Finite Group Databasebeta

← #49132860

Subgroups — #49132860

Order 36224.

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

normal subgroups only

This group has 546 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×#22583637C212yesmaximal, complemented
2#31827118C211yesmaximal, characteristic, complemented
3#31827100C211yesmaximal, characteristic, complemented
4#31826708C212yesmaximal, complemented
5#31826667C211yesmaximal, characteristic, complemented
6#31826657C211yesmaximal, characteristic, complemented
7#31826614C212yesmaximal, complemented
8#31826609C212yesmaximal, complemented
9C4×D2264C211yesmaximal, characteristic, complemented
10#31826544C211yesmaximal, characteristic, complemented
11#492×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2212yescomplemented
13C2×D4×D283—21no—
14C2×D4×D283C2212yes—
15#22583637—28no—
16C2×#17985379C2212yes—
17C2×D2264C2211yescharacteristic
18C2×#17985377C2212yescomplemented
19C2×#17985375C2212yescomplemented
20#22583595—22no—
21#22583586C2212yescomplemented
22#22583585—21no—
23#22583581—21no—
24#22583581C2212yes—
25#22583578—22no—
26C4×D1132C2211yescharacteristic
27C4×D1132C2211yescharacteristic
28C4×D1132C2211yescharacteristic, complemented
29C4⋊C4×D283—21no—
30#22583544C2211yescharacteristic, complemented
31C4⋊(C283⋊C8)C2211yescharacteristic, complemented
32#22583534C2212yescomplemented
33#22583532C2211yescharacteristic
34#22583532C2211yescharacteristic, complemented
35#22583531C2212yescomplemented
36(C2×D283)⋊C8C2212yescomplemented
37C2×SD16×C283C2212yescomplemented
38C2×SD16×C283C2212yescomplemented
39C4⋊Q8×C283C2211yescharacteristic, complemented
40#127×C283C2211yescharacteristic
41C4×C8×C283C2211yescharacteristic, complemented
42C23×D283—22no—
43#17985393—24no—
44#17985393—42no—
45C2×D1132—22no—
46C2×D1132D411yescharacteristic
47C2×D1132C2311yescharacteristic
48C2×D1132C2311yescharacteristic
49C2×C22⋊D283—22no—
50C2×C22⋊D283—22no—
51D4×D283—24no—
52D4×D283—42no—
53D4×D283—44no—
54C2×C4×D283—22no—
55C2×C4×D283—21no—
56C2×C4×D283—21no—
57C2×C4×D283C2312yes—
58#17985379—24no—
59D2264D414yescomplemented
60#17985377—24no—
61#17985375—24no—
62C2×C283⋊C8C2312yes—
63C4⋊Dic283D411yescharacteristic
64C4⋊Dic283C2311yescharacteristic
65C4⋊Dic283C2311yescharacteristic
66Dic283⋊C4—22no—
67C22⋊Dic283—22no—
68(C2×D283)⋊C4—22no—
69(C2×D283)⋊C4—21no—
70(C2×D283)⋊C4—21no—
71(C2×D283)⋊C4D412yes—
72C2×Q8×C283C2312yes—
73C2×D4×C283—21no—
74C2×D4×C283—21no—
75C2×D4×C283C2312yes—
76SD16×C283—24no—
77SD16×C283—42no—
78C2×C8×C283C2311yescharacteristic
79C2×C8×C283C2311yescharacteristic
80C4⋊C4×C283—21no—
81C4⋊C4×C283—21no—
82C42×C283C2311yescharacteristic
83C22×D283—21no—
84C22×D283—21no—
85C22×D283—44no—
86C22×D283—44no—
87C22×D283—82no—
88C22×D283C2×D412yes—
89D1132—22no—
90D1132—21no—
91D1132—21no—
92D1132—44no—
93D1132C2×D412yes—
94D1132C2×D412yescomplemented
95C22⋊D283—44no—
96C22⋊D283—44no—
97C2×Dic283—21no—
98C2×Dic283—21no—
99C2×Dic283C2×D412yes—
100C4×D283—22no—
101C4×D283—22no—
102C4×D283—44no—
103C4×D283—42no—
104C4×D283—42no—
105C283⋊C8—22no—
106C23×C283—22no—
107Q8×C283—22no—
108Q8×C283—22no—
109D4×C283—22no—
110D4×C283—22no—
111D4×C283—42no—
112D4×C283—81no—
113C2×C4×C283—22no—
114C2×C4×C283C2×D411yescharacteristic
115C2×C4×C283C2×D411yescharacteristic
116C2×C4×C283C2411yescharacteristic
117C8×C283—21no—
118C8×C283C2×D412yescomplemented
119C22×C283—42no—
120C22×C283—42no—
121C22×C283—42no—
122C22×C283C22×D411yescharacteristic
123C2×D283—22no—
124C2×D283—22no—
125C2×D283—22no—
126C2×D283—22no—
127C2×D283—22no—
128C2×D283—41no—
129C2×D283—41no—
130C2×D283—41no—
131C2×D283—41no—
132C2×D283—88no—
133C4×C283—21no—
134C4×C283—42no—
135C4×C283AGL(1,ℤ/8)12yescomplemented
136C4×C283C22×D411yescharacteristic
137C4×C283H(5,𝔽2)11yescharacteristic
138Dic283—22no—
139Dic283—41no—
140Dic283—41no—
141C2×C283—82no—
142C2×C283D4211yescharacteristic
143C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C283#57711yescharacteristic
145D283—42no—
146D283—42no—
147D283—42no—
148C283#327211yescharacteristic, complemented, Sylow
149#3272—2831nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2832no—
151#547—2831no—
152D42—2831no—
153#492D28311yescharacteristic, complemented
154#468—2832no—
155#461—2831no—
156#459—2831no—
157#450—2832no—
158#449—2832no—
159C4×D8—2831no—
160#436—2831no—
161C2×#41—2832no—
162C22×D4—2832no—
163C22×D4—5661no—
164AGL(1,ℤ/8)—5668no—
165C2×SD16—2832no—
166C2×SD16C2×D28312yescomplemented
167C2×SD16C2×D28312yescomplemented
168C2×D8—2832no—
169C2×D8—2831no—
170C2×M16—2832no—
171C4⋊Q8C2×D28311yescharacteristic, complemented
172#127C2×D28311yescharacteristic
173#123—5662no—
174C22⋊Q8—2832no—
175#121—2832no—
176#121—5661no—
177#121—5661no—
178#120—5662no—
179C4×D4—2831no—
180C4×D4—2831no—
181C4×D4—2831no—
182C2×C4⋊C4—5661no—
183#107—2831no—
184C4⋊C8—2831no—
185#103—2832no—
186#102—2832no—
187#102—2831no—
188#102—2831no—
189C22⋊C8—2832no—
190C4×C8C2×D28311yescharacteristic, complemented
191C24—5662no—
192#41—5664no—
193#41—11322no—
194C2×Q8C22×D28312yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—2831no—
198C2×D4—2831no—
199C2×D4—2831no—
200C2×D4—5662no—
201C2×D4—5662no—
202C2×D4—5664no—
203C2×D4—5662no—
204C2×D4—11322no—
205C2×D4—11324no—
206C2×D4C22×D28312yes—
207C22×C4—2832no—
208C22×C4—5662no—
209C22×C4—5661no—
210C22×C4—5661no—
211SD16—24no—
212SD16—42no—
213SD16—5664no—
214D8—2834no—
215D8—5664no—
216M16—5664no—
217C2×C8—2832no—
218C2×C8C22×D28311yescharacteristic
219C2×C8C22×D28311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—2831no—
223C4⋊C4—2831no—
224C4⋊C4—2831no—
225C4⋊C4—5662no—
226C22⋊C4—2832no—
227C22⋊C4—5662no—
228C22⋊C4—5662no—
229C22⋊C4—5661no—
230C22⋊C4—5661no—
231C42C22×D28311yescharacteristic
232C23—22no—
233C23—2832no—
234C23—5661no—
235C23—5661no—
236C23—11324no—
237C23—11324no—
238C23—22642no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—2832no—
246D4—2832no—
247D4—5662no—
248D4—5661no—
249D4—5661no—
250D4—11324no—
251D4—11324no—
252D4—11324no—
253C2×C4—22no—
254C2×C4—2832no—
255C2×C4—5662no—
256C2×C4—5662no—
257C2×C4—5661no—
258C2×C4—5661no—
259C2×C4—11324no—
260C2×C4—11322no—
261C2×C4—11322no—
262C2×C4D4×D28311yescharacteristic
263C2×C4D4×D28311yescharacteristic
264C2×C4C23×D28311yescharacteristic
265C8—21no—
266C8—5662no—
267C8D4×D28312yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—5662no—
272C22—5662no—
273C22—5662no—
274C22—5662no—
275C22—5662no—
276C22—11321no—
277C22—11321no—
278C22—11321no—
279C22—11321no—
280C22—22648no—
281C22C2×D4×D28311yescharacteristic
282C4—21no—
283C4—42no—
284C4—5662no—
285C4—11321no—
286C4—11321no—
287C4#2258363712yescomplemented
288C4C2×D4×D28311yescharacteristic
289C4#2258365511yescharacteristic
290C2—82no—
291C2—11322no—
292C2—11322no—
293C2—11322no—
294C2#3182710011yescharacteristic
295C2C2×#2258363711yescharacteristic
296C2#3182727111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#492×C283C2
socleC22×C283C22×D4
hypercenter#492D283
hypocenterC283#3272
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. #31827118C2semidirect
3. #31827100C2semidirect
4. #31826708C2semidirect
5. #31826667C2semidirect
6. #31826657C2semidirect
7. #31826614C2semidirect
8. #31826609C2semidirect
9. C4×D2264C2semidirect
10. #31826544C2semidirect
11. #492×C283C2semidirect
12. C2×#17985393C22semidirect
18. C2×#17985377C22semidirect
19. C2×#17985375C22semidirect
21. #22583586C22semidirect
28. C4×D1132C22semidirect
30. #22583544C22semidirect
31. C4⋊(C283⋊C8)C22semidirect
32. #22583534C22semidirect
34. #22583532C22semidirect
35. #22583531C22semidirect
36. (C2×D283)⋊C8C22semidirect
37. C2×SD16×C283C22semidirect
38. C2×SD16×C283C22semidirect
39. C4⋊Q8×C283C22semidirect
41. C4×C8×C283C22semidirect
59. D2264D4semidirect
94. D1132C2×D4semidirect
118. C8×C283C2×D4semidirect
135. C4×C283AGL(1,ℤ/8)semidirect
148. C283#3272semidirect
153. #492D283semidirect
166. C2×SD16C2×D283semidirect
167. C2×SD16C2×D283semidirect
171. C4⋊Q8C2×D283semidirect
190. C4×C8C2×D283semidirect
267. C8D4×D283semidirect
287. C4#22583637semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
#318271182
#318271002
#318267082
#318266672
#318266572
#318266142
#318266092
C4×D22642
#318265442
#492×C2832
#3272283