Finite Group Databasebeta

← #46847839

Subgroups — #46847839

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×#22576184C211yesmaximal, characteristic, complemented
2C2×D8×D271C211yesmaximal, characteristic, complemented
3#31774542C211yesmaximal, characteristic, complemented
4#31774532C211yesmaximal, characteristic, complemented
5#31774284C211yesmaximal, characteristic, complemented
6#31774149C211yesmaximal, characteristic, complemented
7#31774125C211yesmaximal, characteristic, complemented
8#31774113C211yesmaximal, characteristic, complemented
9#31774049C211yesmaximal, characteristic, complemented
10#31774039C211yesmaximal, characteristic, complemented
11#31774032C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13#31774007C211yesmaximal, characteristic, complemented
14#31773967C211yesmaximal, characteristic, complemented
15#459×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic, complemented
17C2×D4×D271—21no—
18C2×D4×D271C2211yescharacteristic
19C2×D4×D271C2211yescharacteristic, complemented
20C2×D4×D271C2211yescharacteristic, complemented
21#22576184—24no—
22C2×#16864643C2211yescharacteristic, complemented
23C2×#16864642C2211yescharacteristic, complemented
24C2×#16864642C2211yescharacteristic, complemented
25C2×D2168C2211yescharacteristic
26C2×#16864639C2211yescharacteristic
27C2×#16864639C2211yescharacteristic, complemented
28C2×#16864639C2211yescharacteristic, complemented
29D8×D271—24no—
30C2×#16864637C2211yescharacteristic, complemented
31C2×C8×D271C2211yescharacteristic, complemented
32#22576148C2211yescharacteristic, complemented
33#22576144—21no—
34#22576140C2211yescharacteristic, complemented
35#22576135C2211yescharacteristic, complemented
36#22576134—21no—
37#22576131—21no—
38#22576131C2211yescharacteristic
39#22576131C2211yescharacteristic, complemented
40#22576130—21no—
41#22576128—21no—
42#22576128—21no—
43C4×D1084C2211yescharacteristic
44C4×D1084C2211yescharacteristic, complemented
45C2×(C2×D271)⋊C4—21no—
46#22576091C2211yescharacteristic, complemented
47#22576085C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic, complemented
49#22576081C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic, complemented
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53#22576071C2211yescharacteristic, complemented
54C2×D8×C271C2211yescharacteristic, complemented
55C2×D8×C271C2211yescharacteristic, complemented
56#127×C271C2211yescharacteristic, complemented
57C4×D4×C271C2211yescharacteristic, complemented
58C4⋊C8×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63C23×D271—21no—
64#16864653—22no—
65#16864653—41no—
66C2×Dic542C2311yescharacteristic
67C2×D1084—21no—
68C2×D1084D411yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×D1084C2311yescharacteristic
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77D4×D271—22no—
78D4×D271—22no—
79D4×D271—22no—
80D4×D271—41no—
81D4×D271—41no—
82D4×D271—41no—
83D4×D271—44no—
84C22×Dic271—21no—
85C2×C4×D271—21no—
86C2×C4×D271C2311yescharacteristic
87C2×C4×D271C2311yescharacteristic
88#16864643—22no—
89#16864642—22no—
90#16864642—41no—
91D2168—22no—
92#16864639—22no—
93#16864639—41no—
94#16864639D414yescomplemented
95#16864637—22no—
96C2×C271⋊C8C2311yescharacteristic
97C2×C271⋊C8C2311yescharacteristic
98C8×D271—22no—
99C4⋊Dic271—21no—
100C4⋊Dic271D411yescharacteristic
101C4⋊Dic271C2311yescharacteristic
102C22⋊Dic271—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)⋊C4D411yescharacteristic
110(C2×D271)⋊C4D411yescharacteristic
111C2×D4×C271—21no—
112C2×D4×C271—21no—
113C2×D4×C271C2311yescharacteristic
114C2×D4×C271C2311yescharacteristic
115C2×D4×C271C2311yescharacteristic
116C22×C4×C271—21no—
117D8×C271—22no—
118D8×C271—22no—
119C2×C8×C271C2311yescharacteristic
120C2×C8×C271C2311yescharacteristic
121C4⋊C4×C271C2311yescharacteristic
122C22⋊C4×C271—21no—
123C42×C271C2311yescharacteristic
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271—42no—
129C22×D271—42no—
130C22×D271—42no—
131C22×D271—42no—
132C22×D271—81no—
133C22×D271—81no—
134C22×D271—81no—
135C22×D271C2×D411yescharacteristic
136C22×D271C2×D411yescharacteristic
137Dic542—21no—
138Dic542—21no—
139D1084—22no—
140D1084—21no—
141D1084—21no—
142D1084—21no—
143D1084—81no—
144D1084C2×D412yes—
145C22⋊D271—42no—
146C22⋊D271—42no—
147C22⋊D271—42no—
148C22⋊D271—42no—
149C22⋊D271—42no—
150C22⋊D271—42no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—42no—
154C2×Dic271C2×D411yescharacteristic
155C2×Dic271C2×D411yescharacteristic
156C4×D271—21no—
157C4×D271—21no—
158C4×D271—21no—
159C4×D271—21no—
160C4×D271—42no—
161C271⋊C8—21no—
162C271⋊C8C2×D412yescomplemented
163C23×C271—21no—
164C23×C271—21no—
165C23×C271—21no—
166D4×C271—21no—
167D4×C271—21no—
168D4×C271—21no—
169D4×C271—21no—
170D4×C271—21no—
171D4×C271—42no—
172D4×C271—81no—
173D4×C271C2×D412yescomplemented
174C2×C4×C271—21no—
175C2×C4×C271—81no—
176C2×C4×C271C2×D411yescharacteristic
177C2×C4×C271C2×D411yescharacteristic
178C2×C4×C271C2411yescharacteristic
179C8×C271—21no—
180C8×C271—21no—
181C22×C271—22no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—41no—
188C22×C271—41no—
189C22×C271—41no—
190C22×C271C22×D411yescharacteristic
191C2×D271—21no—
192C2×D271—21no—
193C2×D271—21no—
194C2×D271—22no—
195C2×D271—21no—
196C2×D271—21no—
197C2×D271—41no—
198C2×D271—41no—
199C2×D271—41no—
200C2×D271—41no—
201C2×D271—41no—
202C2×D271—84no—
203C2×D271—84no—
204C2×D271—84no—
205C2×D271AGL(1,ℤ/8)12yescomplemented
206C4×C271—21no—
207C4×C271—21no—
208C4×C271—41no—
209C4×C271C22×D411yescharacteristic
210C4×C271H(5,𝔽2)11yescharacteristic
211Dic271—21no—
212Dic271—21no—
213Dic271—41no—
214Dic271—41no—
215C2×C271—42no—
216C2×C271—81no—
217C2×C271—81no—
218C2×C271D4211yescharacteristic
219C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C271#57611yescharacteristic
221D271—22no—
222D271—41no—
223D271—42no—
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#459D27111yescharacteristic, complemented
236#450—2711no—
237#449—2711no—
238#447—2711no—
239#447—2711no—
240#442—2711no—
241#435—2711no—
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—5424no—
256C2×D8C2×D27111yescharacteristic, complemented
257C2×D8C2×D27111yescharacteristic, complemented
258C2×M16—2711no—
259C22×C8—2711no—
260#127C2×D27111yescharacteristic, complemented
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, complemented
275C2×C22⋊C4—5421no—
276#106—2711no—
277C4⋊C8C2×D27111yescharacteristic, complemented
278#103—2711no—
279#102—2711no—
280#102—2711no—
281#102—2711no—
282#102C2×D27111yescharacteristic, complemented
283#102C2×D27111yescharacteristic, complemented
284C22⋊C8—2711no—
285C22⋊C8—2711no—
286#97—2711no—
287C24—5421no—
288C24—5421no—
289C24—5421no—
290#41—5422no—
291#41—10841no—
292C2×Q8—2711no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2711no—
296C2×D4—2711no—
297C2×D4—2711no—
298C2×D4—5421no—
299C2×D4—5421no—
300C2×D4—5421no—
301C2×D4—5421no—
302C2×D4—5422no—
303C2×D4—5422no—
304C2×D4—5422no—
305C2×D4—5421no—
306C2×D4—5421no—
307C2×D4—5421no—
308C2×D4—10841no—
309C2×D4—10841no—
310C2×D4—10841no—
311C2×D4—10844no—
312C2×D4C22×D27111yescharacteristic
313C2×D4C22×D27111yescharacteristic
314C2×D4C22×D27111yescharacteristic
315C22×C4—21no—
316C22×C4—2711no—
317C22×C4—2711no—
318C22×C4—5421no—
319C22×C4—5421no—
320SD16—5422no—
321SD16—5422no—
322SD16—10841no—
323D8—22no—
324D8—22no—
325D8—2714no—
326D8—5422no—
327D8—5422no—
328D8—10841no—
329M16—5422no—
330C2×C8—2711no—
331C2×C8—2711no—
332C2×C8—5422no—
333C2×C8C22×D27111yescharacteristic
334C2×C8C22×D27111yescharacteristic
335C4⋊C4—2711no—
336C4⋊C4—2711no—
337C4⋊C4—5421no—
338C4⋊C4C22×D27111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—2711no—
341C22⋊C4—2711no—
342C22⋊C4—5421no—
343C22⋊C4—5421no—
344C22⋊C4—5421no—
345C22⋊C4—5421no—
346C22⋊C4—5421no—
347C22⋊C4—5421no—
348C22⋊C4—5421no—
349C42C22×D27111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—2711no—
354C23—2711no—
355C23—5421no—
356C23—5421no—
357C23—10842no—
358C23—10842no—
359C23—10842no—
360C23—10842no—
361C23—10842no—
362C23—10842no—
363C23—21681no—
364C23—21681no—
365C23—21681no—
366Q8—5421no—
367Q8—5421no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—2712no—
376D4—5422no—
377D4—5421no—
378D4—5421no—
379D4—5421no—
380D4—10842no—
381D4—10842no—
382D4—10842no—
383D4—10842no—
384D4—10842no—
385D4—10842no—
386D4—21681no—
387D4D4×D27112yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—2711no—
391C2×C4—2711no—
392C2×C4—5421no—
393C2×C4—5421no—
394C2×C4—5421no—
395C2×C4—5421no—
396C2×C4—5421no—
397C2×C4—5421no—
398C2×C4—10842no—
399C2×C4—10842no—
400C2×C4D4×D27111yescharacteristic
401C2×C4D4×D27111yescharacteristic
402C2×C4C23×D27111yescharacteristic
403C8—21no—
404C8—21no—
405C8—2712no—
406C8—5421no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2712no—
417C22—5421no—
418C22—5421no—
419C22—5421no—
420C22—5422no—
421C22—5421no—
422C22—5421no—
423C22—10841no—
424C22—10841no—
425C22—10841no—
426C22—10841no—
427C22—10841no—
428C22—21684no—
429C22—21684no—
430C22—21684no—
431C22C2×D4×D27111yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—5421no—
436C4—5421no—
437C4—10841no—
438C4—10841no—
439C4C2×D4×D27111yescharacteristic
440C4#2257620611yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—5422no—
445C2—10841no—
446C2—10842no—
447C2—21681no—
448C2#3177453111yescharacteristic
449C2AGL(1,ℤ/8)×D27111yescharacteristic
450C2#3177469811yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#459×C271C2
socleC22×C271C22×D4
hypercenter#459D271
hypocenterC271#3264
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. C2×D8×D271C2semidirect
3. #31774542C2semidirect
4. #31774532C2semidirect
5. #31774284C2semidirect
6. #31774149C2semidirect
7. #31774125C2semidirect
8. #31774113C2semidirect
9. #31774049C2semidirect
10. #31774039C2semidirect
11. #31774032C2semidirect
12. #31774029C2semidirect
13. #31774007C2semidirect
14. #31773967C2semidirect
15. #459×C271C2semidirect
16. C2×#16864653C22semidirect
19. C2×D4×D271C22semidirect
20. C2×D4×D271C22semidirect
22. C2×#16864643C22semidirect
23. C2×#16864642C22semidirect
24. C2×#16864642C22semidirect
27. C2×#16864639C22semidirect
28. C2×#16864639C22semidirect
30. C2×#16864637C22semidirect
31. C2×C8×D271C22semidirect
32. #22576148C22semidirect
34. #22576140C22semidirect
35. #22576135C22semidirect
39. #22576131C22semidirect
44. C4×D1084C22semidirect
46. #22576091C22semidirect
47. #22576085C22semidirect
48. #22576082C22semidirect
49. #22576081C22semidirect
50. #22576080C22semidirect
51. (C2×D271)⋊C8C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. #22576071C22semidirect
54. C2×D8×C271C22semidirect
55. C2×D8×C271C22semidirect
56. #127×C271C22semidirect
57. C4×D4×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #102×C271C22semidirect
60. #102×C271C22semidirect
94. #16864639D4semidirect
162. C271⋊C8C2×D4semidirect
173. D4×C271C2×D4semidirect
205. C2×D271AGL(1,ℤ/8)semidirect
225. C271#3264semidirect
235. #459D271semidirect
256. C2×D8C2×D271semidirect
257. C2×D8C2×D271semidirect
260. #127C2×D271semidirect
274. C4×D4C2×D271semidirect
277. C4⋊C8C2×D271semidirect
282. #102C2×D271semidirect
283. #102C2×D271semidirect
387. D4D4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
C2×D8×D2712
#317745422
#317745322
#317742842
#317741492
#317741252
#317741132
#317740492
#317740392
#317740322
#317740292
#317740072
#317739672
#459×C2712
#3264271