Finite Group Databasebeta

← #46848141

Subgroups — #46848141

Order 34688.

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

normal subgroups only

This group has 448 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#31774712C212yesmaximal, complemented
2#31774707C212yesmaximal, complemented
3#31774292C211yesmaximal, characteristic, complemented
4#31774171C212yesmaximal, complemented
5#31774158C212yesmaximal, complemented
6#31774081C212yesmaximal, complemented
7#31774074C211yesmaximal, characteristic, complemented
8#31774016C212yesmaximal, complemented
9#454×C271C211yesmaximal, characteristic, complemented
10#22576207C2212yes—
11#22576205C2211yescharacteristic
12#22576193C2212yes—
13#22576191—22no—
14#22576191C2212yes—
15#22576190—22no—
16#22576190C2212yescomplemented
17#22576189—22no—
18#22576184—22no—
19#22576180—22no—
20#22576180C2212yes—
21#22576179—22no—
22C2×#16864644—22no—
23C2×#16864644C2212yescomplemented
24C2×#16864642C2212yes—
25SD16×D271—22no—
26#22576168C2212yes—
27#22576167C2212yescomplemented
28#22576160C2211yescharacteristic, complemented
29#22576095C2212yescomplemented
30#22576088C2212yes—
31#22576078C2212yescomplemented
32#22576077C2211yescharacteristic
33#22576071C2211yescharacteristic, complemented
342−1+4×C271C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C271C2212yes—
36#124×C271C2211yescharacteristic
37#104×C271C2212yescomplemented
38#101×C271C2211yescharacteristic, complemented
39#16864654—22no—
40#16864654C2312yes—
41#16864653—22no—
42#16864653—22no—
43#16864653—42no—
44C2×Dic542—22no—
45C2×Dic542C2312yes—
46Q8×D271—22no—
47C2×C22⋊D271—41no—
48D4×D271—22no—
49Dic1084—22no—
50#16864644—22no—
51#16864644—22no—
52#16864644—22no—
53#16864644—22no—
54#16864644D412yes—
55#16864643—22no—
56#16864642—22no—
57#16864642—22no—
58#16864642—22no—
59#16864642D412yes—
60#16864641—22no—
61#16864641D412yes—
62#16864639D412yes—
63#16864638—22no—
64#16864638C2312yes—
65#16864637—22no—
66C2×C271⋊C8—22no—
67C2×C271⋊C8C2312yes—
68C8×D271—22no—
69C4⋊Dic271—41no—
70#41×C271—22no—
71#41×C271—41no—
72#41×C271C2312yes—
73C2×Q8×C271—22no—
74C2×Q8×C271C2311yescharacteristic
75C2×D4×C271C2311yescharacteristic
76SD16×C271—22no—
77D8×C271—22no—
78M16×C271C2312yes—
79C22⋊C4×C271—41no—
80C42×C271C2311yescharacteristic
81C22×D271—41no—
82Dic542—22no—
83Dic542—22no—
84Dic542—42no—
85Dic542C2×D412yes—
86D1084C2×D412yes—
87C22⋊D271—22no—
88C22⋊D271—42no—
89C22⋊D271—42no—
90C2×Dic271—22no—
91C2×Dic271—41no—
92C2×Dic271—42no—
93C4×D271—22no—
94C4×D271—42no—
95C271⋊C8—22no—
96C271⋊C8C2×D412yes—
97C271⋊C8C2×D412yes—
98C23×C271—21no—
99Q8×C271—22no—
100Q8×C271—41no—
101Q8×C271C2×D412yes—
102D4×C271—22no—
103D4×C271—22no—
104D4×C271—41no—
105D4×C271C2×D412yes—
106C2×C4×C271—22no—
107C2×C4×C271—21no—
108C2×C4×C271—21no—
109C2×C4×C271—42no—
110C2×C4×C271C2411yescharacteristic
111C8×C271—22no—
112C22×C271—22no—
113C22×C271—41no—
114C22×C271—81no—
115C22×C271H(5,𝔽2)11yescharacteristic
116C2×D271—22no—
117C2×D271—161no—
118C4×C271—22no—
119C4×C271—41no—
120C4×C271—41no—
121C4×C271C22×D412yes—
122Dic271—22no—
123Dic271—42no—
124C2×C271—21no—
125C2×C271—42no—
126C2×C271—81no—
127C2×C271D4211yescharacteristic
128D271—42no—
129C271#328911yescharacteristic, complemented, Sylow
130#3289—2711nomaximal, Sylow
131#578—2712no—
132#577—2712no—
133#497—2711no—
134#473—2712no—
135#471—2712no—
136#456—2712no—
137#455—2711no—
138#454D27111yescharacteristic, complemented
139#444—2712no—
1402−1+4—2712no—
1412−1+4C2×D27111yescharacteristic, complemented
142H(5,𝔽2)—2711no—
143#137—2712no—
144#137—2712no—
145#137—2712no—
146#137—5422no—
147#137—5422no—
148#137—5422no—
149AGL(1,ℤ/8)—5422no—
150AGL(1,ℤ/8)C2×D27112yes—
151#135—2712no—
152#135—5422no—
153#135—5422no—
154C2×Q16—2712no—
155C2×Q16—5422no—
156C2×SD16—2712no—
157C2×SD16—5422no—
158#131—2712no—
159#131—2712no—
160C4⋊Q8—2711no—
161#124C2×D27111yescharacteristic
162#108—2712no—
163#104—2712no—
164#104C2×D27112yescomplemented
165#101—2712no—
166#101C2×D27111yescharacteristic, complemented
167#100—2711no—
168#97—2711no—
169#41—22no—
170#41—41no—
171#41—2712no—
172#41—5422no—
173#41—5422no—
174#41—5422no—
175#41—10842no—
176#41C22×D27112yes—
177C2×Q8—22no—
178C2×Q8—2712no—
179C2×Q8—5422no—
180C2×Q8—5422no—
181C2×Q8C22×D27111yescharacteristic
182C2×D4—5422no—
183C2×D4—10841no—
184C2×D4C22×D27111yescharacteristic
185Q16—2712no—
186Q16—5422no—
187Q16—5422no—
188Q16—5422no—
189Q16—5422no—
190Q16—5422no—
191SD16—22no—
192SD16—2712no—
193SD16—2712no—
194SD16—5422no—
195SD16—5422no—
196SD16—5422no—
197SD16—5422no—
198SD16—5422no—
199D8—22no—
200D8—2712no—
201M16—2712no—
202M16—5422no—
203M16—5422no—
204M16C22×D27112yes—
205C2×C8—2712no—
206C2×C8—5422no—
207C2×C8—5422no—
208C4⋊C4—10841no—
209C22⋊C4—41no—
210C42C22×D27111yescharacteristic
211C23—21no—
212C23—10841no—
213Q8—22no—
214Q8—41no—
215Q8—2712no—
216Q8—5422no—
217Q8—5422no—
218Q8—10842no—
219Q8D4×D27112yes—
220D4—22no—
221D4—22no—
222D4—41no—
223D4—2712no—
224D4—5422no—
225D4—10842no—
226D4—10842no—
227D4D4×D27112yes—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—5422no—
233C2×C4—5422no—
234C2×C4—10842no—
235C2×C4—10841no—
236C2×C4—10842no—
237C2×C4C23×D27111yescharacteristic
238C8—22no—
239C8—2712no—
240C8—2712no—
241C8—5422no—
242C22—22no—
243C22—41no—
244C22—81no—
245C22—5422no—
246C22—43361no—
247C22#2257620511yescharacteristic
248C4—22no—
249C4—41no—
250C4—41no—
251C4—5422no—
252C4—10842no—
253C4C2×D4×D27112yes—
254C2—21no—
255C2—42no—
256C2—81no—
257C2—10842no—
258C2#3177452911yescharacteristic
subgroupG/H
centerC2#31774529
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#454×C271C2
socleC2×C271D42
hypercenter#454D271
hypocenterC271#3289
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #31774712C2semidirect
2. #31774707C2semidirect
3. #31774292C2semidirect
4. #31774171C2semidirect
5. #31774158C2semidirect
6. #31774081C2semidirect
7. #31774074C2semidirect
8. #31774016C2semidirect
9. #454×C271C2semidirect
16. #22576190C22semidirect
23. C2×#16864644C22semidirect
27. #22576167C22semidirect
28. #22576160C22semidirect
29. #22576095C22semidirect
31. #22576078C22semidirect
33. #22576071C22semidirect
34. 2−1+4×C271C22semidirect
37. #104×C271C22semidirect
38. #101×C271C22semidirect
129. C271#3289semidirect
138. #454D271semidirect
141. 2−1+4C2×D271semidirect
164. #104C2×D271semidirect
166. #101C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747122
#317747072
#317742922
#317741712
#317741582
#317740812
#317740742
#317740162
#454×C2712
#3289271