Finite Group Databasebeta

← #46847865

Subgroups — #46847865

Order 34688.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576184C211yesmaximal, characteristic, complemented
2C2×SD16×D271C211yesmaximal, characteristic, complemented
3#31774555C211yesmaximal, characteristic, complemented
4D4×D1084C211yesmaximal, characteristic, complemented
5#31774287C211yesmaximal, characteristic, complemented
6#31774146C211yesmaximal, characteristic, complemented
7#31774132C211yesmaximal, characteristic, complemented
8#31774098C211yesmaximal, characteristic, complemented
9#31774046C211yesmaximal, characteristic, complemented
10#31774043C211yesmaximal, characteristic, complemented
11#31774037C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13#31773997C211yesmaximal, characteristic, complemented
14#31773967C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic, complemented
17C2×Q8×D271C2211yescharacteristic, complemented
18C22×D1084—21no—
19C2×D4×D271C2211yescharacteristic
20C2×D4×D271C2211yescharacteristic
21#22576184—24no—
22C2×#16864643C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic, complemented
24C2×#16864642C2211yescharacteristic
25C2×#16864641C2211yescharacteristic, complemented
26C2×#16864641C2211yescharacteristic, complemented
27SD16×D271—24no—
28C2×#16864639C2211yescharacteristic, complemented
29C2×#16864639C2211yescharacteristic, complemented
30C2×#16864637C2211yescharacteristic, complemented
31C2×C8×D271C2211yescharacteristic, complemented
32#22576157C2211yescharacteristic
33#22576146—21no—
34#22576137—21no—
35#22576136C2211yescharacteristic, complemented
36#22576135C2211yescharacteristic
37#22576135C2211yescharacteristic
38#22576134—21no—
39#22576133—21no—
40#22576131C2211yescharacteristic, complemented
41#22576130—21no—
42#22576130—21no—
43C4×Dic542C2211yescharacteristic, complemented
44C4×D1084C2211yescharacteristic
45(C22⋊D271)⋊C4—21no—
46#22576093C2211yescharacteristic, complemented
47#22576083C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53#22576071C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic
55C2×D8×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic, complemented
57C4×D4×C271C2211yescharacteristic
58C4⋊C8×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864653—22no—
64#16864653—41no—
65C2×Dic542—21no—
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—41no—
69C2×D1084—21no—
70C2×D1084—21no—
71C2×D1084—21no—
72C2×D1084—21no—
73C2×D1084—44no—
74C2×D1084D411yescharacteristic
75C2×D1084C2311yescharacteristic
76C2×D1084C2311yescharacteristic
77C2×C22⋊D271—21no—
78C2×C22⋊D271—21no—
79C2×C22⋊D271—21no—
80C2×C22⋊D271—21no—
81D4×D271—22no—
82D4×D271—22no—
83D4×D271—41no—
84D4×D271—41no—
85C22×Dic271—21no—
86C2×C4×D271—21no—
87C2×C4×D271C2311yescharacteristic
88C2×C4×D271C2311yescharacteristic
89#16864643—22no—
90#16864643—22no—
91#16864642D414yescomplemented
92#16864641—22no—
93#16864641—41no—
94#16864639—22no—
95#16864639—41no—
96#16864637—22no—
97C2×C271⋊C8C2311yescharacteristic
98C2×C271⋊C8C2311yescharacteristic
99C8×D271—22no—
100C4⋊Dic271D411yescharacteristic
101C4⋊Dic271C2311yescharacteristic
102Dic271⋊C4—21no—
103Dic271⋊C4—21no—
104C22⋊Dic271—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108(C2×D271)⋊C4D411yescharacteristic
109(C2×D271)⋊C4D411yescharacteristic
110C4×Dic271—21no—
111C2×Q8×C271C2311yescharacteristic
112C2×D4×C271C2311yescharacteristic
113C2×D4×C271C2311yescharacteristic
114C22×C4×C271—21no—
115SD16×C271—22no—
116D8×C271—22no—
117C2×C8×C271C2311yescharacteristic
118C2×C8×C271C2311yescharacteristic
119C4⋊C4×C271C2311yescharacteristic
120C22⋊C4×C271—21no—
121C22⋊C4×C271—21no—
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—81no—
131C22×D271—81no—
132C22×D271C2×D411yescharacteristic
133C22×D271C2×D411yescharacteristic
134Dic542—21no—
135Dic542—42no—
136Dic542C2×D412yescomplemented
137D1084—22no—
138D1084—21no—
139D1084—21no—
140D1084—21no—
141D1084—21no—
142D1084—42no—
143D1084—42no—
144D1084—42no—
145D1084—81no—
146C22⋊D271—42no—
147C22⋊D271—42no—
148C22⋊D271—42no—
149C22⋊D271—42no—
150C2×Dic271—21no—
151C2×Dic271—21no—
152C2×Dic271—42no—
153C2×Dic271C2×D411yescharacteristic
154C2×Dic271C2×D411yescharacteristic
155C4×D271—21no—
156C4×D271—21no—
157C4×D271—21no—
158C4×D271—21no—
159C4×D271—42no—
160C271⋊C8—21no—
161C271⋊C8C2×D412yescomplemented
162C23×C271—21no—
163C23×C271—21no—
164Q8×C271—21no—
165Q8×C271—21no—
166D4×C271—21no—
167D4×C271—21no—
168D4×C271—21no—
169D4×C271C2×D412yes—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—42no—
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2×D411yescharacteristic
175C2×C4×C271C2411yescharacteristic
176C8×C271—21no—
177C8×C271—21no—
178C22×C271—22no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271C22×D411yescharacteristic
185C2×D271—21no—
186C2×D271—21no—
187C2×D271—21no—
188C2×D271—21no—
189C2×D271—21no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—84no—
197C2×D271—84no—
198C2×D271AGL(1,ℤ/8)12yescomplemented
199C4×C271—21no—
200C4×C271—21no—
201C4×C271—41no—
202C4×C271—41no—
203C4×C271C22×D411yescharacteristic
204C4×C271H(5,𝔽2)11yescharacteristic
205Dic271—21no—
206Dic271—21no—
207Dic271—22no—
208Dic271—41no—
209C2×C271—42no—
210C2×C271—81no—
211C2×C271D4211yescharacteristic
212C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C271#57711yescharacteristic
214D271—22no—
215D271—41no—
216D271—81no—
217D271—81no—
218C271#326511yescharacteristic, complemented, Sylow
219#3265—2711nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2711no—
221C22×SD16—2711no—
222#548—2711no—
223D42—2711no—
224#496—2711no—
225#468—2711no—
226#466—2711no—
227#463D27111yescharacteristic, complemented
228#460—2711no—
229#450—2711no—
230#449—2711no—
231#448—2711no—
232#447—2711no—
233#440—2711no—
234#435—2711no—
235C2×#41—2711no—
236C22×Q8—2711no—
237C22×D4—2711no—
238C22×D4—2711no—
239C22×D4—5421no—
240AGL(1,ℤ/8)—5424no—
241C2×SD16—2711no—
242C2×SD16—2711no—
243C2×SD16—2711no—
244C2×SD16—2711no—
245C2×SD16—2711no—
246C2×SD16—5424no—
247C2×SD16C2×D27111yescharacteristic
248C2×D8—2711no—
249C2×D8—2711no—
250C2×D8C2×D27111yescharacteristic
251C2×M16—2711no—
252C22×C8—2711no—
253#127—2711no—
254#124—5421no—
255#124C2×D27111yescharacteristic, complemented
256C22⋊Q8—2711no—
257C22⋊Q8—5421no—
258#121—2711no—
259#121—2711no—
260#121—2711no—
261#121—5421no—
262#121—5421no—
263#120—5421no—
264#120—5421no—
265C4×Q8—2711no—
266C4×D4—2711no—
267C4×D4—5421no—
268C4×D4C2×D27111yescharacteristic
269#107—2711no—
270C4⋊C8C2×D27111yescharacteristic, complemented
271#103—2711no—
272#103C2×D27111yescharacteristic, complemented
273#102—2711no—
274#102—2711no—
275#102—2711no—
276#102C2×D27111yescharacteristic, complemented
277C22⋊C8—2711no—
278C22⋊C8—2711no—
279#97—2711no—
280C24—5421no—
281C24—5421no—
282#41—5422no—
283#41—10841no—
284C2×Q8—2711no—
285C2×Q8—5421no—
286C2×Q8—5422no—
287C2×Q8—10841no—
288C2×Q8C22×D27111yescharacteristic
289C2×D4—2711no—
290C2×D4—2711no—
291C2×D4—2711no—
292C2×D4—5421no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5422no—
297C2×D4—5422no—
298C2×D4—5421no—
299C2×D4—5421no—
300C2×D4—5421no—
301C2×D4—5421no—
302C2×D4—10844no—
303C2×D4—10841no—
304C2×D4—10841no—
305C2×D4C22×D27111yescharacteristic
306C2×D4C22×D27111yescharacteristic
307C22×C4—21no—
308C22×C4—2711no—
309C22×C4—2711no—
310C22×C4—5421no—
311C22×C4—5421no—
312SD16—22no—
313SD16—2714no—
314SD16—5422no—
315SD16—5422no—
316SD16—5422no—
317SD16—10841no—
318D8—22no—
319D8—5422no—
320D8—10841no—
321M16—5422no—
322C2×C8—2711no—
323C2×C8—2711no—
324C2×C8—5422no—
325C2×C8C22×D27111yescharacteristic
326C2×C8C22×D27111yescharacteristic
327C4⋊C4—2711no—
328C4⋊C4—2711no—
329C4⋊C4—5421no—
330C4⋊C4—5421no—
331C4⋊C4C22×D27111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—2711no—
336C22⋊C4—2711no—
337C22⋊C4—5421no—
338C22⋊C4—5421no—
339C22⋊C4—5421no—
340C22⋊C4—5421no—
341C42—5421no—
342C42C22×D27111yescharacteristic
343C23—21no—
344C23—21no—
345C23—2711no—
346C23—2711no—
347C23—5421no—
348C23—5421no—
349C23—10842no—
350C23—10842no—
351C23—10842no—
352C23—10842no—
353C23—21681no—
354C23—21681no—
355Q8—21no—
356Q8—21no—
357Q8—2712no—
358Q8—5421no—
359Q8—10842no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—5422no—
364D4—5421no—
365D4—5421no—
366D4—5421no—
367D4—5421no—
368D4—10842no—
369D4—10842no—
370D4—10842no—
371D4—10842no—
372D4—10842no—
373D4—10842no—
374D4—10842no—
375D4—21681no—
376D4D4×D27112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—2711no—
381C2×C4—2711no—
382C2×C4—5421no—
383C2×C4—5421no—
384C2×C4—5421no—
385C2×C4—5421no—
386C2×C4—5421no—
387C2×C4—5421no—
388C2×C4—10842no—
389C2×C4—10842no—
390C2×C4D4×D27111yescharacteristic
391C2×C4D4×D27111yescharacteristic
392C2×C4C23×D27111yescharacteristic
393C8—21no—
394C8—21no—
395C8—2712no—
396C8—5421no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—2712no—
404C22—5421no—
405C22—5421no—
406C22—5421no—
407C22—5421no—
408C22—5421no—
409C22—10841no—
410C22—10841no—
411C22—10841no—
412C22—10841no—
413C22—10841no—
414C22—10841no—
415C22—21684no—
416C22—21684no—
417C22C2×D4×D27111yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—5421no—
423C4—5421no—
424C4—5422no—
425C4—10841no—
426C4C2×D4×D27111yescharacteristic
427C4#2257620611yescharacteristic
428C2—42no—
429C2—81no—
430C2—5422no—
431C2—10841no—
432C2—21681no—
433C2—21681no—
434C2#3177453111yescharacteristic
435C2AGL(1,ℤ/8)×D27111yescharacteristic
436C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
hypercenter#463D271
hypocenterC271#3265
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. C2×SD16×D271C2semidirect
3. #31774555C2semidirect
4. D4×D1084C2semidirect
5. #31774287C2semidirect
6. #31774146C2semidirect
7. #31774132C2semidirect
8. #31774098C2semidirect
9. #31774046C2semidirect
10. #31774043C2semidirect
11. #31774037C2semidirect
12. #31774029C2semidirect
13. #31773997C2semidirect
14. #31773967C2semidirect
15. #463×C271C2semidirect
16. C2×#16864653C22semidirect
17. C2×Q8×D271C22semidirect
22. C2×#16864643C22semidirect
23. C2×#16864643C22semidirect
25. C2×#16864641C22semidirect
26. C2×#16864641C22semidirect
28. C2×#16864639C22semidirect
29. C2×#16864639C22semidirect
30. C2×#16864637C22semidirect
31. C2×C8×D271C22semidirect
35. #22576136C22semidirect
40. #22576131C22semidirect
43. C4×Dic542C22semidirect
46. #22576093C22semidirect
47. #22576083C22semidirect
48. #22576082C22semidirect
49. #22576082C22semidirect
51. (C2×D271)⋊C8C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. #22576071C22semidirect
56. #124×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #103×C271C22semidirect
60. #102×C271C22semidirect
91. #16864642D4semidirect
136. Dic542C2×D4semidirect
161. C271⋊C8C2×D4semidirect
198. C2×D271AGL(1,ℤ/8)semidirect
218. C271#3265semidirect
227. #463D271semidirect
255. #124C2×D271semidirect
270. C4⋊C8C2×D271semidirect
272. #103C2×D271semidirect
276. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
C2×SD16×D2712
#317745552
D4×D10842
#317742872
#317741462
#317741322
#317740982
#317740462
#317740432
#317740372
#317740292
#317739972
#317739672
#463×C2712
#3265271