Finite Group Databasebeta

← #46847849

Subgroups — #46847849

Order 34688.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576188C211yesmaximal, characteristic, complemented
2C22×D2168C211yesmaximal, characteristic, complemented
3#31774543C211yesmaximal, characteristic, complemented
4D4×D1084C211yesmaximal, characteristic, complemented
5#31774286C211yesmaximal, characteristic, complemented
6#31774150C211yesmaximal, characteristic, complemented
7#31774126C211yesmaximal, characteristic, complemented
8#31774113C211yesmaximal, characteristic, complemented
9#31774093C211yesmaximal, characteristic, complemented
10#31774050C211yesmaximal, characteristic, complemented
11#31774044C211yesmaximal, characteristic, complemented
12#31774033C211yesmaximal, characteristic, complemented
13#31774033C211yesmaximal, characteristic, complemented
14#31774010C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic, complemented
17C22×D1084C2211yescharacteristic
18C22×D1084C2211yescharacteristic, complemented
19C22×D1084C2211yescharacteristic, complemented
20C2×D4×D271—21no—
21#22576188—24no—
22C2×#16864643C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic, complemented
24C2×#16864643C2211yescharacteristic, complemented
25C2×D2168—24no—
26C2×D2168C2211yescharacteristic
27C2×D2168C2211yescharacteristic
28C2×D2168C2211yescharacteristic, complemented
29C2×D2168C2211yescharacteristic, complemented
30C2×D2168C2211yescharacteristic, complemented
31C2×D2168C2211yescharacteristic, complemented
32#22576157C2211yescharacteristic, complemented
33#22576148C2211yescharacteristic, complemented
34#22576145—21no—
35#22576139C2211yescharacteristic, complemented
36#22576135—21no—
37#22576134C2211yescharacteristic
38#22576134C2211yescharacteristic, complemented
39#22576134C2211yescharacteristic, complemented
40#22576133—21no—
41#22576130—21no—
42#22576130—21no—
43#22576128—21no—
44C4×D1084C2211yescharacteristic, complemented
45C4×D1084C2211yescharacteristic, complemented
46C22⋊C4×D271—21no—
47#22576092C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic, complemented
51#22576082C2211yescharacteristic, complemented
52#22576082C2211yescharacteristic, complemented
53#22576082C2211yescharacteristic, complemented
54C2×M16×C271C2211yescharacteristic, complemented
55C22×C8×C271C2211yescharacteristic, complemented
56C4×D4×C271C2211yescharacteristic
57C4⋊C8×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic, complemented
59C22⋊C8×C271C2211yescharacteristic, complemented
60#97×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63C23×D271—21no—
64#16864654—22no—
65#16864654—41no—
66C2×Dic542C2311yescharacteristic
67C2×D1084—22no—
68C2×D1084—22no—
69C2×D1084—22no—
70C2×D1084—21no—
71C2×D1084—21no—
72C2×D1084—21no—
73C2×D1084—21no—
74C2×D1084—21no—
75C2×D1084—41no—
76C2×D1084—41no—
77C2×D1084—41no—
78C2×D1084C2311yescharacteristic
79C2×D1084C2311yescharacteristic
80C2×D1084C2311yescharacteristic
81C2×D1084C2311yescharacteristic
82C2×D1084C2311yescharacteristic
83C2×C22⋊D271—21no—
84C2×C22⋊D271—21no—
85C2×C22⋊D271—21no—
86C2×C22⋊D271—21no—
87D4×D271—44no—
88C2×C4×D271—21no—
89C2×C4×D271—21no—
90C2×C4×D271—21no—
91#16864643—22no—
92#16864643—22no—
93#16864643—41no—
94D2168—22no—
95D2168—22no—
96D2168—22no—
97D2168—22no—
98D2168—41no—
99D2168D414yescomplemented
100C4⋊Dic271C2311yescharacteristic
101C4⋊Dic271C2311yescharacteristic
102Dic271⋊C4—21no—
103C22⋊Dic271—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108(C2×D271)⋊C4—21no—
109(C2×D271)⋊C4—21no—
110(C2×D271)⋊C4—21no—
111C2×D4×C271D411yescharacteristic
112C22×C4×C271C2311yescharacteristic
113C22×C4×C271C2311yescharacteristic
114M16×C271—22no—
115C2×C8×C271—22no—
116C2×C8×C271C2311yescharacteristic
117C2×C8×C271C2311yescharacteristic
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C4⋊C4×C271D411yescharacteristic
121C22⋊C4×C271D411yescharacteristic
122C22⋊C4×C271D411yescharacteristic
123C42×C271C2311yescharacteristic
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—21no—
128C22×D271—21no—
129C22×D271—42no—
130C22×D271—42no—
131C22×D271—42no—
132C22×D271—42no—
133C22×D271—42no—
134C22×D271—42no—
135C22×D271—81no—
136C22×D271—81no—
137C22×D271—81no—
138Dic542—21no—
139Dic542—21no—
140D1084—21no—
141D1084—21no—
142D1084—21no—
143D1084—21no—
144D1084—21no—
145D1084—21no—
146D1084—21no—
147D1084—21no—
148D1084—42no—
149D1084—42no—
150D1084—42no—
151D1084—42no—
152D1084—81no—
153D1084C2×D412yes—
154D1084C2×D412yescomplemented
155C22⋊D271—42no—
156C22⋊D271—42no—
157C22⋊D271—42no—
158C22⋊D271—81no—
159C2×Dic271—21no—
160C2×Dic271—21no—
161C2×Dic271—21no—
162C4×D271—42no—
163C4×D271—42no—
164C4×D271—81no—
165C23×C271C2×D411yescharacteristic
166C23×C271C2×D411yescharacteristic
167D4×C271—22no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271C2×D411yescharacteristic
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2×D411yescharacteristic
175C2×C4×C271C2×D411yescharacteristic
176C2×C4×C271C2411yescharacteristic
177C8×C271—21no—
178C8×C271—21no—
179C8×C271—21no—
180C8×C271C2×D412yescomplemented
181C22×C271—21no—
182C22×C271—21no—
183C22×C271—21no—
184C22×C271—21no—
185C22×C271—21no—
186C22×C271AGL(1,ℤ/8)12yescomplemented
187C22×C271C22×D411yescharacteristic
188C2×D271—22no—
189C2×D271—22no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—41no—
197C2×D271—41no—
198C2×D271—41no—
199C2×D271—41no—
200C2×D271—41no—
201C2×D271—41no—
202C2×D271—41no—
203C2×D271—84no—
204C2×D271—84no—
205C2×D271—84no—
206C4×C271—21no—
207C4×C271—21no—
208C4×C271—21no—
209C4×C271—21no—
210C4×C271C22×D411yescharacteristic
211C4×C271H(5,𝔽2)11yescharacteristic
212Dic271—41no—
213Dic271—41no—
214Dic271—41no—
215C2×C271—22no—
216C2×C271—41no—
217C2×C271D4211yescharacteristic
218C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C271#57611yescharacteristic
220D271—42no—
221D271—42no—
222D271—81no—
223D271—81no—
224D271—81no—
225C271#326411yescharacteristic, complemented, Sylow
226#3264—2711nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2711no—
228C22×D8—2711no—
229#546—2711no—
230D42—2711no—
231#496—2711no—
232#469—2711no—
233#465—2711no—
234#463—2711no—
235#459—2711no—
236#450—2711no—
237#449—2711no—
238#447—2711no—
239#447—2711no—
240#442—2711no—
241#435D27111yescharacteristic, complemented
242C2×#41—2711no—
243C22×D4—2711no—
244C22×D4—2711no—
245C22×D4—2711no—
246C22×D4—5421no—
247AGL(1,ℤ/8)—5424no—
248C2×SD16—2711no—
249C2×SD16—2711no—
250C2×SD16—2711no—
251C2×D8—2711no—
252C2×D8—2711no—
253C2×D8—2711no—
254C2×D8—2711no—
255C2×D8—2711no—
256C2×D8—2711no—
257C2×D8—5424no—
258C2×M16C2×D27111yescharacteristic, complemented
259C22×C8C2×D27111yescharacteristic, complemented
260#127—2711no—
261#124—2711no—
262#123—5421no—
263C22⋊Q8—2711no—
264#121—2711no—
265#121—2711no—
266#121—2711no—
267#121—5421no—
268#121—5421no—
269#120—5421no—
270#120—5421no—
271#120—5421no—
272C4×D4—2711no—
273C4×D4—2711no—
274C4×D4C2×D27111yescharacteristic
275C2×C22⋊C4—5421no—
276#106—2711no—
277C4⋊C8C2×D27111yescharacteristic, complemented
278#103—2711no—
279#102—2711no—
280#102—2711no—
281#102—2711no—
282#102—2711no—
283#102—2711no—
284C22⋊C8C2×D27111yescharacteristic, complemented
285C22⋊C8C2×D27111yescharacteristic, complemented
286#97C2×D27111yescharacteristic, complemented
287C24—5421no—
288C24—5421no—
289C24—5421no—
290#41—5422no—
291#41—10841no—
292C2×Q8—2711no—
293C2×D4—2711no—
294C2×D4—2711no—
295C2×D4—2711no—
296C2×D4—2711no—
297C2×D4—2711no—
298C2×D4—5422no—
299C2×D4—5422no—
300C2×D4—5422no—
301C2×D4—5421no—
302C2×D4—5421no—
303C2×D4—5421no—
304C2×D4—5421no—
305C2×D4—5421no—
306C2×D4—5421no—
307C2×D4—5421no—
308C2×D4—5421no—
309C2×D4—5421no—
310C2×D4—10841no—
311C2×D4—10841no—
312C2×D4—10841no—
313C2×D4—10844no—
314C2×D4D108411yescharacteristic
315C22×C4—5421no—
316C22×C4—5421no—
317C22×C4—5421no—
318C22×C4C22×D27111yescharacteristic
319C22×C4C22×D27111yescharacteristic
320SD16—5422no—
321SD16—5422no—
322SD16—10841no—
323D8—2714no—
324D8—5422no—
325D8—5422no—
326D8—5422no—
327D8—5422no—
328D8—10841no—
329M16—22no—
330C2×C8—22no—
331C2×C8C22×D27111yescharacteristic
332C2×C8C22×D27111yescharacteristic
333C2×C8C22×D27111yescharacteristic
334C2×C8C22×D27111yescharacteristic
335C4⋊C4—2711no—
336C4⋊C4—2711no—
337C4⋊C4—5421no—
338C4⋊C4D108411yescharacteristic
339C22⋊C4—5421no—
340C22⋊C4—5421no—
341C22⋊C4—5421no—
342C22⋊C4—5421no—
343C22⋊C4—5421no—
344C22⋊C4—5421no—
345C22⋊C4—5421no—
346C22⋊C4—5421no—
347C22⋊C4D108411yescharacteristic
348C22⋊C4D108411yescharacteristic
349C42C22×D27111yescharacteristic
350C23—5421no—
351C23—5421no—
352C23—5421no—
353C23—5421no—
354C23—5421no—
355C23—10842no—
356C23—10842no—
357C23—10842no—
358C23—10842no—
359C23—10842no—
360C23—10842no—
361C23—21681no—
362C23—21681no—
363C23—21681no—
364C23C2×D108411yescharacteristic
365C23C2×D108411yescharacteristic
366Q8—5421no—
367Q8—5421no—
368D4—22no—
369D4—2712no—
370D4—2712no—
371D4—5421no—
372D4—5421no—
373D4—5421no—
374D4—5421no—
375D4—5421no—
376D4—5421no—
377D4—5421no—
378D4—5421no—
379D4—10842no—
380D4—10842no—
381D4—10842no—
382D4—10842no—
383D4—10842no—
384D4—10842no—
385D4—10842no—
386D4—21681no—
387D4—21681no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—5421no—
393C2×C4—5421no—
394C2×C4—5421no—
395C2×C4—10842no—
396C2×C4—10842no—
397C2×C4—21681no—
398C2×C4C2×D108411yescharacteristic
399C2×C4C2×D108411yescharacteristic
400C2×C4C2×D108411yescharacteristic
401C2×C4C2×D108411yescharacteristic
402C2×C4C23×D27111yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D27112yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—5422no—
413C22—5422no—
414C22—10841no—
415C22—10841no—
416C22—10841no—
417C22—10841no—
418C22—10841no—
419C22—10841no—
420C22—10841no—
421C22—10841no—
422C22—10841no—
423C22—10841no—
424C22—10841no—
425C22—10841no—
426C22—10841no—
427C22—21684no—
428C22—21684no—
429C22—21684no—
430C22#2257618812yescomplemented
431C22C22×D108411yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—10841no—
437C4—10841no—
438C4—10841no—
439C4C2×D4×D27111yescharacteristic
440C4#2257620611yescharacteristic
441C2—22no—
442C2—41no—
443C2—10842no—
444C2—10842no—
445C2—21681no—
446C2—21681no—
447C2—21681no—
448C2D4×D108411yescharacteristic
449C2C2×#2257618811yescharacteristic
450C2#3177470211yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#435×C271C2
socleC22×C271C22×D4
hypercenter#435D271
hypocenterC271#3264
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. C22×D2168C2semidirect
3. #31774543C2semidirect
4. D4×D1084C2semidirect
5. #31774286C2semidirect
6. #31774150C2semidirect
7. #31774126C2semidirect
8. #31774113C2semidirect
9. #31774093C2semidirect
10. #31774050C2semidirect
11. #31774044C2semidirect
12. #31774033C2semidirect
13. #31774033C2semidirect
14. #31774010C2semidirect
15. #435×C271C2semidirect
16. C2×#16864654C22semidirect
18. C22×D1084C22semidirect
19. C22×D1084C22semidirect
22. C2×#16864643C22semidirect
23. C2×#16864643C22semidirect
24. C2×#16864643C22semidirect
28. C2×D2168C22semidirect
29. C2×D2168C22semidirect
30. C2×D2168C22semidirect
31. C2×D2168C22semidirect
32. #22576157C22semidirect
33. #22576148C22semidirect
35. #22576139C22semidirect
38. #22576134C22semidirect
39. #22576134C22semidirect
44. C4×D1084C22semidirect
45. C4×D1084C22semidirect
47. #22576092C22semidirect
48. #22576085C22semidirect
49. #22576082C22semidirect
50. #22576082C22semidirect
51. #22576082C22semidirect
52. #22576082C22semidirect
53. #22576082C22semidirect
54. C2×M16×C271C22semidirect
55. C22×C8×C271C22semidirect
57. C4⋊C8×C271C22semidirect
58. C22⋊C8×C271C22semidirect
59. C22⋊C8×C271C22semidirect
60. #97×C271C22semidirect
99. D2168D4semidirect
154. D1084C2×D4semidirect
180. C8×C271C2×D4semidirect
186. C22×C271AGL(1,ℤ/8)semidirect
225. C271#3264semidirect
241. #435D271semidirect
258. C2×M16C2×D271semidirect
259. C22×C8C2×D271semidirect
277. C4⋊C8C2×D271semidirect
284. C22⋊C8C2×D271semidirect
285. C22⋊C8C2×D271semidirect
286. #97C2×D271semidirect
406. C8D4×D271semidirect
430. C22#22576188semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
C22×D21682
#317745432
D4×D10842
#317742862
#317741502
#317741262
#317741132
#317740932
#317740502
#317740442
#317740332
#317740102
#435×C2712
#3264271