Finite Group Databasebeta

← #16858851

Subgroups — #16858851

Order 3968.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#14521851C211yesmaximal, characteristic, complemented
2#14521850C211yesmaximal, characteristic, complemented
3#14521487C211yesmaximal, characteristic, complemented
4#14521486C212yesmaximal, complemented
5#14521483C212yesmaximal, complemented
6#14521480C212yesmaximal, complemented
7#14521479C211yesmaximal, characteristic, complemented
8C2×#11758414C211yesmaximal, characteristic, complemented
9D16×D31C212yesmaximal, complemented
10#14521467C211yesmaximal, characteristic, complemented
11C31×#509C211yesmaximal, characteristic, complemented
12#11758566—21no—
13#11758561—21no—
14C31×#135C2211yescharacteristic, complemented
15C2×D8×C31C2211yescharacteristic, complemented
16#11758526—21no—
17#11758509—21no—
18#11758496C2212yescomplemented
19#11758491—22no—
20#11758489C2212yescomplemented
21#11758488—22no—
22D8×D31C2212yescomplemented
23D8×D31C2212yescomplemented
24#11758481C2211yescharacteristic, complemented
25C2×D248C2211yescharacteristic, complemented
26#11758478C2211yescharacteristic
27C31×SD32C2212yescomplemented
28C31×D16C2212yescomplemented
29C31×M32C2211yescharacteristic, complemented
30#11758417C2211yescharacteristic, complemented
31#11758416C2211yescharacteristic, complemented
32#11758415C2211yescharacteristic, complemented
33#11758414C2211yescharacteristic, complemented
34#11758414C2212yescomplemented
35#11758414C2212yescomplemented
36C2×C31⋊C16C2211yescharacteristic, complemented
37#11758388C2212yescomplemented
38D496C2212yescomplemented
39#11758386C2212yescomplemented
40C16×D31C2212yescomplemented
41#41×C31—21no—
42C2×D4×C31—21no—
43C2×C22⋊D31—42no—
44#92499—22no—
45#92497—22no—
46#92497—41no—
47D4×D31—22no—
48D4×D31—22no—
49D4×D31—42no—
50D4×D31—41no—
51#92495—21no—
52#92495—41no—
53#92495D411yescharacteristic
54C2×D124—21no—
55C2×D124—41no—
56Q16×C31C2311yescharacteristic
57SD16×C31—21no—
58D8×C31—21no—
59D8×C31C2312yes—
60D8×C31C2311yescharacteristic
61C2×C8×C31C2311yescharacteristic
62#92481—22no—
63#92480—22no—
64#92479—22no—
65#92479—22no—
66#92474D411yescharacteristic
67Dic124C2311yescharacteristic
68D248—21no—
69D248C2311yescharacteristic
70D248C2312yes—
71#92470—21no—
72#92469D412yes—
73C8×D31C2312yes—
74C16×C31C2312yes—
75C31⋊C16C2311yescharacteristic
76C31⋊C16C2311yescharacteristic
77C23×C31—41no—
78C22×D31—41no—
79C22×D31—42no—
80C22×D31—42no—
81Q8×C31—21no—
82D4×C31—21no—
83D4×C31—22no—
84D4×C31—41no—
85D4×C31—41no—
86C2×C4×C31—41no—
87C2×C4×C31C2×D411yescharacteristic
88C22⋊D31—41no—
89C22⋊D31—42no—
90C22⋊D31—42no—
91C22⋊D31—42no—
92C22⋊D31D812yes—
93C2×Dic31—42no—
94D124—21no—
95D124—22no—
96D124—42no—
97D124—41no—
98D124D811yescharacteristic
99C4×D31—41no—
100C4×D31—42no—
101C4×D31C2×D412yes—
102Dic62—21no—
103Dic62D811yescharacteristic
104C8×C31C2×D411yescharacteristic
105C8×C31C2411yescharacteristic
106C31⋊C8C2×D412yes—
107#3415—311nomaximal, Sylow
108C22×C31—42no—
109C22×C31—41no—
110C22×C31—161no—
111C22×C31C2×D811yescharacteristic
112C2×D31—42no—
113C2×D31—41no—
114C2×D31—162no—
115C2×D31—162no—
116C2×D31—161no—
117C2×D31C2×D812yes—
118C4×C31—41no—
119C4×C31C2×D811yescharacteristic
120C4×C31C22×D411yescharacteristic
121Dic31—41no—
122Dic31C2×D812yes—
123#576—311no—
124#576—311no—
125#509—312no—
126#509—312no—
127#509—311no—
128#509D3111yescharacteristic, complemented
129#508—311no—
130#508—312no—
131C2×D16—311no—
132C2×D16—312no—
133#504—311no—
134C2×C31—21no—
135C2×C31—81no—
136C2×C31—82no—
137C2×C31C22×D811yescharacteristic
138D31—22no—
139D31—82no—
140D31—81no—
141H(5,𝔽2)—621no—
142H(5,𝔽2)—621no—
143AGL(1,ℤ/8)—622no—
144AGL(1,ℤ/8)—622no—
145AGL(1,ℤ/8)—621no—
146AGL(1,ℤ/8)—621no—
147#135—312no—
148#135—312no—
149#135—311no—
150#135C2×D3111yescharacteristic, complemented
151C2×D8—312no—
152C2×D8—312no—
153C2×D8—311no—
154C2×D8C2×D3111yescharacteristic, complemented
155#131—311no—
156Q32—311no—
157SD32—311no—
158SD32—311no—
159SD32—312no—
160SD32C2×D3112yescomplemented
161D16—312no—
162D16—311no—
163D16—312no—
164D16—312no—
165D16C2×D3112yescomplemented
166M32—312no—
167M32C2×D3111yescharacteristic, complemented
168C2×C16—311no—
169C2×C16—312no—
170C31#341511yescharacteristic, complemented, Sylow
171#41—21no—
172#41—311no—
173#41—622no—
174#41—622no—
175#41—621no—
176#41—1241no—
177#41—1241no—
178C2×D4—21no—
179C2×D4—622no—
180C2×D4—622no—
181C2×D4—621no—
182C2×D4—1241no—
183C2×D4—1242no—
184C2×D4—1241no—
185C2×D4—1242no—
186Q16—311no—
187Q16C22×D3111yescharacteristic
188SD16—21no—
189SD16—622no—
190SD16—622no—
191SD16—621no—
192D8—21no—
193D8—311no—
194D8—312no—
195D8—622no—
196D8—622no—
197D8—621no—
198D8C22×D3112yes—
199D8C22×D3111yescharacteristic
200M16—311no—
201M16—312no—
202C2×C8—312no—
203C2×C8C22×D3111yescharacteristic
204C16—311no—
205C16—311no—
206C16C22×D3112yes—
207C23—41no—
208C23—1242no—
209C23—1242no—
210C23—1241no—
211Q8—21no—
212Q8—311no—
213Q8—621no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—312no—
219D4—311no—
220D4—621no—
221D4—622no—
222D4—1242no—
223D4—1242no—
224D4—1242no—
225D4—1242no—
226D4—1241no—
227D4—1241no—
228C2×C4—41no—
229C2×C4—312no—
230C2×C4—1242no—
231C2×C4—1242no—
232C2×C4—1241no—
233C2×C4D4×D3111yescharacteristic
234C8—312no—
235C8D4×D3111yescharacteristic
236C8C23×D3111yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—312no—
241C22—1242no—
242C22—1241no—
243C22—4961no—
244C22—4962no—
245C22—4962no—
246C22D8×D3111yescharacteristic
247C4—41no—
248C4—312no—
249C4—1241no—
250C4D8×D3111yescharacteristic
251C4C2×D4×D3111yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—622no—
256C2—2482no—
257C2—2481no—
258C2C2×D8×D3111yescharacteristic
subgroupG/H
centerC2C2×D8×D31
commutator, cosocleC8×C31C24
FrattiniC8C23×D31
FittingC31×#509C2
socleC2×C31C22×D8
hypercenter#509D31
hypocenterC31#3415
LCSG ▷ C8×C31 ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #14521851C2semidirect
2. #14521850C2semidirect
3. #14521487C2semidirect
4. #14521486C2semidirect
5. #14521483C2semidirect
6. #14521480C2semidirect
7. #14521479C2semidirect
8. C2×#11758414C2semidirect
9. D16×D31C2semidirect
10. #14521467C2semidirect
11. C31×#509C2semidirect
14. C31×#135C22semidirect
15. C2×D8×C31C22semidirect
18. #11758496C22semidirect
20. #11758489C22semidirect
22. D8×D31C22semidirect
23. D8×D31C22semidirect
24. #11758481C22semidirect
25. C2×D248C22semidirect
27. C31×SD32C22semidirect
28. C31×D16C22semidirect
29. C31×M32C22semidirect
30. #11758417C22semidirect
31. #11758416C22semidirect
32. #11758415C22semidirect
33. #11758414C22semidirect
34. #11758414C22semidirect
35. #11758414C22semidirect
36. C2×C31⋊C16C22semidirect
37. #11758388C22semidirect
38. D496C22semidirect
39. #11758386C22semidirect
40. C16×D31C22semidirect
128. #509D31semidirect
150. #135C2×D31semidirect
154. C2×D8C2×D31semidirect
160. SD32C2×D31semidirect
165. D16C2×D31semidirect
167. M32C2×D31semidirect
170. C31#3415semidirect

Maximal subgroups

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

subgroupindex
#145218512
#145218502
#145214872
#145214862
#145214832
#145214802
#145214792
C2×#117584142
D16×D312
#145214672
C31×#5092
#341531