Finite Group Databasebeta

← #16857496

Subgroups — #16857496

Order 3968.

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

normal subgroups only

This group has 454 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#14521862C211yesmaximal, characteristic, complemented
2#14521855C212yesmaximal, complemented
3#14521850C211yesmaximal, characteristic, complemented
4#14521321C211yesmaximal, characteristic, complemented
5#14521316C211yesmaximal, characteristic, complemented
6#14521309C212yesmaximal, complemented
7#14521226C212yesmaximal, complemented
8#14521221C212yesmaximal, complemented
9#14521167C211yesmaximal, characteristic, complemented
10#14521167C211yesmaximal, characteristic, complemented
11C31×#495C211yesmaximal, characteristic, complemented
12#11758564C2212yescomplemented
13#11758561C2212yes—
14C2×Q16×C31C2211yescharacteristic, complemented
15C2×SD16×C31C2212yescomplemented
16C2×D8×C31C2211yescharacteristic
17C31×#124C2212yescomplemented
18#11758519C2212yes—
19#11758519C2212yescomplemented
20#11758509C2212yes—
21#11758509C2212yescomplemented
22#11758496—21no—
23#11758495—22no—
24Q16×D31—21no—
25#11758493—22no—
26#11758492—22no—
27#11758491—22no—
28SD16×D31—22no—
29#11758489—21no—
30#11758488—22no—
31D8×D31—21no—
32#11758481C2211yescharacteristic
33#11758481C2211yescharacteristic
34#11758478C2211yescharacteristic
35#11758478C2211yescharacteristic, complemented
36#11758478C2212yescomplemented
37C4×C8×C31C2211yescharacteristic, complemented
38#11758424C2212yescomplemented
39#11758421C2212yescomplemented
40#11758407C2211yescharacteristic, complemented
41#11758407C2211yescharacteristic, complemented
42#11758393C2211yescharacteristic
43#11758393C2211yescharacteristic, complemented
44#11758393C2212yescomplemented
45C2×Q8×C31C2312yes—
46C2×D4×C31C2312yes—
47C2×C22⋊D31—42no—
48#92499—22no—
49#92499—22no—
50Q8×D31—22no—
51Q8×D31—22no—
52#92497—22no—
53#92497—22no—
54D4×D31—22no—
55D4×D31—22no—
56#92495—42no—
57#92495C2311yescharacteristic
58#92495C2312yes—
59#92495C2311yescharacteristic
60Q16×C31—21no—
61Q16×C31—21no—
62SD16×C31—22no—
63SD16×C31—22no—
64D8×C31—21no—
65D8×C31—21no—
66C2×C8×C31C2311yescharacteristic
67C2×C8×C31C2311yescharacteristic
68C22⋊C4×C31—42no—
69C42×C31C2311yescharacteristic
70#92482—22no—
71#92482—22no—
72#92481—22no—
73#92481—22no—
74#92480—22no—
75#92480—22no—
76#92479—22no—
77#92479—22no—
78#92474C2311yescharacteristic
79#92474C2312yes—
80#92474C2311yescharacteristic
81Dic124D411yescharacteristic
82Dic124D411yescharacteristic
83D248D411yescharacteristic
84D248D411yescharacteristic
85#92470D412yes—
86#92470D411yescharacteristic
87#92470D411yescharacteristic
88#92469—21no—
89#92469—22no—
90#92469—21no—
91C8×D31—21no—
92C8×D31—22no—
93C8×D31—21no—
94C23×C31—22no—
95C22×D31—42no—
96Q8×C31—22no—
97Q8×C31—22no—
98D4×C31—22no—
99D4×C31—22no—
100C2×C4×C31—21no—
101C2×C4×C31—22no—
102C2×C4×C31C2411yescharacteristic
103C22⋊D31—21no—
104C22⋊D31—22no—
105C22⋊D31—21no—
106C22⋊D31—42no—
107C22⋊D31—42no—
108C2×Dic31—42no—
109D124—42no—
110D124C2×D411yescharacteristic
111D124C2×D412yes—
112D124C2×D411yescharacteristic
113C4×D31—21no—
114C4×D31—22no—
115C4×D31—21no—
116C4×D31—42no—
117C4×D31—42no—
118Dic62—42no—
119Dic62C2×D411yescharacteristic
120Dic62C2×D412yes—
121Dic62C2×D411yescharacteristic
122C8×C31C2×D411yescharacteristic
123C8×C31C2×D411yescharacteristic
124C8×C31C2×D411yescharacteristic
125C8×C31C2×D411yescharacteristic
126C31⋊C8—21no—
127C31⋊C8—22no—
128C31⋊C8—21no—
129#3290—311nomaximal, Sylow
130C22×C31—42no—
131C22×C31—82no—
132C22×C31H(5,𝔽2)11yescharacteristic
133C2×D31—21no—
134C2×D31—22no—
135C2×D31—21no—
136C2×D31—162no—
137C4×C31—41no—
138C4×C31—42no—
139C4×C31C22×D411yescharacteristic
140C4×C31C22×D411yescharacteristic
141Dic31—21no—
142Dic31—22no—
143Dic31—21no—
144#578—311no—
145#577—312no—
146#576—311no—
147#495D3111yescharacteristic, complemented
148#473—311no—
149#472—311no—
150#471—312no—
151#455—312no—
152#454—312no—
153#443—311no—
154#443—311no—
155C2×C31—21no—
156C2×C31—82no—
157C2×C31D4211yescharacteristic
158D31—41no—
159D31—42no—
160D31—41no—
1612−1+4—312no—
162H(5,𝔽2)—312no—
163#137—312no—
164#137—312no—
165#137—622no—
166#137—622no—
167AGL(1,ℤ/8)—312no—
168AGL(1,ℤ/8)—312no—
169AGL(1,ℤ/8)—622no—
170AGL(1,ℤ/8)—622no—
171#135—311no—
172#135—311no—
173#135—621no—
174#135—621no—
175#135—622no—
176C2×Q16—621no—
177C2×Q16C2×D3111yescharacteristic, complemented
178C2×SD16—622no—
179C2×SD16C2×D3112yescomplemented
180C2×D8—621no—
181C2×D8C2×D3111yescharacteristic
182#131—311no—
183#131—312no—
184#131—311no—
185#124C2×D3112yescomplemented
186#108—311no—
187#108—311no—
188#104—311no—
189#104—311no—
190#104—312no—
191#101—312no—
192#100—312no—
193C4×C8C2×D3111yescharacteristic, complemented
194C31#329011yescharacteristic, complemented, Sylow
195#41—311no—
196#41—312no—
197#41—311no—
198#41—622no—
199#41—622no—
200#41—622no—
201#41—622no—
202#41—1242no—
203C2×Q8—622no—
204C2×Q8—622no—
205C2×Q8C22×D3112yes—
206C2×D4—622no—
207C2×D4—622no—
208C2×D4—1242no—
209C2×D4C22×D3112yes—
210Q16—21no—
211Q16—21no—
212Q16—311no—
213Q16—311no—
214Q16—622no—
215Q16—622no—
216SD16—22no—
217SD16—22no—
218SD16—311no—
219SD16—311no—
220SD16—312no—
221SD16—622no—
222SD16—622no—
223SD16—622no—
224SD16—622no—
225D8—21no—
226D8—21no—
227D8—311no—
228D8—311no—
229D8—622no—
230D8—622no—
231M16—311no—
232M16—312no—
233M16—311no—
234M16—621no—
235M16—622no—
236M16—621no—
237C2×C8—621no—
238C2×C8—622no—
239C2×C8—621no—
240C2×C8C22×D3111yescharacteristic
241C2×C8C22×D3111yescharacteristic
242C22⋊C4—42no—
243C42C22×D3111yescharacteristic
244C23—22no—
245C23—1242no—
246Q8—22no—
247Q8—22no—
248Q8—311no—
249Q8—312no—
250Q8—311no—
251Q8—1242no—
252D4—22no—
253D4—22no—
254D4—311no—
255D4—312no—
256D4—311no—
257D4—621no—
258D4—622no—
259D4—621no—
260D4—1242no—
261D4—1242no—
262D4—1242no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—621no—
266C2×C4—622no—
267C2×C4—621no—
268C2×C4—1242no—
269C2×C4—1242no—
270C2×C4—1242no—
271C2×C4C23×D3111yescharacteristic
272C8—621no—
273C8—622no—
274C8—621no—
275C8D4×D3111yescharacteristic
276C8D4×D3111yescharacteristic
277C8D4×D3111yescharacteristic
278C8D4×D3111yescharacteristic
279C22—42no—
280C22—82no—
281C22—621no—
282C22—622no—
283C22—621no—
284C22—4962no—
285C22#1175856111yescharacteristic
286C4—41no—
287C4—42no—
288C4—621no—
289C4—622no—
290C4—621no—
291C4C2×D4×D3111yescharacteristic
292C4C2×D4×D3111yescharacteristic
293C2—21no—
294C2—82no—
295C2—1241no—
296C2—1242no—
297C2—1241no—
298C2#1452168411yescharacteristic
subgroupG/H
centerC2#14521684
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#495C2
socleC2×C31D42
hypercenter#495D31
hypocenterC31#3290
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #14521862C2semidirect
2. #14521855C2semidirect
3. #14521850C2semidirect
4. #14521321C2semidirect
5. #14521316C2semidirect
6. #14521309C2semidirect
7. #14521226C2semidirect
8. #14521221C2semidirect
9. #14521167C2semidirect
10. #14521167C2semidirect
11. C31×#495C2semidirect
12. #11758564C22semidirect
14. C2×Q16×C31C22semidirect
15. C2×SD16×C31C22semidirect
17. C31×#124C22semidirect
19. #11758519C22semidirect
21. #11758509C22semidirect
35. #11758478C22semidirect
36. #11758478C22semidirect
37. C4×C8×C31C22semidirect
38. #11758424C22semidirect
39. #11758421C22semidirect
40. #11758407C22semidirect
41. #11758407C22semidirect
43. #11758393C22semidirect
44. #11758393C22semidirect
147. #495D31semidirect
177. C2×Q16C2×D31semidirect
179. C2×SD16C2×D31semidirect
185. #124C2×D31semidirect
193. C4×C8C2×D31semidirect
194. C31#3290semidirect

Maximal subgroups

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

subgroupindex
#145218622
#145218552
#145218502
#145213212
#145213162
#145213092
#145212262
#145212212
#145211672
C31×#4952
#329031