Finite Group Databasebeta

← #46846090

Subgroups — #46846090

Order 34688.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774693C211yesmaximal, characteristic, complemented
2C2×#22576184C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D271C211yesmaximal, characteristic, complemented
4#31774662C211yesmaximal, characteristic, complemented
5#31774166C211yesmaximal, characteristic, complemented
6#31774165C211yesmaximal, characteristic, complemented
7#31774164C211yesmaximal, characteristic, complemented
8#31774161C211yesmaximal, characteristic, complemented
9#31774160C211yesmaximal, characteristic, complemented
10#31774159C211yesmaximal, characteristic, complemented
11#31774158C211yesmaximal, characteristic, complemented
12#31773944C211yesmaximal, characteristic, complemented
13#31773878C211yesmaximal, characteristic, complemented
14#100×D271C211yesmaximal, characteristic, complemented
15#472×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17#41×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576193C2211yescharacteristic, complemented
20#22576190C2211yescharacteristic, complemented
21#22576189—21no—
22#22576188C2211yescharacteristic
23#22576187—21no—
24#22576186C2211yescharacteristic
25#22576186C2211yescharacteristic, complemented
26#22576185C2211yescharacteristic
27#22576184—21no—
28#22576184—21no—
29#22576184—21no—
30#22576184—21no—
31#22576184—21no—
32#22576184—21no—
33#22576181—21no—
34#22576179—21no—
35C2×#16864643C2211yescharacteristic, complemented
36C2×#16864642C2211yescharacteristic, complemented
37SD16×D271—21no—
38C2×#16864639C2211yescharacteristic, complemented
39D8×D271—21no—
40#22576168—21no—
41#22576168C2211yescharacteristic, complemented
42#22576167C2211yescharacteristic, complemented
43#22576166—21no—
44#22576166C2211yescharacteristic, complemented
45C2×#16864637—21no—
46C2×#16864637C2211yescharacteristic
47M16×D271—21no—
48M16×D271C2211yescharacteristic
49M16×D271C2211yescharacteristic
50M16×D271C2211yescharacteristic
51#22576096C2211yescharacteristic, complemented
52#22576095C2211yescharacteristic, complemented
53#22576095C2211yescharacteristic, complemented
54#22576078C2211yescharacteristic, complemented
55#22576078C2211yescharacteristic, complemented
56#22576077C2211yescharacteristic, complemented
57#22576077C2211yescharacteristic, complemented
58#22576076C2211yescharacteristic, complemented
59#22576076C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C271C2211yescharacteristic
61AGL(1,ℤ/8)×C271C2211yescharacteristic, complemented
62C2×D8×C271C2211yescharacteristic, complemented
63#131×C271C2211yescharacteristic, complemented
64#108×C271C2211yescharacteristic, complemented
65#100×C271C2211yescharacteristic, complemented
66#100×C271C2211yescharacteristic, complemented
67C23×D271—21no—
68#16864655D411yescharacteristic
69#16864654—21no—
70#16864654C2311yescharacteristic
71#16864653—21no—
72#16864653—21no—
73#16864653—21no—
74#16864653—21no—
75#16864653—21no—
76#16864653D411yescharacteristic
77C2×Dic542C2311yescharacteristic
78Q8×D271D411yescharacteristic
79C2×D1084C2311yescharacteristic
80C2×C22⋊D271—21no—
81C2×C22⋊D271—21no—
82D4×D271—21no—
83D4×D271—21no—
84D4×D271—21no—
85D4×D271—21no—
86D4×D271—21no—
87D4×D271D411yescharacteristic
88C22×Dic271—21no—
89C2×C4×D271—21no—
90C2×C4×D271C2311yescharacteristic
91Dic1084—21no—
92#16864644—21no—
93#16864643—21no—
94#16864643—21no—
95#16864643—21no—
96#16864643—21no—
97#16864642—21no—
98#16864642—21no—
99#16864642—21no—
100#16864642—21no—
101#16864642—21no—
102#16864641—21no—
103D2168—21no—
104#16864639—21no—
105#16864639—21no—
106#16864639—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—
118#16864637D411yescharacteristic
119#16864637D411yescharacteristic
120#16864637D411yescharacteristic
121#16864637D411yescharacteristic
122C2×C271⋊C8—21no—
123C2×C271⋊C8C2311yescharacteristic
124C8×D271—21no—
125C8×D271—21no—
126C8×D271—21no—
127C8×D271—21no—
128C8×D271—21no—
129#41×C271C2311yescharacteristic
130C2×D4×C271C2311yescharacteristic
131C2×D4×C271C2311yescharacteristic
132SD16×C271—21no—
133SD16×C271—21no—
134D8×C271—21no—
135D8×C271—21no—
136D8×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—
153Dic542C2×D411yescharacteristic
154D1084—21no—
155D1084—21no—
156D1084—21no—
157D1084C2×D411yescharacteristic
158C22⋊D271—21no—
159C22⋊D271—21no—
160C22⋊D271—21no—
161C22⋊D271—41no—
162C22⋊D271—41no—
163C22⋊D271—41no—
164C22⋊D271—41no—
165C2×Dic271—21no—
166C2×Dic271—21no—
167C2×Dic271—41no—
168C2×Dic271—41no—
169C2×Dic271#4111yescharacteristic
170C4×D271—21no—
171C4×D271—21no—
172C4×D271—21no—
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—
183C23×C271—21no—
184Q8×C271C2×D411yescharacteristic
185D4×C271—21no—
186D4×C271—21no—
187D4×C271—21no—
188D4×C271—21no—
189D4×C271—21no—
190D4×C271C2×D411yescharacteristic
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—41no—
201C22×C271—81no—
202C22×C271—81no—
203C22×C271C2×#4111yescharacteristic
204C2×D271—21no—
205C2×D271—41no—
206C2×D271—41no—
207C2×D271—81no—
208C2×D271—82no—
209C2×D271—82no—
210C2×D271—81no—
211C2×D271#12111yescharacteristic
212C2×D271#12111yescharacteristic
213C4×C271—21no—
214C4×C271C22×D411yescharacteristic
215C4×C271C22×D411yescharacteristic
216Dic271—21no—
217Dic271—41no—
218Dic271#12111yescharacteristic
219Dic271#12111yescharacteristic
220C2×C271—21no—
221C2×C271—41no—
222C2×C271—81no—
223C2×C271—81no—
224C2×C271C2×#12111yescharacteristic
225D271—21no—
226D271—21no—
227D271—41no—
228D271—81no—
229C271#306811yescharacteristic, complemented, Sylow
230#3068—2711nomaximal, Sylow
231#575—2711no—
232C2×AGL(1,ℤ/8)—2711no—
233C2×AGL(1,ℤ/8)—2711no—
234#568—2711no—
235#472—2711no—
236#472—2711no—
237#472—2711no—
238#472D27111yescharacteristic, complemented
239#471—2711no—
240#471—2711no—
241#471—2711no—
242#471—2711no—
243#430—2711no—
244#413—2711no—
245C2×#100—2711no—
246C2×#41—2711no—
247C2×#41—2711no—
248C22×D4—2711no—
249#137—2711no—
250#137—2711no—
251#137—5421no—
252AGL(1,ℤ/8)—2711no—
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)—5421no—
259AGL(1,ℤ/8)—5421no—
260AGL(1,ℤ/8)—5421no—
261AGL(1,ℤ/8)—5421no—
262AGL(1,ℤ/8)—5421no—
263AGL(1,ℤ/8)C2×D27111yescharacteristic
264AGL(1,ℤ/8)C2×D27111yescharacteristic, complemented
265#135—5421no—
266#135—5421no—
267C2×SD16—2711no—
268C2×SD16—2711no—
269C2×SD16—5421no—
270C2×D8—2711no—
271C2×D8—5421no—
272C2×D8C2×D27111yescharacteristic, complemented
273#131—2711no—
274#131—2711no—
275#131—2711no—
276#131—5421no—
277#131—5421no—
278#131C2×D27111yescharacteristic, complemented
279C2×M16—2711no—
280C2×M16—2711no—
281C2×M16—2711no—
282C2×M16—2711no—
283C2×M16—5421no—
284C2×M16—5421no—
285#108—2711no—
286#108—2711no—
287#108—2711no—
288#108C2×D27111yescharacteristic, complemented
289#101—2711no—
290#101—2711no—
291#100—2711no—
292#100—2711no—
293#100—2711no—
294#100—2711no—
295#100C2×D27111yescharacteristic, complemented
296#100C2×D27111yescharacteristic, complemented
297C24—5421no—
298#41—2711no—
299#41—2711no—
300#41—2711no—
301#41—5421no—
302#41—5421no—
303#41—5421no—
304#41—5421no—
305#41—5421no—
306#41—5421no—
307#41C22×D27111yescharacteristic
308C2×Q8—2711no—
309C2×Q8—2711no—
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×D4—5421no—
319C2×D4C22×D27111yescharacteristic
320C2×D4C22×D27111yescharacteristic
321C22×C4—2711no—
322C22×C4—5421no—
323C22×C4—5421no—
324Q16—5421no—
325Q16—5421no—
326SD16—21no—
327SD16—21no—
328SD16—5421no—
329SD16—5421no—
330SD16—5421no—
331SD16—5421no—
332SD16—5421no—
333SD16—5421no—
334SD16—5421no—
335SD16—5421no—
336SD16—5421no—
337SD16—5421no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—5421no—
343D8—5421no—
344D8—5421no—
345D8—5421no—
346D8—5421no—
347D8—5421no—
348M16—21no—
349M16—2711no—
350M16—2711no—
351M16—2711no—
352M16—2711no—
353M16—2711no—
354M16—2711no—
355M16—2711no—
356M16—5421no—
357M16—5421no—
358M16—5421no—
359M16—5421no—
360M16—5421no—
361M16—5421no—
362M16C22×D27111yescharacteristic
363M16C22×D27111yescharacteristic
364M16C22×D27111yescharacteristic
365C2×C8—21no—
366C2×C8—2711no—
367C2×C8—5421no—
368C2×C8—5421no—
369C2×C8—5421no—
370C2×C8—5421no—
371C2×C8—5421no—
372C2×C8—5421no—
373C2×C8C22×D27111yescharacteristic
374C23—21no—
375C23—21no—
376C23—2711no—
377C23—5421no—
378C23—5421no—
379C23—10841no—
380C23—10841no—
381C23—21682no—
382Q8—2711no—
383Q8—5421no—
384Q8—5421no—
385Q8—5421no—
386Q8D4×D27111yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—2711no—
393D4—5421no—
394D4—5421no—
395D4—5421no—
396D4—5421no—
397D4—5421no—
398D4—5421no—
399D4—10841no—
400D4—10841no—
401D4—10841no—
402D4—10841no—
403D4D4×D27111yescharacteristic
404C2×C4—21no—
405C2×C4—2711no—
406C2×C4—2711no—
407C2×C4—2711no—
408C2×C4—2711no—
409C2×C4—2711no—
410C2×C4—5421no—
411C2×C4—5421no—
412C2×C4—5421no—
413C2×C4—5421no—
414C2×C4—5421no—
415C2×C4—10841no—
416C2×C4—10841no—
417C2×C4C23×D27111yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—2711no—
422C8—2711no—
423C8—5421no—
424C8—5421no—
425C8—5421no—
426C8D4×D27111yescharacteristic
427C8D4×D27111yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—2711no—
434C22—2711no—
435C22—5421no—
436C22—10841no—
437C22—10841no—
438C22—21681no—
439C22—21681no—
440C22—21682no—
441C22—21682no—
442C22#41×D27111yescharacteristic
443C4—21no—
444C4—2711no—
445C4—2711no—
446C4—5421no—
447C4—10841no—
448C4C2×D4×D27111yescharacteristic
449C4C2×D4×D27111yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—5421no—
455C2—5421no—
456C2—10841no—
457C2—21681no—
458C2#121×D27111yescharacteristic
subgroupG/H
centerC2#121×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#472×C271C2
socleC2×C271C2×#121
hypercenter#472D271
hypocenterC271#3068
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #31774693C2semidirect
2. C2×#22576184C2semidirect
3. AGL(1,ℤ/8)×D271C2semidirect
4. #31774662C2semidirect
5. #31774166C2semidirect
6. #31774165C2semidirect
7. #31774164C2semidirect
8. #31774161C2semidirect
9. #31774160C2semidirect
10. #31774159C2semidirect
11. #31774158C2semidirect
12. #31773944C2semidirect
13. #31773878C2semidirect
14. #100×D271C2semidirect
15. #472×C271C2semidirect
19. #22576193C22semidirect
20. #22576190C22semidirect
25. #22576186C22semidirect
35. C2×#16864643C22semidirect
36. C2×#16864642C22semidirect
38. C2×#16864639C22semidirect
41. #22576168C22semidirect
42. #22576167C22semidirect
44. #22576166C22semidirect
51. #22576096C22semidirect
52. #22576095C22semidirect
53. #22576095C22semidirect
54. #22576078C22semidirect
55. #22576078C22semidirect
56. #22576077C22semidirect
57. #22576077C22semidirect
58. #22576076C22semidirect
59. #22576076C22semidirect
61. AGL(1,ℤ/8)×C271C22semidirect
62. C2×D8×C271C22semidirect
63. #131×C271C22semidirect
64. #108×C271C22semidirect
65. #100×C271C22semidirect
66. #100×C271C22semidirect
229. C271#3068semidirect
238. #472D271semidirect
264. AGL(1,ℤ/8)C2×D271semidirect
272. C2×D8C2×D271semidirect
278. #131C2×D271semidirect
288. #108C2×D271semidirect
295. #100C2×D271semidirect
296. #100C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746932
C2×#225761842
AGL(1,ℤ/8)×D2712
#317746622
#317741662
#317741652
#317741642
#317741612
#317741602
#317741592
#317741582
#317739442
#317738782
#100×D2712
#472×C2712
#3068271