Finite Group Databasebeta

← #46848144

Subgroups — #46848144

Order 34688.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774712C211yesmaximal, characteristic, complemented
2#31774704C211yesmaximal, characteristic, complemented
3#31774700C211yesmaximal, characteristic, complemented
4#31774278C211yesmaximal, characteristic, complemented
5#31774170C211yesmaximal, characteristic, complemented
6#31774165C211yesmaximal, characteristic, complemented
7#31774160C211yesmaximal, characteristic, complemented
8#31774157C211yesmaximal, characteristic, complemented
9#31774075C211yesmaximal, characteristic, complemented
10#31774073C211yesmaximal, characteristic, complemented
11#31774070C211yesmaximal, characteristic, complemented
12#31774068C211yesmaximal, characteristic, complemented
13#31774014C211yesmaximal, characteristic, complemented
14#31774012C211yesmaximal, characteristic, complemented
15#577×C271C211yesmaximal, characteristic, complemented
16#22576207C2211yescharacteristic, complemented
17#22576206C2211yescharacteristic
18#22576192—21no—
19#22576191—21no—
20#22576191C2211yescharacteristic, complemented
21#22576190—21no—
22#22576190C2211yescharacteristic
23#22576190C2211yescharacteristic, complemented
24#22576186—21no—
25#22576186C2211yescharacteristic
26#22576185—21no—
27#22576185C2211yescharacteristic
28#22576185C2211yescharacteristic, complemented
29#22576184—21no—
30#22576181—21no—
31#22576180—21no—
32#22576180—21no—
33#22576180C2211yescharacteristic
34C2×#16864644—21no—
35C2×#16864644C2211yescharacteristic, complemented
36C2×#16864643C2211yescharacteristic, complemented
37SD16×D271—21no—
38C2×#16864639—21no—
39C2×#16864639C2211yescharacteristic
40#22576167C2211yescharacteristic
41#22576167C2211yescharacteristic, complemented
42#22576166C2211yescharacteristic, complemented
43#22576149C2211yescharacteristic, complemented
44#22576147C2211yescharacteristic, complemented
45#22576096C2211yescharacteristic, complemented
46#22576095C2211yescharacteristic, complemented
47#22576087C2211yescharacteristic
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, complemented
57H(5,𝔽2)×C271C2211yescharacteristic
58#137×C271—21no—
59#137×C271C2211yescharacteristic
60AGL(1,ℤ/8)×C271—21no—
61AGL(1,ℤ/8)×C271C2211yescharacteristic, complemented
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
78D4×D271—21no—
79D4×D271—41no—
80#16864644—21no—
81#16864644—21no—
82#16864644—21no—
83#16864644—21no—
84#16864644—21no—
85#16864644D411yescharacteristic
86#16864643—21no—
87#16864643—21no—
88#16864642—21no—
89#16864642—21no—
90#16864642—21no—
91#16864642D411yescharacteristic
92#16864641—21no—
93#16864641—21no—
94#16864641—21no—
95#16864641D411yescharacteristic
96#16864639—21no—
97#16864639—21no—
98#16864639—21no—
99#16864639—21no—
100#16864639—21no—
101#16864639D411yescharacteristic
102#16864638—21no—
103#16864638—21no—
104#16864638C2311yescharacteristic
105#16864638C2311yescharacteristic
106#16864638C2311yescharacteristic
107#16864637—21no—
108C2×C271⋊C8—21no—
109C2×C271⋊C8—21no—
110C2×C271⋊C8C2311yescharacteristic
111C8×D271—21no—
112C22⋊Dic271—41no—
113(C2×D271)⋊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
132SD16×C271—21no—
133SD16×C271—21no—
134SD16×C271—21no—
135SD16×C271—21no—
136SD16×C271D411yescharacteristic
137SD16×C271D411yescharacteristic
138D8×C271—21no—
139D8×C271D411yescharacteristic
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—
157C2×Dic271—21no—
158C2×Dic271—21no—
159C2×Dic271—41no—
160C4×D271—21no—
161C4×D271—41no—
162C4×D271—41no—
163C271⋊C8—21no—
164C271⋊C8—21no—
165C271⋊C8—21no—
166C271⋊C8C2×D411yescharacteristic
167C271⋊C8C2×D411yescharacteristic
168C23×C271—21no—
169C23×C271—41no—
170Q8×C271—21no—
171Q8×C271—21no—
172Q8×C271—41no—
173Q8×C271C2×D411yescharacteristic
174Q8×C271C2×D411yescharacteristic
175Q8×C271C2×D411yescharacteristic
176D4×C271—21no—
177D4×C271—21no—
178D4×C271—21no—
179D4×C271—21no—
180D4×C271—21no—
181D4×C271—41no—
182D4×C271—41no—
183D4×C271—41no—
184D4×C271C2×D411yescharacteristic
185D4×C271C2×D411yescharacteristic
186D4×C271C2×D411yescharacteristic
187C2×C4×C271—21no—
188C2×C4×C271—21no—
189C2×C4×C271—21no—
190C2×C4×C271—21no—
191C2×C4×C271—41no—
192C2×C4×C271—41no—
193C2×C4×C271—41no—
194C2×C4×C271C2411yescharacteristic
195C8×C271—21no—
196C8×C271C2×D411yescharacteristic
197C8×C271C2×D411yescharacteristic
198C22×C271—21no—
199C22×C271—21no—
200C22×C271—21no—
201C22×C271—41no—
202C22×C271—81no—
203C22×C271—161no—
204C22×C271H(5,𝔽2)11yescharacteristic
205C2×D271—21no—
206C2×D271—41no—
207C2×D271—81no—
208C2×D271—161no—
209C4×C271—21no—
210C4×C271—21no—
211C4×C271—21no—
212C4×C271—41no—
213C4×C271C22×D411yescharacteristic
214C4×C271C22×D411yescharacteristic
215Dic271—21no—
216Dic271—41no—
217Dic271—41no—
218C2×C271—21no—
219C2×C271—41no—
220C2×C271—41no—
221C2×C271—41no—
222C2×C271—81no—
223C2×C271D4211yescharacteristic
224D271—41no—
225D271—81no—
226C271#329011yescharacteristic, complemented, Sylow
227#3290—2711nomaximal, Sylow
228#578—2711no—
229#577—2711no—
230#577D27111yescharacteristic, complemented
231#576—2711no—
232#495—2711no—
233#473—2711no—
234#472—2711no—
235#471—2711no—
236#471—2711no—
237#455—2711no—
238#455—2711no—
239#454—2711no—
240#454—2711no—
241#443—2711no—
242#443—2711no—
2432−1+4—2711no—
2442−1+4C2×D27111yescharacteristic, complemented
245H(5,𝔽2)—2711no—
246H(5,𝔽2)C2×D27111yescharacteristic
247#137—21no—
248#137—2711no—
249#137—2711no—
250#137—2711no—
251#137—5421no—
252#137—5421no—
253#137—5421no—
254#137C2×D27111yescharacteristic
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—2711no—
257AGL(1,ℤ/8)—2711no—
258AGL(1,ℤ/8)—2711no—
259AGL(1,ℤ/8)—5421no—
260AGL(1,ℤ/8)—5421no—
261AGL(1,ℤ/8)—5421no—
262AGL(1,ℤ/8)C2×D27111yescharacteristic, complemented
263#135—21no—
264#135—2711no—
265#135—5421no—
266#135—5421no—
267#135—5421no—
268#135C2×D27111yescharacteristic
269C2×Q16—2711no—
270C2×Q16—5421no—
271C2×SD16—21no—
272C2×SD16—2711no—
273C2×SD16—5421no—
274C2×SD16C2×D27111yescharacteristic, complemented
275C2×D8—2711no—
276C2×D8—5421no—
277#131—2711no—
278#131—2711no—
279#131—2711no—
280#131C2×D27111yescharacteristic, complemented
281#124—2711no—
282#124—2711no—
283#108—2711no—
284#108—2711no—
285#104—2711no—
286#104—2711no—
287#104—2711no—
288#104—2711no—
289#101—2711no—
290#101—2711no—
291#100—2711no—
292#100—2711no—
293C4×C8—2711no—
294#41—21no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—41no—
299#41—2711no—
300#41—5421no—
301#41—5421no—
302#41—5421no—
303#41—5421no—
304#41—10841no—
305#41C22×D27111yescharacteristic
306#41C22×D27111yescharacteristic
307#41C22×D27111yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—2711no—
311C2×Q8—5421no—
312C2×Q8—5421no—
313C2×Q8C22×D27111yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—2711no—
318C2×D4—5421no—
319C2×D4—5421no—
320C2×D4—10841no—
321C2×D4C22×D27111yescharacteristic
322Q16—21no—
323Q16—2711no—
324Q16—5421no—
325Q16—5421no—
326Q16—5421no—
327Q16—5421no—
328Q16—5421no—
329Q16C22⋊D27111yescharacteristic
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—
344SD16C22⋊D27111yescharacteristic
345SD16C22⋊D27111yescharacteristic
346D8—21no—
347D8—2711no—
348D8—5421no—
349D8—5421no—
350D8—5421no—
351D8—5421no—
352D8—5421no—
353D8C22⋊D27111yescharacteristic
354M16—21no—
355M16—2711no—
356M16—2711no—
357M16—2711no—
358M16—5421no—
359M16—5421no—
360M16—5421no—
361M16C22×D27111yescharacteristic
362C2×C8—21no—
363C2×C8—2711no—
364C2×C8—5421no—
365C2×C8—5421no—
366C2×C8—5421no—
367C2×C8C22×D27111yescharacteristic
368C22⋊C4—10841no—
369C22⋊C4—10841no—
370C42—2711no—
371C23—21no—
372C23—41no—
373C23—5421no—
374C23—10841no—
375Q8—21no—
376Q8—21no—
377Q8—41no—
378Q8—2711no—
379Q8—5421no—
380Q8—5421no—
381Q8—10841no—
382Q8D4×D27111yescharacteristic
383Q8C2×C22⋊D27111yescharacteristic
384Q8C2×C22⋊D27111yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—2711no—
394D4—5421no—
395D4—5421no—
396D4—5421no—
397D4—10841no—
398D4—10841no—
399D4—10841no—
400D4D4×D27111yescharacteristic
401D4C2×C22⋊D27111yescharacteristic
402D4C2×C22⋊D27111yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—5421no—
411C2×C4—5421no—
412C2×C4—5421no—
413C2×C4—10841no—
414C2×C4—10841no—
415C2×C4—10841no—
416C2×C4C23×D27111yescharacteristic
417C8—21no—
418C8—2711no—
419C8—2711no—
420C8—5421no—
421C8—5421no—
422C8—5421no—
423C8C2×C22⋊D27111yescharacteristic
424C8C2×C22⋊D27111yescharacteristic
425C22—21no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—81no—
430C22—161no—
431C22—5421no—
432C22—10841no—
433C22—21681no—
434C22—43361no—
435C22#2257620511yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—5421no—
441C4—10841no—
442C4—10841no—
443C4C2×D4×D27111yescharacteristic
444C4C22×C22⋊D27111yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—81no—
450C2—10841no—
451C2—21681no—
452C2D4×C22⋊D27111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#577×C271C2
socleC2×C271D42
hypercenter#577D271
hypocenterC271#3290
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. #31774712C2semidirect
2. #31774704C2semidirect
3. #31774700C2semidirect
4. #31774278C2semidirect
5. #31774170C2semidirect
6. #31774165C2semidirect
7. #31774160C2semidirect
8. #31774157C2semidirect
9. #31774075C2semidirect
10. #31774073C2semidirect
11. #31774070C2semidirect
12. #31774068C2semidirect
13. #31774014C2semidirect
14. #31774012C2semidirect
15. #577×C271C2semidirect
16. #22576207C22semidirect
20. #22576191C22semidirect
23. #22576190C22semidirect
28. #22576185C22semidirect
35. C2×#16864644C22semidirect
36. C2×#16864643C22semidirect
41. #22576167C22semidirect
42. #22576166C22semidirect
43. #22576149C22semidirect
44. #22576147C22semidirect
45. #22576096C22semidirect
46. #22576095C22semidirect
48. #22576087C22semidirect
49. #22576087C22semidirect
50. #22576086C22semidirect
51. #22576079C22semidirect
52. #22576078C22semidirect
53. #22576077C22semidirect
54. #22576076C22semidirect
55. C8×Dic271C22semidirect
56. 2−1+4×C271C22semidirect
61. AGL(1,ℤ/8)×C271C22semidirect
65. C2×SD16×C271C22semidirect
66. #131×C271C22semidirect
226. C271#3290semidirect
230. #577D271semidirect
244. 2−1+4C2×D271semidirect
262. AGL(1,ℤ/8)C2×D271semidirect
274. C2×SD16C2×D271semidirect
280. #131C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747122
#317747042
#317747002
#317742782
#317741702
#317741652
#317741602
#317741572
#317740752
#317740732
#317740702
#317740682
#317740142
#317740122
#577×C2712
#3290271