Finite Group Databasebeta

← #46848149

Subgroups — #46848149

Order 34688.

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

normal subgroups only

This group has 454 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#31774710C211yesmaximal, characteristic, complemented
2#31774703C212yesmaximal, complemented
3#31774698C211yesmaximal, characteristic, complemented
4#31774169C211yesmaximal, characteristic, complemented
5#31774164C211yesmaximal, characteristic, complemented
6#31774157C212yesmaximal, complemented
7#31774074C212yesmaximal, complemented
8#31774069C212yesmaximal, complemented
9#31774015C211yesmaximal, characteristic, complemented
10#31774015C211yesmaximal, characteristic, complemented
11#495×C271C211yesmaximal, characteristic, complemented
12#22576208C2212yescomplemented
13#22576205C2212yes—
14#22576192—22no—
15#22576191C2212yes—
16#22576191C2212yescomplemented
17#22576189—22no—
18#22576187—22no—
19#22576186C2212yes—
20#22576186C2212yescomplemented
21#22576184—22no—
22#22576183—21no—
23#22576182C2211yescharacteristic
24#22576182C2211yescharacteristic
25#22576181—22no—
26#22576179—21no—
27Q16×D271—21no—
28SD16×D271—22no—
29D8×D271—21no—
30#22576166C2211yescharacteristic
31#22576166C2212yescomplemented
32#22576166C2211yescharacteristic, complemented
33#22576096C2211yescharacteristic, complemented
34#22576096C2211yescharacteristic, complemented
35#22576088C2211yescharacteristic
36#22576088C2211yescharacteristic, complemented
37#22576088C2212yescomplemented
38#22576079C2212yescomplemented
39#22576077C2212yescomplemented
40C2×Q16×C271C2211yescharacteristic, complemented
41C2×SD16×C271C2212yescomplemented
42C2×D8×C271C2211yescharacteristic
43#124×C271C2212yescomplemented
44C4×C8×C271C2211yescharacteristic, complemented
45#16864655—22no—
46#16864655—22no—
47#16864654—42no—
48#16864654C2311yescharacteristic
49#16864654C2312yes—
50#16864654C2311yescharacteristic
51#16864653—22no—
52#16864653—22no—
53Q8×D271—22no—
54Q8×D271—22no—
55C2×C22⋊D271—42no—
56D4×D271—22no—
57D4×D271—22no—
58Dic1084D411yescharacteristic
59Dic1084D411yescharacteristic
60#16864644—22no—
61#16864644—22no—
62#16864643D411yescharacteristic
63#16864643D411yescharacteristic
64#16864643D412yes—
65#16864642—22no—
66#16864642—22no—
67#16864641—22no—
68#16864641—22no—
69D2168D411yescharacteristic
70D2168D411yescharacteristic
71#16864639—22no—
72#16864639—22no—
73#16864638C2311yescharacteristic
74#16864638C2312yes—
75#16864638C2311yescharacteristic
76#16864637—21no—
77#16864637—22no—
78#16864637—21no—
79C8×D271—21no—
80C8×D271—22no—
81C8×D271—21no—
82C2×Q8×C271C2312yes—
83C2×D4×C271C2312yes—
84Q16×C271—21no—
85Q16×C271—21no—
86SD16×C271—22no—
87SD16×C271—22no—
88D8×C271—21no—
89D8×C271—21no—
90C2×C8×C271C2311yescharacteristic
91C2×C8×C271C2311yescharacteristic
92C22⋊C4×C271—42no—
93C42×C271C2311yescharacteristic
94C22×D271—42no—
95Dic542—42no—
96Dic542C2×D411yescharacteristic
97Dic542C2×D412yes—
98Dic542C2×D411yescharacteristic
99D1084—42no—
100D1084C2×D411yescharacteristic
101D1084C2×D412yes—
102D1084C2×D411yescharacteristic
103C22⋊D271—21no—
104C22⋊D271—22no—
105C22⋊D271—21no—
106C22⋊D271—42no—
107C22⋊D271—42no—
108C2×Dic271—42no—
109C4×D271—21no—
110C4×D271—22no—
111C4×D271—21no—
112C4×D271—42no—
113C4×D271—42no—
114C271⋊C8—21no—
115C271⋊C8—22no—
116C271⋊C8—21no—
117C23×C271—22no—
118Q8×C271—22no—
119Q8×C271—22no—
120D4×C271—22no—
121D4×C271—22no—
122C2×C4×C271—21no—
123C2×C4×C271—22no—
124C2×C4×C271C2411yescharacteristic
125C8×C271C2×D411yescharacteristic
126C8×C271C2×D411yescharacteristic
127C8×C271C2×D411yescharacteristic
128C8×C271C2×D411yescharacteristic
129C22×C271—42no—
130C22×C271—82no—
131C22×C271H(5,𝔽2)11yescharacteristic
132C2×D271—21no—
133C2×D271—22no—
134C2×D271—21no—
135C2×D271—162no—
136C4×C271—41no—
137C4×C271—42no—
138C4×C271C22×D411yescharacteristic
139C4×C271C22×D411yescharacteristic
140Dic271—21no—
141Dic271—22no—
142Dic271—21no—
143C2×C271—21no—
144C2×C271—82no—
145C2×C271D4211yescharacteristic
146D271—41no—
147D271—42no—
148D271—41no—
149C271#329011yescharacteristic, complemented, Sylow
150#3290—2711nomaximal, Sylow
151#578—2711no—
152#577—2712no—
153#576—2711no—
154#495D27111yescharacteristic, complemented
155#473—2711no—
156#472—2711no—
157#471—2712no—
158#455—2712no—
159#454—2712no—
160#443—2711no—
161#443—2711no—
1622−1+4—2712no—
163H(5,𝔽2)—2712no—
164#137—2712no—
165#137—2712no—
166#137—5422no—
167#137—5422no—
168AGL(1,ℤ/8)—2712no—
169AGL(1,ℤ/8)—2712no—
170AGL(1,ℤ/8)—5422no—
171AGL(1,ℤ/8)—5422no—
172#135—2711no—
173#135—2711no—
174#135—5421no—
175#135—5421no—
176#135—5422no—
177C2×Q16—5421no—
178C2×Q16C2×D27111yescharacteristic, complemented
179C2×SD16—5422no—
180C2×SD16C2×D27112yescomplemented
181C2×D8—5421no—
182C2×D8C2×D27111yescharacteristic
183#131—2711no—
184#131—2712no—
185#131—2711no—
186#124C2×D27112yescomplemented
187#108—2711no—
188#108—2711no—
189#104—2711no—
190#104—2712no—
191#104—2711no—
192#101—2712no—
193#100—2712no—
194C4×C8C2×D27111yescharacteristic, complemented
195#41—2711no—
196#41—2712no—
197#41—2711no—
198#41—5422no—
199#41—5422no—
200#41—5422no—
201#41—5422no—
202#41—10842no—
203C2×Q8—5422no—
204C2×Q8—5422no—
205C2×Q8C22×D27112yes—
206C2×D4—5422no—
207C2×D4—5422no—
208C2×D4—10842no—
209C2×D4C22×D27112yes—
210Q16—21no—
211Q16—21no—
212Q16—2711no—
213Q16—2711no—
214Q16—5422no—
215Q16—5422no—
216SD16—22no—
217SD16—22no—
218SD16—2711no—
219SD16—2711no—
220SD16—2712no—
221SD16—5422no—
222SD16—5422no—
223SD16—5422no—
224SD16—5422no—
225D8—21no—
226D8—21no—
227D8—2711no—
228D8—2711no—
229D8—5422no—
230D8—5422no—
231M16—2711no—
232M16—2712no—
233M16—2711no—
234M16—5421no—
235M16—5422no—
236M16—5421no—
237C2×C8—5421no—
238C2×C8—5422no—
239C2×C8—5421no—
240C2×C8C22×D27111yescharacteristic
241C2×C8C22×D27111yescharacteristic
242C22⋊C4—42no—
243C42C22×D27111yescharacteristic
244C23—22no—
245C23—10842no—
246Q8—22no—
247Q8—22no—
248Q8—2711no—
249Q8—2712no—
250Q8—2711no—
251Q8—10842no—
252D4—22no—
253D4—22no—
254D4—2711no—
255D4—2712no—
256D4—2711no—
257D4—5421no—
258D4—5422no—
259D4—5421no—
260D4—10842no—
261D4—10842no—
262D4—10842no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—5421no—
266C2×C4—5422no—
267C2×C4—5421no—
268C2×C4—10842no—
269C2×C4—10842no—
270C2×C4—10842no—
271C2×C4C23×D27111yescharacteristic
272C8—5421no—
273C8—5422no—
274C8—5421no—
275C8D4×D27111yescharacteristic
276C8D4×D27111yescharacteristic
277C8D4×D27111yescharacteristic
278C8D4×D27111yescharacteristic
279C22—42no—
280C22—82no—
281C22—5421no—
282C22—5422no—
283C22—5421no—
284C22—43362no—
285C22#2257620511yescharacteristic
286C4—41no—
287C4—42no—
288C4—5421no—
289C4—5422no—
290C4—5421no—
291C4C2×D4×D27111yescharacteristic
292C4C2×D4×D27111yescharacteristic
293C2—21no—
294C2—82no—
295C2—10841no—
296C2—10842no—
297C2—10841no—
298C2#3177453211yescharacteristic
subgroupG/H
centerC2#31774532
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#495×C271C2
socleC2×C271D42
hypercenter#495D271
hypocenterC271#3290
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #31774710C2semidirect
2. #31774703C2semidirect
3. #31774698C2semidirect
4. #31774169C2semidirect
5. #31774164C2semidirect
6. #31774157C2semidirect
7. #31774074C2semidirect
8. #31774069C2semidirect
9. #31774015C2semidirect
10. #31774015C2semidirect
11. #495×C271C2semidirect
12. #22576208C22semidirect
16. #22576191C22semidirect
20. #22576186C22semidirect
31. #22576166C22semidirect
32. #22576166C22semidirect
33. #22576096C22semidirect
34. #22576096C22semidirect
36. #22576088C22semidirect
37. #22576088C22semidirect
38. #22576079C22semidirect
39. #22576077C22semidirect
40. C2×Q16×C271C22semidirect
41. C2×SD16×C271C22semidirect
43. #124×C271C22semidirect
44. C4×C8×C271C22semidirect
149. C271#3290semidirect
154. #495D271semidirect
178. C2×Q16C2×D271semidirect
180. C2×SD16C2×D271semidirect
186. #124C2×D271semidirect
194. C4×C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747102
#317747032
#317746982
#317741692
#317741642
#317741572
#317740742
#317740692
#317740152
#495×C2712
#3290271