Finite Group Databasebeta

← #46845640

Subgroups — #46845640

Order 34688.

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

normal subgroups only

This group has 752 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
1C2×#22576206C211yesmaximal, characteristic
2C22×#16864641C211yesmaximal, characteristic, complemented
3#31774471C211yesmaximal, characteristic, complemented
4#31774058C214yesmaximal, complemented
5#31774046C214yesmaximal, complemented
6#31773903C211yesmaximal, characteristic, complemented
7#31773893C211yesmaximal, characteristic, complemented
8#31773865C211yesmaximal, characteristic, complemented
9C2×#137×C271C211yesmaximal, characteristic, complemented
10#22576206—24no—
11#22576206—24no—
12C2×#16864655C2212yes—
13C2×#16864654—21no—
14C2×#16864654C2211yescharacteristic
15C22×D1084—21no—
16C22×D1084C2211yescharacteristic
17C2×D4×D271—22no—
18C2×D4×D271C2212yes—
19C2×#16864641—24no—
20C2×#16864641C2214yescomplemented
21C22×C271⋊C8C2211yescharacteristic, complemented
22#22576158—21no—
23#22576149—21no—
24#22576143—22no—
25#22576136C2214yescomplemented
26#22576119C2211yescharacteristic, complemented
27#22576084C2212yescomplemented
28#22576082C2214yescomplemented
29#22576081C2212yescomplemented
30(C2×D271)⋊C8C2214yescomplemented
31C2×#41×C271C2211yescharacteristic
32C22×Q8×C271C2211yescharacteristic, complemented
33#137×C271—24no—
34C2×Q16×C271C2212yescomplemented
35C2×SD16×C271C2212yescomplemented
36C2×M16×C271C2211yescharacteristic, complemented
37C23×D271—22no—
38C23×D271—21no—
39#16864655—24no—
40#16864655—24no—
41#16864654—28no—
42#16864654—22no—
43#16864654—22no—
44C2×Dic542D411yescharacteristic
45C2×D1084—28no—
46C2×D1084—22no—
47C2×D1084—21no—
48C2×D1084—22no—
49C2×D1084D411yescharacteristic
50C2×D1084D414yes—
51C2×D1084C2312yes—
52C2×C22⋊D271—22no—
53C2×C22⋊D271D412yes—
54D4×D271—216no—
55D4×D271—24no—
56D4×D271—24no—
57C2×C4×D271—22no—
58C2×C4×D271C2312yes—
59#16864641—28no—
60C2×C271⋊C8—22no—
61C2×C271⋊C8C2312yes—
62C4⋊Dic271C2312yes—
63Dic271⋊C4—24no—
64C22⋊Dic271—21no—
65(C2×D271)⋊C4—24no—
66C4×Dic271—21no—
67#41×C271—22no—
68#41×C271D414yes—
69C2×Q8×C271—21no—
70C2×Q8×C271—22no—
71C2×Q8×C271—22no—
72C2×Q8×C271C2312yes—
73C2×Q8×C271C2311yescharacteristic
74C2×D4×C271—21no—
75C2×D4×C271C2311yescharacteristic
76C22×C4×C271—21no—
77C22×C4×C271—21no—
78C22×C4×C271C2311yescharacteristic
79Q16×C271—24no—
80SD16×C271—24no—
81M16×C271—22no—
82C2×C8×C271C2312yes—
83C22×D271—22no—
84C22×D271—28no—
85C22×D271—24no—
86C22×D271—84no—
87C22×D271—82no—
88C22×D271C2×D412yes—
89Dic542—22no—
90D1084—28no—
91D1084—22no—
92D1084—24no—
93D1084—22no—
94D1084C2×D414yes—
95C22⋊D271—28no—
96C22⋊D271—24no—
97C2×Dic271—22no—
98C2×Dic271C2×D412yes—
99C4×D271—28no—
100C4×D271—22no—
101C4×D271—22no—
102C271⋊C8—22no—
103C23×C271—21no—
104C23×C271C2×D411yescharacteristic
105Q8×C271—21no—
106Q8×C271—22no—
107Q8×C271—22no—
108Q8×C271—42no—
109Q8×C271C2×D412yes—
110D4×C271—24no—
111D4×C271—21no—
112D4×C271C2×D412yes—
113C2×C4×C271—21no—
114C2×C4×C271—24no—
115C2×C4×C271—22no—
116C2×C4×C271—42no—
117C2×C4×C271C2×D411yescharacteristic
118C2×C4×C271C2×D412yes—
119C2×C4×C271C2×D412yes—
120C2×C4×C271C2411yescharacteristic
121C8×C271—22no—
122C22×C271—22no—
123C22×C271—22no—
124C22×C271—42no—
125C22×C271#12012yes—
126C22×C271C22×D411yescharacteristic
127C2×D271—22no—
128C2×D271—24no—
129C2×D271—24no—
130C2×D271—44no—
131C2×D271—84no—
132C2×D271—88no—
133C4×C271—22no—
134C4×C271—42no—
135C4×C271#12012yes—
136C4×C271C22×D411yescharacteristic
137C4×C271C22×D411yescharacteristic
138Dic271—22no—
139Dic271—42no—
140C2×C271—22no—
141C2×C271—42no—
142C2×C271C2×#12011yescharacteristic
143C2×C271#57712yes—
144D271—42no—
145D271—44no—
146C271#300911yescharacteristic, complemented, Sylow
147#3009—2711nomaximal, Sylow
148C2×H(5,𝔽2)—2711no—
149C2×#137D27111yescharacteristic, complemented
150C22×SD16—2711no—
151#536—2711no—
152#452—2714no—
153#450—2714no—
154#418—2711no—
155#416—2711no—
156#408—2711no—
157H(5,𝔽2)—5424no—
158H(5,𝔽2)—5424no—
159C2×#41—2711no—
160C2×#41—2712no—
161C2×#41—5421no—
162C2×#41C2×D27111yescharacteristic
163C22×Q8C2×D27111yescharacteristic, complemented
164C22×D4—2712no—
165C22×D4—2711no—
166C22×D4—5421no—
167C22×D4—5422no—
168#137—24no—
169C2×Q16C2×D27112yescomplemented
170C2×SD16—2714no—
171C2×SD16—5424no—
172C2×SD16C2×D27112yescomplemented
173C2×M16C2×D27111yescharacteristic, complemented
174C22×C8—2711no—
175C4⋊Q8—5421no—
176#124—5421no—
177#123—5422no—
178C22⋊Q8—2714no—
179#117—2711no—
180#103—2712no—
181#102—2712no—
182#102—2714no—
183C22⋊C8—2714no—
184C24—5422no—
185C24—5421no—
186#41—22no—
187#41—5428no—
188#41—5424no—
189#41—5424no—
190#41—5422no—
191#41—5422no—
192#41C22⋊D27114yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—2711no—
197C2×Q8C22×D27112yes—
198C2×Q8C22×D27111yescharacteristic
199C2×D4—21no—
200C2×D4—2712no—
201C2×D4—2714no—
202C2×D4—2711no—
203C2×D4—2712no—
204C2×D4—5422no—
205C2×D4—5428no—
206C2×D4—5422no—
207C2×D4—5424no—
208C2×D4—54216no—
209C2×D4—5424no—
210C2×D4—5421no—
211C2×D4—5422no—
212C2×D4C22×D27111yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2712no—
216C22×C4—5422no—
217C22×C4C22×D27111yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—5428no—
221M16—22no—
222C2×C8—2712no—
223C2×C8—5422no—
224C2×C8C22×D27112yes—
225C4⋊C4—2712no—
226C4⋊C4—5424no—
227C22⋊C4—5424no—
228C22⋊C4—5421no—
229C42—5421no—
230C23—21no—
231C23—2712no—
232C23—5428no—
233C23—5422no—
234C23—5424no—
235C23—21682no—
236C23—21684no—
237C23D4×D27111yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—5422no—
243Q8C2×C22⋊D27112yes—
244D4—24no—
245D4—21no—
246D4—2714no—
247D4—5428no—
248D4—5428no—
249D4—5424no—
250D4—5424no—
251D4—5422no—
252D4—5422no—
253D4C2×C22⋊D27112yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—2712no—
259C2×C4—5428no—
260C2×C4—5422no—
261C2×C4—5422no—
262C2×C4—5422no—
263C2×C4D4×D27111yescharacteristic
264C2×C4D4×D27112yes—
265C2×C4C2×C22⋊D27112yes—
266C2×C4C23×D27111yescharacteristic
267C8—22no—
268C8—5422no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—5424no—
273C22—5422no—
274C22—5424no—
275C22—10844no—
276C22—21684no—
277C22—21688no—
278C22#2257612812yes—
279C22C2×D4×D27111yescharacteristic
280C4—22no—
281C4—42no—
282C4—5422no—
283C4—10842no—
284C4#2257612812yes—
285C4C2×D4×D27111yescharacteristic
286C4C22×C22⋊D27111yescharacteristic
287C2—22no—
288C2—42no—
289C2—10842no—
290C2—10844no—
291C2C2×#2257612811yescharacteristic
292C2#3177470612yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#137×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576206C2
hypercenterC2×#137D271
hypocenterC271#3009
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#16864641C2semidirect
3. #31774471C2semidirect
4. #31774058C2semidirect
5. #31774046C2semidirect
6. #31773903C2semidirect
7. #31773893C2semidirect
8. #31773865C2semidirect
9. C2×#137×C271C2semidirect
20. C2×#16864641C22semidirect
21. C22×C271⋊C8C22semidirect
25. #22576136C22semidirect
26. #22576119C22semidirect
27. #22576084C22semidirect
28. #22576082C22semidirect
29. #22576081C22semidirect
30. (C2×D271)⋊C8C22semidirect
32. C22×Q8×C271C22semidirect
34. C2×Q16×C271C22semidirect
35. C2×SD16×C271C22semidirect
36. C2×M16×C271C22semidirect
146. C271#3009semidirect
149. C2×#137D271semidirect
163. C22×Q8C2×D271semidirect
169. C2×Q16C2×D271semidirect
172. C2×SD16C2×D271semidirect
173. C2×M16C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225762062
C22×#168646412
#317744712
#317740582
#317740462
#317739032
#317738932
#317738652
C2×#137×C2712
#3009271