Finite Group Databasebeta

← #49133050

Subgroups — #49133050

Order 36224.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#31827278C211yesmaximal, characteristic, complemented
2#31827274C212yesmaximal, complemented
3#31827270C211yesmaximal, characteristic, complemented
4#31826850C211yesmaximal, characteristic, complemented
5#31826736C211yesmaximal, characteristic, complemented
6#31826731C212yesmaximal, complemented
7#31826641C212yesmaximal, complemented
8#31826640C212yesmaximal, complemented
9#31826581C211yesmaximal, characteristic, complemented
10#31826580C211yesmaximal, characteristic, complemented
11#473×C283C211yesmaximal, characteristic, complemented
12#22583658C2212yescomplemented
13#22583656C2212yes—
14#22583642—22no—
15#22583642—22no—
16#22583641C2212yescomplemented
17#22583638—22no—
18#22583638C2212yes—
19#22583638C2212yescomplemented
20#22583637—22no—
21#22583633—22no—
22#22583633—21no—
23#22583632—21no—
24#22583632C2211yescharacteristic
25#22583630C2211yescharacteristic
26Q16×D283—21no—
27C2×#17985379C2212yescomplemented
28SD16×D283—22no—
29C2×D2264—21no—
30C2×D2264C2211yescharacteristic
31#22583618C2212yescomplemented
32#22583616C2211yescharacteristic, complemented
33#22583599C2212yescomplemented
34#22583545C2211yescharacteristic, complemented
35#22583537C2211yescharacteristic
36#22583536C2211yescharacteristic, complemented
37#22583536C2212yescomplemented
38#22583526C2212yescomplemented
39C8×Dic283C2211yescharacteristic, complemented
40#137×C283C2212yes—
41C2×Q16×C283C2211yescharacteristic, complemented
42#131×C283C2211yescharacteristic
43#108×C283C2211yescharacteristic, complemented
44#101×C283C2212yescomplemented
45#17985393—22no—
46#17985393—22no—
47#17985393—22no—
48#17985392—22no—
49#17985392—42no—
50#17985392C2311yescharacteristic
51#17985392C2312yes—
52Q8×D283—22no—
53Q8×D283—22no—
54C2×D1132—22no—
55C2×D1132C2312yes—
56D4×D283—22no—
57D4×D283—42no—
58Dic1132D411yescharacteristic
59#17985382—22no—
60#17985382D411yescharacteristic
61#17985381—22no—
62#17985381—22no—
63#17985381D412yes—
64#17985380D411yescharacteristic
65#17985379—22no—
66#17985379—22no—
67#17985379—22no—
68#17985379D411yescharacteristic
69D2264—22no—
70D2264—22no—
71D2264—21no—
72D2264—21no—
73D2264D411yescharacteristic
74#17985377D411yescharacteristic
75#17985376C2311yescharacteristic
76#17985375—22no—
77#17985375—21no—
78C2×C283⋊C8C2311yescharacteristic
79C8×D283—22no—
80C8×D283—21no—
81(C2×D283)⋊C4—42no—
82C4×Dic283C2311yescharacteristic
83#41×C283C2311yescharacteristic
84C2×Q8×C283C2312yes—
85Q16×C283—22no—
86Q16×C283—21no—
87Q16×C283—21no—
88SD16×C283—22no—
89M16×C283—21no—
90M16×C283C2312yes—
91M16×C283C2311yescharacteristic
92C2×C8×C283—21no—
93C2×C8×C283C2311yescharacteristic
94C22×D283—22no—
95C22×D283—42no—
96Dic566—42no—
97Dic566C2×D411yescharacteristic
98Dic566C2×D412yes—
99D1132—22no—
100D1132—22no—
101D1132—42no—
102D1132—42no—
103D1132C2×D411yescharacteristic
104D1132C2×D412yes—
105C22⋊D283—22no—
106C22⋊D283—21no—
107C22⋊D283—42no—
108C2×Dic283—21no—
109C4×D283—22no—
110C4×D283—21no—
111C4×D283—42no—
112C4×D283—42no—
113C4×D283—42no—
114C283⋊C8—21no—
115C283⋊C8C2×D411yescharacteristic
116C283⋊C8C2×D411yescharacteristic
117Q8×C283—22no—
118Q8×C283—22no—
119Q8×C283C2×D411yescharacteristic
120D4×C283—21no—
121D4×C283C2×D411yescharacteristic
122C2×C4×C283—21no—
123C2×C4×C283—22no—
124C2×C4×C283C2411yescharacteristic
125C8×C283—21no—
126C8×C283—22no—
127C8×C283C2×D411yescharacteristic
128C8×C283C2×D411yescharacteristic
129C22×C283—21no—
130C22×C283H(5,𝔽2)11yescharacteristic
131C2×D283—21no—
132C2×D283—22no—
133C2×D283—42no—
134C2×D283—82no—
135C2×D283—162no—
136C4×C283—21no—
137C4×C283—42no—
138C4×C283C22×D411yescharacteristic
139C4×C283C22×D411yescharacteristic
140Dic283—22no—
141Dic283—21no—
142Dic283—41no—
143C2×C283—21no—
144C2×C283—41no—
145C2×C283D4211yescharacteristic
146D283—41no—
147D283—42no—
148D283—82no—
149C283#329011yescharacteristic, complemented, Sylow
150#3290—2831nomaximal, Sylow
151#578—2831no—
152#577—2832no—
153#576—2831no—
154#495—2831no—
155#473D28311yescharacteristic, complemented
156#472—2831no—
157#471—2832no—
158#455—2832no—
159#454—2832no—
160#443—2831no—
161#443—2831no—
1622−1+4—2832no—
163H(5,𝔽2)—2832no—
164#137—2832no—
165#137—5662no—
166#137—5662no—
167#137C2×D28312yes—
168AGL(1,ℤ/8)—2832no—
169AGL(1,ℤ/8)—2832no—
170AGL(1,ℤ/8)—5662no—
171AGL(1,ℤ/8)—5662no—
172#135—2831no—
173#135—2831no—
174#135—5661no—
175#135—5661no—
176#135—5662no—
177C2×Q16—5661no—
178C2×Q16C2×D28311yescharacteristic, complemented
179C2×SD16—2832no—
180C2×SD16—5662no—
181C2×D8—2831no—
182C2×D8—5661no—
183#131—2832no—
184#131—2831no—
185#131C2×D28311yescharacteristic
186#124—2832no—
187#108—2831no—
188#108C2×D28311yescharacteristic, complemented
189#104—2831no—
190#104—2831no—
191#104—2832no—
192#101C2×D28312yescomplemented
193#100—2832no—
194C4×C8—2831no—
195#41—2831no—
196#41—2832no—
197#41—5662no—
198#41—5662no—
199#41—5662no—
200#41—5662no—
201#41—11322no—
202#41C22×D28311yescharacteristic
203C2×Q8—5662no—
204C2×Q8—5662no—
205C2×Q8C22×D28312yes—
206C2×D4—2832no—
207C2×D4—5662no—
208C2×D4—5662no—
209C2×D4—11322no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—2831no—
214Q16—2831no—
215Q16—5662no—
216SD16—22no—
217SD16—2831no—
218SD16—2831no—
219SD16—2832no—
220SD16—5662no—
221SD16—5662no—
222SD16—5662no—
223SD16—5662no—
224SD16—5662no—
225D8—2831no—
226D8—2831no—
227D8—5662no—
228D8—5662no—
229D8—5661no—
230D8—5661no—
231M16—21no—
232M16—2831no—
233M16—5662no—
234M16—5661no—
235M16C22×D28312yes—
236M16C22×D28311yescharacteristic
237C2×C8—21no—
238C2×C8—2831no—
239C2×C8—5662no—
240C2×C8—5661no—
241C2×C8C22×D28311yescharacteristic
242C22⋊C4—11322no—
243C42—2831no—
244C23—5662no—
245C23—11322no—
246Q8—22no—
247Q8—22no—
248Q8—2831no—
249Q8—2832no—
250Q8—11322no—
251Q8D4×D28311yescharacteristic
252D4—21no—
253D4—2831no—
254D4—2832no—
255D4—5662no—
256D4—5661no—
257D4—5662no—
258D4—5662no—
259D4—11322no—
260D4—11322no—
261D4—11322no—
262D4D4×D28311yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—5662no—
266C2×C4—5661no—
267C2×C4—5661no—
268C2×C4—11322no—
269C2×C4—11322no—
270C2×C4—11322no—
271C2×C4C23×D28311yescharacteristic
272C8—21no—
273C8—22no—
274C8—2831no—
275C8—2831no—
276C8—5661no—
277C8D4×D28311yescharacteristic
278C8D4×D28311yescharacteristic
279C22—21no—
280C22—5661no—
281C22—5662no—
282C22—11322no—
283C22—45282no—
284C22—22642no—
285C22#2258365611yescharacteristic
286C4—21no—
287C4—42no—
288C4—5662no—
289C4—5661no—
290C4—11321no—
291C4C2×D4×D28311yescharacteristic
292C4C2×D4×D28311yescharacteristic
293C2—21no—
294C2—41no—
295C2—11322no—
296C2—11321no—
297C2—22642no—
298C2#3182709911yescharacteristic
subgroupG/H
centerC2#31827099
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#473×C283C2
socleC2×C283D42
hypercenter#473D283
hypocenterC283#3290
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #31827278C2semidirect
2. #31827274C2semidirect
3. #31827270C2semidirect
4. #31826850C2semidirect
5. #31826736C2semidirect
6. #31826731C2semidirect
7. #31826641C2semidirect
8. #31826640C2semidirect
9. #31826581C2semidirect
10. #31826580C2semidirect
11. #473×C283C2semidirect
12. #22583658C22semidirect
16. #22583641C22semidirect
19. #22583638C22semidirect
27. C2×#17985379C22semidirect
31. #22583618C22semidirect
32. #22583616C22semidirect
33. #22583599C22semidirect
34. #22583545C22semidirect
36. #22583536C22semidirect
37. #22583536C22semidirect
38. #22583526C22semidirect
39. C8×Dic283C22semidirect
41. C2×Q16×C283C22semidirect
43. #108×C283C22semidirect
44. #101×C283C22semidirect
149. C283#3290semidirect
155. #473D283semidirect
178. C2×Q16C2×D283semidirect
188. #108C2×D283semidirect
192. #101C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272782
#318272742
#318272702
#318268502
#318267362
#318267312
#318266412
#318266402
#318265812
#318265802
#473×C2832
#3290283