Finite Group Databasebeta

← #27154876

Subgroups — #27154876

Order 12416.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#20261537C211yesmaximal, characteristic, complemented
2#20261536C211yesmaximal, characteristic, complemented
3#20261173C211yesmaximal, characteristic, complemented
4#20261172C212yesmaximal, complemented
5#20261169C212yesmaximal, complemented
6#20261166C212yesmaximal, complemented
7#20261165C211yesmaximal, characteristic, complemented
8C2×#15690751C211yesmaximal, characteristic, complemented
9D16×D97C212yesmaximal, complemented
10#20261153C211yesmaximal, characteristic, complemented
11#509×C97C211yesmaximal, characteristic, complemented
12#15690856—21no—
13#15690855—21no—
14#15690837—22no—
15#15690836—21no—
16#15690835—21no—
17#15690834—22no—
18#15690833C2212yescomplemented
19#15690832C2211yescharacteristic, complemented
20#15690829C2212yescomplemented
21C2×D776C2211yescharacteristic, complemented
22D8×D97C2212yescomplemented
23D8×D97C2212yescomplemented
24#15690816C2211yescharacteristic
25#15690756C2211yescharacteristic, complemented
26#15690755C2212yescomplemented
27#15690754C2211yescharacteristic, complemented
28#15690753C2211yescharacteristic, complemented
29D1552C2212yescomplemented
30#15690751C2211yescharacteristic, complemented
31#15690751C2212yescomplemented
32#15690751C2212yescomplemented
33#15690749C2212yescomplemented
34C2×#13095486C2211yescharacteristic, complemented
35C16×D97C2212yescomplemented
36#135×C97C2211yescharacteristic, complemented
37C2×D8×C97C2211yescharacteristic, complemented
38SD32×C97C2212yescomplemented
39D16×C97C2212yescomplemented
40M32×C97C2211yescharacteristic, complemented
41#13095512—22no—
42#13095511—21no—
43#13095511—41no—
44#13095511D411yescharacteristic
45#13095510—22no—
46#13095510—41no—
47C2×D388—21no—
48C2×D388—41no—
49C2×C22⋊D97—42no—
50D4×D97—22no—
51D4×D97—22no—
52D4×D97—42no—
53D4×D97—41no—
54Dic388C2311yescharacteristic
55#13095500—21no—
56#13095499—22no—
57#13095498—22no—
58D776—21no—
59D776C2311yescharacteristic
60D776C2312yes—
61#13095496—22no—
62#13095496—22no—
63#13095495D411yescharacteristic
64#13095494D412yes—
65C8×D97C2312yes—
66#13095486C2311yescharacteristic
67#13095486C2311yescharacteristic
68#41×C97—21no—
69C2×D4×C97—21no—
70Q16×C97C2311yescharacteristic
71SD16×C97—21no—
72D8×C97—21no—
73D8×C97C2311yescharacteristic
74D8×C97C2312yes—
75C2×C8×C97C2311yescharacteristic
76C16×C97C2312yes—
77C23×C97—41no—
78C22×D97—41no—
79C22×D97—42no—
80C22×D97—42no—
81Q8×C97—21no—
82D4×C97—22no—
83D4×C97—21no—
84D4×C97—41no—
85D4×C97—41no—
86C2×C4×C97—41no—
87C2×C4×C97C2×D411yescharacteristic
88C22⋊D97—41no—
89C22⋊D97—42no—
90C22⋊D97—42no—
91C22⋊D97—42no—
92C22⋊D97D812yes—
93C2×Dic97—42no—
94D388—21no—
95D388—22no—
96D388—42no—
97D388—41no—
98D388D811yescharacteristic
99C4×D97—41no—
100C4×D97—42no—
101C4×D97C2×D412yes—
102Dic194—21no—
103Dic194D811yescharacteristic
104C8×C97C2×D411yescharacteristic
105C8×C97C2411yescharacteristic
106#11715895C2×D412yes—
107C22×C97—42no—
108C22×C97—41no—
109C22×C97—161no—
110C22×C97C2×D811yescharacteristic
111C2×D97—42no—
112C2×D97—41no—
113C2×D97—162no—
114C2×D97—162no—
115C2×D97—161no—
116C2×D97C2×D812yes—
117C4×C97—41no—
118C4×C97C2×D811yescharacteristic
119C4×C97C22×D411yescharacteristic
120Dic97—41no—
121Dic97C2×D812yes—
122C2×C97—21no—
123C2×C97—81no—
124C2×C97—82no—
125C2×C97C22×D811yescharacteristic
126D97—22no—
127D97—82no—
128D97—81no—
129#3415—971nomaximal, Sylow
130C97#341511yescharacteristic, complemented, Sylow
131#576—971no—
132#576—971no—
133#509—972no—
134#509—972no—
135#509—971no—
136#509D9711yescharacteristic, complemented
137#508—971no—
138#508—972no—
139C2×D16—971no—
140C2×D16—972no—
141#504—971no—
142H(5,𝔽2)—1941no—
143H(5,𝔽2)—1941no—
144AGL(1,ℤ/8)—1942no—
145AGL(1,ℤ/8)—1942no—
146AGL(1,ℤ/8)—1941no—
147AGL(1,ℤ/8)—1941no—
148#135—972no—
149#135—972no—
150#135—971no—
151#135C2×D9711yescharacteristic, complemented
152C2×D8—972no—
153C2×D8—972no—
154C2×D8—971no—
155C2×D8C2×D9711yescharacteristic, complemented
156#131—971no—
157Q32—971no—
158SD32—971no—
159SD32—971no—
160SD32—972no—
161SD32C2×D9712yescomplemented
162D16—972no—
163D16—971no—
164D16—972no—
165D16—972no—
166D16C2×D9712yescomplemented
167M32—972no—
168M32C2×D9711yescharacteristic, complemented
169C2×C16—971no—
170C2×C16—972no—
171#41—21no—
172#41—971no—
173#41—3881no—
174#41—3881no—
175#41—1942no—
176#41—1942no—
177#41—1941no—
178C2×D4—21no—
179C2×D4—3881no—
180C2×D4—3882no—
181C2×D4—3881no—
182C2×D4—3882no—
183C2×D4—1942no—
184C2×D4—1942no—
185C2×D4—1941no—
186Q16—971no—
187Q16C22×D9711yescharacteristic
188SD16—21no—
189SD16—1942no—
190SD16—1942no—
191SD16—1941no—
192D8—21no—
193D8—971no—
194D8—972no—
195D8—1942no—
196D8—1942no—
197D8—1941no—
198D8C22×D9712yes—
199D8C22×D9711yescharacteristic
200M16—971no—
201M16—972no—
202C2×C8—972no—
203C2×C8C22×D9711yescharacteristic
204C16—971no—
205C16—971no—
206C16C22×D9712yes—
207C23—41no—
208C23—3882no—
209C23—3882no—
210C23—3881no—
211Q8—21no—
212Q8—971no—
213Q8—1941no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—972no—
219D4—971no—
220D4—3882no—
221D4—3882no—
222D4—3882no—
223D4—3882no—
224D4—3881no—
225D4—3881no—
226D4—1941no—
227D4—1942no—
228C2×C4—41no—
229C2×C4—972no—
230C2×C4—3882no—
231C2×C4—3882no—
232C2×C4—3881no—
233C2×C4D4×D9711yescharacteristic
234C8—972no—
235C8D4×D9711yescharacteristic
236C8C23×D9711yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—15521no—
241C22—15522no—
242C22—15522no—
243C22—972no—
244C22—3882no—
245C22—3881no—
246C22D8×D9711yescharacteristic
247C4—41no—
248C4—972no—
249C4—3881no—
250C4D8×D9711yescharacteristic
251C4C2×D4×D9711yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—7762no—
256C2—7761no—
257C2—1942no—
258C2C2×D8×D9711yescharacteristic
subgroupG/H
centerC2C2×D8×D97
commutator, cosocleC8×C97C24
FrattiniC8C23×D97
Fitting#509×C97C2
socleC2×C97C22×D8
hypercenter#509D97
hypocenterC97#3415
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20261537C2semidirect
2. #20261536C2semidirect
3. #20261173C2semidirect
4. #20261172C2semidirect
5. #20261169C2semidirect
6. #20261166C2semidirect
7. #20261165C2semidirect
8. C2×#15690751C2semidirect
9. D16×D97C2semidirect
10. #20261153C2semidirect
11. #509×C97C2semidirect
18. #15690833C22semidirect
19. #15690832C22semidirect
20. #15690829C22semidirect
21. C2×D776C22semidirect
22. D8×D97C22semidirect
23. D8×D97C22semidirect
25. #15690756C22semidirect
26. #15690755C22semidirect
27. #15690754C22semidirect
28. #15690753C22semidirect
29. D1552C22semidirect
30. #15690751C22semidirect
31. #15690751C22semidirect
32. #15690751C22semidirect
33. #15690749C22semidirect
34. C2×#13095486C22semidirect
35. C16×D97C22semidirect
36. #135×C97C22semidirect
37. C2×D8×C97C22semidirect
38. SD32×C97C22semidirect
39. D16×C97C22semidirect
40. M32×C97C22semidirect
130. C97#3415semidirect
136. #509D97semidirect
151. #135C2×D97semidirect
155. C2×D8C2×D97semidirect
161. SD32C2×D97semidirect
166. D16C2×D97semidirect
168. M32C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615372
#202615362
#202611732
#202611722
#202611692
#202611662
#202611652
C2×#156907512
D16×D972
#202611532
#509×C972
#341597