Finite Group Databasebeta

← #49131743

Subgroups — #49131743

Order 36224.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#22583637C212yesmaximal, complemented
2C2×#22583633C211yesmaximal, characteristic, complemented
3C2×D8×D283C211yesmaximal, characteristic, complemented
4#31827033C212yesmaximal, complemented
5#31826855C212yesmaximal, complemented
6#31826851C212yesmaximal, complemented
7#31826850C211yesmaximal, characteristic, complemented
8#31826842C211yesmaximal, characteristic, complemented
9#31826840C211yesmaximal, characteristic, complemented
10#31826419C211yesmaximal, characteristic, complemented
11#495×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2212yes—
13C2×D4×D283C2212yes—
14#22583637—28no—
15#22583633—24no—
16C2×#17985379C2212yescomplemented
17C2×#17985379C2212yescomplemented
18C2×D2264C2211yescharacteristic
19C2×D2264C2211yescharacteristic, complemented
20C2×D2264C2212yescomplemented
21C2×#17985377C2212yes—
22C2×#17985377C2212yescomplemented
23D8×D283—24no—
24C2×#17985375C2211yescharacteristic
25C2×C8×D283C2211yescharacteristic
26#22583607C2212yescomplemented
27#22583606C2212yescomplemented
28#22583599C2212yescomplemented
29#22583583—24no—
30#22583580—24no—
31#22583567C2211yescharacteristic
32#22583521C2211yescharacteristic, complemented
33#22583520C2211yescharacteristic, complemented
34C8×Dic283C2211yescharacteristic, complemented
35C2×Q16×C283C2211yescharacteristic, complemented
36C2×SD16×C283C2212yescomplemented
37C2×D8×C283C2211yescharacteristic
38#124×C283C2212yescomplemented
39C4×C8×C283C2211yescharacteristic, complemented
40C23×D283—22no—
41#17985393—24no—
42#17985393—24no—
43C2×D1132—22no—
44C2×D1132—24no—
45C2×D1132C2312yes—
46C2×D1132C2312yes—
47C2×C22⋊D283—22no—
48C2×C22⋊D283—24no—
49D4×D283—24no—
50D4×D283—24no—
51C2×C4×D283—22no—
52C2×C4×D283C2311yescharacteristic
53#17985379—24no—
54#17985379—24no—
55D2264—22no—
56D2264—22no—
57D2264—24no—
58#17985377—24no—
59#17985377—24no—
60#17985375D414yescomplemented
61C2×C283⋊C8C2311yescharacteristic
62C2×C283⋊C8C2311yescharacteristic
63C8×D283D414yescomplemented
64Dic283⋊C4D412yes—
65(C2×D283)⋊C4—24no—
66(C2×D283)⋊C4D412yes—
67C4×Dic283C2311yescharacteristic
68C2×Q8×C283C2312yes—
69C2×D4×C283C2312yes—
70Q16×C283—22no—
71SD16×C283—24no—
72D8×C283—22no—
73C2×C8×C283C2311yescharacteristic
74C2×C8×C283C2311yescharacteristic
75C22⋊C4×C283—24no—
76C42×C283C2311yescharacteristic
77C22×D283—22no—
78C22×D283—22no—
79C22×D283—44no—
80C22×D283—84no—
81C22×D283C2×D411yescharacteristic
82D1132—22no—
83D1132—22no—
84D1132—22no—
85D1132—22no—
86D1132—44no—
87D1132—84no—
88C22⋊D283—44no—
89C22⋊D283—84no—
90C2×Dic283C2×D412yes—
91C2×Dic283C2×D411yescharacteristic
92C4×D283—44no—
93C4×D283C2×D412yes—
94C4×D283C2×D412yes—
95C283⋊C8C2×D412yes—
96C283⋊C8C2×D412yes—
97C23×C283—22no—
98Q8×C283—22no—
99Q8×C283—22no—
100D4×C283—22no—
101D4×C283—22no—
102C2×C4×C283—22no—
103C2×C4×C283C2×D412yes—
104C2×C4×C283C2411yescharacteristic
105C8×C283C2×D412yes—
106C8×C283C2×D412yes—
107C22×C283—42no—
108C22×C283—44no—
109C22×C283C22×D411yescharacteristic
110C2×D283—22no—
111C2×D283—42no—
112C2×D283—42no—
113C2×D283—44no—
114C2×D283—44no—
115C2×D283—88no—
116C2×D283#12712yes—
117C4×C283—22no—
118C4×C283—42no—
119C4×C283C22×D411yescharacteristic
120C4×C283C22×D411yescharacteristic
121Dic283—22no—
122Dic283#12712yes—
123C2×C283—82no—
124C2×C283C2×#12711yescharacteristic
125C2×C283#57612yes—
126D283—22no—
127D283—82no—
128D283—82no—
129C283#315511yescharacteristic, complemented, Sylow
130#3155—2831nomaximal, Sylow
131C2×AGL(1,ℤ/8)—2832no—
132C2×#135—2831no—
133C22×D8—2831no—
134#535—2832no—
135#496—2832no—
136#496—2832no—
137#495—2831no—
138#495D28311yescharacteristic, complemented
139#493—2831no—
140#493—2831no—
141#405—2831no—
142C2×#41—2832no—
143C22×D4—2832no—
144AGL(1,ℤ/8)—5668no—
145#135—5664no—
146C2×Q16C2×D28311yescharacteristic, complemented
147C2×SD16—2832no—
148C2×SD16—2832no—
149C2×SD16C2×D28312yescomplemented
150C2×D8—2832no—
151C2×D8—2832no—
152C2×D8—2831no—
153C2×D8—2831no—
154C2×D8—2832no—
155C2×D8—5664no—
156C2×D8C2×D28311yescharacteristic
157C2×M16—2831no—
158C22×C8—2831no—
159#127—2832no—
160#127—2832no—
161#124—2832no—
162#124C2×D28312yescomplemented
163#121—5664no—
164#120—5664no—
165#117—2831no—
166#97—2831no—
167#97—2831no—
168C4×C8—2831no—
169C4×C8C2×D28311yescharacteristic, complemented
170C24—5662no—
171#41—5664no—
172#41—5664no—
173C2×Q8C22×D28312yes—
174C2×D4—2832no—
175C2×D4—2832no—
176C2×D4—5662no—
177C2×D4—5664no—
178C2×D4—5664no—
179C2×D4—5662no—
180C2×D4—5664no—
181C2×D4—5664no—
182C2×D4C22×D28312yes—
183C22×C4—2831no—
184C22×C4—5662no—
185Q16—22no—
186SD16—24no—
187SD16—5664no—
188SD16—5664no—
189D8—22no—
190D8—5664no—
191D8—5664no—
192D8—5662no—
193D8—5662no—
194D8—5664no—
195M16—2834no—
196C2×C8—2831no—
197C2×C8—2831no—
198C2×C8—2834no—
199C2×C8C22×D28311yescharacteristic
200C2×C8C22×D28311yescharacteristic
201C4⋊C4—2832no—
202C22⋊C4—24no—
203C22⋊C4—2832no—
204C22⋊C4—5664no—
205C42—2831no—
206C42C22×D28311yescharacteristic
207C23—22no—
208C23—2831no—
209C23—5662no—
210C23—5662no—
211C23—11324no—
212C23—22644no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—5662no—
218D4—5662no—
219D4—5662no—
220D4—5662no—
221D4—11324no—
222D4—11324no—
223D4—22644no—
224D4—22644no—
225C2×C4—22no—
226C2×C4—2831no—
227C2×C4—2832no—
228C2×C4—2832no—
229C2×C4—2832no—
230C2×C4—11324no—
231C2×C4D4×D28312yes—
232C2×C4C23×D28311yescharacteristic
233C8—2832no—
234C8—2832no—
235C8D4×D28312yes—
236C8D4×D28312yes—
237C22—42no—
238C22—44no—
239C22—2832no—
240C22—5662no—
241C22—11322no—
242C22—11322no—
243C22—11324no—
244C22—11324no—
245C22—22648no—
246C22C2×D4×D28311yescharacteristic
247C4—22no—
248C4—42no—
249C4—2832no—
250C4—5662no—
251C4C2×D4×D28311yescharacteristic
252C4C2×D4×D28311yescharacteristic
253C2—82no—
254C2—5662no—
255C2—22642no—
256C2—22642no—
257C2#127×D28311yescharacteristic
258C2#3182726712yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#495×C283C2
socleC22×C283C22×D4
hypercenter#495D283
hypocenterC283#3155
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. C2×#22583633C2semidirect
3. C2×D8×D283C2semidirect
4. #31827033C2semidirect
5. #31826855C2semidirect
6. #31826851C2semidirect
7. #31826850C2semidirect
8. #31826842C2semidirect
9. #31826840C2semidirect
10. #31826419C2semidirect
11. #495×C283C2semidirect
16. C2×#17985379C22semidirect
17. C2×#17985379C22semidirect
19. C2×D2264C22semidirect
20. C2×D2264C22semidirect
22. C2×#17985377C22semidirect
26. #22583607C22semidirect
27. #22583606C22semidirect
28. #22583599C22semidirect
32. #22583521C22semidirect
33. #22583520C22semidirect
34. C8×Dic283C22semidirect
35. C2×Q16×C283C22semidirect
36. C2×SD16×C283C22semidirect
38. #124×C283C22semidirect
39. C4×C8×C283C22semidirect
60. #17985375D4semidirect
63. C8×D283D4semidirect
129. C283#3155semidirect
138. #495D283semidirect
146. C2×Q16C2×D283semidirect
149. C2×SD16C2×D283semidirect
162. #124C2×D283semidirect
169. C4×C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
C2×#225836332
C2×D8×D2832
#318270332
#318268552
#318268512
#318268502
#318268422
#318268402
#318264192
#495×C2832
#3155283