Finite Group Databasebeta

← #471×D271 #46846059

Subgroups — #471×D271 #46846059

Order 34688.

468 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#137×D271C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D271C211yesmaximal, characteristic
3C2×SD16×D271C211yesmaximal, characteristic, complemented
4#131×D271C211yesmaximal, characteristic, complemented
5#31774163C211yesmaximal, characteristic, complemented
6#31774162C211yesmaximal, characteristic, complemented
7#31774161C211yesmaximal, characteristic, complemented
8#31774160C211yesmaximal, characteristic, complemented
9#31774159C211yesmaximal, characteristic, complemented
10#31774158C211yesmaximal, characteristic, complemented
11#31774157C211yesmaximal, characteristic, complemented
12#108×D271C211yesmaximal, characteristic, complemented
13#101×D271C211yesmaximal, characteristic, complemented
14#100×D271C211yesmaximal, characteristic, complemented
15#471×C271C211yesmaximal, characteristic, complemented
16#41×D271C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576193C2211yescharacteristic, complemented
20#22576192—21no—
21#22576191C2211yescharacteristic, complemented
22#22576190C2211yescharacteristic, complemented
23#22576189—21no—
24#22576188C2211yescharacteristic
25#22576187—21no—
26#22576186C2211yescharacteristic
27#22576185C2211yescharacteristic
28#22576184—21no—
29Q16×D271—21no—
30C2×#16864643C2211yescharacteristic, complemented
31C2×#16864642C2211yescharacteristic, complemented
32C2×#16864641C2211yescharacteristic, complemented
33SD16×D271—21no—
34SD16×D271—21no—
35SD16×D271—21no—
36SD16×D271—21no—
37SD16×D271—21no—
38SD16×D271—21no—
39D8×D271—21no—
40#22576168—21no—
41#22576168C2211yescharacteristic, complemented
42#22576167C2211yescharacteristic, complemented
43#22576166—21no—
44#22576166C2211yescharacteristic, complemented
45M16×D271—21no—
46M16×D271C2211yescharacteristic
47M16×D271C2211yescharacteristic
48M16×D271C2211yescharacteristic
49C2×C8×D271—21no—
50C2×C8×D271C2211yescharacteristic
51#22576096C2211yescharacteristic, complemented
52#22576095C2211yescharacteristic, complemented
53#22576095C2211yescharacteristic, complemented
54#22576079C2211yescharacteristic, complemented
55#22576078C2211yescharacteristic, complemented
56#22576078C2211yescharacteristic, complemented
57#22576077C2211yescharacteristic, complemented
58#22576076C2211yescharacteristic, complemented
59#22576076C2211yescharacteristic, complemented
60#137×C271C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C271C2211yescharacteristic
62C2×SD16×C271C2211yescharacteristic, complemented
63#131×C271C2211yescharacteristic, complemented
64#108×C271C2211yescharacteristic, complemented
65#101×C271C2211yescharacteristic, complemented
66#100×C271C2211yescharacteristic, complemented
67C23×D271—21no—
68#16864655D411yescharacteristic
69#16864654—21no—
70#16864654C2311yescharacteristic
71#16864653—21no—
72#16864653D411yescharacteristic
73C2×Dic542—21no—
74C2×Dic542C2311yescharacteristic
75Q8×D271—21no—
76Q8×D271—21no—
77Q8×D271—21no—
78Q8×D271—21no—
79Q8×D271D411yescharacteristic
80C2×D1084C2311yescharacteristic
81C2×C22⋊D271—21no—
82D4×D271—21no—
83D4×D271—21no—
84D4×D271—21no—
85D4×D271—21no—
86D4×D271—21no—
87D4×D271D411yescharacteristic
88C2×C4×D271—21no—
89C2×C4×D271—21no—
90C2×C4×D271C2311yescharacteristic
91Dic1084—21no—
92#16864644—21no—
93#16864644—21no—
94#16864643—21no—
95#16864643—21no—
96#16864643—21no—
97#16864643—21no—
98#16864642—21no—
99#16864642—21no—
100#16864642—21no—
101#16864642—21no—
102#16864641—21no—
103#16864641—21no—
104#16864641—21no—
105#16864641—21no—
106D2168—21no—
107#16864639—21no—
108#16864639—21no—
109#16864638—21no—
110#16864638C2311yescharacteristic
111#16864638C2311yescharacteristic
112#16864638C2311yescharacteristic
113#16864637—21no—
114#16864637—21no—
115#16864637—21no—
116#16864637—21no—
117#16864637—21no—
118C2×C271⋊C8—21no—
119C2×C271⋊C8C2311yescharacteristic
120C8×D271—21no—
121C8×D271—21no—
122C8×D271—21no—
123C8×D271—21no—
124C8×D271—21no—
125C8×D271D411yescharacteristic
126C8×D271D411yescharacteristic
127C8×D271D411yescharacteristic
128C8×D271D411yescharacteristic
129#41×C271C2311yescharacteristic
130C2×Q8×C271C2311yescharacteristic
131C2×D4×C271C2311yescharacteristic
132Q16×C271—21no—
133SD16×C271—21no—
134SD16×C271—21no—
135SD16×C271—21no—
136SD16×C271—21no—
137D8×C271—21no—
138M16×C271—21no—
139M16×C271C2311yescharacteristic
140M16×C271C2311yescharacteristic
141M16×C271C2311yescharacteristic
142C2×C8×C271—21no—
143C2×C8×C271C2311yescharacteristic
144C22×D271—21no—
145C22×D271—21no—
146C22×D271—41no—
147C22×D271—41no—
148C22×D271—82no—
149C22×D271#4111yescharacteristic
150Dic542—21no—
151Dic542—21no—
152Dic542—21no—
153Dic542—41no—
154Dic542—41no—
155Dic542C2×D411yescharacteristic
156D1084—21no—
157D1084—21no—
158D1084—21no—
159D1084C2×D411yescharacteristic
160C22⋊D271—21no—
161C22⋊D271—21no—
162C22⋊D271—21no—
163C22⋊D271—41no—
164C22⋊D271—41no—
165C2×Dic271—21no—
166C2×Dic271—21no—
167C2×Dic271#4111yescharacteristic
168C4×D271—21no—
169C4×D271—21no—
170C4×D271—21no—
171C4×D271—41no—
172C4×D271—41no—
173C4×D271C2×D411yescharacteristic
174C4×D271C2×D411yescharacteristic
175C4×D271C2×D411yescharacteristic
176C4×D271C2×D411yescharacteristic
177C271⋊C8—21no—
178C271⋊C8—21no—
179C271⋊C8—21no—
180C271⋊C8C2×D411yescharacteristic
181C271⋊C8C2×D411yescharacteristic
182C23×C271—21no—
183Q8×C271—21no—
184Q8×C271—21no—
185Q8×C271C2×D411yescharacteristic
186D4×C271—21no—
187D4×C271—21no—
188D4×C271—21no—
189D4×C271C2×D411yescharacteristic
190C2×C4×C271—21no—
191C2×C4×C271—21no—
192C2×C4×C271C2411yescharacteristic
193C8×C271—21no—
194C8×C271—21no—
195C8×C271—21no—
196C8×C271C2×D411yescharacteristic
197C8×C271C2×D411yescharacteristic
198C22×C271—21no—
199C22×C271—41no—
200C22×C271—81no—
201C22×C271C2×#4111yescharacteristic
202C2×D271—22no—
203C2×D271—21no—
204C2×D271—42no—
205C2×D271—41no—
206C2×D271—82no—
207C2×D271—82no—
208C2×D271—81no—
209C2×D271#12111yescharacteristic
210C2×D271#12111yescharacteristic
211C4×C271—21no—
212C4×C271—41no—
213C4×C271C22×D411yescharacteristic
214C4×C271C22×D411yescharacteristic
215Dic271—21no—
216Dic271—41no—
217Dic271#12111yescharacteristic
218Dic271#12111yescharacteristic
219C2×C271—21no—
220C2×C271—41no—
221C2×C271—81no—
222C2×C271C2×#12111yescharacteristic
223D271—21no—
224D271—41no—
225D271—81no—
226D271#47112yesdirect product factor
227C271C2×#47111yescharacteristic, complemented, Sylow
228C2×#471—2711nomaximal, Sylow
229C2×#137—2711no—
230C2×AGL(1,ℤ/8)—2711no—
231C22×SD16—2711no—
232C2×#131—2711no—
233#471—2711no—
234#471—2711no—
235#471—2711no—
236#471—2711no—
237#471—2711no—
238#471—2711no—
239#471—2711no—
240#471D27111yescharacteristic, direct product factor
241C2×#108—2711no—
242C2×#101—2711no—
243C2×#100—2711no—
244C2×#41—2711no—
245C22×Q8—2711no—
246C22×D4—2711no—
247#137—2711no—
248#137—2711no—
249#137—2711no—
250#137—5421no—
251#137—5421no—
252#137C2×D27111yescharacteristic, complemented
253AGL(1,ℤ/8)—2711no—
254AGL(1,ℤ/8)—2711no—
255AGL(1,ℤ/8)—2711no—
256AGL(1,ℤ/8)—5421no—
257AGL(1,ℤ/8)—5421no—
258AGL(1,ℤ/8)C2×D27111yescharacteristic
259C2×Q16—5421no—
260C2×SD16—2711no—
261C2×SD16—2711no—
262C2×SD16—2711no—
263C2×SD16—5421no—
264C2×SD16—5421no—
265C2×SD16—5421no—
266C2×SD16—5421no—
267C2×SD16—5421no—
268C2×SD16—5421no—
269C2×SD16C2×D27111yescharacteristic, complemented
270C2×D8—5421no—
271#131—2711no—
272#131—2711no—
273#131—2711no—
274#131—5421no—
275#131—5421no—
276#131C2×D27111yescharacteristic, complemented
277C2×M16—2711no—
278C2×M16—2711no—
279C2×M16—2711no—
280C2×M16—5421no—
281C22×C8—2711no—
282C22×C8—5421no—
283#108—2711no—
284#108—2711no—
285#108—2711no—
286#108C2×D27111yescharacteristic, complemented
287#101—2711no—
288#101—2711no—
289#101—2711no—
290#101C2×D27111yescharacteristic, complemented
291#100—2711no—
292#100—2711no—
293#100—2711no—
294#100C2×D27111yescharacteristic, complemented
295C24—5421no—
296#41—2711no—
297#41—2711no—
298#41—2711no—
299#41—5421no—
300#41—5421no—
301#41C22×D27111yescharacteristic
302C2×Q8—2711no—
303C2×Q8—2711no—
304C2×Q8—5421no—
305C2×Q8—5421no—
306C2×Q8—5421no—
307C2×Q8—5421no—
308C2×Q8—5421no—
309C2×Q8C22×D27111yescharacteristic
310C2×D4—2711no—
311C2×D4—2711no—
312C2×D4—5421no—
313C2×D4—5421no—
314C2×D4—5421no—
315C2×D4—5421no—
316C2×D4—5421no—
317C2×D4—5421no—
318C2×D4C22×D27111yescharacteristic
319C22×C4—2711no—
320C22×C4—5421no—
321C22×C4—5421no—
322Q16—21no—
323Q16—5421no—
324Q16—5421no—
325Q16—5421no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—5421no—
331SD16—5421no—
332SD16—5421no—
333SD16—5421no—
334SD16—5421no—
335SD16—5421no—
336SD16—5421no—
337SD16—5421no—
338SD16—5421no—
339SD16—5421no—
340SD16—5421no—
341SD16—5421no—
342D8—21no—
343D8—5421no—
344D8—5421no—
345D8—5421no—
346M16—21no—
347M16—2711no—
348M16—2711no—
349M16—2711no—
350M16—5421no—
351M16—5421no—
352M16—5421no—
353M16—5421no—
354M16—5421no—
355M16—5421no—
356M16C22×D27111yescharacteristic
357M16C22×D27111yescharacteristic
358M16C22×D27111yescharacteristic
359C2×C8—21no—
360C2×C8—2711no—
361C2×C8—2711no—
362C2×C8—2711no—
363C2×C8—2711no—
364C2×C8—2711no—
365C2×C8—5421no—
366C2×C8—5421no—
367C2×C8—5421no—
368C2×C8—5421no—
369C2×C8—5421no—
370C2×C8—5421no—
371C2×C8C22×D27111yescharacteristic
372C23—21no—
373C23—2711no—
374C23—5421no—
375C23—5421no—
376C23—10841no—
377C23—10841no—
378C23—21682no—
379Q8—21no—
380Q8—21no—
381Q8—2711no—
382Q8—5421no—
383Q8—5421no—
384Q8—5421no—
385Q8—10841no—
386Q8—10841no—
387Q8D4×D27111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—2711no—
392D4—5421no—
393D4—5421no—
394D4—5421no—
395D4—5421no—
396D4—5421no—
397D4—5421no—
398D4—10841no—
399D4—10841no—
400D4D4×D27111yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—2711no—
404C2×C4—2711no—
405C2×C4—2711no—
406C2×C4—2711no—
407C2×C4—2711no—
408C2×C4—5421no—
409C2×C4—5421no—
410C2×C4—5421no—
411C2×C4—5421no—
412C2×C4—5421no—
413C2×C4—10841no—
414C2×C4—10841no—
415C2×C4C23×D27111yescharacteristic
416C8—21no—
417C8—21no—
418C8—21no—
419C8—2711no—
420C8—2711no—
421C8—5421no—
422C8—5421no—
423C8—5421no—
424C8D4×D27111yescharacteristic
425C8D4×D27111yescharacteristic
426C22—21no—
427C22—41no—
428C22—81no—
429C22—2711no—
430C22—2711no—
431C22—5422no—
432C22—5421no—
433C22—10842no—
434C22—10841no—
435C22—21681no—
436C22—21682no—
437C22—21682no—
438C22#41×D27111yescharacteristic
439C4—21no—
440C4—41no—
441C4—2711no—
442C4—2711no—
443C4—5421no—
444C4—10841no—
445C4C2×D4×D27111yescharacteristic
446C4C2×D4×D27111yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—2712no—
451C2—5421no—
452C2—10841no—
453C2—21681no—
454C2#121×D27111yescharacteristic
subgroupG/H
centerC2#121×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#471×C271C2
socleC2×C271C2×#121
prime radicalAGL(1,ℤ/8)×D271C2
hypercenter#471D271
hypocenterC271C2×#471
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #137×D271C2semidirect
3. C2×SD16×D271C2semidirect
4. #131×D271C2semidirect
5. #31774163C2semidirect
6. #31774162C2semidirect
7. #31774161C2semidirect
8. #31774160C2semidirect
9. #31774159C2semidirect
10. #31774158C2semidirect
11. #31774157C2semidirect
12. #108×D271C2semidirect
13. #101×D271C2semidirect
14. #100×D271C2semidirect
15. #471×C271C2semidirect
19. #22576193C22semidirect
21. #22576191C22semidirect
22. #22576190C22semidirect
30. C2×#16864643C22semidirect
31. C2×#16864642C22semidirect
32. C2×#16864641C22semidirect
41. #22576168C22semidirect
42. #22576167C22semidirect
44. #22576166C22semidirect
51. #22576096C22semidirect
52. #22576095C22semidirect
53. #22576095C22semidirect
54. #22576079C22semidirect
55. #22576078C22semidirect
56. #22576078C22semidirect
57. #22576077C22semidirect
58. #22576076C22semidirect
59. #22576076C22semidirect
60. #137×C271C22semidirect
62. C2×SD16×C271C22semidirect
63. #131×C271C22semidirect
64. #108×C271C22semidirect
65. #101×C271C22semidirect
66. #100×C271C22semidirect
226. D271#471direct (N × H)
227. C271C2×#471semidirect
240. #471D271direct (N × H)
252. #137C2×D271semidirect
269. C2×SD16C2×D271semidirect
276. #131C2×D271semidirect
286. #108C2×D271semidirect
290. #101C2×D271semidirect
294. #100C2×D271semidirect

Maximal subgroups

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

subgroupindex
#137×D2712
AGL(1,ℤ/8)×D2712
C2×SD16×D2712
#131×D2712
#317741632
#317741622
#317741612
#317741602
#317741592
#317741582
#317741572
#108×D2712
#101×D2712
#100×D2712
#471×C2712
C2×#471271