Finite Group Databasebeta

← #46848156

Subgroups — #46848156

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#31774709C211yesmaximal, characteristic, complemented
2#31774707C211yesmaximal, characteristic, complemented
3#31774706C211yesmaximal, characteristic, complemented
4#31774703C211yesmaximal, characteristic, complemented
5#31774268C211yesmaximal, characteristic, complemented
6#31774163C211yesmaximal, characteristic, complemented
7#31774161C211yesmaximal, characteristic, complemented
8#31774159C211yesmaximal, characteristic, complemented
9#31774082C211yesmaximal, characteristic, complemented
10#31774078C211yesmaximal, characteristic, complemented
11#31774067C211yesmaximal, characteristic, complemented
12#31774063C211yesmaximal, characteristic, complemented
13#31774013C211yesmaximal, characteristic, complemented
14#31774012C211yesmaximal, characteristic, complemented
15#471×C271C211yesmaximal, characteristic, complemented
16#22576208C2211yescharacteristic
17#22576207C2211yescharacteristic
18#22576206C2211yescharacteristic, complemented
19#22576205C2211yescharacteristic, complemented
20#22576193—21no—
21#22576193C2211yescharacteristic, complemented
22#22576193C2211yescharacteristic, complemented
23#22576192—21no—
24#22576191C2211yescharacteristic, complemented
25#22576189—21no—
26#22576189—21no—
27#22576188—21no—
28#22576188C2211yescharacteristic
29#22576188C2211yescharacteristic
30#22576187—21no—
31#22576187—21no—
32#22576186C2211yescharacteristic
33#22576184—21no—
34#22576183—21no—
35#22576182—21no—
36#22576182C2211yescharacteristic
37#22576181—21no—
38#22576180C2211yescharacteristic
39#22576179—21no—
40C2×#16864643—21no—
41C2×#16864643C2211yescharacteristic, complemented
42C2×#16864642C2211yescharacteristic, complemented
43C2×#16864641C2211yescharacteristic, complemented
44SD16×D271—21no—
45SD16×D271—21no—
46SD16×D271—21no—
47#22576168C2211yescharacteristic, complemented
48#22576168C2211yescharacteristic, complemented
49#22576166C2211yescharacteristic, complemented
50#22576158C2211yescharacteristic, complemented
51#22576156C2211yescharacteristic
52#22576095C2211yescharacteristic, complemented
53#22576087C2211yescharacteristic, complemented
54#22576086C2211yescharacteristic, complemented
55#22576086C2211yescharacteristic, complemented
56#22576086C2211yescharacteristic, complemented
57#22576078C2211yescharacteristic, complemented
58#22576076C2211yescharacteristic, complemented
59C8×Dic271C2211yescharacteristic, 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
67#16864655—21no—
68#16864655—21no—
69#16864655—21no—
70#16864654—21no—
71#16864654—21no—
72#16864654—41no—
73#16864654C2311yescharacteristic
74#16864654C2311yescharacteristic
75#16864654C2311yescharacteristic
76#16864653—21no—
77#16864653—21no—
78#16864653—21no—
79#16864653—41no—
80C2×Dic542—21no—
81C2×Dic542C2311yescharacteristic
82Q8×D271—21no—
83Q8×D271—21no—
84Q8×D271—21no—
85C2×D1084—21no—
86C2×D1084C2311yescharacteristic
87C2×C22⋊D271—41no—
88C2×C22⋊D271—41no—
89D4×D271—21no—
90D4×D271—21no—
91D4×D271—21no—
92D4×D271—41no—
93Dic1084—21no—
94Dic1084—21no—
95Dic1084D411yescharacteristic, complemented
96#16864644—21no—
97#16864644D411yescharacteristic, complemented
98#16864643—21no—
99#16864643—21no—
100#16864643—21no—
101#16864643—21no—
102#16864643—21no—
103#16864643—21no—
104#16864643D411yescharacteristic, complemented
105#16864643D411yescharacteristic, complemented
106#16864642—21no—
107#16864642—21no—
108#16864642—21no—
109#16864642D411yescharacteristic, complemented
110#16864641—21no—
111#16864641—21no—
112#16864641—21no—
113#16864641D411yescharacteristic, complemented
114D2168—21no—
115D2168—21no—
116D2168D411yescharacteristic, complemented
117#16864639—21no—
118#16864639D411yescharacteristic, complemented
119#16864638C2311yescharacteristic
120#16864637—21no—
121#16864637—21no—
122#16864637—21no—
123C2×C271⋊C8C2311yescharacteristic
124C8×D271—21no—
125C8×D271—21no—
126C8×D271—21no—
127Dic271⋊C4—41no—
128C4×Dic271C2311yescharacteristic
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—41no—
146C22×D271—41no—
147Dic542—21no—
148Dic542—21no—
149Dic542—41no—
150Dic542—41no—
151Dic542C2×D411yescharacteristic
152Dic542C2×D411yescharacteristic
153Dic542C2×D411yescharacteristic
154D1084—21no—
155D1084—21no—
156D1084—41no—
157D1084—41no—
158D1084C2×D411yescharacteristic
159D1084C2×D411yescharacteristic
160D1084C2×D411yescharacteristic
161C22⋊D271—21no—
162C22⋊D271—21no—
163C22⋊D271—21no—
164C22⋊D271—41no—
165C22⋊D271—41no—
166C22⋊D271—41no—
167C22⋊D271—41no—
168C22⋊D271—82no—
169C2×Dic271—21no—
170C2×Dic271—21no—
171C2×Dic271—41no—
172C2×Dic271—41no—
173C4×D271—21no—
174C4×D271—21no—
175C4×D271—21no—
176C4×D271—41no—
177C4×D271—41no—
178C4×D271—41no—
179C4×D271—41no—
180C271⋊C8—21no—
181C271⋊C8C2×D411yescharacteristic
182C271⋊C8C2×D411yescharacteristic
183C23×C271—21no—
184Q8×C271—21no—
185Q8×C271—21no—
186Q8×C271C2×D411yescharacteristic
187D4×C271—21no—
188D4×C271—21no—
189D4×C271—21no—
190D4×C271C2×D411yescharacteristic
191C2×C4×C271—21no—
192C2×C4×C271—21no—
193C2×C4×C271C2411yescharacteristic
194C8×C271—21no—
195C8×C271—21no—
196C8×C271—21no—
197C8×C271C2×D411yescharacteristic
198C8×C271C2×D411yescharacteristic
199C22×C271—21no—
200C22×C271—41no—
201C22×C271—81no—
202C22×C271H(5,𝔽2)11yescharacteristic
203C2×D271—21no—
204C2×D271—21no—
205C2×D271—21no—
206C2×D271—41no—
207C2×D271—81no—
208C2×D271—161no—
209C2×D271—161no—
210C4×C271—21no—
211C4×C271—41no—
212C4×C271C22×D411yescharacteristic
213C4×C271C22×D411yescharacteristic
214Dic271—22no—
215Dic271—21no—
216Dic271—21no—
217Dic271—21no—
218Dic271—41no—
219C2×C271—21no—
220C2×C271—41no—
221C2×C271—81no—
222C2×C271D4211yescharacteristic
223D271—41no—
224D271—41no—
225D271—41no—
226D271—81no—
227C271#329111yescharacteristic, complemented, Sylow
228#3291—2711nomaximal, Sylow
229#577—2711no—
230#577—2711no—
231#577—2711no—
232#577—2711no—
233#492—2711no—
234#471—2711no—
235#471—2711no—
236#471—2711no—
237#471D27111yescharacteristic, complemented
238#456—2711no—
239#456—2711no—
240#453—2711no—
241#453—2711no—
242#443—2711no—
243#443—2711no—
2442−1+4—2711no—
2452−1+4—2711no—
246H(5,𝔽2)—2711no—
247H(5,𝔽2)—2711no—
248#137—2711no—
249#137—2711no—
250#137—2711no—
251#137—5421no—
252#137—5421no—
253#137—5421no—
254#137—5421no—
255#137C2×D27111yescharacteristic, complemented
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)—5421no—
263AGL(1,ℤ/8)C2×D27111yescharacteristic
264#135—2711no—
265#135—2711no—
266#135—5421no—
267#135—5421no—
268#135—5421no—
269#135—5421no—
270C2×SD16—2711no—
271C2×SD16—2711no—
272C2×SD16—2711no—
273C2×SD16—5421no—
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#108C2×D27111yescharacteristic, complemented
286#104—2711no—
287#104—2711no—
288#104—2711no—
289#104—2711no—
290#101—2711no—
291#101C2×D27111yescharacteristic, complemented
292#100—2711no—
293#100C2×D27111yescharacteristic, complemented
294C4×C8—2711no—
295#41—2711no—
296#41—2711no—
297#41—2711no—
298#41—5421no—
299#41—5421no—
300#41—5421no—
301#41—5421no—
302#41—5421no—
303#41—5421no—
304#41—5421no—
305#41—5421no—
306#41—10841no—
307#41—10841no—
308#41C22×D27111yescharacteristic
309C2×Q8—2711no—
310C2×Q8—5421no—
311C2×Q8—5421no—
312C2×Q8—5421no—
313C2×Q8—5421no—
314C2×Q8C22×D27111yescharacteristic
315C2×D4—2711no—
316C2×D4—5421no—
317C2×D4—5421no—
318C2×D4—5421no—
319C2×D4—5421no—
320C2×D4—10841no—
321C2×D4—10841no—
322C2×D4—10841no—
323C2×D4C22×D27111yescharacteristic
324Q16—21no—
325Q16—2711no—
326Q16—2711no—
327Q16—5421no—
328Q16—5421no—
329Q16—5421no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—2711no—
335SD16—2711no—
336SD16—2711no—
337SD16—2711no—
338SD16—5421no—
339SD16—5421no—
340SD16—5421no—
341SD16—5421no—
342SD16—5421no—
343SD16—5421no—
344SD16—5421no—
345SD16—5421no—
346SD16—5421no—
347SD16—5421no—
348SD16—5421no—
349SD16—5421no—
350D8—21no—
351D8—2711no—
352D8—2711no—
353D8—5421no—
354D8—5421no—
355D8—5421no—
356M16—21no—
357M16—2711no—
358M16—5421no—
359M16—5421no—
360M16—5421no—
361M16C22×D27111yescharacteristic
362M16C22×D27111yescharacteristic
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—5421no—
374C23—10841no—
375C23—10841no—
376Q8—21no—
377Q8—21no—
378Q8—2711no—
379Q8—2711no—
380Q8—2711no—
381Q8—5421no—
382Q8—5421no—
383Q8—10841no—
384Q8—10841no—
385Q8D4×D27111yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—2711no—
390D4—2711no—
391D4—2711no—
392D4—5421no—
393D4—5421no—
394D4—5421no—
395D4—5421no—
396D4—5421no—
397D4—10841no—
398D4—10841no—
399D4—10841no—
400D4—10841no—
401D4—10841no—
402D4—10841no—
403D4—21682no—
404D4D4×D27111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—5421no—
408C2×C4—5421no—
409C2×C4—5421no—
410C2×C4—5421no—
411C2×C4—5421no—
412C2×C4—10841no—
413C2×C4—10841no—
414C2×C4—10841no—
415C2×C4—10841no—
416C2×C4—10841no—
417C2×C4—10841no—
418C2×C4C23×D27111yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—2711no—
423C8—2711no—
424C8—5421no—
425C8D4×D27111yescharacteristic
426C8D4×D27111yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—5421no—
431C22—5421no—
432C22—5421no—
433C22—10841no—
434C22—21681no—
435C22—43361no—
436C22—43361no—
437C22#2257620611yescharacteristic
438C4—21no—
439C4—41no—
440C4—5422no—
441C4—5421no—
442C4—5421no—
443C4—5421no—
444C4—10841no—
445C4C2×D4×D27111yescharacteristic
446C4C2×D4×D27111yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—10841no—
451C2—10841no—
452C2—10841no—
453C2—21681no—
454C2#3177453111yescharacteristic
subgroupG/H
centerC2#31774531
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#471×C271C2
socleC2×C271D42
hypercenter#471D271
hypocenterC271#3291
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. #31774709C2semidirect
2. #31774707C2semidirect
3. #31774706C2semidirect
4. #31774703C2semidirect
5. #31774268C2semidirect
6. #31774163C2semidirect
7. #31774161C2semidirect
8. #31774159C2semidirect
9. #31774082C2semidirect
10. #31774078C2semidirect
11. #31774067C2semidirect
12. #31774063C2semidirect
13. #31774013C2semidirect
14. #31774012C2semidirect
15. #471×C271C2semidirect
18. #22576206C22semidirect
19. #22576205C22semidirect
21. #22576193C22semidirect
22. #22576193C22semidirect
24. #22576191C22semidirect
41. C2×#16864643C22semidirect
42. C2×#16864642C22semidirect
43. C2×#16864641C22semidirect
47. #22576168C22semidirect
48. #22576168C22semidirect
49. #22576166C22semidirect
50. #22576158C22semidirect
52. #22576095C22semidirect
53. #22576087C22semidirect
54. #22576086C22semidirect
55. #22576086C22semidirect
56. #22576086C22semidirect
57. #22576078C22semidirect
58. #22576076C22semidirect
59. C8×Dic271C22semidirect
60. #137×C271C22semidirect
62. C2×SD16×C271C22semidirect
63. #131×C271C22semidirect
64. #108×C271C22semidirect
65. #101×C271C22semidirect
66. #100×C271C22semidirect
95. Dic1084D4semidirect
97. #16864644D4semidirect
104. #16864643D4semidirect
105. #16864643D4semidirect
109. #16864642D4semidirect
113. #16864641D4semidirect
116. D2168D4semidirect
118. #16864639D4semidirect
227. C271#3291semidirect
237. #471D271semidirect
255. #137C2×D271semidirect
277. C2×SD16C2×D271semidirect
281. #131C2×D271semidirect
285. #108C2×D271semidirect
291. #101C2×D271semidirect
293. #100C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747092
#317747072
#317747062
#317747032
#317742682
#317741632
#317741612
#317741592
#317740822
#317740782
#317740672
#317740632
#317740132
#317740122
#471×C2712
#3291271