Finite Group Databasebeta

← #49133029

Subgroups — #49133029

Order 36224.

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

normal subgroups only

This group has 464 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#31827273C212yesmaximal, complemented
2#31827269C212yesmaximal, complemented
3#31826855C211yesmaximal, characteristic, complemented
4#31826733C212yesmaximal, complemented
5#31826731C212yesmaximal, complemented
6#31826642C211yesmaximal, characteristic, complemented
7#31826634C212yesmaximal, complemented
8#31826584C212yesmaximal, complemented
9#454×C283C211yesmaximal, characteristic, complemented
10#22583656C2212yescomplemented
11#22583655C2211yescharacteristic
12#22583641—22no—
13#22583641C2212yescomplemented
14#22583638C2212yes—
15#22583637—22no—
16#22583635—22no—
17#22583635C2212yescomplemented
18#22583634—22no—
19#22583631—22no—
20#22583630—22no—
21#22583630C2212yes—
22C2×#17985379—22no—
23C2×#17985379C2212yescomplemented
24C2×#17985377C2212yescomplemented
25D8×D283—22no—
26#22583618C2212yescomplemented
27#22583617C2212yescomplemented
28#22583607C2211yescharacteristic, complemented
29#22583545C2212yescomplemented
30#22583538C2212yescomplemented
31#22583527C2211yescharacteristic, complemented
32#22583526C2212yescomplemented
33#22583521C2211yescharacteristic, complemented
342−1+4×C283C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C283C2212yescomplemented
36#124×C283C2211yescharacteristic, complemented
37#104×C283C2212yescomplemented
38#101×C283C2211yescharacteristic, complemented
39#17985393—22no—
40#17985392—22no—
41#17985392C2312yes—
42#17985391—22no—
43C2×D1132—22no—
44C2×D1132—41no—
45C2×D1132C2312yes—
46C2×C22⋊D283—41no—
47D4×D283—22no—
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#17985379D412yescomplemented
59D2264—22no—
60#17985377—22no—
61#17985377—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×D4×C283C2311yescharacteristic
76SD16×C283—22no—
77D8×C283—22no—
78M16×C283C2312yes—
79C22⋊C4×C283—41no—
80C42×C283C2311yescharacteristic
81C22×D283—22no—
82C22×D283—41no—
83C22×D283—42no—
84Dic566C2×D412yes—
85D1132—22no—
86D1132—22no—
87D1132—42no—
88D1132—82no—
89D1132C2×D412yes—
90C22⋊D283—22no—
91C22⋊D283—42no—
92C22⋊D283—42no—
93C2×Dic283—41no—
94C4×D283—22no—
95C4×D283—42no—
96C283⋊C8—22no—
97C283⋊C8C2×D412yes—
98C283⋊C8C2×D412yes—
99C23×C283—21no—
100Q8×C283—22no—
101Q8×C283—41no—
102Q8×C283C2×D412yes—
103D4×C283—22no—
104D4×C283—22no—
105D4×C283—41no—
106D4×C283C2×D412yes—
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×C283—41no—
115C22×C283—81no—
116C22×C283H(5,𝔽2)11yescharacteristic
117C2×D283—22no—
118C2×D283—42no—
119C2×D283—82no—
120C2×D283—161no—
121C2×D283—162no—
122C4×C283—22no—
123C4×C283—41no—
124C4×C283—41no—
125C4×C283C22×D412yes—
126Dic283—22no—
127C2×C283—21no—
128C2×C283—42no—
129C2×C283—81no—
130C2×C283D4211yescharacteristic
131D283—42no—
132D283—82no—
133C283#328811yescharacteristic, complemented, Sylow
134#3288—2831nomaximal, Sylow
135#577—2832no—
136#576—2832no—
137#496—2831no—
138#472—2832no—
139#471—2832no—
140#455—2831no—
141#454D28311yescharacteristic, complemented
142#453—2832no—
143#444—2832no—
1442−1+4C2×D28311yescharacteristic, complemented
145H(5,𝔽2)—2832no—
146H(5,𝔽2)—2831no—
147#137—2832no—
148#137—5662no—
149AGL(1,ℤ/8)—2832no—
150AGL(1,ℤ/8)—2832no—
151AGL(1,ℤ/8)—5662no—
152AGL(1,ℤ/8)—5662no—
153AGL(1,ℤ/8)—5662no—
154AGL(1,ℤ/8)C2×D28312yescomplemented
155#135—2832no—
156#135—5662no—
157#135—5662no—
158C2×SD16—2832no—
159C2×SD16—5662no—
160C2×D8—2832no—
161C2×D8—5662no—
162#131—2832no—
163#131—2832no—
164#127—2831no—
165#124C2×D28311yescharacteristic, complemented
166#108—2832no—
167#104—2832no—
168#104C2×D28312yescomplemented
169#101C2×D28311yescharacteristic, complemented
170#100—2831no—
171#100—2832no—
172#97—2831no—
173#41—22no—
174#41—41no—
175#41—2832no—
176#41—5662no—
177#41—5662no—
178#41—5662no—
179#41C22×D28312yes—
180C2×Q8—22no—
181C2×Q8C22×D28311yescharacteristic
182C2×D4—2832no—
183C2×D4—5662no—
184C2×D4—5662no—
185C2×D4—5662no—
186C2×D4—11321no—
187C2×D4—11322no—
188C2×D4—11321no—
189C2×D4C22×D28311yescharacteristic
190Q16—2832no—
191Q16—5662no—
192SD16—22no—
193SD16—2832no—
194SD16—2832no—
195SD16—5662no—
196SD16—5662no—
197SD16—5662no—
198SD16—5662no—
199SD16—5662no—
200D8—22no—
201D8—2832no—
202D8—5662no—
203D8—5662no—
204D8—5662no—
205D8—5662no—
206M16—2832no—
207M16—5662no—
208M16—5662no—
209M16C22×D28312yes—
210C2×C8—2832no—
211C2×C8—5662no—
212C2×C8—5662no—
213C22⋊C4—41no—
214C42C22×D28311yescharacteristic
215C23—21no—
216C23—5662no—
217C23—11322no—
218C23—11321no—
219Q8—22no—
220Q8—41no—
221Q8—2832no—
222Q8D4×D28312yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—2832no—
227D4—5662no—
228D4—5662no—
229D4—5662no—
230D4—11322no—
231D4—11322no—
232D4—11322no—
233D4—22642no—
234D4D4×D28312yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—5662no—
240C2×C4—11322no—
241C2×C4—11321no—
242C2×C4C23×D28311yescharacteristic
243C8—22no—
244C8—2832no—
245C8—2832no—
246C8—5662no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—5662no—
251C22—11322no—
252C22—45282no—
253C22—45281no—
254C22—22642no—
255C22#2258365511yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—5662no—
260C4C2×D4×D28312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—11322no—
265C2—22642no—
266C2#3182709711yescharacteristic
subgroupG/H
centerC2#31827097
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#454×C283C2
socleC2×C283D42
hypercenter#454D283
hypocenterC283#3288
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #31827273C2semidirect
2. #31827269C2semidirect
3. #31826855C2semidirect
4. #31826733C2semidirect
5. #31826731C2semidirect
6. #31826642C2semidirect
7. #31826634C2semidirect
8. #31826584C2semidirect
9. #454×C283C2semidirect
10. #22583656C22semidirect
13. #22583641C22semidirect
17. #22583635C22semidirect
23. C2×#17985379C22semidirect
24. C2×#17985377C22semidirect
26. #22583618C22semidirect
27. #22583617C22semidirect
28. #22583607C22semidirect
29. #22583545C22semidirect
30. #22583538C22semidirect
31. #22583527C22semidirect
32. #22583526C22semidirect
33. #22583521C22semidirect
34. 2−1+4×C283C22semidirect
35. AGL(1,ℤ/8)×C283C22semidirect
36. #124×C283C22semidirect
37. #104×C283C22semidirect
38. #101×C283C22semidirect
51. #17985382D4semidirect
53. #17985380D4semidirect
58. #17985379D4semidirect
63. #17985377D4semidirect
133. C283#3288semidirect
141. #454D283semidirect
144. 2−1+4C2×D283semidirect
154. AGL(1,ℤ/8)C2×D283semidirect
165. #124C2×D283semidirect
168. #104C2×D283semidirect
169. #101C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272732
#318272692
#318268552
#318267332
#318267312
#318266422
#318266342
#318265842
#454×C2832
#3288283