Finite Group Databasebeta

← #16857509

Subgroups — #16857509

Order 3968.

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

normal subgroups only

This group has 476 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#14521854C211yesmaximal, characteristic, complemented
2#14521853C212yesmaximal, complemented
3#14521850C211yesmaximal, characteristic, complemented
4#14521424C211yesmaximal, characteristic, complemented
5#14521320C211yesmaximal, characteristic, complemented
6#14521318C212yesmaximal, complemented
7#14521219C212yesmaximal, complemented
8#14521215C212yesmaximal, complemented
9#14521165C211yesmaximal, characteristic, complemented
10#14521164C211yesmaximal, characteristic, complemented
11C31×#472C211yesmaximal, characteristic, complemented
12#11758566C2212yescomplemented
13#11758561C2212yescomplemented
14C31×AGL(1,ℤ/8)C2212yescomplemented
15C2×D8×C31C2211yescharacteristic, complemented
16C31×#131C2211yescharacteristic, complemented
17#11758527C2211yescharacteristic
18#11758517C2212yescomplemented
19#11758509C2212yescomplemented
20C2×#92479C2212yescomplemented
21#11758493—22no—
22#11758491—22no—
23#11758489—21no—
24#11758488—22no—
25#11758488—22no—
26D8×D31—22no—
27D8×D31—21no—
28#11758485—22no—
29#11758485C2212yescomplemented
30#11758485C2212yescomplemented
31#11758484C2212yescomplemented
32#11758481—21no—
33#11758481C2211yescharacteristic
34C2×D248—21no—
35C2×D248C2211yescharacteristic, complemented
36#11758478C2211yescharacteristic, complemented
37C31×#108C2211yescharacteristic, complemented
38C31×#100C2212yescomplemented
39#11758425C2211yescharacteristic, complemented
40#11758413C2211yescharacteristic, complemented
41#11758413C2212yescomplemented
42#11758411C2212yescomplemented
43#11758410C2211yescharacteristic, complemented
44C8×Dic31C2211yescharacteristic, complemented
45#41×C31C2311yescharacteristic
46C2×D4×C31C2312yes—
47C2×C22⋊D31—42no—
48C2×C22⋊D31—42no—
49#92499—22no—
50#92497—22no—
51#92497—22no—
52D4×D31—22no—
53D4×D31—22no—
54D4×D31—22no—
55D4×D31—42no—
56#92495—22no—
57#92495C2311yescharacteristic
58#92495C2312yes—
59C2×D124—22no—
60C2×D124C2312yes—
61SD16×C31—22no—
62D8×C31—22no—
63D8×C31—21no—
64D8×C31—21no—
65M16×C31—21no—
66M16×C31C2311yescharacteristic
67M16×C31C2312yes—
68C2×C8×C31—21no—
69C2×C8×C31C2311yescharacteristic
70#92482D411yescharacteristic, complemented
71#92481D411yescharacteristic, complemented
72#92480—22no—
73#92480D411yescharacteristic, complemented
74#92479—22no—
75#92479—22no—
76#92479—22no—
77#92479D411yescharacteristic, complemented
78C4×Dic31C2311yescharacteristic
79#92474C2311yescharacteristic
80C2×C31⋊C8C2311yescharacteristic
81Dic124D411yescharacteristic, complemented
82D248—22no—
83D248—22no—
84D248—21no—
85D248—21no—
86D248D411yescharacteristic, complemented
87#92470—22no—
88#92470—22no—
89#92470D412yescomplemented
90#92469—21no—
91#92469—22no—
92C8×D31—21no—
93C8×D31—22no—
94C23×C31—22no—
95C22×D31—22no—
96C22×D31—42no—
97C22×D31—42no—
98Q8×C31C2×D411yescharacteristic
99D4×C31—21no—
100D4×C31—22no—
101D4×C31—22no—
102D4×C31C2×D411yescharacteristic
103C2×C4×C31—21no—
104C2×C4×C31C2411yescharacteristic
105C22⋊D31—22no—
106C22⋊D31—21no—
107C22⋊D31—42no—
108C22⋊D31—42no—
109C22⋊D31—42no—
110C22⋊D31—84no—
111C2×Dic31—21no—
112C2×Dic31—42no—
113D124—22no—
114D124—22no—
115D124—42no—
116D124C2×D412yes—
117D124C2×D411yescharacteristic
118C4×D31—21no—
119C4×D31—22no—
120C4×D31—42no—
121Dic62C2×D411yescharacteristic
122Dic62C2×D412yes—
123C8×C31—22no—
124C8×C31—21no—
125C8×C31C2×D411yescharacteristic
126C8×C31C2×D411yescharacteristic
127C31⋊C8—21no—
128C31⋊C8C2×D411yescharacteristic
129C31⋊C8C2×D411yescharacteristic
130#3292—311nomaximal, Sylow
131C22×C31—21no—
132C22×C31—42no—
133C22×C31—82no—
134C22×C31H(5,𝔽2)11yescharacteristic
135C2×D31—22no—
136C2×D31—21no—
137C2×D31—42no—
138C2×D31—82no—
139C2×D31—162no—
140C2×D31—162no—
141C4×C31—21no—
142C4×C31C22×D411yescharacteristic
143C4×C31C22×D411yescharacteristic
144Dic31—22no—
145Dic31—22no—
146Dic31—21no—
147#576—312no—
148#576—311no—
149#576—311no—
150#493—311no—
151#472—311no—
152#472—312no—
153#472D3111yescharacteristic, complemented
154#453—312no—
155#453—312no—
156#443—311no—
157#443—311no—
158C2×C31—21no—
159C2×C31—41no—
160C2×C31—82no—
161C2×C31D4211yescharacteristic
162D31—42no—
163D31—41no—
164D31—82no—
165H(5,𝔽2)—312no—
166H(5,𝔽2)—312no—
167AGL(1,ℤ/8)—312no—
168AGL(1,ℤ/8)—312no—
169AGL(1,ℤ/8)—312no—
170AGL(1,ℤ/8)—622no—
171AGL(1,ℤ/8)—622no—
172AGL(1,ℤ/8)—622no—
173AGL(1,ℤ/8)—622no—
174AGL(1,ℤ/8)C2×D3112yescomplemented
175#135—311no—
176#135—311no—
177#135—621no—
178#135—621no—
179#135—622no—
180C2×D8—312no—
181C2×D8—311no—
182C2×D8—621no—
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#108C2×D3111yescharacteristic, complemented
192#104—311no—
193#104—312no—
194#104—311no—
195#100—312no—
196#100C2×D3112yescomplemented
197C4×C8—311no—
198C31#329211yescharacteristic, complemented, Sylow
199#41—311no—
200#41—312no—
201#41—622no—
202#41—622no—
203#41—622no—
204#41—622no—
205#41C22×D3111yescharacteristic
206C2×D4—312no—
207C2×D4—622no—
208C2×D4—622no—
209C2×D4—622no—
210C2×D4—622no—
211C2×D4—1242no—
212C2×D4—1242no—
213C2×D4—1242no—
214C2×D4C22×D3112yes—
215Q16—311no—
216Q16—311no—
217SD16—22no—
218SD16—311no—
219SD16—311no—
220SD16—312no—
221SD16—622no—
222SD16—622no—
223SD16—622no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—311no—
228D8—311no—
229D8—622no—
230D8—622no—
231D8—622no—
232D8—622no—
233D8—622no—
234D8—621no—
235D8—621no—
236M16—21no—
237M16—311no—
238M16—622no—
239M16—621no—
240M16C22×D3112yes—
241M16C22×D3111yescharacteristic
242C2×C8—21no—
243C2×C8—311no—
244C2×C8—622no—
245C2×C8—621no—
246C2×C8C22×D3111yescharacteristic
247C42—311no—
248C23—22no—
249C23—622no—
250C23—1242no—
251C23—1242no—
252Q8—311no—
253Q8—312no—
254Q8D4×D3111yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—311no—
259D4—312no—
260D4—622no—
261D4—621no—
262D4—622no—
263D4—622no—
264D4—1242no—
265D4—1242no—
266D4—1242no—
267D4—1242no—
268D4—2484no—
269D4D4×D3111yescharacteristic
270C2×C4—21no—
271C2×C4—621no—
272C2×C4—622no—
273C2×C4—621no—
274C2×C4—1242no—
275C2×C4—1242no—
276C2×C4C23×D3111yescharacteristic
277C8—21no—
278C8—22no—
279C8—311no—
280C8—311no—
281C8—621no—
282C8D4×D3111yescharacteristic
283C8D4×D3111yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—622no—
288C22—621no—
289C22—1242no—
290C22—4962no—
291C22—4962no—
292C22—2482no—
293C22#1175856611yescharacteristic
294C4—21no—
295C4—622no—
296C4—622no—
297C4—621no—
298C4C2×D4×D3111yescharacteristic
299C4C2×D4×D3111yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—1242no—
304C2—1241no—
305C2—2482no—
306C2#1452168311yescharacteristic
subgroupG/H
centerC2#14521683
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#472C2
socleC2×C31D42
hypercenter#472D31
hypocenterC31#3292
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #14521854C2semidirect
2. #14521853C2semidirect
3. #14521850C2semidirect
4. #14521424C2semidirect
5. #14521320C2semidirect
6. #14521318C2semidirect
7. #14521219C2semidirect
8. #14521215C2semidirect
9. #14521165C2semidirect
10. #14521164C2semidirect
11. C31×#472C2semidirect
12. #11758566C22semidirect
13. #11758561C22semidirect
14. C31×AGL(1,ℤ/8)C22semidirect
15. C2×D8×C31C22semidirect
16. C31×#131C22semidirect
18. #11758517C22semidirect
19. #11758509C22semidirect
20. C2×#92479C22semidirect
29. #11758485C22semidirect
30. #11758485C22semidirect
31. #11758484C22semidirect
35. C2×D248C22semidirect
36. #11758478C22semidirect
37. C31×#108C22semidirect
38. C31×#100C22semidirect
39. #11758425C22semidirect
40. #11758413C22semidirect
41. #11758413C22semidirect
42. #11758411C22semidirect
43. #11758410C22semidirect
44. C8×Dic31C22semidirect
70. #92482D4semidirect
71. #92481D4semidirect
73. #92480D4semidirect
77. #92479D4semidirect
81. Dic124D4semidirect
86. D248D4semidirect
89. #92470D4semidirect
153. #472D31semidirect
174. AGL(1,ℤ/8)C2×D31semidirect
185. C2×D8C2×D31semidirect
188. #131C2×D31semidirect
191. #108C2×D31semidirect
196. #100C2×D31semidirect
198. C31#3292semidirect

Maximal subgroups

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

subgroupindex
#145218542
#145218532
#145218502
#145214242
#145213202
#145213182
#145212192
#145212152
#145211652
#145211642
C31×#4722
#329231