Finite Group Databasebeta

← #16857479

Subgroups — #16857479

Order 3968.

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#14521857C212yesmaximal, complemented
2#14521853C212yesmaximal, complemented
3#14521439C211yesmaximal, characteristic, complemented
4#14521317C212yesmaximal, complemented
5#14521315C212yesmaximal, complemented
6#14521226C211yesmaximal, characteristic, complemented
7#14521218C212yesmaximal, complemented
8#14521168C212yesmaximal, complemented
9C31×#454C211yesmaximal, characteristic, complemented
10C31×2−1+4C2211yescharacteristic, complemented
11#11758566C2212yescomplemented
12#11758561C2211yescharacteristic
13C31×AGL(1,ℤ/8)C2212yescomplemented
14C31×#124C2211yescharacteristic, complemented
15#11758527—22no—
16#11758527C2212yes—
17#11758526—22no—
18#11758526C2212yescomplemented
19#11758525C2212yescomplemented
20#11758519—22no—
21#11758519C2212yescomplemented
22C2×#92481—22no—
23C2×#92481C2212yescomplemented
24C2×#92479C2212yescomplemented
25#11758493—22no—
26#11758491—22no—
27#11758488—22no—
28D8×D31—22no—
29#11758485C2212yes—
30#11758484C2212yescomplemented
31#11758451C2211yescharacteristic, complemented
32C31×#104C2212yescomplemented
33C31×#101C2211yescharacteristic, complemented
34#11758421C2211yescharacteristic, complemented
35#11758411C2212yescomplemented
36#11758410C2212yescomplemented
37#11758393C2212yescomplemented
38#11758391C2211yescharacteristic, complemented
39#41×C31—22no—
40#41×C31—41no—
41#41×C31C2312yes—
42C2×Q8×C31—22no—
43C2×Q8×C31C2311yescharacteristic
44C2×D4×C31C2311yescharacteristic
45C2×C22⋊D31—41no—
46#92499—22no—
47#92497—22no—
48D4×D31—22no—
49D4×D31—22no—
50D4×D31—42no—
51#92495—22no—
52#92495C2312yes—
53C2×D124—22no—
54C2×D124—41no—
55C2×D124C2312yes—
56SD16×C31—22no—
57D8×C31—22no—
58M16×C31C2312yes—
59C22⋊C4×C31—41no—
60C42×C31C2311yescharacteristic
61#92482—22no—
62#92482D412yescomplemented
63#92481—22no—
64#92481—22no—
65#92481—22no—
66#92481—22no—
67#92481D412yescomplemented
68#92480D412yescomplemented
69#92479—22no—
70#92479—22no—
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—
81C23×C31—21no—
82C22×D31—22no—
83C22×D31—41no—
84C22×D31—42no—
85Q8×C31—22no—
86Q8×C31—41no—
87Q8×C31C2×D412yes—
88D4×C31—22no—
89D4×C31—22no—
90D4×C31—41no—
91D4×C31C2×D412yes—
92C2×C4×C31—22no—
93C2×C4×C31—21no—
94C2×C4×C31—21no—
95C2×C4×C31—42no—
96C2×C4×C31C2411yescharacteristic
97C22⋊D31—22no—
98C22⋊D31—42no—
99C22⋊D31—42no—
100C2×Dic31—41no—
101D124—22no—
102D124—22no—
103D124—42no—
104D124—82no—
105D124C2×D412yes—
106C4×D31—22no—
107C4×D31—42no—
108Dic62C2×D412yes—
109C8×C31—22no—
110C31⋊C8—22no—
111C31⋊C8C2×D412yes—
112C31⋊C8C2×D412yes—
113#3288—311nomaximal, Sylow
114C22×C31—22no—
115C22×C31—41no—
116C22×C31—81no—
117C22×C31H(5,𝔽2)11yescharacteristic
118C2×D31—22no—
119C2×D31—42no—
120C2×D31—82no—
121C2×D31—161no—
122C2×D31—162no—
123C4×C31—22no—
124C4×C31—41no—
125C4×C31—41no—
126C4×C31C22×D412yes—
127Dic31—22no—
128#577—312no—
129#576—312no—
130#496—311no—
131#472—312no—
132#471—312no—
133#455—311no—
134#454D3111yescharacteristic, complemented
135#453—312no—
136#444—312no—
137C2×C31—21no—
138C2×C31—42no—
139C2×C31—81no—
140C2×C31D4211yescharacteristic
141D31—42no—
142D31—82no—
1432−1+4C2×D3111yescharacteristic, complemented
144H(5,𝔽2)—312no—
145H(5,𝔽2)—311no—
146#137—312no—
147#137—622no—
148AGL(1,ℤ/8)—312no—
149AGL(1,ℤ/8)—312no—
150AGL(1,ℤ/8)—622no—
151AGL(1,ℤ/8)—622no—
152AGL(1,ℤ/8)—622no—
153AGL(1,ℤ/8)C2×D3112yescomplemented
154#135—312no—
155#135—622no—
156#135—622no—
157C2×SD16—312no—
158C2×SD16—622no—
159C2×D8—312no—
160C2×D8—622no—
161#131—312no—
162#131—312no—
163#127—311no—
164#124C2×D3111yescharacteristic, complemented
165#108—312no—
166#104—312no—
167#104C2×D3112yescomplemented
168#101C2×D3111yescharacteristic, complemented
169#100—311no—
170#100—312no—
171#97—311no—
172C31#328811yescharacteristic, complemented, Sylow
173#41—22no—
174#41—41no—
175#41—312no—
176#41—622no—
177#41—622no—
178#41—622no—
179#41C22×D3112yes—
180C2×Q8—22no—
181C2×Q8C22×D3111yescharacteristic
182C2×D4—312no—
183C2×D4—622no—
184C2×D4—622no—
185C2×D4—622no—
186C2×D4—1241no—
187C2×D4—1242no—
188C2×D4—1241no—
189C2×D4C22×D3111yescharacteristic
190Q16—312no—
191Q16—622no—
192SD16—22no—
193SD16—312no—
194SD16—312no—
195SD16—622no—
196SD16—622no—
197SD16—622no—
198SD16—622no—
199SD16—622no—
200D8—22no—
201D8—312no—
202D8—622no—
203D8—622no—
204D8—622no—
205D8—622no—
206M16—312no—
207M16—622no—
208M16—622no—
209M16C22×D3112yes—
210C2×C8—312no—
211C2×C8—622no—
212C2×C8—622no—
213C22⋊C4—41no—
214C42C22×D3111yescharacteristic
215C23—21no—
216C23—622no—
217C23—1242no—
218C23—1241no—
219Q8—22no—
220Q8—41no—
221Q8—312no—
222Q8D4×D3112yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—312no—
227D4—622no—
228D4—622no—
229D4—622no—
230D4—1242no—
231D4—1242no—
232D4—1242no—
233D4—2482no—
234D4D4×D3112yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—622no—
240C2×C4—1242no—
241C2×C4—1241no—
242C2×C4C23×D3111yescharacteristic
243C8—22no—
244C8—312no—
245C8—312no—
246C8—622no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—622no—
251C22—1242no—
252C22—4962no—
253C22—4961no—
254C22—2482no—
255C22#1175856111yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—622no—
260C4C2×D4×D3112yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—1242no—
265C2—2482no—
266C2#1452168111yescharacteristic
subgroupG/H
centerC2#14521681
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#454C2
socleC2×C31D42
hypercenter#454D31
hypocenterC31#3288
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #14521857C2semidirect
2. #14521853C2semidirect
3. #14521439C2semidirect
4. #14521317C2semidirect
5. #14521315C2semidirect
6. #14521226C2semidirect
7. #14521218C2semidirect
8. #14521168C2semidirect
9. C31×#454C2semidirect
10. C31×2−1+4C22semidirect
11. #11758566C22semidirect
13. C31×AGL(1,ℤ/8)C22semidirect
14. C31×#124C22semidirect
18. #11758526C22semidirect
19. #11758525C22semidirect
21. #11758519C22semidirect
23. C2×#92481C22semidirect
24. C2×#92479C22semidirect
30. #11758484C22semidirect
31. #11758451C22semidirect
32. C31×#104C22semidirect
33. C31×#101C22semidirect
34. #11758421C22semidirect
35. #11758411C22semidirect
36. #11758410C22semidirect
37. #11758393C22semidirect
38. #11758391C22semidirect
62. #92482D4semidirect
67. #92481D4semidirect
68. #92480D4semidirect
72. #92479D4semidirect
134. #454D31semidirect
143. 2−1+4C2×D31semidirect
153. AGL(1,ℤ/8)C2×D31semidirect
164. #124C2×D31semidirect
167. #104C2×D31semidirect
168. #101C2×D31semidirect
172. C31#3288semidirect

Maximal subgroups

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

subgroupindex
#145218572
#145218532
#145214392
#145213172
#145213152
#145212262
#145212182
#145211682
C31×#4542
#328831