Finite Group Databasebeta

← #46848154

Subgroups — #46848154

Order 34688.

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#31774706C212yesmaximal, complemented
2#31774705C212yesmaximal, complemented
3#31774271C211yesmaximal, characteristic, complemented
4#31774163C212yesmaximal, complemented
5#31774162C212yesmaximal, complemented
6#31774079C211yesmaximal, characteristic, complemented
7#31774066C212yesmaximal, complemented
8#31774011C212yesmaximal, complemented
9#456×C271C211yesmaximal, characteristic, complemented
10#22576208C2211yescharacteristic
11#22576206C2212yescomplemented
12#22576192—22no—
13#22576191—22no—
14#22576191C2212yescomplemented
15#22576188C2212yes—
16#22576187—22no—
17#22576185—22no—
18#22576185C2212yes—
19#22576183—22no—
20#22576180—22no—
21#22576180C2212yes—
22C2×#16864641—22no—
23C2×#16864641C2212yescomplemented
24C2×#16864641C2212yescomplemented
25SD16×D271—22no—
26#22576168C2212yescomplemented
27#22576167C2212yescomplemented
28#22576157C2211yescharacteristic
29#22576095C2212yescomplemented
30#22576088C2212yescomplemented
31#22576079C2211yescharacteristic, complemented
32#22576076C2212yescomplemented
33C4×C271⋊C8C2211yescharacteristic, complemented
342−1+4×C271C2211yescharacteristic
35#137×C271C2212yescomplemented
36C4⋊Q8×C271C2211yescharacteristic, complemented
37#104×C271C2212yescomplemented
38#101×C271C2211yescharacteristic, complemented
39#16864655—22no—
40#16864655—22no—
41#16864654—22no—
42#16864654—41no—
43#16864654C2312yes—
44Q8×D271—22no—
45C2×D1084—22no—
46C2×D1084—41no—
47C2×D1084C2312yes—
48D4×D271—22no—
49D4×D271—42no—
50#16864644—22no—
51#16864644D412yescomplemented
52#16864643—22no—
53#16864642D412yescomplemented
54#16864641—22no—
55#16864641—22no—
56#16864641—22no—
57#16864641—22no—
58#16864641—22no—
59#16864641—22no—
60#16864641D412yescomplemented
61D2168—22no—
62#16864639—22no—
63#16864639D412yescomplemented
64#16864638—22no—
65#16864638C2312yes—
66#16864637—22no—
67C2×C271⋊C8—22no—
68C2×C271⋊C8C2312yes—
69C8×D271—22no—
70#41×C271—22no—
71#41×C271—41no—
72#41×C271C2312yes—
73C2×Q8×C271—22no—
74C2×Q8×C271C2311yescharacteristic
75C2×Q8×C271C2311yescharacteristic
76Q16×C271—22no—
77SD16×C271—22no—
78M16×C271C2312yes—
79C4⋊C4×C271—41no—
80C42×C271C2311yescharacteristic
81C22×D271—22no—
82C22×D271—42no—
83Dic542—41no—
84Dic542C2×D412yes—
85D1084—22no—
86D1084—22no—
87D1084—42no—
88D1084—41no—
89D1084—82no—
90D1084C2×D412yes—
91C22⋊D271—22no—
92C22⋊D271—42no—
93C4×D271—22no—
94C4×D271—42no—
95C4×D271—42no—
96C271⋊C8—22no—
97C271⋊C8C2×D412yes—
98C271⋊C8C2×D412yes—
99Q8×C271—22no—
100Q8×C271—22no—
101Q8×C271—41no—
102Q8×C271C2×D412yes—
103D4×C271—22no—
104D4×C271—41no—
105D4×C271C2×D412yes—
106C2×C4×C271—21no—
107C2×C4×C271—22no—
108C2×C4×C271—21no—
109C2×C4×C271—21no—
110C2×C4×C271—42no—
111C2×C4×C271C2411yescharacteristic
112C8×C271—22no—
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×C271—41no—
122C4×C271—41no—
123C4×C271C22×D412yes—
124Dic271—22no—
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—2711no—
138#456D27111yescharacteristic, complemented
139#453—2712no—
140#443—2712no—
1412−1+4—2711no—
1422−1+4C2×D27111yescharacteristic
143H(5,𝔽2)—2712no—
144#137—2712no—
145#137—5422no—
146#137—5422no—
147#137C2×D27112yescomplemented
148AGL(1,ℤ/8)—2712no—
149AGL(1,ℤ/8)—2712no—
150AGL(1,ℤ/8)—5422no—
151AGL(1,ℤ/8)—5422no—
152#135—2712no—
153#135—5422no—
154#135—5422no—
155C2×SD16—2712no—
156C2×SD16—2712no—
157C2×SD16—5422no—
158C2×SD16—5422no—
159#131—2712no—
160#131—2712no—
161C4⋊Q8C2×D27111yescharacteristic, complemented
162#127—2711no—
163#108—2712no—
164#104—2712no—
165#104C2×D27112yescomplemented
166#101—2711no—
167#101C2×D27111yescharacteristic, complemented
168#100—2712no—
169C4×C8—2711no—
170#41—22no—
171#41—41no—
172#41—2712no—
173#41—5422no—
174#41—5422no—
175#41—5422no—
176#41—10841no—
177#41C22×D27112yes—
178C2×Q8—22no—
179C2×Q8—5422no—
180C2×Q8C22×D27111yescharacteristic
181C2×Q8C22×D27111yescharacteristic
182C2×D4—2712no—
183C2×D4—5422no—
184C2×D4—5422no—
185C2×D4—10841no—
186C2×D4—10842no—
187Q16—22no—
188Q16—2712no—
189Q16—5422no—
190SD16—22no—
191SD16—2712no—
192SD16—2712no—
193SD16—5422no—
194SD16—5422no—
195SD16—5422no—
196SD16—5422no—
197SD16—5422no—
198SD16—5422no—
199SD16—5422no—
200D8—2712no—
201D8—5422no—
202D8—5422no—
203M16—2712no—
204M16—5422no—
205M16—5422no—
206M16C22×D27112yes—
207C2×C8—2712no—
208C2×C8—5422no—
209C2×C8—5422no—
210C4⋊C4—41no—
211C42C22×D27111yescharacteristic
212C23—5422no—
213C23—10842no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—2712no—
218Q8—10841no—
219Q8D4×D27112yes—
220D4—22no—
221D4—41no—
222D4—2712no—
223D4—5422no—
224D4—5422no—
225D4—5422no—
226D4—10842no—
227D4—10841no—
228D4—10842no—
229D4—21682no—
230D4D4×D27112yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—5422no—
237C2×C4—10842no—
238C2×C4—10842no—
239C2×C4C23×D27111yescharacteristic
240C8—22no—
241C8—2712no—
242C8—2712no—
243C8—5422no—
244C22—22no—
245C22—5422no—
246C22—10842no—
247C22—21682no—
248C22—43362no—
249C22#2257620511yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—5422no—
255C4C2×D4×D27112yes—
256C2—21no—
257C2—42no—
258C2—10842no—
259C2—21682no—
260C2#3177452911yescharacteristic
subgroupG/H
centerC2#31774529
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#456×C271C2
socleC2×C271D42
hypercenter#456D271
hypocenterC271#3291
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #31774706C2semidirect
2. #31774705C2semidirect
3. #31774271C2semidirect
4. #31774163C2semidirect
5. #31774162C2semidirect
6. #31774079C2semidirect
7. #31774066C2semidirect
8. #31774011C2semidirect
9. #456×C271C2semidirect
11. #22576206C22semidirect
14. #22576191C22semidirect
23. C2×#16864641C22semidirect
24. C2×#16864641C22semidirect
26. #22576168C22semidirect
27. #22576167C22semidirect
29. #22576095C22semidirect
30. #22576088C22semidirect
31. #22576079C22semidirect
32. #22576076C22semidirect
33. C4×C271⋊C8C22semidirect
35. #137×C271C22semidirect
36. C4⋊Q8×C271C22semidirect
37. #104×C271C22semidirect
38. #101×C271C22semidirect
51. #16864644D4semidirect
53. #16864642D4semidirect
60. #16864641D4semidirect
63. #16864639D4semidirect
130. C271#3291semidirect
138. #456D271semidirect
147. #137C2×D271semidirect
161. C4⋊Q8C2×D271semidirect
165. #104C2×D271semidirect
167. #101C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747062
#317747052
#317742712
#317741632
#317741622
#317740792
#317740662
#317740112
#456×C2712
#3291271