Finite Group Databasebeta

← #16857508

Subgroups — #16857508

Order 3968.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#14521852C212yesmaximal, complemented
2#14521851C211yesmaximal, characteristic, complemented
3#14521424C211yesmaximal, characteristic, complemented
4#14521319C211yesmaximal, characteristic, complemented
5#14521317C212yesmaximal, complemented
6#14521316C211yesmaximal, characteristic, complemented
7#14521217C212yesmaximal, complemented
8#14521215C212yesmaximal, complemented
9#14521166C211yesmaximal, characteristic, complemented
10#14521164C211yesmaximal, characteristic, complemented
11C31×#576C211yesmaximal, characteristic, complemented
12C31×H(5,𝔽2)C2212yescomplemented
13#11758566C2212yescomplemented
14C31×AGL(1,ℤ/8)—22no—
15C31×AGL(1,ℤ/8)C2212yescomplemented
16C31×#135—21no—
17C31×#135C2211yescharacteristic
18C2×D8×C31—21no—
19C2×D8×C31C2211yescharacteristic, complemented
20C31×#131C2211yescharacteristic, complemented
21#11758527—22no—
22#11758527C2211yescharacteristic
23#11758526—22no—
24#11758526C2212yescomplemented
25#11758526C2212yescomplemented
26#11758525C2212yescomplemented
27#11758517C2212yescomplemented
28#11758509—22no—
29#11758509C2212yescomplemented
30C2×#92479—22no—
31C2×#92479C2212yescomplemented
32#11758496—21no—
33#11758491—22no—
34D8×D31—21no—
35C2×D248C2211yescharacteristic, complemented
36#11758478C2211yescharacteristic, complemented
37#11758425C2211yescharacteristic, complemented
38#11758425C2212yescomplemented
39#11758421C2212yescomplemented
40#11758413C2211yescharacteristic, complemented
41#11758411C2212yescomplemented
42#11758410C2211yescharacteristic, complemented
43#11758407C2211yescharacteristic, complemented
44C8×Dic31C2211yescharacteristic, complemented
45#41×C31—22no—
46#41×C31—22no—
47#41×C31C2312yes—
48#41×C31C2311yescharacteristic
49C2×D4×C31—22no—
50C2×D4×C31—22no—
51C2×D4×C31—42no—
52C2×D4×C31C2312yes—
53C2×C22⋊D31—42no—
54#92499—22no—
55D4×D31—22no—
56D4×D31—42no—
57#92495—22no—
58#92495C2311yescharacteristic
59C2×D124—22no—
60C2×D124C2312yes—
61Q16×C31D411yescharacteristic, complemented
62SD16×C31—22no—
63SD16×C31D412yescomplemented
64D8×C31—22no—
65D8×C31—21no—
66D8×C31—21no—
67D8×C31D411yescharacteristic, complemented
68M16×C31—21no—
69M16×C31C2311yescharacteristic
70C2×C8×C31—21no—
71C2×C8×C31C2311yescharacteristic
72#92482D411yescharacteristic, complemented
73#92481—22no—
74#92481—22no—
75#92481D411yescharacteristic, complemented
76#92480—22no—
77#92480D411yescharacteristic, complemented
78#92479—22no—
79#92479—22no—
80#92479—22no—
81#92479—22no—
82#92479—22no—
83#92479D411yescharacteristic, complemented
84C4×Dic31C2311yescharacteristic
85#92474—22no—
86#92474C2312yes—
87#92474C2311yescharacteristic
88C2×C31⋊C8—22no—
89C2×C31⋊C8C2311yescharacteristic
90D248—21no—
91D248—21no—
92#92469—21no—
93C8×D31—21no—
94C23×C31—22no—
95C23×C31—42no—
96C22×D31—22no—
97C22×D31—42no—
98Q8×C31C2×D411yescharacteristic
99Q8×C31C2×D412yes—
100D4×C31—22no—
101D4×C31—21no—
102D4×C31—22no—
103D4×C31—22no—
104D4×C31—42no—
105D4×C31—42no—
106D4×C31C2×D411yescharacteristic
107D4×C31C2×D412yes—
108C2×C4×C31—22no—
109C2×C4×C31—21no—
110C2×C4×C31—42no—
111C2×C4×C31C2411yescharacteristic
112C22⋊D31—21no—
113C22⋊D31—42no—
114C22⋊D31—84no—
115C2×Dic31—21no—
116D124—22no—
117D124—22no—
118D124—42no—
119D124C2×D411yescharacteristic
120C4×D31—21no—
121C4×D31—42no—
122Dic62C2×D411yescharacteristic
123C8×C31—21no—
124C8×C31C2×D411yescharacteristic
125C8×C31C2×D411yescharacteristic
126C31⋊C8—21no—
127C31⋊C8—22no—
128C31⋊C8C2×D411yescharacteristic
129C31⋊C8C2×D411yescharacteristic
130#3292—311nomaximal, Sylow
131C22×C31—22no—
132C22×C31—21no—
133C22×C31—42no—
134C22×C31—82no—
135C22×C31—162no—
136C22×C31H(5,𝔽2)11yescharacteristic
137C2×D31—21no—
138C2×D31—42no—
139C2×D31—82no—
140C2×D31—162no—
141C4×C31—22no—
142C4×C31—21no—
143C4×C31C22×D411yescharacteristic
144C4×C31C22×D411yescharacteristic
145Dic31—22no—
146Dic31—21no—
147#576—312no—
148#576—311no—
149#576D3111yescharacteristic, complemented
150#493—311no—
151#472—312no—
152#472—311no—
153#472—311no—
154#453—312no—
155#453—312no—
156#443—311no—
157#443—311no—
158C2×C31—21no—
159C2×C31—42no—
160C2×C31—41no—
161C2×C31—82no—
162C2×C31D4211yescharacteristic
163D31—41no—
164D31—82no—
165H(5,𝔽2)—312no—
166H(5,𝔽2)C2×D3112yescomplemented
167AGL(1,ℤ/8)—22no—
168AGL(1,ℤ/8)—312no—
169AGL(1,ℤ/8)—312no—
170AGL(1,ℤ/8)—312no—
171AGL(1,ℤ/8)—622no—
172AGL(1,ℤ/8)—622no—
173AGL(1,ℤ/8)—622no—
174AGL(1,ℤ/8)C2×D3112yescomplemented
175#135—21no—
176#135—311no—
177#135—622no—
178#135—621no—
179#135C2×D3111yescharacteristic
180C2×D8—21no—
181C2×D8—312no—
182C2×D8—311no—
183C2×D8—622no—
184C2×D8—621no—
185C2×D8C2×D3111yescharacteristic, complemented
186#131—312no—
187#131—311no—
188#131C2×D3111yescharacteristic, complemented
189#127—312no—
190#108—311no—
191#108—311no—
192#104—311no—
193#104—312no—
194#104—311no—
195#100—312no—
196#100—312no—
197C4×C8—311no—
198C31#329211yescharacteristic, complemented, Sylow
199#41—22no—
200#41—22no—
201#41—311no—
202#41—622no—
203#41—622no—
204#41C22×D3111yescharacteristic
205#41C22×D3112yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—312no—
210C2×D4—622no—
211C2×D4—622no—
212C2×D4—1242no—
213C2×D4—1242no—
214C2×D4C22×D3112yes—
215Q16—311no—
216Q16C22⋊D3111yescharacteristic, complemented
217SD16—22no—
218SD16—311no—
219SD16—311no—
220SD16—622no—
221SD16—622no—
222SD16—622no—
223SD16C22⋊D3112yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—311no—
228D8—622no—
229D8—622no—
230D8—622no—
231D8—622no—
232D8—622no—
233D8—621no—
234D8—621no—
235D8C22⋊D3111yescharacteristic, complemented
236M16—21no—
237M16—312no—
238M16—311no—
239M16—622no—
240M16—621no—
241M16C22×D3111yescharacteristic
242C2×C8—21no—
243C2×C8—311no—
244C2×C8—622no—
245C2×C8—621no—
246C2×C8C22×D3111yescharacteristic
247C42—311no—
248C23—22no—
249C23—42no—
250C23—622no—
251C23—1242no—
252Q8—311no—
253Q8D4×D3111yescharacteristic
254Q8C2×C22⋊D3112yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—311no—
262D4—621no—
263D4—622no—
264D4—622no—
265D4—1242no—
266D4—1242no—
267D4—2484no—
268D4D4×D3111yescharacteristic
269D4C2×C22⋊D3112yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—621no—
274C2×C4—621no—
275C2×C4—1242no—
276C2×C4C23×D3111yescharacteristic
277C8—21no—
278C8—311no—
279C8—311no—
280C8—622no—
281C8—621no—
282C8C2×C22⋊D3111yescharacteristic
283C8C2×C22⋊D3111yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—621no—
290C22—1242no—
291C22—4962no—
292C22—2482no—
293C22#1175856111yescharacteristic
294C4—21no—
295C4—22no—
296C4—622no—
297C4—621no—
298C4C2×D4×D3111yescharacteristic
299C4C22×C22⋊D3111yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—1241no—
305C2—2482no—
306C2D4×C22⋊D3111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#576C2
socleC2×C31D42
hypercenter#576D31
hypocenterC31#3292
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #14521852C2semidirect
2. #14521851C2semidirect
3. #14521424C2semidirect
4. #14521319C2semidirect
5. #14521317C2semidirect
6. #14521316C2semidirect
7. #14521217C2semidirect
8. #14521215C2semidirect
9. #14521166C2semidirect
10. #14521164C2semidirect
11. C31×#576C2semidirect
12. C31×H(5,𝔽2)C22semidirect
13. #11758566C22semidirect
15. C31×AGL(1,ℤ/8)C22semidirect
19. C2×D8×C31C22semidirect
20. C31×#131C22semidirect
24. #11758526C22semidirect
25. #11758526C22semidirect
26. #11758525C22semidirect
27. #11758517C22semidirect
29. #11758509C22semidirect
31. C2×#92479C22semidirect
35. C2×D248C22semidirect
36. #11758478C22semidirect
37. #11758425C22semidirect
38. #11758425C22semidirect
39. #11758421C22semidirect
40. #11758413C22semidirect
41. #11758411C22semidirect
42. #11758410C22semidirect
43. #11758407C22semidirect
44. C8×Dic31C22semidirect
61. Q16×C31D4semidirect
63. SD16×C31D4semidirect
67. D8×C31D4semidirect
72. #92482D4semidirect
75. #92481D4semidirect
77. #92480D4semidirect
83. #92479D4semidirect
149. #576D31semidirect
166. H(5,𝔽2)C2×D31semidirect
174. AGL(1,ℤ/8)C2×D31semidirect
185. C2×D8C2×D31semidirect
188. #131C2×D31semidirect
198. C31#3292semidirect
216. Q16C22⋊D31semidirect
223. SD16C22⋊D31semidirect
235. D8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145218522
#145218512
#145214242
#145213192
#145213172
#145213162
#145212172
#145212152
#145211662
#145211642
C31×#5762
#329231