Finite Group Databasebeta

← #49131710

Subgroups — #49131710

Order 36224.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22583634C212yesmaximal, complemented
3#31827014C211yesmaximal, characteristic, complemented
4#127×D283C211yesmaximal, characteristic, complemented
5#31826855C211yesmaximal, characteristic, complemented
6#31826854C211yesmaximal, characteristic, complemented
7#31826852C211yesmaximal, characteristic, complemented
8#31826840C212yesmaximal, complemented
9#31826836C212yesmaximal, complemented
10#31826417C211yesmaximal, characteristic, complemented
11#496×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2211yescharacteristic
13C2×#17985391C2211yescharacteristic
14C2×D4×D283—22no—
15C2×D4×D283C2212yes—
16#22583637—28no—
17#22583634—28no—
18C2×#17985381C2212yescomplemented
19C2×#17985380C2212yescomplemented
20C2×#17985379C2212yescomplemented
21C2×D2264C2212yescomplemented
22C2×#17985377C2212yescomplemented
23C2×#17985377C2212yescomplemented
24C2×#17985375C2212yes—
25#22583608C2211yescharacteristic, complemented
26#22583607C2211yescharacteristic
27#22583606—22no—
28#22583606C2212yes—
29#22583606C2211yescharacteristic, complemented
30#22583598C2211yescharacteristic, complemented
31#22583583—22no—
32(C22⋊D283)⋊C4—22no—
33C42×D283C2211yescharacteristic
34#22583521C2211yescharacteristic, complemented
35C8×Dic283C2212yescomplemented
36C2×SD16×C283C2212yescomplemented
37C2×D8×C283C2212yescomplemented
38#127×C283C2211yescharacteristic
39#124×C283C2211yescharacteristic, complemented
40#97×C283C2211yescharacteristic, complemented
41C23×D283—22no—
42#17985393—24no—
43#17985391—24no—
44C2×Dic566C2311yescharacteristic
45C2×D1132—21no—
46C2×D1132—22no—
47C2×D1132C2311yescharacteristic
48C2×D1132C2312yes—
49C2×C22⋊D283—21no—
50C2×C22⋊D283—22no—
51C2×C22⋊D283—22no—
52C2×C22⋊D283—24no—
53D4×D283—24no—
54D4×D283—24no—
55D4×D283—48no—
56C22×Dic283—21no—
57C2×C4×D283—21no—
58C2×C4×D283D412yes—
59C2×C4×D283C2311yescharacteristic
60#17985381—24no—
61#17985380—24no—
62#17985379—24no—
63D2264—24no—
64#17985377—24no—
65#17985377—24no—
66#17985375D418yescomplemented
67C2×C283⋊C8C2312yes—
68Dic283⋊C4—22no—
69(C2×D283)⋊C4—22no—
70C4×Dic283D412yes—
71C4×Dic283C2311yescharacteristic
72C2×Q8×C283C2311yescharacteristic
73C2×D4×C283—22no—
74C2×D4×C283C2312yes—
75C2×D4×C283C2311yescharacteristic
76SD16×C283—24no—
77D8×C283—24no—
78C2×C8×C283C2312yes—
79C22⋊C4×C283—22no—
80C42×C283C2311yescharacteristic
81C22×D283—21no—
82C22×D283—22no—
83C22×D283—48no—
84C22×D283—44no—
85C22×D283C2×D411yescharacteristic
86Dic566—22no—
87D1132—22no—
88D1132—22no—
89D1132—22no—
90D1132—44no—
91D1132—42no—
92C22⋊D283—42no—
93C22⋊D283—44no—
94C22⋊D283—48no—
95C22⋊D283—44no—
96C2×Dic283—21no—
97C2×Dic283—42no—
98C2×Dic283C2×D411yescharacteristic
99C2×Dic283C2×D412yes—
100C4×D283—24no—
101C4×D283—42no—
102C4×D283C2×D414yes—
103C283⋊C8C2×D414yescomplemented
104C23×C283—21no—
105C23×C283—22no—
106Q8×C283—22no—
107D4×C283—22no—
108D4×C283—22no—
109D4×C283—22no—
110D4×C283—44no—
111C2×C4×C283—21no—
112C2×C4×C283C2×D412yes—
113C2×C4×C283C2411yescharacteristic
114C8×C283C2×D414yescomplemented
115C22×C283—41no—
116C22×C283—42no—
117C22×C283—44no—
118C22×C283—42no—
119C22×C283C22×D411yescharacteristic
120C2×D283—22no—
121C2×D283—41no—
122C2×D283—42no—
123C2×D283—44no—
124C2×D283—42no—
125C2×D283—88no—
126C2×D283#12712yes—
127C4×C283—22no—
128C4×C283—41no—
129C4×C283C22×D412yes—
130Dic283—41no—
131Dic283#12712yes—
132Dic283AGL(1,ℤ/8)14yescomplemented
133C2×C283—82no—
134C2×C283—81no—
135C2×C283C2×#12711yescharacteristic
136C2×C283C2×AGL(1,ℤ/8)12yes—
137D283—22no—
138D283—81no—
139D283—82no—
140C283#315111yescharacteristic, complemented, Sylow
141#3151—2831nomaximal, Sylow
142C2×AGL(1,ℤ/8)—2832no—
143C2×AGL(1,ℤ/8)—2832no—
144#532—2831no—
145C2×#127—2831no—
146#496—2831no—
147#496—2831no—
148#496—2831no—
149#496D28311yescharacteristic, complemented
150#493—2832no—
151#492—2832no—
152C4×M16—2831no—
153C2×#41—2831no—
154C2×#41—2831no—
155C22×D4—2832no—
156C22×D4—5662no—
157AGL(1,ℤ/8)—5668no—
158AGL(1,ℤ/8)—5668no—
159C2×SD16—2832no—
160C2×SD16—2832no—
161C2×SD16—2832no—
162C2×SD16C2×D28312yescomplemented
163C2×D8—2832no—
164C2×D8—2832no—
165C2×D8—2832no—
166C2×D8C2×D28312yescomplemented
167C2×M16—2832no—
168C4⋊Q8—2831no—
169#127—2832no—
170#127—2831no—
171#127—2831no—
172#127—5662no—
173#127C2×D28311yescharacteristic
174#124—2831no—
175#124C2×D28311yescharacteristic, complemented
176#121—5662no—
177C4×D4—5662no—
178C2×C42—2831no—
179#97—2831no—
180#97C2×D28311yescharacteristic, complemented
181C4×C8—2832no—
182C24—5662no—
183#41—5664no—
184#41—5664no—
185C2×Q8—2831no—
186C2×Q8C22×D28311yescharacteristic
187C2×D4—22no—
188C2×D4—2831no—
189C2×D4—2832no—
190C2×D4—5661no—
191C2×D4—5664no—
192C2×D4—5664no—
193C2×D4—5662no—
194C2×D4—5664no—
195C2×D4—5662no—
196C2×D4—5661no—
197C2×D4—5662no—
198C2×D4—11328no—
199C2×D4C22×D28311yescharacteristic
200C2×D4C22×D28312yes—
201C22×C4—2831no—
202C22×C4—2832no—
203C22×C4—5661no—
204C22×C4—5661no—
205SD16—24no—
206SD16—5664no—
207SD16—5664no—
208SD16—5664no—
209D8—24no—
210D8—5664no—
211D8—5664no—
212D8—5664no—
213M16—2838no—
214C2×C8—2832no—
215C2×C8C22×D28312yes—
216C4⋊C4—5662no—
217C22⋊C4—22no—
218C22⋊C4—5662no—
219C42—2832no—
220C42—2831no—
221C42C22×D28311yescharacteristic
222C23—21no—
223C23—22no—
224C23—2831no—
225C23—5661no—
226C23—5662no—
227C23—11324no—
228C23—11328no—
229Q8—22no—
230Q8—5662no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—5662no—
236D4—5662no—
237D4—5662no—
238D4—11324no—
239D4—11322no—
240D4—11324no—
241D4—11328no—
242D4—11324no—
243D4—11322no—
244C2×C4—21no—
245C2×C4—2832no—
246C2×C4—2831no—
247C2×C4—2834no—
248C2×C4—5664no—
249C2×C4—5661no—
250C2×C4—11322no—
251C2×C4—11322no—
252C2×C4D4×D28312yes—
253C2×C4C23×D28311yescharacteristic
254C8—2834no—
255C8D4×D28314yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—2832no—
261C22—5662no—
262C22—11324no—
263C22—11322no—
264C22—11321no—
265C22—11322no—
266C22—22648no—
267C22C2×D4×D28311yescharacteristic
268C4—22no—
269C4—41no—
270C4—2834no—
271C4—2832no—
272C4—11321no—
273C4C2×D4×D28312yes—
274C2—82no—
275C2—81no—
276C2—5662no—
277C2—22642no—
278C2—22641no—
279C2#127×D28311yescharacteristic
280C2AGL(1,ℤ/8)×D28312yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#496×C283C2
socleC22×C283C22×D4
hypercenter#496D283
hypocenterC283#3151
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. C2×#22583634C2semidirect
3. #31827014C2semidirect
4. #127×D283C2semidirect
5. #31826855C2semidirect
6. #31826854C2semidirect
7. #31826852C2semidirect
8. #31826840C2semidirect
9. #31826836C2semidirect
10. #31826417C2semidirect
11. #496×C283C2semidirect
18. C2×#17985381C22semidirect
19. C2×#17985380C22semidirect
20. C2×#17985379C22semidirect
21. C2×D2264C22semidirect
22. C2×#17985377C22semidirect
23. C2×#17985377C22semidirect
25. #22583608C22semidirect
29. #22583606C22semidirect
30. #22583598C22semidirect
34. #22583521C22semidirect
35. C8×Dic283C22semidirect
36. C2×SD16×C283C22semidirect
37. C2×D8×C283C22semidirect
39. #124×C283C22semidirect
40. #97×C283C22semidirect
66. #17985375D4semidirect
103. C283⋊C8C2×D4semidirect
114. C8×C283C2×D4semidirect
132. Dic283AGL(1,ℤ/8)semidirect
140. C283#3151semidirect
149. #496D283semidirect
162. C2×SD16C2×D283semidirect
166. C2×D8C2×D283semidirect
175. #124C2×D283semidirect
180. #97C2×D283semidirect
255. C8D4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
C2×#225836342
#318270142
#127×D2832
#318268552
#318268542
#318268522
#318268402
#318268362
#318264172
#496×C2832
#3151283