Finite Group Databasebeta

← #49133051

Subgroups — #49133051

Order 36224.

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

normal subgroups only

This group has 460 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#31827274C212yesmaximal, complemented
2#31827273C212yesmaximal, complemented
3#31826839C211yesmaximal, characteristic, complemented
4#31826731C212yesmaximal, complemented
5#31826730C212yesmaximal, complemented
6#31826647C211yesmaximal, characteristic, complemented
7#31826634C212yesmaximal, complemented
8#31826579C212yesmaximal, complemented
9#456×C283C211yesmaximal, characteristic, complemented
10#22583658C2211yescharacteristic
11#22583656C2212yescomplemented
12#22583642—22no—
13#22583641—22no—
14#22583641C2212yescomplemented
15#22583638C2212yes—
16#22583637—22no—
17#22583635—22no—
18#22583635C2212yes—
19#22583633—22no—
20#22583630—22no—
21#22583630C2212yes—
22C2×#17985379—22no—
23C2×#17985379C2212yescomplemented
24C2×#17985379C2212yescomplemented
25SD16×D283—22no—
26#22583618C2212yescomplemented
27#22583617C2212yescomplemented
28#22583607C2211yescharacteristic
29#22583545C2212yescomplemented
30#22583538C2212yescomplemented
31#22583529C2211yescharacteristic, complemented
32#22583526C2212yescomplemented
33C4×C283⋊C8C2211yescharacteristic, complemented
342−1+4×C283C2211yescharacteristic
35#137×C283C2212yescomplemented
36C4⋊Q8×C283C2211yescharacteristic, complemented
37#104×C283C2212yescomplemented
38#101×C283C2211yescharacteristic, complemented
39#17985393—22no—
40#17985393—22no—
41#17985392—22no—
42#17985392—41no—
43#17985392C2312yes—
44Q8×D283—22no—
45C2×D1132—22no—
46C2×D1132—41no—
47C2×D1132C2312yes—
48D4×D283—22no—
49D4×D283—42no—
50#17985382—22no—
51#17985382D412yescomplemented
52#17985381—22no—
53#17985380D412yescomplemented
54#17985379—22no—
55#17985379—22no—
56#17985379—22no—
57#17985379—22no—
58#17985379—22no—
59#17985379—22no—
60#17985379D412yescomplemented
61D2264—22no—
62#17985377—22no—
63#17985377D412yescomplemented
64#17985376—22no—
65#17985376C2312yes—
66#17985375—22no—
67C2×C283⋊C8—22no—
68C2×C283⋊C8C2312yes—
69C8×D283—22no—
70#41×C283—22no—
71#41×C283—41no—
72#41×C283C2312yes—
73C2×Q8×C283—22no—
74C2×Q8×C283C2311yescharacteristic
75C2×Q8×C283C2311yescharacteristic
76Q16×C283—22no—
77SD16×C283—22no—
78M16×C283C2312yes—
79C4⋊C4×C283—41no—
80C42×C283C2311yescharacteristic
81C22×D283—22no—
82C22×D283—42no—
83Dic566—41no—
84Dic566C2×D412yes—
85D1132—22no—
86D1132—22no—
87D1132—42no—
88D1132—41no—
89D1132—82no—
90D1132C2×D412yes—
91C22⋊D283—22no—
92C22⋊D283—42no—
93C4×D283—22no—
94C4×D283—42no—
95C4×D283—42no—
96C283⋊C8—22no—
97C283⋊C8C2×D412yes—
98C283⋊C8C2×D412yes—
99Q8×C283—22no—
100Q8×C283—22no—
101Q8×C283—41no—
102Q8×C283C2×D412yes—
103D4×C283—22no—
104D4×C283—41no—
105D4×C283C2×D412yes—
106C2×C4×C283—21no—
107C2×C4×C283—22no—
108C2×C4×C283—21no—
109C2×C4×C283—21no—
110C2×C4×C283—42no—
111C2×C4×C283C2411yescharacteristic
112C8×C283—22no—
113C22×C283—22no—
114C22×C283H(5,𝔽2)11yescharacteristic
115C2×D283—22no—
116C2×D283—42no—
117C2×D283—82no—
118C2×D283—162no—
119C4×C283—22no—
120C4×C283—22no—
121C4×C283—41no—
122C4×C283—41no—
123C4×C283C22×D412yes—
124Dic283—22no—
125C2×C283—21no—
126C2×C283—42no—
127C2×C283D4211yescharacteristic
128D283—42no—
129D283—82no—
130C283#329111yescharacteristic, complemented, Sylow
131#3291—2831nomaximal, Sylow
132#577—2832no—
133#577—2832no—
134#492—2831no—
135#471—2832no—
136#471—2832no—
137#456—2831no—
138#456D28311yescharacteristic, complemented
139#453—2832no—
140#443—2832no—
1412−1+4—2831no—
1422−1+4C2×D28311yescharacteristic
143H(5,𝔽2)—2832no—
144#137—2832no—
145#137—5662no—
146#137—5662no—
147#137C2×D28312yescomplemented
148AGL(1,ℤ/8)—2832no—
149AGL(1,ℤ/8)—2832no—
150AGL(1,ℤ/8)—5662no—
151AGL(1,ℤ/8)—5662no—
152#135—2832no—
153#135—5662no—
154#135—5662no—
155C2×SD16—2832no—
156C2×SD16—2832no—
157C2×SD16—5662no—
158C2×SD16—5662no—
159#131—2832no—
160#131—2832no—
161C4⋊Q8C2×D28311yescharacteristic, complemented
162#127—2831no—
163#108—2832no—
164#104—2832no—
165#104C2×D28312yescomplemented
166#101—2831no—
167#101C2×D28311yescharacteristic, complemented
168#100—2832no—
169C4×C8—2831no—
170#41—22no—
171#41—41no—
172#41—2832no—
173#41—5662no—
174#41—5662no—
175#41—5662no—
176#41—11321no—
177#41C22×D28312yes—
178C2×Q8—22no—
179C2×Q8—5662no—
180C2×Q8C22×D28311yescharacteristic
181C2×Q8C22×D28311yescharacteristic
182C2×D4—2832no—
183C2×D4—5662no—
184C2×D4—5662no—
185C2×D4—11321no—
186C2×D4—11322no—
187Q16—22no—
188Q16—2832no—
189Q16—5662no—
190SD16—22no—
191SD16—2832no—
192SD16—2832no—
193SD16—5662no—
194SD16—5662no—
195SD16—5662no—
196SD16—5662no—
197SD16—5662no—
198SD16—5662no—
199SD16—5662no—
200D8—2832no—
201D8—5662no—
202D8—5662no—
203M16—2832no—
204M16—5662no—
205M16—5662no—
206M16C22×D28312yes—
207C2×C8—2832no—
208C2×C8—5662no—
209C2×C8—5662no—
210C4⋊C4—41no—
211C42C22×D28311yescharacteristic
212C23—5662no—
213C23—11322no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—2832no—
218Q8—11321no—
219Q8D4×D28312yes—
220D4—22no—
221D4—41no—
222D4—2832no—
223D4—5662no—
224D4—5662no—
225D4—5662no—
226D4—11322no—
227D4—11321no—
228D4—11322no—
229D4—22642no—
230D4D4×D28312yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—5662no—
237C2×C4—11322no—
238C2×C4—11322no—
239C2×C4C23×D28311yescharacteristic
240C8—22no—
241C8—2832no—
242C8—2832no—
243C8—5662no—
244C22—22no—
245C22—5662no—
246C22—11322no—
247C22—45282no—
248C22—22642no—
249C22#2258365511yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—5662no—
255C4C2×D4×D28312yes—
256C2—21no—
257C2—42no—
258C2—11322no—
259C2—22642no—
260C2#3182709711yescharacteristic
subgroupG/H
centerC2#31827097
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#456×C283C2
socleC2×C283D42
hypercenter#456D283
hypocenterC283#3291
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #31827274C2semidirect
2. #31827273C2semidirect
3. #31826839C2semidirect
4. #31826731C2semidirect
5. #31826730C2semidirect
6. #31826647C2semidirect
7. #31826634C2semidirect
8. #31826579C2semidirect
9. #456×C283C2semidirect
11. #22583656C22semidirect
14. #22583641C22semidirect
23. C2×#17985379C22semidirect
24. C2×#17985379C22semidirect
26. #22583618C22semidirect
27. #22583617C22semidirect
29. #22583545C22semidirect
30. #22583538C22semidirect
31. #22583529C22semidirect
32. #22583526C22semidirect
33. C4×C283⋊C8C22semidirect
35. #137×C283C22semidirect
36. C4⋊Q8×C283C22semidirect
37. #104×C283C22semidirect
38. #101×C283C22semidirect
51. #17985382D4semidirect
53. #17985380D4semidirect
60. #17985379D4semidirect
63. #17985377D4semidirect
130. C283#3291semidirect
138. #456D283semidirect
147. #137C2×D283semidirect
161. C4⋊Q8C2×D283semidirect
165. #104C2×D283semidirect
167. #101C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272742
#318272732
#318268392
#318267312
#318267302
#318266472
#318266342
#318265792
#456×C2832
#3291283