Finite Group Databasebeta

← #46848158

Subgroups — #46848158

Order 34688.

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#31774709C212yesmaximal, complemented
2#31774704C212yesmaximal, complemented
3#31774270C211yesmaximal, characteristic, complemented
4#31774161C212yesmaximal, complemented
5#31774160C212yesmaximal, complemented
6#31774081C212yesmaximal, complemented
7#31774066C212yesmaximal, complemented
8#31774015C211yesmaximal, characteristic, complemented
9#443×C271C211yesmaximal, characteristic, complemented
10#22576207C2212yes—
11#22576206C2212yescomplemented
12#22576193—22no—
13#22576193C2212yescomplemented
14#22576192—22no—
15#22576190C2212yescomplemented
16#22576188—22no—
17#22576188C2212yes—
18#22576185C2212yes—
19#22576184—22no—
20#22576182—22no—
21#22576182C2211yescharacteristic
22#22576181—22no—
23C2×#16864643—22no—
24C2×#16864643C2212yescomplemented
25C2×#16864643C2212yescomplemented
26SD16×D271—22no—
27#22576166C2212yescomplemented
28#22576160C2211yescharacteristic, complemented
29#22576157C2211yescharacteristic
30#22576096C2211yescharacteristic, complemented
31#22576088C2212yescomplemented
32#22576078C2212yescomplemented
33#22576076C2212yescomplemented
34#135×C271C2211yescharacteristic
35#131×C271C2212yescomplemented
36#108×C271C2211yescharacteristic, complemented
37#104×C271C2212yescomplemented
38C4×C8×C271C2211yescharacteristic, complemented
39#16864655—22no—
40#16864654—22no—
41#16864654—22no—
42#16864654C2312yes—
43#16864653—22no—
44#16864653—42no—
45C2×Dic542—22no—
46C2×Dic542C2312yes—
47Q8×D271—22no—
48C2×D1084—22no—
49C2×D1084—41no—
50C2×D1084C2312yes—
51D4×D271—22no—
52D4×D271—42no—
53Dic1084—22no—
54Dic1084D411yescharacteristic, complemented
55#16864644—22no—
56#16864643—22no—
57#16864643—22no—
58#16864643—22no—
59#16864643—22no—
60#16864643—22no—
61#16864643—22no—
62#16864643D412yescomplemented
63#16864642—22no—
64#16864641—22no—
65D2168—22no—
66D2168D411yescharacteristic, complemented
67#16864639—22no—
68#16864638C2312yes—
69#16864637—22no—
70C8×D271—22no—
71C4⋊Dic271—41no—
72#41×C271C2312yes—
73Q16×C271D411yescharacteristic, complemented
74SD16×C271D412yescomplemented
75D8×C271D411yescharacteristic, complemented
76M16×C271—22no—
77M16×C271C2312yes—
78C2×C8×C271—22no—
79C2×C8×C271C2311yescharacteristic
80C2×C8×C271C2311yescharacteristic
81C42×C271C2311yescharacteristic
82C22×D271—22no—
83C22×D271—42no—
84Dic542—22no—
85Dic542—22no—
86Dic542—42no—
87Dic542C2×D412yes—
88D1084—22no—
89D1084—22no—
90D1084—42no—
91D1084—82no—
92D1084C2×D412yes—
93C22⋊D271—22no—
94C22⋊D271—42no—
95C22⋊D271—42no—
96C2×Dic271—22no—
97C2×Dic271—42no—
98C4×D271—22no—
99C4×D271—42no—
100C4×D271—42no—
101C271⋊C8—22no—
102Q8×C271C2×D412yes—
103D4×C271—22no—
104D4×C271C2×D412yes—
105C2×C4×C271—21no—
106C2×C4×C271—22no—
107C2×C4×C271C2411yescharacteristic
108C8×C271—22no—
109C8×C271C2×D411yescharacteristic
110C8×C271C2×D411yescharacteristic
111C8×C271C2×D411yescharacteristic
112C8×C271C2×D411yescharacteristic
113C22×C271—22no—
114C22×C271H(5,𝔽2)11yescharacteristic
115C2×D271—22no—
116C2×D271—42no—
117C2×D271—82no—
118C2×D271—162no—
119C4×C271—22no—
120C4×C271—22no—
121C4×C271C22×D411yescharacteristic
122C4×C271C22×D411yescharacteristic
123Dic271—22no—
124Dic271—42no—
125C2×C271—21no—
126C2×C271—42no—
127C2×C271D4211yescharacteristic
128D271—42no—
129D271—82no—
130C271#329111yescharacteristic, complemented, Sylow
131#3291—2711nomaximal, Sylow
132#577—2712no—
133#577—2712no—
134#492—2711no—
135#471—2712no—
136#471—2712no—
137#456—2712no—
138#453—2712no—
139#443—2711no—
140#443D27111yescharacteristic, complemented
1412−1+4—2712no—
142H(5,𝔽2)—2712no—
143#137—2712no—
144#137—2712no—
145#137—5422no—
146#137—5422no—
147AGL(1,ℤ/8)—2712no—
148AGL(1,ℤ/8)—2712no—
149AGL(1,ℤ/8)—5422no—
150AGL(1,ℤ/8)—5422no—
151#135—2711no—
152#135—5422no—
153#135—5422no—
154#135C2×D27111yescharacteristic
155C2×SD16—2712no—
156C2×SD16—2712no—
157C2×SD16—5422no—
158C2×SD16—5422no—
159#131—2712no—
160#131C2×D27112yescomplemented
161C4⋊Q8—2711no—
162#127—2711no—
163#108—2711no—
164#108C2×D27111yescharacteristic, complemented
165#104—2712no—
166#104C2×D27112yescomplemented
167#101—2712no—
168#100—2712no—
169C4×C8C2×D27111yescharacteristic, complemented
170#41—2712no—
171#41—5422no—
172#41—5422no—
173#41—5422no—
174#41—5422no—
175#41—10842no—
176#41C22×D27112yes—
177C2×Q8—2712no—
178C2×Q8—5422no—
179C2×Q8—5422no—
180C2×D4—2712no—
181C2×D4—5422no—
182C2×D4—5422no—
183C2×D4—10841no—
184C2×D4—10842no—
185Q16—2711no—
186Q16—5422no—
187Q16—5422no—
188Q16D108411yescharacteristic, complemented
189SD16—2712no—
190SD16—5422no—
191SD16—5422no—
192SD16—5422no—
193SD16—5422no—
194SD16—5422no—
195SD16—5422no—
196SD16—5422no—
197SD16—5422no—
198SD16D108412yescomplemented
199D8—2711no—
200D8—5422no—
201D8—5422no—
202D8D108411yescharacteristic, complemented
203M16—22no—
204M16—2712no—
205M16—5422no—
206M16C22×D27112yes—
207C2×C8—22no—
208C2×C8—5422no—
209C2×C8C22×D27111yescharacteristic
210C2×C8C22×D27111yescharacteristic
211C4⋊C4—10841no—
212C42C22×D27111yescharacteristic
213C23—5422no—
214C23—10842no—
215Q8—2712no—
216Q8—5422no—
217Q8—5422no—
218Q8—10842no—
219Q8C2×D108412yes—
220D4—22no—
221D4—2712no—
222D4—5422no—
223D4—5422no—
224D4—5422no—
225D4—10842no—
226D4—10842no—
227D4—10842no—
228D4—21682no—
229D4C2×D108412yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—5422no—
233C2×C4—5422no—
234C2×C4—10842no—
235C2×C4—10842no—
236C2×C4—10842no—
237C2×C4C23×D27111yescharacteristic
238C8—22no—
239C8—5422no—
240C8D4×D27111yescharacteristic
241C8D4×D27111yescharacteristic
242C8C2×D108411yescharacteristic
243C8C2×D108411yescharacteristic
244C22—22no—
245C22—5422no—
246C22—10842no—
247C22—21682no—
248C22—43362no—
249C22#2257620611yescharacteristic
250C4—22no—
251C4—22no—
252C4—5422no—
253C4—10842no—
254C4C2×D4×D27111yescharacteristic
255C4C22×D108411yescharacteristic
256C2—21no—
257C2—42no—
258C2—10842no—
259C2—21682no—
260C2D4×D108411yescharacteristic
subgroupG/H
centerC2D4×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#443×C271C2
socleC2×C271D42
hypercenter#443D271
hypocenterC271#3291
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #31774709C2semidirect
2. #31774704C2semidirect
3. #31774270C2semidirect
4. #31774161C2semidirect
5. #31774160C2semidirect
6. #31774081C2semidirect
7. #31774066C2semidirect
8. #31774015C2semidirect
9. #443×C271C2semidirect
11. #22576206C22semidirect
13. #22576193C22semidirect
15. #22576190C22semidirect
24. C2×#16864643C22semidirect
25. C2×#16864643C22semidirect
27. #22576166C22semidirect
28. #22576160C22semidirect
30. #22576096C22semidirect
31. #22576088C22semidirect
32. #22576078C22semidirect
33. #22576076C22semidirect
35. #131×C271C22semidirect
36. #108×C271C22semidirect
37. #104×C271C22semidirect
38. C4×C8×C271C22semidirect
54. Dic1084D4semidirect
62. #16864643D4semidirect
66. D2168D4semidirect
73. Q16×C271D4semidirect
74. SD16×C271D4semidirect
75. D8×C271D4semidirect
130. C271#3291semidirect
140. #443D271semidirect
160. #131C2×D271semidirect
164. #108C2×D271semidirect
166. #104C2×D271semidirect
169. C4×C8C2×D271semidirect
188. Q16D1084semidirect
198. SD16D1084semidirect
202. D8D1084semidirect

Maximal subgroups

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

subgroupindex
#317747092
#317747042
#317742702
#317741612
#317741602
#317740812
#317740662
#317740152
#443×C2712
#3291271