Finite Group Databasebeta

← #46846325

Subgroups — #46846325

Order 34688.

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

normal subgroups only

This group has 418 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#31774466C211yesmaximal, characteristic
2#31774377C211yesmaximal, characteristic, complemented
3#31774224C212yesmaximal, complemented
4#31774223C212yesmaximal, complemented
5#31774217C212yesmaximal, complemented
6#31774212C212yesmaximal, complemented
7#31773940C211yesmaximal, characteristic, complemented
8#31773901C211yesmaximal, characteristic, complemented
9C2×#22576082C211yesmaximal, characteristic, complemented
10#31773863C211yesmaximal, characteristic, complemented
11#427×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13C22×D1084C2211yescharacteristic
14C2×#16864638C2211yescharacteristic
15#22576157—21no—
16#22576148—21no—
17#22576135C2212yes—
18#22576135C2212yes—
19#22576130—22no—
20C4×Dic542—21no—
21C4×D1084—21no—
22(C22⋊D271)⋊C4—22no—
23#22576118C2212yescomplemented
24C2×C4⋊Dic271C2211yescharacteristic
25C4⋊C4×D271C2212yescomplemented
26#22576093C2212yescomplemented
27#22576091C2212yescomplemented
28#22576082C2212yes—
29#22576082C2212yes—
30#22576080C2212yes—
31#22576080C2212yes—
32(C2×D271)⋊C8C2212yescomplemented
33(C2×D271)⋊C8C2212yescomplemented
34C22×C8×C271C2211yescharacteristic
35#117×C271C2211yescharacteristic
36#117×C271C2211yescharacteristic
37#107×C271C2211yescharacteristic, complemented
38#107×C271C2211yescharacteristic, complemented
39#106×C271C2211yescharacteristic, complemented
40#106×C271C2211yescharacteristic, complemented
41C23×D271—21no—
42#16864654—22no—
43#16864654—22no—
44C2×Dic542D411yescharacteristic
45C2×D1084—22no—
46C2×D1084—22no—
47C2×D1084—21no—
48C2×D1084—22no—
49C2×D1084—22no—
50C2×D1084D411yescharacteristic
51C2×D1084C2312yes—
52C2×C22⋊D271D412yes—
53C22×Dic271—21no—
54C2×C4×D271—22no—
55C2×C4×D271C2312yes—
56#16864638—22no—
57C2×C271⋊C8C2312yes—
58C4⋊Dic271—21no—
59C4⋊Dic271C2312yes—
60Dic271⋊C4—22no—
61(C2×D271)⋊C4—22no—
62(C2×D271)⋊C4—22no—
63(C2×D271)⋊C4—22no—
64C4×Dic271—22no—
65C22×C4×C271C2311yescharacteristic
66C2×C8×C271—22no—
67C2×C8×C271C2311yescharacteristic
68C2×C8×C271C2311yescharacteristic
69C4⋊C4×C271C2311yescharacteristic
70C4⋊C4×C271C2311yescharacteristic
71C4⋊C4×C271C2311yescharacteristic
72C4⋊C4×C271C2311yescharacteristic
73C22⋊C4×C271—21no—
74C22⋊C4×C271—21no—
75C42×C271—21no—
76C42×C271—21no—
77C22×D271—22no—
78C22×D271—42no—
79C22×D271—81no—
80C22×D271—81no—
81C22×D271C2×D412yes—
82Dic542—22no—
83D1084—22no—
84D1084—22no—
85D1084—22no—
86D1084—42no—
87D1084—82no—
88D1084—82no—
89C22⋊D271—24no—
90C2×Dic271—22no—
91C2×Dic271—81no—
92C2×Dic271C2×D412yes—
93C4×D271—22no—
94C4×D271—22no—
95C4×D271—82no—
96C271⋊C8—22no—
97C23×C271C2×D411yescharacteristic
98C2×C4×C271—21no—
99C2×C4×C271—21no—
100C2×C4×C271—21no—
101C2×C4×C271—21no—
102C2×C4×C271C2×D411yescharacteristic
103C2×C4×C271#4112yes—
104C2×C4×C271#4112yes—
105C2×C4×C271C2411yescharacteristic
106C8×C271—21no—
107C8×C271—21no—
108C22×C271—21no—
109C22×C271—21no—
110C22×C271#12312yes—
111C22×C271C22×D411yescharacteristic
112C2×D271—22no—
113C2×D271—22no—
114C2×D271—22no—
115C2×D271—42no—
116C2×D271—42no—
117C2×D271—42no—
118C2×D271—84no—
119C4×C271—41no—
120C4×C271—41no—
121C4×C271—41no—
122C4×C271—41no—
123C4×C271#12312yes—
124C4×C271C2×#4111yescharacteristic
125C4×C271C2×#4111yescharacteristic
126Dic271—22no—
127Dic271—42no—
128C2×C271—22no—
129C2×C271C2×#12311yescharacteristic
130C2×C271#57611yescharacteristic
131C2×C271#57711yescharacteristic
132D271—42no—
133D271—82no—
134C271#309611yescharacteristic, complemented, Sylow
135#3096—2711nomaximal, Sylow
136#535—2711no—
137#520—2711no—
138#482—2712no—
139#482—2712no—
140#481—2712no—
141#480—2712no—
142#428—2711no—
143#427D27111yescharacteristic, complemented
144#418—2711no—
145C2×#102—2711no—
146#408—2711no—
147C2×#41—2711no—
148C22×D4—2711no—
149C2×M16—2711no—
150C22×C8C2×D27111yescharacteristic
151#127—5421no—
152#124—5421no—
153#121—2712no—
154#121—2712no—
155#120—5422no—
156C4×Q8—5421no—
157C4×D4—5421no—
158C4×D4—5422no—
159#117—2712no—
160#117C2×D27111yescharacteristic
161#117C2×D27111yescharacteristic
162C2×C4⋊C4—2711no—
163C2×C4⋊C4—2712no—
164#107—2712no—
165#107C2×D27111yescharacteristic, complemented
166#107C2×D27111yescharacteristic, complemented
167#106—2712no—
168#106C2×D27111yescharacteristic, complemented
169#106C2×D27111yescharacteristic, complemented
170#102—2712no—
171#102—2712no—
172#102—2712no—
173#102—2712no—
174C22⋊C8—2712no—
175C22⋊C8—2712no—
176C24—5421no—
177#41—5422no—
178#41—5422no—
179C2×Q8—2711no—
180C2×D4—2712no—
181C2×D4—2711no—
182C2×D4—2712no—
183C2×D4—5422no—
184C2×D4—5422no—
185C2×D4—5421no—
186C2×D4—5422no—
187C2×D4—5422no—
188C22×C4—2712no—
189C22×C4—5422no—
190C22×C4—5421no—
191C22×C4C22×D27111yescharacteristic
192M16—5422no—
193C2×C8—22no—
194C2×C8—2712no—
195C2×C8C22×D27111yescharacteristic
196C2×C8C22×D27111yescharacteristic
197C4⋊C4—2712no—
198C4⋊C4—5422no—
199C4⋊C4—5421no—
200C4⋊C4C22×D27111yescharacteristic
201C4⋊C4C22×D27111yescharacteristic
202C4⋊C4C22×D27111yescharacteristic
203C4⋊C4C22×D27111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—5422no—
207C22⋊C4—5422no—
208C22⋊C4—5422no—
209C42—21no—
210C42—21no—
211C42—5422no—
212C23—2712no—
213C23—5422no—
214C23—10842no—
215C23—21681no—
216C23—21681no—
217C23D4×D27111yescharacteristic
218Q8—5422no—
219D4—5424no—
220D4—5422no—
221D4—5422no—
222D4—5422no—
223D4—10842no—
224D4—21682no—
225D4—21682no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—2712no—
231C2×C4—5422no—
232C2×C4—5422no—
233C2×C4—5422no—
234C2×C4—21682no—
235C2×C4—21681no—
236C2×C4D4×D27111yescharacteristic
237C2×C4#1686465412yes—
238C2×C4#1686465512yes—
239C2×C4C23×D27111yescharacteristic
240C8—21no—
241C8—21no—
242C8—5422no—
243C22—21no—
244C22—21no—
245C22—5422no—
246C22—5422no—
247C22—5422no—
248C22—10842no—
249C22—10842no—
250C22—10842no—
251C22—21684no—
252C22#2257614512yes—
253C22C2×D4×D27111yescharacteristic
254C4—41no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—5422no—
259C4—10842no—
260C4#2257614512yes—
261C4C2×#1686465411yescharacteristic
262C4C2×#1686465511yescharacteristic
263C2—22no—
264C2—10842no—
265C2—21682no—
266C2C2×#2257614511yescharacteristic
267C2#3177469911yescharacteristic
268C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#427×C271C2
socleC22×C271C22×D4
prime radical#31774466C2
hypercenter#427D271
hypocenterC271#3096
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #31774377C2semidirect
3. #31774224C2semidirect
4. #31774223C2semidirect
5. #31774217C2semidirect
6. #31774212C2semidirect
7. #31773940C2semidirect
8. #31773901C2semidirect
9. C2×#22576082C2semidirect
10. #31773863C2semidirect
11. #427×C271C2semidirect
23. #22576118C22semidirect
25. C4⋊C4×D271C22semidirect
26. #22576093C22semidirect
27. #22576091C22semidirect
32. (C2×D271)⋊C8C22semidirect
33. (C2×D271)⋊C8C22semidirect
37. #107×C271C22semidirect
38. #107×C271C22semidirect
39. #106×C271C22semidirect
40. #106×C271C22semidirect
134. C271#3096semidirect
143. #427D271semidirect
165. #107C2×D271semidirect
166. #107C2×D271semidirect
168. #106C2×D271semidirect
169. #106C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744662
#317743772
#317742242
#317742232
#317742172
#317742122
#317739402
#317739012
C2×#225760822
#317738632
#427×C2712
#3096271