Finite Group Databasebeta

← #46845609

Subgroups — #46845609

Order 34688.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#16864654C211yesmaximal, characteristic, complemented
2C2×#22576193C211yesmaximal, characteristic, complemented
3C2×#22576188C211yesmaximal, characteristic, complemented
4#31774408C211yesmaximal, characteristic, complemented
5#31774061C214yesmaximal, complemented
6#31774044C214yesmaximal, complemented
7#31773907C211yesmaximal, characteristic, complemented
8#31773903C211yesmaximal, characteristic, complemented
9#407×C271C211yesmaximal, characteristic, complemented
10C2×#16864654—24no—
11C2×#16864654—24no—
12C2×#16864654—22no—
13C2×#16864654C2214yes—
14C2×#16864654C2211yescharacteristic
15C22×Dic542C2211yescharacteristic
16C22×D1084C2211yescharacteristic
17C22×C22⋊D271—21no—
18C22×C4×D271—21no—
19#22576193—24no—
20#22576188—24no—
21C2×Dic1084C2212yescomplemented
22C2×#16864643C2212yescomplemented
23C2×#16864643C2212yescomplemented
24C2×D2168C2212yescomplemented
25#22576143—21no—
26#22576139C2212yescomplemented
27#22576134C2212yescomplemented
28#22576129—21no—
29C4×C22⋊D271—22no—
30#22576119C2211yescharacteristic
31#22576085C2214yescomplemented
32#22576082C2214yescomplemented
33C23×C4×C271C2211yescharacteristic
34C2×M16×C271C2211yescharacteristic
35C2×M16×C271C2211yescharacteristic
36C22⋊C8×C271C2214yescomplemented
37C23×D271—21no—
38#16864654—216no—
39#16864654—28no—
40#16864654—28no—
41#16864654—22no—
42#16864654—22no—
43C2×Dic542—22no—
44C2×Dic542—21no—
45C2×Dic542D414yescomplemented
46C2×Dic542C2311yescharacteristic
47C2×Dic542C2312yes—
48C2×D1084—22no—
49C2×D1084—21no—
50C2×D1084D414yescomplemented
51C2×D1084C2311yescharacteristic
52C2×D1084C2312yes—
53C2×C22⋊D271—28no—
54C2×C22⋊D271—24no—
55C2×C22⋊D271—22no—
56C2×C22⋊D271—21no—
57C22×Dic271—21no—
58C2×C4×D271—28no—
59C2×C4×D271—24no—
60C2×C4×D271—21no—
61Dic1084—24no—
62#16864643—24no—
63#16864643—24no—
64D2168—24no—
65C4⋊Dic271C2312yes—
66Dic271⋊C4—22no—
67C22⋊Dic271—22no—
68C22⋊Dic271—21no—
69(C2×D271)⋊C4—22no—
70C4×Dic271—21no—
71C24×C271D411yescharacteristic
72C22×C4×C271—22no—
73C22×C4×C271—22no—
74C22×C4×C271D411yescharacteristic
75C22×C4×C271D412yes—
76C22×C4×C271C2312yes—
77C22×C4×C271C2311yescharacteristic
78M16×C271—22no—
79M16×C271—22no—
80C2×C8×C271C2312yes—
81C2×C8×C271C2312yes—
82C22×D271—24no—
83C22×D271—22no—
84C22×D271—21no—
85C22×D271—44no—
86Dic542—24no—
87Dic542—22no—
88Dic542—21no—
89Dic542—21no—
90Dic542C2×D414yes—
91D1084—24no—
92D1084—22no—
93D1084—21no—
94D1084—21no—
95D1084C2×D414yes—
96C22⋊D271—216no—
97C22⋊D271—44no—
98C22⋊D271—42no—
99C2×Dic271—24no—
100C2×Dic271—22no—
101C2×Dic271—22no—
102C2×Dic271—21no—
103C4×D271—216no—
104C4×D271—42no—
105C23×C271—22no—
106C23×C271—42no—
107C23×C271C2×D412yes—
108C23×C271C2×D411yescharacteristic
109C2×C4×C271—22no—
110C2×C4×C271—24no—
111C2×C4×C271—22no—
112C2×C4×C271—22no—
113C2×C4×C271C2×D412yes—
114C2×C4×C271C2×D412yes—
115C2×C4×C271C2×D412yes—
116C2×C4×C271C2×D411yescharacteristic
117C2×C4×C271C2411yescharacteristic
118C8×C271—22no—
119C8×C271—22no—
120C22×C271—22no—
121C22×C271—22no—
122C22×C271—22no—
123C22×C271—22no—
124C22×C271—44no—
125C22×C271#12012yes—
126C22×C271C22×D411yescharacteristic
127C2×D271—24no—
128C2×D271—48no—
129C2×D271—44no—
130C2×D271—41no—
131C2×D271—42no—
132C4×C271—22no—
133C4×C271#12012yes—
134C4×C271C22×D411yescharacteristic
135C4×C271C22×D411yescharacteristic
136Dic271—24no—
137Dic271—41no—
138Dic271—42no—
139C2×C271—22no—
140C2×C271—42no—
141C2×C271C2×#12011yescharacteristic
142C2×C271#57512yes—
143D271—44no—
144D271—81no—
145C271#300511yescharacteristic, complemented, Sylow
146#3005—2711nomaximal, Sylow
147C22×#41—2711no—
148C2×#137—2711no—
149C2×AGL(1,ℤ/8)—2711no—
150#525—2711no—
151#452—2714no—
152#449—2714no—
153#419—2711no—
154#418—2711no—
155#407D27111yescharacteristic, complemented
156C2×#41—2711no—
157C2×#41—2714no—
158C2×#41—5424no—
159C2×#41—5424no—
160C2×#41—5422no—
161C22×Q8—2711no—
162C22×D4—2711no—
163C22×D4—5421no—
164C23×C4—5421no—
165C23×C4C2×D27111yescharacteristic
166#137—5424no—
167AGL(1,ℤ/8)—5424no—
168C2×Q16—2712no—
169C2×SD16—2712no—
170C2×SD16—2712no—
171C2×D8—2712no—
172C2×M16C2×D27111yescharacteristic
173C2×M16C2×D27111yescharacteristic
174#123—5421no—
175C22⋊Q8—2712no—
176#121—2712no—
177#120—5421no—
178C4×D4—5422no—
179#117—2711no—
180#103—2714no—
181#102—2714no—
182C22⋊C8C2×D27114yescomplemented
183C24—5421no—
184C24D108411yescharacteristic
185#41—5428no—
186#41—5428no—
187#41—54216no—
188#41—5422no—
189#41—5422no—
190C2×Q8—2714no—
191C2×Q8—2712no—
192C2×Q8—2711no—
193C2×Q8—5422no—
194C2×Q8—5421no—
195C2×D4—2714no—
196C2×D4—2712no—
197C2×D4—2711no—
198C2×D4—5428no—
199C2×D4—5424no—
200C2×D4—5422no—
201C2×D4—5421no—
202C2×D4—5422no—
203C2×D4—5421no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—5428no—
207C22×C4—5421no—
208C22×C4—5424no—
209C22×C4—5421no—
210C22×C4D108412yes—
211C22×C4D108411yescharacteristic
212C22×C4C22×D27112yes—
213C22×C4C22×D27111yescharacteristic
214Q16—5424no—
215SD16—5424no—
216SD16—5424no—
217D8—5424no—
218M16—22no—
219M16—22no—
220C2×C8C22×D27112yes—
221C2×C8C22×D27112yes—
222C4⋊C4—2712no—
223C4⋊C4—5422no—
224C22⋊C4—5421no—
225C22⋊C4—5422no—
226C22⋊C4—5422no—
227C42—5421no—
228C23—22no—
229C23—42no—
230C23—5424no—
231C23—5422no—
232C23—5421no—
233C23—10844no—
234C23C2×D108412yes—
235C23C2×D108411yescharacteristic
236Q8—2714no—
237Q8—5424no—
238Q8—5422no—
239Q8—5421no—
240Q8—5421no—
241D4—2714no—
242D4—54216no—
243D4—5424no—
244D4—5422no—
245D4—5421no—
246D4—5421no—
247D4—10844no—
248D4—10842no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—54216no—
254C2×C4—5424no—
255C2×C4—5422no—
256C2×C4—5422no—
257C2×C4—5421no—
258C2×C4—10842no—
259C2×C4D4×D27112yes—
260C2×C4D4×D27112yes—
261C2×C4C2×D108412yes—
262C2×C4C2×D108411yescharacteristic
263C2×C4C23×D27111yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—5424no—
272C22—10848no—
273C22—10844no—
274C22—10841no—
275C22—10842no—
276C22#2257613012yes—
277C22C22×D108411yescharacteristic
278C4—22no—
279C4—5424no—
280C4—10841no—
281C4—10842no—
282C4#2257613012yes—
283C4C2×D4×D27111yescharacteristic
284C4C2×D4×D27111yescharacteristic
285C2—22no—
286C2—42no—
287C2—10844no—
288C2—21681no—
289C2C2×#2257613011yescharacteristic
290C2#3177469712yes—
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#407×C271C2
socleC22×C271C22×D4
hypercenter#407D271
hypocenterC271#3005
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#16864654C2semidirect
2. C2×#22576193C2semidirect
3. C2×#22576188C2semidirect
4. #31774408C2semidirect
5. #31774061C2semidirect
6. #31774044C2semidirect
7. #31773907C2semidirect
8. #31773903C2semidirect
9. #407×C271C2semidirect
21. C2×Dic1084C22semidirect
22. C2×#16864643C22semidirect
23. C2×#16864643C22semidirect
24. C2×D2168C22semidirect
26. #22576139C22semidirect
27. #22576134C22semidirect
31. #22576085C22semidirect
32. #22576082C22semidirect
36. C22⋊C8×C271C22semidirect
45. C2×Dic542D4semidirect
50. C2×D1084D4semidirect
145. C271#3005semidirect
155. #407D271semidirect
182. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646542
C2×#225761932
C2×#225761882
#317744082
#317740612
#317740442
#317739072
#317739032
#407×C2712
#3005271