Finite Group Databasebeta

← #49131741

Subgroups — #49131741

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×#22583634C212yesmaximal, complemented
2C2×#22583629C211yesmaximal, characteristic, complemented
3C2×D8×D283C211yesmaximal, characteristic, complemented
4#31827028C212yesmaximal, complemented
5#31826853C212yesmaximal, complemented
6#31826852C212yesmaximal, complemented
7#31826849C211yesmaximal, characteristic, complemented
8#31826848C211yesmaximal, characteristic, complemented
9#31826840C211yesmaximal, characteristic, complemented
10#31826419C211yesmaximal, characteristic, complemented
11#493×C283C211yesmaximal, characteristic, complemented
12C2×#17985391C2212yes—
13C2×D4×D283C2212yes—
14#22583634—28no—
15#22583629—24no—
16C2×Dic1132C2211yescharacteristic, complemented
17C2×#17985381C2212yescomplemented
18C2×#17985380C2212yescomplemented
19C2×#17985380C2212yescomplemented
20C2×D2264C2211yescharacteristic
21C2×#17985377C2212yes—
22C2×#17985377C2212yescomplemented
23D8×D283—24no—
24C2×#17985375C2211yescharacteristic
25C2×C8×D283C2211yescharacteristic
26#22583606C2212yescomplemented
27#22583598C2212yescomplemented
28#22583597C2212yescomplemented
29#22583582—24no—
30#22583578—24no—
31#22583567C2211yescharacteristic
32#22583521C2211yescharacteristic, complemented
33#22583520C2211yescharacteristic, complemented
34C8×Dic283C2211yescharacteristic, complemented
35C2×D8×C283C2211yescharacteristic
36C2×D8×C283C2211yescharacteristic, complemented
37C2×D8×C283C2212yescomplemented
38#127×C283C2212yescomplemented
39C4×C8×C283C2211yescharacteristic, complemented
40C23×D283—22no—
41#17985391—24no—
42#17985391—24no—
43C2×Dic566C2312yes—
44C2×D1132C2312yes—
45C2×C22⋊D283—22no—
46C2×C22⋊D283—22no—
47C2×C22⋊D283—24no—
48D4×D283—24no—
49D4×D283—24no—
50C22×Dic283—22no—
51C2×C4×D283C2311yescharacteristic
52Dic1132—22no—
53#17985381—24no—
54#17985380—24no—
55#17985380—24no—
56D2264—22no—
57#17985377—24no—
58#17985377—24no—
59#17985375D414yescomplemented
60C2×C283⋊C8C2311yescharacteristic
61C2×C283⋊C8C2311yescharacteristic
62C8×D283D414yescomplemented
63Dic283⋊C4D412yes—
64C22⋊Dic283—24no—
65(C2×D283)⋊C4—24no—
66(C2×D283)⋊C4D412yes—
67C4×Dic283C2311yescharacteristic
68C2×D4×C283—24no—
69C2×D4×C283C2312yes—
70C2×D4×C283C2312yes—
71D8×C283—22no—
72D8×C283—22no—
73D8×C283—24no—
74C2×C8×C283C2311yescharacteristic
75C2×C8×C283C2311yescharacteristic
76C42×C283C2311yescharacteristic
77C22×D283—22no—
78C22×D283—44no—
79C22×D283—84no—
80C22×D283C2×D411yescharacteristic
81Dic566—22no—
82Dic566—22no—
83D1132—22no—
84D1132—22no—
85C22⋊D283—44no—
86C22⋊D283—44no—
87C22⋊D283—84no—
88C2×Dic283—22no—
89C2×Dic283—44no—
90C2×Dic283C2×D412yes—
91C2×Dic283C2×D411yescharacteristic
92C4×D283C2×D412yes—
93C4×D283C2×D412yes—
94C283⋊C8C2×D412yes—
95C283⋊C8C2×D412yes—
96C23×C283—22no—
97C23×C283—22no—
98D4×C283—22no—
99D4×C283—22no—
100D4×C283—22no—
101D4×C283—22no—
102D4×C283—84no—
103C2×C4×C283C2×D412yes—
104C2×C4×C283C2411yescharacteristic
105C8×C283C2×D412yes—
106C8×C283C2×D412yes—
107C22×C283—42no—
108C22×C283—42no—
109C22×C283—44no—
110C22×C283—44no—
111C22×C283C22×D411yescharacteristic
112C2×D283—22no—
113C2×D283—42no—
114C2×D283—44no—
115C2×D283—88no—
116C2×D283#12712yes—
117C4×C283—22no—
118C4×C283C22×D411yescharacteristic
119C4×C283C22×D411yescharacteristic
120Dic283—22no—
121Dic283—42no—
122Dic283#12712yes—
123C2×C283—82no—
124C2×C283—82no—
125C2×C283C2×#12711yescharacteristic
126C2×C283#57612yes—
127D283—22no—
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#495—2831no—
139#493—2831no—
140#493D28311yescharacteristic, complemented
141#405—2831no—
142C2×#41—2832no—
143C22×D4—2832no—
144AGL(1,ℤ/8)—5668no—
145#135—5664no—
146C2×Q16—2831no—
147C2×SD16—2832no—
148C2×SD16—2832no—
149C2×SD16—2832no—
150C2×D8—2832no—
151C2×D8—2832no—
152C2×D8—2831no—
153C2×D8—5664no—
154C2×D8C2×D28311yescharacteristic
155C2×D8C2×D28311yescharacteristic, complemented
156C2×D8C2×D28312yescomplemented
157C2×M16—2831no—
158C22×C8—2831no—
159#127—2832no—
160#127C2×D28312yescomplemented
161#124—2832no—
162#124—2832no—
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×Q8—2832no—
174C2×D4—24no—
175C2×D4—2832no—
176C2×D4—5664no—
177C2×D4—5664no—
178C2×D4—5662no—
179C2×D4—5664no—
180C2×D4—5662no—
181C2×D4C22×D28312yes—
182C2×D4C22×D28312yes—
183C22×C4—2831no—
184C22×C4—5662no—
185Q16—5662no—
186SD16—5664no—
187SD16—5664no—
188SD16—5664no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—5664no—
193D8—5664no—
194D8—5662no—
195M16—2834no—
196C2×C8—2831no—
197C2×C8—2831no—
198C2×C8—2834no—
199C2×C8C22×D28311yescharacteristic
200C2×C8C22×D28311yescharacteristic
201C4⋊C4—2832no—
202C22⋊C4—2832no—
203C22⋊C4—5664no—
204C22⋊C4—5664no—
205C42—2831no—
206C42C22×D28311yescharacteristic
207C23—22no—
208C23—22no—
209C23—2831no—
210C23—5662no—
211C23—11324no—
212C23—22644no—
213Q8—5662no—
214Q8—5662no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—5662no—
221D4—5662no—
222D4—11324no—
223D4—11324no—
224D4—22644no—
225C2×C4—2831no—
226C2×C4—2832no—
227C2×C4—2832no—
228C2×C4—2832no—
229C2×C4—5662no—
230C2×C4—11324no—
231C2×C4D4×D28312yes—
232C2×C4C23×D28311yescharacteristic
233C8—2832no—
234C8—2832no—
235C8D4×D28312yes—
236C8D4×D28312yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—2832no—
242C22—5662no—
243C22—11322no—
244C22—11324no—
245C22—22648no—
246C22C2×D4×D28311yescharacteristic
247C4—22no—
248C4—2832no—
249C4—5662no—
250C4—11322no—
251C4C2×D4×D28311yescharacteristic
252C4C2×D4×D28311yescharacteristic
253C2—82no—
254C2—82no—
255C2—5662no—
256C2—22642no—
257C2#127×D28311yescharacteristic
258C2#3182726612yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#493×C283C2
socleC22×C283C22×D4
hypercenter#493D283
hypocenterC283#3155
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#22583634C2semidirect
2. C2×#22583629C2semidirect
3. C2×D8×D283C2semidirect
4. #31827028C2semidirect
5. #31826853C2semidirect
6. #31826852C2semidirect
7. #31826849C2semidirect
8. #31826848C2semidirect
9. #31826840C2semidirect
10. #31826419C2semidirect
11. #493×C283C2semidirect
16. C2×Dic1132C22semidirect
17. C2×#17985381C22semidirect
18. C2×#17985380C22semidirect
19. C2×#17985380C22semidirect
22. C2×#17985377C22semidirect
26. #22583606C22semidirect
27. #22583598C22semidirect
28. #22583597C22semidirect
32. #22583521C22semidirect
33. #22583520C22semidirect
34. C8×Dic283C22semidirect
36. C2×D8×C283C22semidirect
37. C2×D8×C283C22semidirect
38. #127×C283C22semidirect
39. C4×C8×C283C22semidirect
59. #17985375D4semidirect
62. C8×D283D4semidirect
129. C283#3155semidirect
140. #493D283semidirect
155. C2×D8C2×D283semidirect
156. C2×D8C2×D283semidirect
160. #127C2×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×#225836342
C2×#225836292
C2×D8×D2832
#318270282
#318268532
#318268522
#318268492
#318268482
#318268402
#318264192
#493×C2832
#3155283