Finite Group Databasebeta

← #46820611

Subgroups — #46820611

Order 34432.

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#31771401C211yesmaximal, characteristic, complemented
2C2×#22575477C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D269C211yesmaximal, characteristic, complemented
4#31771370C211yesmaximal, characteristic, complemented
5#31770874C211yesmaximal, characteristic, complemented
6#31770873C211yesmaximal, characteristic, complemented
7#31770872C211yesmaximal, characteristic, complemented
8#31770869C211yesmaximal, characteristic, complemented
9#31770868C211yesmaximal, characteristic, complemented
10#31770867C211yesmaximal, characteristic, complemented
11#31770866C211yesmaximal, characteristic, complemented
12#31770652C211yesmaximal, characteristic, complemented
13#31770586C211yesmaximal, characteristic, complemented
14#100×D269C211yesmaximal, characteristic, complemented
15#472×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17#41×D269C2211yescharacteristic
18C2×D4×D269C2211yescharacteristic
19#22575486C2211yescharacteristic, complemented
20#22575483C2211yescharacteristic, complemented
21#22575482—21no—
22#22575481C2211yescharacteristic
23#22575480—21no—
24#22575479C2211yescharacteristic
25#22575479C2211yescharacteristic, complemented
26#22575478C2211yescharacteristic
27#22575477—21no—
28#22575477—21no—
29#22575477—21no—
30#22575477—21no—
31#22575477—21no—
32#22575477—21no—
33#22575474—21no—
34#22575472—21no—
35C2×#16864391C2211yescharacteristic, complemented
36C2×#16864390C2211yescharacteristic, complemented
37SD16×D269—21no—
38C2×#16864387C2211yescharacteristic, complemented
39D8×D269—21no—
40#22575461—21no—
41#22575461C2211yescharacteristic, complemented
42#22575460C2211yescharacteristic, complemented
43#22575459—21no—
44#22575459C2211yescharacteristic, complemented
45C2×#16864385—21no—
46C2×#16864385C2211yescharacteristic
47M16×D269—21no—
48M16×D269C2211yescharacteristic
49M16×D269C2211yescharacteristic
50M16×D269C2211yescharacteristic
51#22575389C2211yescharacteristic, complemented
52#22575388C2211yescharacteristic, complemented
53#22575388C2211yescharacteristic, complemented
54#22575371C2211yescharacteristic, complemented
55#22575371C2211yescharacteristic, complemented
56#22575370C2211yescharacteristic, complemented
57#22575370C2211yescharacteristic, complemented
58#22575369C2211yescharacteristic, complemented
59#22575369C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C269C2211yescharacteristic
61AGL(1,ℤ/8)×C269C2211yescharacteristic, complemented
62C2×D8×C269C2211yescharacteristic, complemented
63#131×C269C2211yescharacteristic, complemented
64#108×C269C2211yescharacteristic, complemented
65#100×C269C2211yescharacteristic, complemented
66#100×C269C2211yescharacteristic, complemented
67C23×D269—21no—
68#16864403D411yescharacteristic
69#16864402—21no—
70#16864402C2311yescharacteristic
71#16864401—21no—
72#16864401—21no—
73#16864401—21no—
74#16864401—21no—
75#16864401—21no—
76#16864401D411yescharacteristic
77C2×Dic538C2311yescharacteristic
78Q8×D269D411yescharacteristic
79C2×D1076C2311yescharacteristic
80C2×C22⋊D269—21no—
81C2×C22⋊D269—21no—
82D4×D269—21no—
83D4×D269—21no—
84D4×D269—21no—
85D4×D269—21no—
86D4×D269—21no—
87D4×D269D411yescharacteristic
88C22×Dic269—21no—
89C2×C4×D269—21no—
90C2×C4×D269C2311yescharacteristic
91Dic1076—21no—
92#16864392—21no—
93#16864391—21no—
94#16864391—21no—
95#16864391—21no—
96#16864391—21no—
97#16864390—21no—
98#16864390—21no—
99#16864390—21no—
100#16864390—21no—
101#16864390—21no—
102#16864389—21no—
103D2152—21no—
104#16864387—21no—
105#16864387—21no—
106#16864387—21no—
107#16864387—21no—
108#16864387—21no—
109#16864386—21no—
110#16864386C2311yescharacteristic
111#16864386C2311yescharacteristic
112#16864386C2311yescharacteristic
113#16864385—21no—
114#16864385—21no—
115#16864385—21no—
116#16864385—21no—
117#16864385—21no—
118#16864385D411yescharacteristic
119#16864385D411yescharacteristic
120#16864385D411yescharacteristic
121#16864385D411yescharacteristic
122C2×#14524130—21no—
123C2×#14524130C2311yescharacteristic
124C8×D269—21no—
125C8×D269—21no—
126C8×D269—21no—
127C8×D269—21no—
128C8×D269—21no—
129#41×C269C2311yescharacteristic
130C2×D4×C269C2311yescharacteristic
131C2×D4×C269C2311yescharacteristic
132SD16×C269—21no—
133SD16×C269—21no—
134D8×C269—21no—
135D8×C269—21no—
136D8×C269—21no—
137D8×C269—21no—
138M16×C269—21no—
139M16×C269C2311yescharacteristic
140M16×C269C2311yescharacteristic
141M16×C269C2311yescharacteristic
142C2×C8×C269—21no—
143C2×C8×C269C2311yescharacteristic
144C22×D269—21no—
145C22×D269—21no—
146C22×D269—41no—
147C22×D269—41no—
148C22×D269—82no—
149C22×D269#4111yescharacteristic
150Dic538—21no—
151Dic538—21no—
152Dic538—21no—
153Dic538C2×D411yescharacteristic
154D1076—21no—
155D1076—21no—
156D1076—21no—
157D1076C2×D411yescharacteristic
158C22⋊D269—21no—
159C22⋊D269—21no—
160C22⋊D269—21no—
161C22⋊D269—41no—
162C22⋊D269—41no—
163C22⋊D269—41no—
164C22⋊D269—41no—
165C2×Dic269—21no—
166C2×Dic269—21no—
167C2×Dic269—41no—
168C2×Dic269—41no—
169C2×Dic269#4111yescharacteristic
170C4×D269—21no—
171C4×D269—21no—
172C4×D269—21no—
173C4×D269C2×D411yescharacteristic
174C4×D269C2×D411yescharacteristic
175C4×D269C2×D411yescharacteristic
176C4×D269C2×D411yescharacteristic
177#14524130—21no—
178#14524130—21no—
179#14524130—21no—
180#14524130C2×D411yescharacteristic
181#14524130C2×D411yescharacteristic
182C23×C269—21no—
183C23×C269—21no—
184Q8×C269C2×D411yescharacteristic
185D4×C269—21no—
186D4×C269—21no—
187D4×C269—21no—
188D4×C269—21no—
189D4×C269—21no—
190D4×C269C2×D411yescharacteristic
191C2×C4×C269—21no—
192C2×C4×C269C2411yescharacteristic
193C8×C269—21no—
194C8×C269—21no—
195C8×C269—21no—
196C8×C269C2×D411yescharacteristic
197C8×C269C2×D411yescharacteristic
198C22×C269—21no—
199C22×C269—41no—
200C22×C269—41no—
201C22×C269—81no—
202C22×C269—81no—
203C22×C269C2×#4111yescharacteristic
204C2×D269—21no—
205C2×D269—41no—
206C2×D269—41no—
207C2×D269—81no—
208C2×D269—82no—
209C2×D269—82no—
210C2×D269—81no—
211C2×D269#12111yescharacteristic
212C2×D269#12111yescharacteristic
213C4×C269—21no—
214C4×C269C22×D411yescharacteristic
215C4×C269C22×D411yescharacteristic
216Dic269—21no—
217Dic269—41no—
218Dic269#12111yescharacteristic
219Dic269#12111yescharacteristic
220C2×C269—21no—
221C2×C269—41no—
222C2×C269—81no—
223C2×C269—81no—
224C2×C269C2×#12111yescharacteristic
225D269—21no—
226D269—21no—
227D269—41no—
228D269—81no—
229C269#306811yescharacteristic, complemented, Sylow
230#3068—2691nomaximal, Sylow
231#575—2691no—
232C2×AGL(1,ℤ/8)—2691no—
233C2×AGL(1,ℤ/8)—2691no—
234#568—2691no—
235#472—2691no—
236#472—2691no—
237#472—2691no—
238#472D26911yescharacteristic, complemented
239#471—2691no—
240#471—2691no—
241#471—2691no—
242#471—2691no—
243#430—2691no—
244#413—2691no—
245C2×#100—2691no—
246C2×#41—2691no—
247C2×#41—2691no—
248C22×D4—2691no—
249#137—2691no—
250#137—2691no—
251#137—5381no—
252AGL(1,ℤ/8)—2691no—
253AGL(1,ℤ/8)—2691no—
254AGL(1,ℤ/8)—2691no—
255AGL(1,ℤ/8)—2691no—
256AGL(1,ℤ/8)—5381no—
257AGL(1,ℤ/8)—5381no—
258AGL(1,ℤ/8)—5381no—
259AGL(1,ℤ/8)—5381no—
260AGL(1,ℤ/8)—5381no—
261AGL(1,ℤ/8)—5381no—
262AGL(1,ℤ/8)—5381no—
263AGL(1,ℤ/8)C2×D26911yescharacteristic
264AGL(1,ℤ/8)C2×D26911yescharacteristic, complemented
265#135—5381no—
266#135—5381no—
267C2×SD16—2691no—
268C2×SD16—2691no—
269C2×SD16—5381no—
270C2×D8—2691no—
271C2×D8—5381no—
272C2×D8C2×D26911yescharacteristic, complemented
273#131—2691no—
274#131—2691no—
275#131—2691no—
276#131—5381no—
277#131—5381no—
278#131C2×D26911yescharacteristic, complemented
279C2×M16—2691no—
280C2×M16—2691no—
281C2×M16—2691no—
282C2×M16—2691no—
283C2×M16—5381no—
284C2×M16—5381no—
285#108—2691no—
286#108—2691no—
287#108—2691no—
288#108C2×D26911yescharacteristic, complemented
289#101—2691no—
290#101—2691no—
291#100—2691no—
292#100—2691no—
293#100—2691no—
294#100—2691no—
295#100C2×D26911yescharacteristic, complemented
296#100C2×D26911yescharacteristic, complemented
297C24—5381no—
298#41—2691no—
299#41—2691no—
300#41—2691no—
301#41—5381no—
302#41—5381no—
303#41—5381no—
304#41—5381no—
305#41—5381no—
306#41—5381no—
307#41C22×D26911yescharacteristic
308C2×Q8—2691no—
309C2×Q8—2691no—
310C2×D4—2691no—
311C2×D4—2691no—
312C2×D4—5381no—
313C2×D4—5381no—
314C2×D4—5381no—
315C2×D4—5381no—
316C2×D4—5381no—
317C2×D4—5381no—
318C2×D4—5381no—
319C2×D4C22×D26911yescharacteristic
320C2×D4C22×D26911yescharacteristic
321C22×C4—2691no—
322C22×C4—5381no—
323C22×C4—5381no—
324Q16—5381no—
325Q16—5381no—
326SD16—21no—
327SD16—21no—
328SD16—5381no—
329SD16—5381no—
330SD16—5381no—
331SD16—5381no—
332SD16—5381no—
333SD16—5381no—
334SD16—5381no—
335SD16—5381no—
336SD16—5381no—
337SD16—5381no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—5381no—
343D8—5381no—
344D8—5381no—
345D8—5381no—
346D8—5381no—
347D8—5381no—
348M16—21no—
349M16—2691no—
350M16—2691no—
351M16—2691no—
352M16—2691no—
353M16—2691no—
354M16—2691no—
355M16—2691no—
356M16—5381no—
357M16—5381no—
358M16—5381no—
359M16—5381no—
360M16—5381no—
361M16—5381no—
362M16C22×D26911yescharacteristic
363M16C22×D26911yescharacteristic
364M16C22×D26911yescharacteristic
365C2×C8—21no—
366C2×C8—2691no—
367C2×C8—5381no—
368C2×C8—5381no—
369C2×C8—5381no—
370C2×C8—5381no—
371C2×C8—5381no—
372C2×C8—5381no—
373C2×C8C22×D26911yescharacteristic
374C23—21no—
375C23—21no—
376C23—2691no—
377C23—5381no—
378C23—5381no—
379C23—10761no—
380C23—10761no—
381C23—21522no—
382Q8—2691no—
383Q8—5381no—
384Q8—5381no—
385Q8—5381no—
386Q8D4×D26911yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—2691no—
393D4—5381no—
394D4—5381no—
395D4—5381no—
396D4—5381no—
397D4—5381no—
398D4—5381no—
399D4—10761no—
400D4—10761no—
401D4—10761no—
402D4—10761no—
403D4D4×D26911yescharacteristic
404C2×C4—21no—
405C2×C4—2691no—
406C2×C4—2691no—
407C2×C4—2691no—
408C2×C4—2691no—
409C2×C4—2691no—
410C2×C4—5381no—
411C2×C4—5381no—
412C2×C4—5381no—
413C2×C4—5381no—
414C2×C4—5381no—
415C2×C4—10761no—
416C2×C4—10761no—
417C2×C4C23×D26911yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—2691no—
422C8—2691no—
423C8—5381no—
424C8—5381no—
425C8—5381no—
426C8D4×D26911yescharacteristic
427C8D4×D26911yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—2691no—
434C22—2691no—
435C22—5381no—
436C22—10761no—
437C22—10761no—
438C22—21521no—
439C22—21521no—
440C22—21522no—
441C22—21522no—
442C22#41×D26911yescharacteristic
443C4—21no—
444C4—2691no—
445C4—2691no—
446C4—5381no—
447C4—10761no—
448C4C2×D4×D26911yescharacteristic
449C4C2×D4×D26911yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—5381no—
455C2—5381no—
456C2—10761no—
457C2—21521no—
458C2#121×D26911yescharacteristic
subgroupG/H
centerC2#121×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#472×C269C2
socleC2×C269C2×#121
hypercenter#472D269
hypocenterC269#3068
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #31771401C2semidirect
2. C2×#22575477C2semidirect
3. AGL(1,ℤ/8)×D269C2semidirect
4. #31771370C2semidirect
5. #31770874C2semidirect
6. #31770873C2semidirect
7. #31770872C2semidirect
8. #31770869C2semidirect
9. #31770868C2semidirect
10. #31770867C2semidirect
11. #31770866C2semidirect
12. #31770652C2semidirect
13. #31770586C2semidirect
14. #100×D269C2semidirect
15. #472×C269C2semidirect
19. #22575486C22semidirect
20. #22575483C22semidirect
25. #22575479C22semidirect
35. C2×#16864391C22semidirect
36. C2×#16864390C22semidirect
38. C2×#16864387C22semidirect
41. #22575461C22semidirect
42. #22575460C22semidirect
44. #22575459C22semidirect
51. #22575389C22semidirect
52. #22575388C22semidirect
53. #22575388C22semidirect
54. #22575371C22semidirect
55. #22575371C22semidirect
56. #22575370C22semidirect
57. #22575370C22semidirect
58. #22575369C22semidirect
59. #22575369C22semidirect
61. AGL(1,ℤ/8)×C269C22semidirect
62. C2×D8×C269C22semidirect
63. #131×C269C22semidirect
64. #108×C269C22semidirect
65. #100×C269C22semidirect
66. #100×C269C22semidirect
229. C269#3068semidirect
238. #472D269semidirect
264. AGL(1,ℤ/8)C2×D269semidirect
272. C2×D8C2×D269semidirect
278. #131C2×D269semidirect
288. #108C2×D269semidirect
295. #100C2×D269semidirect
296. #100C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714012
C2×#225754772
AGL(1,ℤ/8)×D2692
#317713702
#317708742
#317708732
#317708722
#317708692
#317708682
#317708672
#317708662
#317706522
#317705862
#100×D2692
#472×C2692
#3068269