Finite Group Databasebeta

← #46845593

Subgroups — #46845593

Order 34688.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×D4×D271C211yesmaximal, characteristic, complemented
2C2×#22576188C211yesmaximal, characteristic, complemented
3C2×#22576186C211yesmaximal, characteristic, complemented
4#31774409C211yesmaximal, characteristic, complemented
5#31774049C214yesmaximal, complemented
6#31774032C214yesmaximal, complemented
7#31773901C211yesmaximal, characteristic, complemented
8#31773860C211yesmaximal, characteristic, complemented
9#418×C271C211yesmaximal, characteristic, complemented
10C24×D271—21no—
11C2×#16864654C2211yescharacteristic, complemented
12C22×D1084C2211yescharacteristic
13C22×C22⋊D271—21no—
14C2×D4×D271—24no—
15C2×D4×D271—22no—
16C2×D4×D271—24no—
17C2×D4×D271C2214yes—
18C22×C4×D271C2211yescharacteristic
19#22576188—24no—
20#22576186—24no—
21C2×#16864643C2212yescomplemented
22C2×#16864642C2212yescomplemented
23C2×D2168C2212yescomplemented
24C2×#16864639C2212yescomplemented
25C2×#16864638C2211yescharacteristic, complemented
26#22576144—21no—
27#22576140C2212yescomplemented
28#22576135C2212yescomplemented
29#22576130—21no—
30C4×D1084—22no—
31#22576080C2214yescomplemented
32(C2×D271)⋊C8C2214yescomplemented
33C22×D4×C271C2211yescharacteristic
34C2×M16×C271C2211yescharacteristic, complemented
35#117×C271C2211yescharacteristic, complemented
36#102×C271C2214yescomplemented
37C23×D271—28no—
38C23×D271—24no—
39C23×D271—21no—
40C23×D271—48no—
41C23×D271D411yescharacteristic
42#16864654—22no—
43#16864654—22no—
44C2×Dic542C2311yescharacteristic, complemented
45C2×D1084—22no—
46C2×D1084—21no—
47C2×D1084—22no—
48C2×D1084D414yescomplemented
49C2×D1084C2312yes—
50C2×D1084C2311yescharacteristic, complemented
51C2×C22⋊D271—28no—
52C2×C22⋊D271—24no—
53C2×C22⋊D271—21no—
54D4×D271—216no—
55D4×D271—28no—
56D4×D271—28no—
57C22×Dic271D411yescharacteristic
58C2×C4×D271—22no—
59C2×C4×D271—22no—
60C2×C4×D271—21no—
61C2×C4×D271D412yes—
62C2×C4×D271C2312yes—
63#16864643—24no—
64#16864642—24no—
65D2168—24no—
66#16864639—24no—
67#16864638—22no—
68C2×C271⋊C8C2312yescomplemented
69C4⋊Dic271—22no—
70(C2×D271)⋊C4—22no—
71(C2×D271)⋊C4—22no—
72C24×C271—21no—
73C2×D4×C271—21no—
74C2×D4×C271—22no—
75C2×D4×C271D414yescomplemented
76C2×D4×C271C2312yes—
77C22×C4×C271C2311yescharacteristic
78M16×C271—22no—
79C2×C8×C271C2312yescomplemented
80C4⋊C4×C271C2312yescomplemented
81C22⋊C4×C271—21no—
82C42×C271—21no—
83C22×D271—22no—
84C22×D271—24no—
85C22×D271—216no—
86C22×D271—24no—
87C22×D271—22no—
88C22×D271—21no—
89C22×D271—432no—
90C22×D271—48no—
91C22×D271—44no—
92C22×D271—48no—
93C22×D271C2×D412yes—
94Dic542—21no—
95Dic542—21no—
96D1084—24no—
97D1084—22no—
98D1084—21no—
99D1084—21no—
100D1084—44no—
101D1084C2×D414yes—
102C22⋊D271—216no—
103C22⋊D271—42no—
104C2×Dic271—22no—
105C2×Dic271—21no—
106C2×Dic271C2×D412yes—
107C4×D271—24no—
108C4×D271—22no—
109C4×D271—22no—
110C4×D271—42no—
111C271⋊C8—22no—
112C23×C271—24no—
113C23×C271—22no—
114C23×C271—44no—
115C23×C271C2×D411yescharacteristic
116D4×C271—24no—
117D4×C271—22no—
118D4×C271C2×D414yes—
119C2×C4×C271—22no—
120C2×C4×C271C2×D411yescharacteristic
121C2×C4×C271C2×D412yes—
122C2×C4×C271C2×D412yes—
123C2×C4×C271C2411yescharacteristic
124C8×C271—22no—
125C22×C271—22no—
126C22×C271—24no—
127C22×C271—48no—
128C22×C271—44no—
129C22×C271#12012yes—
130C22×C271C22×D411yescharacteristic
131C2×D271—28no—
132C2×D271—22no—
133C2×D271—22no—
134C2×D271—24no—
135C2×D271—432no—
136C2×D271—48no—
137C2×D271—44no—
138C2×D271—41no—
139C2×D271—42no—
140C2×D271AGL(1,ℤ/8)14yescomplemented
141C4×C271—42no—
142C4×C271#12012yes—
143C4×C271C22×D411yescharacteristic
144C4×C271C22×D411yescharacteristic
145Dic271—22no—
146Dic271—41no—
147C2×C271—22no—
148C2×C271—44no—
149C2×C271C2×#12011yescharacteristic
150C2×C271C2×AGL(1,ℤ/8)12yes—
151D271—24no—
152D271—44no—
153D271—81no—
154C271#300311yescharacteristic, complemented, Sylow
155#3003—2711nomaximal, Sylow
156C23×D4—2711no—
157C2×AGL(1,ℤ/8)—2711no—
158C2×AGL(1,ℤ/8)—2711no—
159#525—2711no—
160#450—2714no—
161#447—2714no—
162#418—2711no—
163#418D27111yescharacteristic, complemented
164#407—2711no—
165C25—5421no—
166C2×#41—2711no—
167C22×D4—2714no—
168C22×D4—2711no—
169C22×D4—5422no—
170C22×D4—5421no—
171C22×D4—5424no—
172C22×D4—5424no—
173C22×D4C2×D27111yescharacteristic
174C23×C4—2711no—
175AGL(1,ℤ/8)—5424no—
176AGL(1,ℤ/8)—5424no—
177C2×SD16—2712no—
178C2×SD16—2712no—
179C2×D8—2712no—
180C2×D8—2712no—
181C2×M16—2711no—
182C2×M16C2×D27111yescharacteristic, complemented
183#123—5421no—
184C22⋊Q8—2712no—
185#121—2712no—
186#120—5421no—
187C4×D4—5422no—
188#117C2×D27111yescharacteristic, complemented
189#102—2714no—
190#102C2×D27114yescomplemented
191C22⋊C8—2714no—
192C24—21no—
193C24—2711no—
194C24—5424no—
195C24—5428no—
196C24—5421no—
197C24—10848no—
198#41—5422no—
199#41—5422no—
200C2×Q8—2711no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—2714no—
204C2×D4—2712no—
205C2×D4—2711no—
206C2×D4—5422no—
207C2×D4—5424no—
208C2×D4—5428no—
209C2×D4—54216no—
210C2×D4—5428no—
211C2×D4—5428no—
212C2×D4—5421no—
213C2×D4—5421no—
214C2×D4—5422no—
215C2×D4D108414yescomplemented
216C2×D4C22×D27112yes—
217C22×C4—2711no—
218C22×C4—2712no—
219C22×C4—2712no—
220C22×C4—5422no—
221C22×C4—5422no—
222C22×C4—5421no—
223C22×C4C22×D27111yescharacteristic
224SD16—5424no—
225SD16—5424no—
226D8—5424no—
227D8—5424no—
228M16—22no—
229M16—5422no—
230C2×C8—2712no—
231C2×C8C22×D27112yescomplemented
232C4⋊C4—5422no—
233C4⋊C4C22×D27112yescomplemented
234C22⋊C4—21no—
235C22⋊C4—5422no—
236C22⋊C4—5422no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—2712no—
242C23—5424no—
243C23—54216no—
244C23—5422no—
245C23—5424no—
246C23—5422no—
247C23—5421no—
248C23—10844no—
249C23—10848no—
250C23—108432no—
251C23—10848no—
252C23D4×D27111yescharacteristic
253Q8—5421no—
254Q8—5421no—
255D4—24no—
256D4—22no—
257D4—2714no—
258D4—54216no—
259D4—5424no—
260D4—5421no—
261D4—5422no—
262D4—5421no—
263D4—10844no—
264D4—10842no—
265D4C2×D108414yes—
266C2×C4—22no—
267C2×C4—2712no—
268C2×C4—5422no—
269C2×C4—5424no—
270C2×C4—5422no—
271C2×C4—5422no—
272C2×C4—5421no—
273C2×C4—10842no—
274C2×C4D4×D27111yescharacteristic
275C2×C4D4×D27112yes—
276C2×C4C2×D108412yes—
277C2×C4C23×D27111yescharacteristic
278C8—22no—
279C8—5422no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—2714no—
285C22—5428no—
286C22—5422no—
287C22—5422no—
288C22—5424no—
289C22—10848no—
290C22—108432no—
291C22—10844no—
292C22—10841no—
293C22—10842no—
294C22#2257613012yes—
295C22C2×D4×D27111yescharacteristic
296C4—42no—
297C4—5422no—
298C4—10841no—
299C4#2257613012yes—
300C4C2×D4×D27111yescharacteristic
301C4C22×D108411yescharacteristic
302C2—22no—
303C2—44no—
304C2—5424no—
305C2—10844no—
306C2—21681no—
307C2C2×#2257613011yescharacteristic
308C2AGL(1,ℤ/8)×D27112yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#418×C271C2
socleC22×C271C22×D4
hypercenter#418D271
hypocenterC271#3003
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D271C2semidirect
2. C2×#22576188C2semidirect
3. C2×#22576186C2semidirect
4. #31774409C2semidirect
5. #31774049C2semidirect
6. #31774032C2semidirect
7. #31773901C2semidirect
8. #31773860C2semidirect
9. #418×C271C2semidirect
11. C2×#16864654C22semidirect
21. C2×#16864643C22semidirect
22. C2×#16864642C22semidirect
23. C2×D2168C22semidirect
24. C2×#16864639C22semidirect
25. C2×#16864638C22semidirect
27. #22576140C22semidirect
28. #22576135C22semidirect
31. #22576080C22semidirect
32. (C2×D271)⋊C8C22semidirect
34. C2×M16×C271C22semidirect
35. #117×C271C22semidirect
36. #102×C271C22semidirect
44. C2×Dic542C23semidirect
48. C2×D1084D4semidirect
50. C2×D1084C23semidirect
68. C2×C271⋊C8C23semidirect
75. C2×D4×C271D4semidirect
79. C2×C8×C271C23semidirect
80. C4⋊C4×C271C23semidirect
140. C2×D271AGL(1,ℤ/8)semidirect
154. C271#3003semidirect
163. #418D271semidirect
182. C2×M16C2×D271semidirect
188. #117C2×D271semidirect
190. #102C2×D271semidirect
215. C2×D4D1084semidirect
231. C2×C8C22×D271semidirect
233. C4⋊C4C22×D271semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2712
C2×#225761882
C2×#225761862
#317744092
#317740492
#317740322
#317739012
#317738602
#418×C2712
#3003271