Finite Group Databasebeta

← #27153530

Subgroups — #27153530

Order 12416.

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

normal subgroups only

This group has 460 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#20261547C212yesmaximal, complemented
2#20261542C212yesmaximal, complemented
3#20261108C211yesmaximal, characteristic, complemented
4#20260999C212yesmaximal, complemented
5#20260998C212yesmaximal, complemented
6#20260919C212yesmaximal, complemented
7#20260904C212yesmaximal, complemented
8#20260853C211yesmaximal, characteristic, complemented
9#443×C97C211yesmaximal, characteristic, complemented
10#15690857C2212yes—
11#15690856C2212yescomplemented
12#15690843—22no—
13#15690843C2212yescomplemented
14#15690842—22no—
15#15690840C2212yescomplemented
16#15690838—22no—
17#15690838C2212yes—
18#15690835C2212yes—
19#15690834—22no—
20#15690832—22no—
21#15690832C2211yescharacteristic
22#15690831—22no—
23C2×#13095500—22no—
24C2×#13095500C2212yescomplemented
25C2×#13095500C2212yescomplemented
26SD16×D97—22no—
27#15690816C2212yescomplemented
28#15690810C2211yescharacteristic, complemented
29#15690807C2211yescharacteristic
30#15690746C2211yescharacteristic, complemented
31#15690738C2212yescomplemented
32#15690728C2212yescomplemented
33#15690726C2212yescomplemented
34#135×C97C2211yescharacteristic
35#131×C97C2212yescomplemented
36#108×C97C2211yescharacteristic, complemented
37#104×C97C2212yescomplemented
38C4×C8×C97C2211yescharacteristic, complemented
39#13095512—22no—
40#13095511—22no—
41#13095511—22no—
42#13095511C2312yes—
43#13095510—22no—
44#13095510—42no—
45C2×Dic194—22no—
46C2×Dic194C2312yes—
47Q8×D97—22no—
48C2×D388—22no—
49C2×D388—41no—
50C2×D388C2312yes—
51D4×D97—22no—
52D4×D97—42no—
53Dic388—22no—
54Dic388D411yescharacteristic, complemented
55#13095501—22no—
56#13095500—22no—
57#13095500—22no—
58#13095500—22no—
59#13095500—22no—
60#13095500—22no—
61#13095500—22no—
62#13095500D412yescomplemented
63#13095499—22no—
64#13095498—22no—
65D776—22no—
66D776D411yescharacteristic, complemented
67#13095496—22no—
68#13095495C2312yes—
69#13095494—22no—
70C8×D97—22no—
71C4⋊Dic97—41no—
72#41×C97C2312yes—
73Q16×C97D411yescharacteristic, complemented
74SD16×C97D412yescomplemented
75D8×C97D411yescharacteristic, complemented
76M16×C97—22no—
77M16×C97C2312yes—
78C2×C8×C97—22no—
79C2×C8×C97C2311yescharacteristic
80C2×C8×C97C2311yescharacteristic
81C42×C97C2311yescharacteristic
82C22×D97—22no—
83C22×D97—42no—
84Q8×C97C2×D412yes—
85D4×C97—22no—
86D4×C97C2×D412yes—
87C2×C4×C97—21no—
88C2×C4×C97—22no—
89C2×C4×C97C2411yescharacteristic
90C22⋊D97—22no—
91C22⋊D97—42no—
92C22⋊D97—42no—
93C2×Dic97—22no—
94C2×Dic97—42no—
95D388—22no—
96D388—22no—
97D388—42no—
98D388—82no—
99D388C2×D412yes—
100C4×D97—22no—
101C4×D97—42no—
102C4×D97—42no—
103Dic194—22no—
104Dic194—22no—
105Dic194—42no—
106Dic194C2×D412yes—
107C8×C97—22no—
108C8×C97C2×D411yescharacteristic
109C8×C97C2×D411yescharacteristic
110C8×C97C2×D411yescharacteristic
111C8×C97C2×D411yescharacteristic
112#11715895—22no—
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×C97C22×D411yescharacteristic
122C4×C97C22×D411yescharacteristic
123Dic97—22no—
124Dic97—42no—
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—972no—
138#453—972no—
139#443—971no—
140#443D9711yescharacteristic, complemented
1412−1+4—972no—
142H(5,𝔽2)—972no—
143#137—972no—
144#137—972no—
145#137—1942no—
146#137—1942no—
147AGL(1,ℤ/8)—972no—
148AGL(1,ℤ/8)—972no—
149AGL(1,ℤ/8)—1942no—
150AGL(1,ℤ/8)—1942no—
151#135—971no—
152#135—1942no—
153#135—1942no—
154#135C2×D9711yescharacteristic
155C2×SD16—972no—
156C2×SD16—972no—
157C2×SD16—1942no—
158C2×SD16—1942no—
159#131—972no—
160#131C2×D9712yescomplemented
161C4⋊Q8—971no—
162#127—971no—
163#108—971no—
164#108C2×D9711yescharacteristic, complemented
165#104—972no—
166#104C2×D9712yescomplemented
167#101—972no—
168#100—972no—
169C4×C8C2×D9711yescharacteristic, complemented
170#41—972no—
171#41—3882no—
172#41—1942no—
173#41—1942no—
174#41—1942no—
175#41—1942no—
176#41C22×D9712yes—
177C2×Q8—972no—
178C2×Q8—1942no—
179C2×Q8—1942no—
180C2×D4—972no—
181C2×D4—3881no—
182C2×D4—3882no—
183C2×D4—1942no—
184C2×D4—1942no—
185Q16—971no—
186Q16—1942no—
187Q16—1942no—
188Q16D38811yescharacteristic, complemented
189SD16—972no—
190SD16—1942no—
191SD16—1942no—
192SD16—1942no—
193SD16—1942no—
194SD16—1942no—
195SD16—1942no—
196SD16—1942no—
197SD16—1942no—
198SD16D38812yescomplemented
199D8—971no—
200D8—1942no—
201D8—1942no—
202D8D38811yescharacteristic, complemented
203M16—22no—
204M16—972no—
205M16—1942no—
206M16C22×D9712yes—
207C2×C8—22no—
208C2×C8—1942no—
209C2×C8C22×D9711yescharacteristic
210C2×C8C22×D9711yescharacteristic
211C4⋊C4—3881no—
212C42C22×D9711yescharacteristic
213C23—3882no—
214C23—1942no—
215Q8—972no—
216Q8—3882no—
217Q8—1942no—
218Q8—1942no—
219Q8C2×D38812yes—
220D4—22no—
221D4—7762no—
222D4—972no—
223D4—3882no—
224D4—3882no—
225D4—3882no—
226D4—1942no—
227D4—1942no—
228D4—1942no—
229D4C2×D38812yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—3882no—
233C2×C4—3882no—
234C2×C4—3882no—
235C2×C4—1942no—
236C2×C4—1942no—
237C2×C4C23×D9711yescharacteristic
238C8—22no—
239C8—1942no—
240C8D4×D9711yescharacteristic
241C8D4×D9711yescharacteristic
242C8C2×D38811yescharacteristic
243C8C2×D38811yescharacteristic
244C22—22no—
245C22—7762no—
246C22—15522no—
247C22—3882no—
248C22—1942no—
249C22#1569085611yescharacteristic
250C4—22no—
251C4—22no—
252C4—3882no—
253C4—1942no—
254C4C2×D4×D9711yescharacteristic
255C4C22×D38811yescharacteristic
256C2—21no—
257C2—42no—
258C2—7762no—
259C2—3882no—
260C2D4×D38811yescharacteristic
subgroupG/H
centerC2D4×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#443×C97C2
socleC2×C97D42
hypercenter#443D97
hypocenterC97#3291
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #20261547C2semidirect
2. #20261542C2semidirect
3. #20261108C2semidirect
4. #20260999C2semidirect
5. #20260998C2semidirect
6. #20260919C2semidirect
7. #20260904C2semidirect
8. #20260853C2semidirect
9. #443×C97C2semidirect
11. #15690856C22semidirect
13. #15690843C22semidirect
15. #15690840C22semidirect
24. C2×#13095500C22semidirect
25. C2×#13095500C22semidirect
27. #15690816C22semidirect
28. #15690810C22semidirect
30. #15690746C22semidirect
31. #15690738C22semidirect
32. #15690728C22semidirect
33. #15690726C22semidirect
35. #131×C97C22semidirect
36. #108×C97C22semidirect
37. #104×C97C22semidirect
38. C4×C8×C97C22semidirect
54. Dic388D4semidirect
62. #13095500D4semidirect
66. D776D4semidirect
73. Q16×C97D4semidirect
74. SD16×C97D4semidirect
75. D8×C97D4semidirect
131. C97#3291semidirect
140. #443D97semidirect
160. #131C2×D97semidirect
164. #108C2×D97semidirect
166. #104C2×D97semidirect
169. C4×C8C2×D97semidirect
188. Q16D388semidirect
198. SD16D388semidirect
202. D8D388semidirect

Maximal subgroups

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

subgroupindex
#202615472
#202615422
#202611082
#202609992
#202609982
#202609192
#202609042
#202608532
#443×C972
#329197