D28311yescharacteristic
251C4C2×D4×D28311yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—5662no—
256C2—22642no—
257C2—22641no—
258C2C2×D8×D28311yescharacteristic
subgroupG/H
centerC2C2×D8×D283
commutator, cosocleC8×C283C24
FrattiniC8C23×D283
Fitting#509×C283C2
socleC2×C283C22×D8
hypercenter#509D283
hypocenterC283#3415
LCSG ▷ C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #31827267C2semidirect
2. #31827266C2semidirect
3. #31826903C2semidirect
4. #31826902C2semidirect
5. #31826899C2semidirect
6. #31826896C2semidirect
7. #31826895C2semidirect
8. C2×#22583551C2semidirect
9. D16×D283C2semidirect
10. #31826883C2semidirect
11. #509×C283C2semidirect
18. #22583633C22semidirect
19. #22583632C22semidirect
20. #22583629C22semidirect
21. C2×D2264C22semidirect
22. D8×D283C22semidirect
23. D8×D283C22semidirect
25. #22583556C22semidirect
26. #22583555C22semidirect
27. #22583554C22semidirect
28. #22583553C22semidirect
29. D4528C22semidirect
30. #22583551C22semidirect
31. #22583551C22semidirect
32. #22583551C22semidirect
33. #22583549C22semidirect
34. C2×C283⋊C16C22semidirect
35. C16×D283C22semidirect
36. #135×C283C22semidirect
37. C2×D8×C283C22semidirect
38. SD32×C283C22semidirect
39. D16×C283C22semidirect
40. M32×C283C22semidirect
129. C283#3415semidirect
136. #509D283semidirect
151. #135C2×D283semidirect
155. C2×D8C2×D283semidirect
161. SD32C2×D283semidirect
166. D16C2×D283semidirect
168. M32C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272672
#318272662
#318269032
#318269022
#318268992
#318268962
#318268952
C2×#225835512
D16×D2832
#318268832
#509×C2832
#3415283