Finite Group Databasebeta

← #27153526

Subgroups — #27153526

Order 12416.

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

normal subgroups only

This group has 460 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#20261544C212yesmaximal, complemented
2#20261543C212yesmaximal, complemented
3#20261109C211yesmaximal, characteristic, complemented
4#20261001C212yesmaximal, complemented
5#20261000C212yesmaximal, complemented
6#20260917C211yesmaximal, characteristic, complemented
7#20260904C212yesmaximal, complemented
8#20260849C212yesmaximal, complemented
9#456×C97C211yesmaximal, characteristic, complemented
10#15690858C2211yescharacteristic
11#15690856C2212yescomplemented
12#15690842—22no—
13#15690841—22no—
14#15690841C2212yescomplemented
15#15690838C2212yes—
16#15690837—22no—
17#15690835—22no—
18#15690835C2212yes—
19#15690833—22no—
20#15690830—22no—
21#15690830C2212yes—
22C2×#13095498—22no—
23C2×#13095498C2212yescomplemented
24C2×#13095498C2212yescomplemented
25SD16×D97—22no—
26#15690818C2212yescomplemented
27#15690817C2212yescomplemented
28#15690807C2211yescharacteristic
29#15690745C2212yescomplemented
30#15690738C2212yescomplemented
31#15690729C2211yescharacteristic, complemented
32#15690726C2212yescomplemented
33C4×#11715895C2211yescharacteristic, complemented
342−1+4×C97C2211yescharacteristic
35#137×C97C2212yescomplemented
36C4⋊Q8×C97C2211yescharacteristic, complemented
37#104×C97C2212yescomplemented
38#101×C97C2211yescharacteristic, complemented
39#13095512—22no—
40#13095512—22no—
41#13095511—22no—
42#13095511—41no—
43#13095511C2312yes—
44Q8×D97—22no—
45C2×D388—22no—
46C2×D388—41no—
47C2×D388C2312yes—
48D4×D97—22no—
49D4×D97—42no—
50#13095501—22no—
51#13095501D412yescomplemented
52#13095500—22no—
53#13095499D412yescomplemented
54#13095498—22no—
55#13095498—22no—
56#13095498—22no—
57#13095498—22no—
58#13095498—22no—
59#13095498—22no—
60#13095498D412yescomplemented
61D776—22no—
62#13095496—22no—
63#13095496D412yescomplemented
64#13095495—22no—
65#13095495C2312yes—
66#13095494—22no—
67C2×#11715895—22no—
68C2×#11715895C2312yes—
69C8×D97—22no—
70#41×C97—22no—
71#41×C97—41no—
72#41×C97C2312yes—
73C2×Q8×C97—22no—
74C2×Q8×C97C2311yescharacteristic
75C2×Q8×C97C2311yescharacteristic
76Q16×C97—22no—
77SD16×C97—22no—
78M16×C97C2312yes—
79C4⋊C4×C97—41no—
80C42×C97C2311yescharacteristic
81C22×D97—22no—
82C22×D97—42no—
83Q8×C97—22no—
84Q8×C97—22no—
85Q8×C97—41no—
86Q8×C97C2×D412yes—
87D4×C97—22no—
88D4×C97—41no—
89D4×C97C2×D412yes—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97—21no—
93C2×C4×C97—21no—
94C2×C4×C97—42no—
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—22no—
97C22⋊D97—42no—
98D388—22no—
99D388—22no—
100D388—42no—
101D388—41no—
102D388—82no—
103D388C2×D412yes—
104C4×D97—22no—
105C4×D97—42no—
106C4×D97—42no—
107Dic194—41no—
108Dic194C2×D412yes—
109C8×C97—22no—
110#11715895—22no—
111#11715895C2×D412yes—
112#11715895C2×D412yes—
113C22×C97—22no—
114C22×C97H(5,𝔽2)11yescharacteristic
115C2×D97—22no—
116C2×D97—42no—
117C2×D97—82no—
118C2×D97—162no—
119C4×C97—22no—
120C4×C97—22no—
121C4×C97—41no—
122C4×C97—41no—
123C4×C97C22×D412yes—
124Dic97—22no—
125C2×C97—21no—
126C2×C97—42no—
127C2×C97D4211yescharacteristic
128D97—42no—
129D97—82no—
130#3291—971nomaximal, Sylow
131C97#329111yescharacteristic, complemented, Sylow
132#577—972no—
133#577—972no—
134#492—971no—
135#471—972no—
136#471—972no—
137#456—971no—
138#456D9711yescharacteristic, complemented
139#453—972no—
140#443—972no—
1412−1+4—971no—
1422−1+4C2×D9711yescharacteristic
143H(5,𝔽2)—972no—
144#137—972no—
145#137—1942no—
146#137—1942no—
147#137C2×D9712yescomplemented
148AGL(1,ℤ/8)—972no—
149AGL(1,ℤ/8)—972no—
150AGL(1,ℤ/8)—1942no—
151AGL(1,ℤ/8)—1942no—
152#135—972no—
153#135—1942no—
154#135—1942no—
155C2×SD16—972no—
156C2×SD16—972no—
157C2×SD16—1942no—
158C2×SD16—1942no—
159#131—972no—
160#131—972no—
161C4⋊Q8C2×D9711yescharacteristic, complemented
162#127—971no—
163#108—972no—
164#104—972no—
165#104C2×D9712yescomplemented
166#101—971no—
167#101C2×D9711yescharacteristic, complemented
168#100—972no—
169C4×C8—971no—
170#41—22no—
171#41—41no—
172#41—972no—
173#41—3881no—
174#41—1942no—
175#41—1942no—
176#41—1942no—
177#41C22×D9712yes—
178C2×Q8—22no—
179C2×Q8—1942no—
180C2×Q8C22×D9711yescharacteristic
181C2×Q8C22×D9711yescharacteristic
182C2×D4—972no—
183C2×D4—3881no—
184C2×D4—3882no—
185C2×D4—1942no—
186C2×D4—1942no—
187Q16—22no—
188Q16—972no—
189Q16—1942no—
190SD16—22no—
191SD16—972no—
192SD16—972no—
193SD16—1942no—
194SD16—1942no—
195SD16—1942no—
196SD16—1942no—
197SD16—1942no—
198SD16—1942no—
199SD16—1942no—
200D8—972no—
201D8—1942no—
202D8—1942no—
203M16—972no—
204M16—1942no—
205M16—1942no—
206M16C22×D9712yes—
207C2×C8—972no—
208C2×C8—1942no—
209C2×C8—1942no—
210C4⋊C4—41no—
211C42C22×D9711yescharacteristic
212C23—3882no—
213C23—1942no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—972no—
218Q8—3881no—
219Q8D4×D9712yes—
220D4—22no—
221D4—41no—
222D4—7762no—
223D4—972no—
224D4—3882no—
225D4—3881no—
226D4—3882no—
227D4—1942no—
228D4—1942no—
229D4—1942no—
230D4D4×D9712yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—3882no—
237C2×C4—3882no—
238C2×C4—1942no—
239C2×C4C23×D9711yescharacteristic
240C8—22no—
241C8—972no—
242C8—972no—
243C8—1942no—
244C22—22no—
245C22—7762no—
246C22—15522no—
247C22—3882no—
248C22—1942no—
249C22#1569085511yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—1942no—
255C4C2×D4×D9712yes—
256C2—21no—
257C2—42no—
258C2—7762no—
259C2—3882no—
260C2#2026136711yescharacteristic
subgroupG/H
centerC2#20261367
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#456×C97C2
socleC2×C97D42
hypercenter#456D97
hypocenterC97#3291
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #20261544C2semidirect
2. #20261543C2semidirect
3. #20261109C2semidirect
4. #20261001C2semidirect
5. #20261000C2semidirect
6. #20260917C2semidirect
7. #20260904C2semidirect
8. #20260849C2semidirect
9. #456×C97C2semidirect
11. #15690856C22semidirect
14. #15690841C22semidirect
23. C2×#13095498C22semidirect
24. C2×#13095498C22semidirect
26. #15690818C22semidirect
27. #15690817C22semidirect
29. #15690745C22semidirect
30. #15690738C22semidirect
31. #15690729C22semidirect
32. #15690726C22semidirect
33. C4×#11715895C22semidirect
35. #137×C97C22semidirect
36. C4⋊Q8×C97C22semidirect
37. #104×C97C22semidirect
38. #101×C97C22semidirect
51. #13095501D4semidirect
53. #13095499D4semidirect
60. #13095498D4semidirect
63. #13095496D4semidirect
131. C97#3291semidirect
138. #456D97semidirect
147. #137C2×D97semidirect
161. C4⋊Q8C2×D97semidirect
165. #104C2×D97semidirect
167. #101C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615442
#202615432
#202611092
#202610012
#202610002
#202609172
#202609042
#202608492
#456×C972
#329197