Finite Group Databasebeta

← #46848155

Subgroups — #46848155

Order 34688.

460 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#31774708C211yesmaximal, characteristic, complemented
2#31774705C211yesmaximal, characteristic, complemented
3#31774704C211yesmaximal, characteristic, complemented
4#31774268C211yesmaximal, characteristic, complemented
5#31774162C211yesmaximal, characteristic, complemented
6#31774160C211yesmaximal, characteristic, complemented
7#31774158C211yesmaximal, characteristic, complemented
8#31774157C211yesmaximal, characteristic, complemented
9#31774080C211yesmaximal, characteristic, complemented
10#31774078C211yesmaximal, characteristic, complemented
11#31774065C211yesmaximal, characteristic, complemented
12#31774063C211yesmaximal, characteristic, complemented
13#31774014C211yesmaximal, characteristic, complemented
14#31774012C211yesmaximal, characteristic, complemented
15#577×C271C211yesmaximal, characteristic, complemented
16#22576207C2211yescharacteristic
17#22576206C2211yescharacteristic, complemented
18#22576192—21no—
19#22576191—21no—
20#22576191C2211yescharacteristic, complemented
21#22576190—21no—
22#22576190C2211yescharacteristic, complemented
23#22576190C2211yescharacteristic, complemented
24#22576186—21no—
25#22576186C2211yescharacteristic
26#22576185—21no—
27#22576185C2211yescharacteristic
28#22576185C2211yescharacteristic
29#22576184—21no—
30#22576181—21no—
31#22576180—21no—
32#22576180—21no—
33#22576180C2211yescharacteristic
34C2×#16864643C2211yescharacteristic, complemented
35C2×#16864642—21no—
36C2×#16864642C2211yescharacteristic, complemented
37C2×#16864641—21no—
38C2×#16864641C2211yescharacteristic, complemented
39SD16×D271—21no—
40#22576167C2211yescharacteristic, complemented
41#22576167C2211yescharacteristic, complemented
42#22576166C2211yescharacteristic, complemented
43#22576158C2211yescharacteristic, complemented
44#22576156C2211yescharacteristic
45#22576096C2211yescharacteristic, complemented
46#22576095C2211yescharacteristic, complemented
47#22576087C2211yescharacteristic, complemented
48#22576087C2211yescharacteristic, complemented
49#22576087C2211yescharacteristic, complemented
50#22576086C2211yescharacteristic, complemented
51#22576079C2211yescharacteristic, complemented
52#22576078C2211yescharacteristic, complemented
53#22576077C2211yescharacteristic, complemented
54#22576076C2211yescharacteristic, complemented
55C8×Dic271C2211yescharacteristic, complemented
562−1+4×C271C2211yescharacteristic
57H(5,𝔽2)×C271C2211yescharacteristic, complemented
58#137×C271—21no—
59#137×C271C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C271—21no—
61AGL(1,ℤ/8)×C271C2211yescharacteristic
62#135×C271—21no—
63#135×C271C2211yescharacteristic
64C2×SD16×C271—21no—
65C2×SD16×C271C2211yescharacteristic, complemented
66#131×C271C2211yescharacteristic, complemented
67#16864655—21no—
68#16864654—21no—
69#16864654—21no—
70#16864654C2311yescharacteristic
71#16864653—21no—
72#16864653—41no—
73C2×Dic542—21no—
74C2×Dic542C2311yescharacteristic
75Q8×D271—21no—
76C2×D1084—21no—
77C2×D1084C2311yescharacteristic
78C2×C22⋊D271—41no—
79D4×D271—21no—
80D4×D271—41no—
81#16864644—21no—
82#16864644—21no—
83#16864644—21no—
84#16864644D411yescharacteristic, complemented
85#16864643—21no—
86#16864643—21no—
87#16864642—21no—
88#16864642—21no—
89#16864642—21no—
90#16864642—21no—
91#16864642—21no—
92#16864642D411yescharacteristic, complemented
93#16864641—21no—
94#16864641—21no—
95#16864641—21no—
96#16864641—21no—
97#16864641—21no—
98#16864641D411yescharacteristic, complemented
99#16864639—21no—
100#16864639—21no—
101#16864639—21no—
102#16864639D411yescharacteristic, complemented
103#16864638—21no—
104#16864638—21no—
105#16864638C2311yescharacteristic
106#16864638C2311yescharacteristic
107#16864638C2311yescharacteristic
108#16864637—21no—
109C2×C271⋊C8—21no—
110C2×C271⋊C8—21no—
111C2×C271⋊C8C2311yescharacteristic
112C8×D271—21no—
113Dic271⋊C4—41no—
114C4×Dic271C2311yescharacteristic
115#41×C271—21no—
116#41×C271—21no—
117#41×C271—21no—
118#41×C271—21no—
119#41×C271—41no—
120#41×C271C2311yescharacteristic
121#41×C271C2311yescharacteristic
122#41×C271C2311yescharacteristic
123C2×Q8×C271—21no—
124C2×Q8×C271—21no—
125C2×Q8×C271C2311yescharacteristic
126C2×D4×C271—21no—
127C2×D4×C271—21no—
128C2×D4×C271—41no—
129C2×D4×C271C2311yescharacteristic
130Q16×C271—21no—
131Q16×C271D411yescharacteristic, complemented
132SD16×C271—21no—
133SD16×C271—21no—
134SD16×C271—21no—
135SD16×C271—21no—
136SD16×C271D411yescharacteristic, complemented
137SD16×C271D411yescharacteristic, complemented
138D8×C271—21no—
139D8×C271D411yescharacteristic, complemented
140M16×C271—21no—
141M16×C271C2311yescharacteristic
142C2×C8×C271—21no—
143C2×C8×C271C2311yescharacteristic
144C22×D271—21no—
145C22×D271—41no—
146Dic542—21no—
147Dic542—21no—
148Dic542—41no—
149Dic542C2×D411yescharacteristic
150D1084—21no—
151D1084—21no—
152D1084—41no—
153D1084C2×D411yescharacteristic
154C22⋊D271—21no—
155C22⋊D271—41no—
156C22⋊D271—41no—
157C22⋊D271—82no—
158C2×Dic271—21no—
159C2×Dic271—21no—
160C2×Dic271—41no—
161C4×D271—21no—
162C4×D271—41no—
163C4×D271—41no—
164C271⋊C8—21no—
165C271⋊C8—21no—
166C271⋊C8—21no—
167C271⋊C8C2×D411yescharacteristic
168C271⋊C8C2×D411yescharacteristic
169C23×C271—21no—
170C23×C271—41no—
171Q8×C271—21no—
172Q8×C271—21no—
173Q8×C271—41no—
174Q8×C271C2×D411yescharacteristic
175Q8×C271C2×D411yescharacteristic
176Q8×C271C2×D411yescharacteristic
177D4×C271—21no—
178D4×C271—21no—
179D4×C271—21no—
180D4×C271—21no—
181D4×C271—21no—
182D4×C271—41no—
183D4×C271—41no—
184D4×C271—41no—
185D4×C271C2×D411yescharacteristic
186D4×C271C2×D411yescharacteristic
187D4×C271C2×D411yescharacteristic
188C2×C4×C271—21no—
189C2×C4×C271—21no—
190C2×C4×C271—21no—
191C2×C4×C271—21no—
192C2×C4×C271—41no—
193C2×C4×C271—41no—
194C2×C4×C271—41no—
195C2×C4×C271C2411yescharacteristic
196C8×C271—21no—
197C8×C271C2×D411yescharacteristic
198C8×C271C2×D411yescharacteristic
199C22×C271—21no—
200C22×C271—21no—
201C22×C271—21no—
202C22×C271—41no—
203C22×C271—81no—
204C22×C271—161no—
205C22×C271H(5,𝔽2)11yescharacteristic
206C2×D271—21no—
207C2×D271—41no—
208C2×D271—81no—
209C2×D271—161no—
210C4×C271—21no—
211C4×C271—21no—
212C4×C271—21no—
213C4×C271—41no—
214C4×C271C22×D411yescharacteristic
215C4×C271C22×D411yescharacteristic
216Dic271—22no—
217Dic271—21no—
218Dic271—41no—
219C2×C271—21no—
220C2×C271—41no—
221C2×C271—41no—
222C2×C271—41no—
223C2×C271—81no—
224C2×C271D4211yescharacteristic
225D271—41no—
226D271—81no—
227C271#329111yescharacteristic, complemented, Sylow
228#3291—2711nomaximal, Sylow
229#577—2711no—
230#577—2711no—
231#577—2711no—
232#577D27111yescharacteristic, complemented
233#492—2711no—
234#471—2711no—
235#471—2711no—
236#471—2711no—
237#471—2711no—
238#456—2711no—
239#456—2711no—
240#453—2711no—
241#453—2711no—
242#443—2711no—
243#443—2711no—
2442−1+4—2711no—
2452−1+4C2×D27111yescharacteristic
246H(5,𝔽2)—2711no—
247H(5,𝔽2)C2×D27111yescharacteristic, complemented
248#137—21no—
249#137—2711no—
250#137—2711no—
251#137—2711no—
252#137—5421no—
253#137—5421no—
254#137—5421no—
255#137C2×D27111yescharacteristic, complemented
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—2711no—
258AGL(1,ℤ/8)—2711no—
259AGL(1,ℤ/8)—2711no—
260AGL(1,ℤ/8)—5421no—
261AGL(1,ℤ/8)—5421no—
262AGL(1,ℤ/8)—5421no—
263AGL(1,ℤ/8)C2×D27111yescharacteristic
264#135—21no—
265#135—2711no—
266#135—5421no—
267#135—5421no—
268#135—5421no—
269#135C2×D27111yescharacteristic
270C2×SD16—21no—
271C2×SD16—2711no—
272C2×SD16—2711no—
273C2×SD16—2711no—
274C2×SD16—5421no—
275C2×SD16—5421no—
276C2×SD16—5421no—
277C2×SD16C2×D27111yescharacteristic, complemented
278#131—2711no—
279#131—2711no—
280#131—2711no—
281#131C2×D27111yescharacteristic, complemented
282C4⋊Q8—2711no—
283#127—2711no—
284#108—2711no—
285#108—2711no—
286#104—2711no—
287#104—2711no—
288#104—2711no—
289#104—2711no—
290#101—2711no—
291#101—2711no—
292#100—2711no—
293#100—2711no—
294C4×C8—2711no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—2711no—
301#41—5421no—
302#41—5421no—
303#41—5421no—
304#41—5421no—
305#41—10841no—
306#41C22×D27111yescharacteristic
307#41C22×D27111yescharacteristic
308#41C22×D27111yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—2711no—
312C2×Q8—5421no—
313C2×Q8—5421no—
314C2×Q8C22×D27111yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—2711no—
319C2×D4—5421no—
320C2×D4—5421no—
321C2×D4—10841no—
322C2×D4—10841no—
323C2×D4C22×D27111yescharacteristic
324Q16—21no—
325Q16—2711no—
326Q16—5421no—
327Q16—5421no—
328Q16—5421no—
329Q16C22⋊D27111yescharacteristic, complemented
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—2711no—
335SD16—2711no—
336SD16—5421no—
337SD16—5421no—
338SD16—5421no—
339SD16—5421no—
340SD16—5421no—
341SD16—5421no—
342SD16—5421no—
343SD16—5421no—
344SD16—5421no—
345SD16—5421no—
346SD16—5421no—
347SD16—5421no—
348SD16C22⋊D27111yescharacteristic, complemented
349SD16C22⋊D27111yescharacteristic, complemented
350D8—21no—
351D8—2711no—
352D8—5421no—
353D8—5421no—
354D8—5421no—
355D8C22⋊D27111yescharacteristic, complemented
356M16—21no—
357M16—2711no—
358M16—2711no—
359M16—2711no—
360M16—5421no—
361M16—5421no—
362M16—5421no—
363M16C22×D27111yescharacteristic
364C2×C8—21no—
365C2×C8—2711no—
366C2×C8—5421no—
367C2×C8—5421no—
368C2×C8—5421no—
369C2×C8C22×D27111yescharacteristic
370C4⋊C4—10841no—
371C42—2711no—
372C23—21no—
373C23—41no—
374C23—5421no—
375C23—10841no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—2711no—
380Q8—5421no—
381Q8—5421no—
382Q8—10841no—
383Q8D4×D27111yescharacteristic
384Q8C2×C22⋊D27111yescharacteristic
385Q8C2×C22⋊D27111yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—2711no—
395D4—5421no—
396D4—5421no—
397D4—5421no—
398D4—10841no—
399D4—10841no—
400D4—10841no—
401D4—21682no—
402D4D4×D27111yescharacteristic
403D4C2×C22⋊D27111yescharacteristic
404D4C2×C22⋊D27111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—5421no—
413C2×C4—5421no—
414C2×C4—5421no—
415C2×C4—10841no—
416C2×C4—10841no—
417C2×C4—10841no—
418C2×C4C23×D27111yescharacteristic
419C8—21no—
420C8—2711no—
421C8—2711no—
422C8—5421no—
423C8—5421no—
424C8—5421no—
425C8C2×C22⋊D27111yescharacteristic
426C8C2×C22⋊D27111yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—5421no—
434C22—10841no—
435C22—21681no—
436C22—43361no—
437C22#2257620511yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—5422no—
443C4—5421no—
444C4—10841no—
445C4C2×D4×D27111yescharacteristic
446C4C22×C22⋊D27111yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—10841no—
453C2—21681no—
454C2D4×C22⋊D27111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#577×C271C2
socleC2×C271D42
hypercenter#577D271
hypocenterC271#3291
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #31774708C2semidirect
2. #31774705C2semidirect
3. #31774704C2semidirect
4. #31774268C2semidirect
5. #31774162C2semidirect
6. #31774160C2semidirect
7. #31774158C2semidirect
8. #31774157C2semidirect
9. #31774080C2semidirect
10. #31774078C2semidirect
11. #31774065C2semidirect
12. #31774063C2semidirect
13. #31774014C2semidirect
14. #31774012C2semidirect
15. #577×C271C2semidirect
17. #22576206C22semidirect
20. #22576191C22semidirect
22. #22576190C22semidirect
23. #22576190C22semidirect
34. C2×#16864643C22semidirect
36. C2×#16864642C22semidirect
38. C2×#16864641C22semidirect
40. #22576167C22semidirect
41. #22576167C22semidirect
42. #22576166C22semidirect
43. #22576158C22semidirect
45. #22576096C22semidirect
46. #22576095C22semidirect
47. #22576087C22semidirect
48. #22576087C22semidirect
49. #22576087C22semidirect
50. #22576086C22semidirect
51. #22576079C22semidirect
52. #22576078C22semidirect
53. #22576077C22semidirect
54. #22576076C22semidirect
55. C8×Dic271C22semidirect
57. H(5,𝔽2)×C271C22semidirect
59. #137×C271C22semidirect
65. C2×SD16×C271C22semidirect
66. #131×C271C22semidirect
84. #16864644D4semidirect
92. #16864642D4semidirect
98. #16864641D4semidirect
102. #16864639D4semidirect
131. Q16×C271D4semidirect
136. SD16×C271D4semidirect
137. SD16×C271D4semidirect
139. D8×C271D4semidirect
227. C271#3291semidirect
232. #577D271semidirect
247. H(5,𝔽2)C2×D271semidirect
255. #137C2×D271semidirect
277. C2×SD16C2×D271semidirect
281. #131C2×D271semidirect
329. Q16C22⋊D271semidirect
348. SD16C22⋊D271semidirect
349. SD16C22⋊D271semidirect
355. D8C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
#317747082
#317747052
#317747042
#317742682
#317741622
#317741602
#317741582
#317741572
#317740802
#317740782
#317740652
#317740632
#317740142
#317740122
#577×C2712
#3291271