Finite Group Databasebeta

← #49132880

Subgroups — #49132880

Order 36224.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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
2#31827135C211yesmaximal, characteristic, complemented
3#31827100C211yesmaximal, characteristic, complemented
4#31826715C212yesmaximal, complemented
5#31826662C211yesmaximal, characteristic, complemented
6#31826657C211yesmaximal, characteristic, complemented
7#31826607C212yesmaximal, complemented
8#31826597C212yesmaximal, complemented
9C4×#17985381C211yesmaximal, characteristic, complemented
10#31826544C211yesmaximal, characteristic, complemented
11#493×C283C211yesmaximal, characteristic, complemented
12C2×#17985391C2212yescomplemented
13C2×D4×D283—21no—
14C2×D4×D283C2212yes—
15#22583634—28no—
16C2×#17985381C2211yescharacteristic
17C2×#17985380C2212yescomplemented
18C2×#17985377C2212yes—
19C2×#17985375C2212yescomplemented
20#22583606—21no—
21#22583582—22no—
22#22583581—21no—
23#22583581C2212yes—
24#22583581C2212yescomplemented
25#22583578—22no—
26C4×Dic566C2211yescharacteristic, complemented
27C4×D1132C2211yescharacteristic
28C4×D1132C2211yescharacteristic
29D4×Dic283—21no—
30#22583542C2211yescharacteristic, complemented
31C4⋊(C283⋊C8)C2211yescharacteristic, complemented
32#22583535C2211yescharacteristic, complemented
33#22583532C2211yescharacteristic
34#22583531C2212yescomplemented
35#22583531C2212yescomplemented
36(C2×D283)⋊C8C2212yescomplemented
37C2×D8×C283C2212yescomplemented
38C2×D8×C283C2212yescomplemented
39#127×C283C2211yescharacteristic
40#127×C283C2211yescharacteristic, complemented
41C4×C8×C283C2211yescharacteristic, complemented
42C23×D283—22no—
43#17985391—24no—
44#17985391—42no—
45C2×Dic566C2311yescharacteristic
46C2×D1132D411yescharacteristic
47C2×D1132C2311yescharacteristic
48C2×C22⋊D283—22no—
49C2×C22⋊D283—22no—
50C2×C22⋊D283—22no—
51C2×C22⋊D283—22no—
52D4×D283—24no—
53D4×D283—42no—
54D4×D283—44no—
55C22×Dic283—22no—
56C2×C4×D283—21no—
57C2×C4×D283C2312yes—
58#17985381D414yescomplemented
59#17985380—24no—
60#17985377—24no—
61#17985375—24no—
62C2×C283⋊C8C2312yes—
63C4⋊Dic283D411yescharacteristic
64C4⋊Dic283C2311yescharacteristic
65C4⋊Dic283C2311yescharacteristic
66Dic283⋊C4—21no—
67C22⋊Dic283—22no—
68C22⋊Dic283—22no—
69(C2×D283)⋊C4—21no—
70(C2×D283)⋊C4D412yes—
71C4×Dic283—21no—
72C2×D4×C283—21no—
73C2×D4×C283—21no—
74C2×D4×C283—21no—
75C2×D4×C283—21no—
76C2×D4×C283C2312yes—
77C2×D4×C283C2312yes—
78D8×C283—24no—
79D8×C283—42no—
80C2×C8×C283C2311yescharacteristic
81C2×C8×C283C2311yescharacteristic
82C42×C283C2311yescharacteristic
83C22×D283—21no—
84C22×D283—44no—
85C22×D283—44no—
86C22×D283—82no—
87C22×D283C2×D412yes—
88Dic566—21no—
89Dic566C2×D412yescomplemented
90D1132—22no—
91D1132—21no—
92D1132C2×D412yes—
93C22⋊D283—44no—
94C22⋊D283—44no—
95C22⋊D283—44no—
96C22⋊D283—82no—
97C2×Dic283—21no—
98C2×Dic283—21no—
99C2×Dic283—21no—
100C2×Dic283—44no—
101C2×Dic283C2×D412yes—
102C4×D283—22no—
103C4×D283—22no—
104C4×D283—42no—
105C283⋊C8—22no—
106C23×C283—22no—
107C23×C283—22no—
108D4×C283—22no—
109D4×C283—22no—
110D4×C283—22no—
111D4×C283—22no—
112D4×C283—42no—
113D4×C283—42no—
114D4×C283—81no—
115D4×C283—81no—
116C2×C4×C283C2×D411yescharacteristic
117C2×C4×C283C2×D411yescharacteristic
118C2×C4×C283C2411yescharacteristic
119C8×C283—21no—
120C8×C283C2×D412yescomplemented
121C22×C283—42no—
122C22×C283—42no—
123C22×C283—42no—
124C22×C283—42no—
125C22×C283—42no—
126C22×C283—42no—
127C22×C283C22×D411yescharacteristic
128C2×D283—22no—
129C2×D283—22no—
130C2×D283—22no—
131C2×D283—22no—
132C2×D283—41no—
133C2×D283—41no—
134C2×D283—88no—
135C4×C283—21no—
136C4×C283AGL(1,ℤ/8)12yescomplemented
137C4×C283C22×D411yescharacteristic
138C4×C283H(5,𝔽2)11yescharacteristic
139Dic283—22no—
140Dic283—22no—
141Dic283—41no—
142Dic283—41no—
143C2×C283—82no—
144C2×C283—82no—
145C2×C283D4211yescharacteristic
146C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C283#57611yescharacteristic
148D283—42no—
149D283—42no—
150C283#327411yescharacteristic, complemented, Sylow
151#3274—2831nomaximal, Sylow
152C2×AGL(1,ℤ/8)—2832no—
153#550—2831no—
154D42—2831no—
155#493D28311yescharacteristic, complemented
156#469—2832no—
157#460—2831no—
158#459—2831no—
159#449—2832no—
160#447—2832no—
161C4×SD16—2831no—
162#436—2831no—
163C2×#41—2832no—
164C22×D4—2832no—
165C22×D4—5661no—
166AGL(1,ℤ/8)—5668no—
167C2×SD16—2832no—
168C2×SD16—2831no—
169C2×D8—2832no—
170C2×D8C2×D28312yescomplemented
171C2×D8C2×D28312yescomplemented
172C2×M16—2832no—
173#127—5661no—
174#127C2×D28311yescharacteristic
175#127C2×D28311yescharacteristic, complemented
176#121—2832no—
177#121—2832no—
178#121—5662no—
179#121—5661no—
180#120—5662no—
181C4×Q8—2831no—
182C4×D4—2831no—
183C4×D4—2831no—
184C4×D4—5661no—
185#106—2831no—
186C4⋊C8—2831no—
187#103—2831no—
188#102—2832no—
189#102—2832no—
190#102—2831no—
191C22⋊C8—2832no—
192C4×C8C2×D28311yescharacteristic, complemented
193C24—5662no—
194#41—5664no—
195#41—11322no—
196C2×Q8—2831no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2831no—
202C2×D4—2831no—
203C2×D4—5662no—
204C2×D4—5664no—
205C2×D4—5662no—
206C2×D4—5662no—
207C2×D4—5662no—
208C2×D4—11322no—
209C2×D4—11324no—
210C2×D4C22×D28312yes—
211C2×D4C22×D28312yes—
212C22×C4—2832no—
213C22×C4—5662no—
214C22×C4—5661no—
215SD16—2834no—
216SD16—5664no—
217D8—24no—
218D8—42no—
219D8—5664no—
220M16—5664no—
221C2×C8—2832no—
222C2×C8C22×D28311yescharacteristic
223C2×C8C22×D28311yescharacteristic
224C4⋊C4—2831no—
225C4⋊C4—2831no—
226C4⋊C4—2831no—
227C4⋊C4—5661no—
228C22⋊C4—2832no—
229C22⋊C4—5662no—
230C22⋊C4—5662no—
231C22⋊C4—5661no—
232C42—5661no—
233C42C22×D28311yescharacteristic
234C23—22no—
235C23—22no—
236C23—2832no—
237C23—5661no—
238C23—11324no—
239C23—11324no—
240C23—22642no—
241Q8—2832no—
242Q8—5661no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—2832no—
252D4—5662no—
253D4—5661no—
254D4—11324no—
255D4—11324no—
256D4—11324no—
257D4—22642no—
258C2×C4—2832no—
259C2×C4—5661no—
260C2×C4—5662no—
261C2×C4—5662no—
262C2×C4—5661no—
263C2×C4—5661no—
264C2×C4—11324no—
265C2×C4—11322no—
266C2×C4D4×D28311yescharacteristic
267C2×C4D4×D28311yescharacteristic
268C2×C4C23×D28311yescharacteristic
269C8—21no—
270C8—5662no—
271C8D4×D28312yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—5662no—
279C22—5662no—
280C22—5662no—
281C22—5662no—
282C22—11321no—
283C22—11321no—
284C22—22648no—
285C22C2×D4×D28311yescharacteristic
286C4—21no—
287C4—5662no—
288C4—5662no—
289C4—11321no—
290C4—11321no—
291C4#2258363412yescomplemented
292C4C2×D4×D28311yescharacteristic
293C4#2258365511yescharacteristic
294C2—82no—
295C2—82no—
296C2—11322no—
297C2—11322no—
298C2#3182710011yescharacteristic
299C2C2×#2258363411yescharacteristic
300C2#3182726611yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#493×C283C2
socleC22×C283C22×D4
hypercenter#493D283
hypocenterC283#3274
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. #31827135C2semidirect
3. #31827100C2semidirect
4. #31826715C2semidirect
5. #31826662C2semidirect
6. #31826657C2semidirect
7. #31826607C2semidirect
8. #31826597C2semidirect
9. C4×#17985381C2semidirect
10. #31826544C2semidirect
11. #493×C283C2semidirect
12. C2×#17985391C22semidirect
17. C2×#17985380C22semidirect
19. C2×#17985375C22semidirect
24. #22583581C22semidirect
26. C4×Dic566C22semidirect
30. #22583542C22semidirect
31. C4⋊(C283⋊C8)C22semidirect
32. #22583535C22semidirect
34. #22583531C22semidirect
35. #22583531C22semidirect
36. (C2×D283)⋊C8C22semidirect
37. C2×D8×C283C22semidirect
38. C2×D8×C283C22semidirect
40. #127×C283C22semidirect
41. C4×C8×C283C22semidirect
58. #17985381D4semidirect
89. Dic566C2×D4semidirect
120. C8×C283C2×D4semidirect
136. C4×C283AGL(1,ℤ/8)semidirect
150. C283#3274semidirect
155. #493D283semidirect
170. C2×D8C2×D283semidirect
171. C2×D8C2×D283semidirect
175. #127C2×D283semidirect
192. C4×C8C2×D283semidirect
271. C8D4×D283semidirect
291. C4#22583634semidirect

Maximal subgroups

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

subgroupindex
C2×#225836342
#318271352
#318271002
#318267152
#318266622
#318266572
#318266072
#318265972
C4×#179853812
#318265442
#493×C2832
#3274283