Finite Group Databasebeta

← #16857501

Subgroups — #16857501

Order 3968.

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#14521858C212yesmaximal, complemented
2#14521857C212yesmaximal, complemented
3#14521423C211yesmaximal, characteristic, complemented
4#14521315C212yesmaximal, complemented
5#14521314C212yesmaximal, complemented
6#14521231C211yesmaximal, characteristic, complemented
7#14521218C212yesmaximal, complemented
8#14521163C212yesmaximal, complemented
9C31×#456C211yesmaximal, characteristic, complemented
10C31×2−1+4C2211yescharacteristic
11#11758566C2212yescomplemented
12#11758564C2211yescharacteristic
13C31×#137C2212yescomplemented
14C31×C4⋊Q8C2211yescharacteristic, complemented
15#11758527—22no—
16#11758527C2212yes—
17#11758526—22no—
18#11758526C2212yes—
19#11758525C2212yescomplemented
20#11758519—22no—
21#11758519C2212yescomplemented
22C2×#92481—22no—
23C2×#92481C2212yescomplemented
24C2×#92481C2212yescomplemented
25#11758496—22no—
26#11758495—22no—
27#11758491—22no—
28SD16×D31—22no—
29#11758485C2212yes—
30#11758484C2212yescomplemented
31#11758451C2211yescharacteristic
32C31×#104C2212yescomplemented
33C31×#101C2211yescharacteristic, complemented
34#11758424C2211yescharacteristic, complemented
35#11758411C2212yescomplemented
36#11758410C2212yescomplemented
37#11758393C2212yescomplemented
38C4×C31⋊C8C2211yescharacteristic, complemented
39#41×C31—22no—
40#41×C31—41no—
41#41×C31C2312yes—
42C2×Q8×C31—22no—
43C2×Q8×C31C2311yescharacteristic
44C2×Q8×C31C2311yescharacteristic
45#92499—22no—
46#92499—22no—
47Q8×D31—22no—
48D4×D31—22no—
49D4×D31—42no—
50#92495—22no—
51#92495—41no—
52#92495C2312yes—
53C2×D124—22no—
54C2×D124—41no—
55C2×D124C2312yes—
56Q16×C31—22no—
57SD16×C31—22no—
58M16×C31C2312yes—
59C4⋊C4×C31—41no—
60C42×C31C2311yescharacteristic
61#92482—22no—
62#92482D412yescomplemented
63#92481—22no—
64#92481—22no—
65#92481—22no—
66#92481—22no—
67#92481—22no—
68#92481—22no—
69#92481D412yescomplemented
70#92480D412yescomplemented
71#92479—22no—
72#92479D412yescomplemented
73#92474—22no—
74#92474C2312yes—
75C2×C31⋊C8—22no—
76C2×C31⋊C8C2312yes—
77D248—22no—
78#92470—22no—
79#92469—22no—
80C8×D31—22no—
81C22×D31—22no—
82C22×D31—42no—
83Q8×C31—22no—
84Q8×C31—22no—
85Q8×C31—41no—
86Q8×C31C2×D412yes—
87D4×C31—22no—
88D4×C31—41no—
89D4×C31C2×D412yes—
90C2×C4×C31—21no—
91C2×C4×C31—22no—
92C2×C4×C31—21no—
93C2×C4×C31—21no—
94C2×C4×C31—42no—
95C2×C4×C31C2411yescharacteristic
96C22⋊D31—22no—
97C22⋊D31—42no—
98D124—22no—
99D124—22no—
100D124—42no—
101D124—41no—
102D124—82no—
103D124C2×D412yes—
104C4×D31—22no—
105C4×D31—42no—
106C4×D31—42no—
107Dic62—41no—
108Dic62C2×D412yes—
109C8×C31—22no—
110C31⋊C8—22no—
111C31⋊C8C2×D412yes—
112C31⋊C8C2×D412yes—
113#3291—311nomaximal, Sylow
114C22×C31—22no—
115C22×C31H(5,𝔽2)11yescharacteristic
116C2×D31—22no—
117C2×D31—42no—
118C2×D31—82no—
119C2×D31—162no—
120C4×C31—22no—
121C4×C31—22no—
122C4×C31—41no—
123C4×C31—41no—
124C4×C31C22×D412yes—
125Dic31—22no—
126#577—312no—
127#577—312no—
128#492—311no—
129#471—312no—
130#471—312no—
131#456—311no—
132#456D3111yescharacteristic, complemented
133#453—312no—
134#443—312no—
135C2×C31—21no—
136C2×C31—42no—
137C2×C31D4211yescharacteristic
138D31—42no—
139D31—82no—
1402−1+4—311no—
1412−1+4C2×D3111yescharacteristic
142H(5,𝔽2)—312no—
143#137—312no—
144#137—622no—
145#137—622no—
146#137C2×D3112yescomplemented
147AGL(1,ℤ/8)—312no—
148AGL(1,ℤ/8)—312no—
149AGL(1,ℤ/8)—622no—
150AGL(1,ℤ/8)—622no—
151#135—312no—
152#135—622no—
153#135—622no—
154C2×SD16—312no—
155C2×SD16—312no—
156C2×SD16—622no—
157C2×SD16—622no—
158#131—312no—
159#131—312no—
160C4⋊Q8C2×D3111yescharacteristic, complemented
161#127—311no—
162#108—312no—
163#104—312no—
164#104C2×D3112yescomplemented
165#101—311no—
166#101C2×D3111yescharacteristic, complemented
167#100—312no—
168C4×C8—311no—
169C31#329111yescharacteristic, complemented, Sylow
170#41—22no—
171#41—41no—
172#41—312no—
173#41—622no—
174#41—622no—
175#41—622no—
176#41—1241no—
177#41C22×D3112yes—
178C2×Q8—22no—
179C2×Q8—622no—
180C2×Q8C22×D3111yescharacteristic
181C2×Q8C22×D3111yescharacteristic
182C2×D4—312no—
183C2×D4—622no—
184C2×D4—622no—
185C2×D4—1241no—
186C2×D4—1242no—
187Q16—22no—
188Q16—312no—
189Q16—622no—
190SD16—22no—
191SD16—312no—
192SD16—312no—
193SD16—622no—
194SD16—622no—
195SD16—622no—
196SD16—622no—
197SD16—622no—
198SD16—622no—
199SD16—622no—
200D8—312no—
201D8—622no—
202D8—622no—
203M16—312no—
204M16—622no—
205M16—622no—
206M16C22×D3112yes—
207C2×C8—312no—
208C2×C8—622no—
209C2×C8—622no—
210C4⋊C4—41no—
211C42C22×D3111yescharacteristic
212C23—622no—
213C23—1242no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—312no—
218Q8—1241no—
219Q8D4×D3112yes—
220D4—22no—
221D4—41no—
222D4—312no—
223D4—622no—
224D4—622no—
225D4—622no—
226D4—1242no—
227D4—1241no—
228D4—1242no—
229D4—2482no—
230D4D4×D3112yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—622no—
237C2×C4—1242no—
238C2×C4—1242no—
239C2×C4C23×D3111yescharacteristic
240C8—22no—
241C8—312no—
242C8—312no—
243C8—622no—
244C22—22no—
245C22—622no—
246C22—1242no—
247C22—4962no—
248C22—2482no—
249C22#1175856111yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—622no—
255C4C2×D4×D3112yes—
256C2—21no—
257C2—42no—
258C2—1242no—
259C2—2482no—
260C2#1452168111yescharacteristic
subgroupG/H
centerC2#14521681
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#456C2
socleC2×C31D42
hypercenter#456D31
hypocenterC31#3291
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #14521858C2semidirect
2. #14521857C2semidirect
3. #14521423C2semidirect
4. #14521315C2semidirect
5. #14521314C2semidirect
6. #14521231C2semidirect
7. #14521218C2semidirect
8. #14521163C2semidirect
9. C31×#456C2semidirect
11. #11758566C22semidirect
13. C31×#137C22semidirect
14. C31×C4⋊Q8C22semidirect
19. #11758525C22semidirect
21. #11758519C22semidirect
23. C2×#92481C22semidirect
24. C2×#92481C22semidirect
30. #11758484C22semidirect
32. C31×#104C22semidirect
33. C31×#101C22semidirect
34. #11758424C22semidirect
35. #11758411C22semidirect
36. #11758410C22semidirect
37. #11758393C22semidirect
38. C4×C31⋊C8C22semidirect
62. #92482D4semidirect
69. #92481D4semidirect
70. #92480D4semidirect
72. #92479D4semidirect
132. #456D31semidirect
146. #137C2×D31semidirect
160. C4⋊Q8C2×D31semidirect
164. #104C2×D31semidirect
166. #101C2×D31semidirect
169. C31#3291semidirect

Maximal subgroups

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

subgroupindex
#145218582
#145218572
#145214232
#145213152
#145213142
#145212312
#145212182
#145211632
C31×#4562
#329131