Finite Group Databasebeta

← #30677753

Subgroups — #30677753

Order 16768.

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#22554124C211yesmaximal, characteristic, complemented
2#22554122C211yesmaximal, characteristic, complemented
3#22554121C211yesmaximal, characteristic, complemented
4#22554118C211yesmaximal, characteristic, complemented
5#22553683C211yesmaximal, characteristic, complemented
6#22553578C211yesmaximal, characteristic, complemented
7#22553576C211yesmaximal, characteristic, complemented
8#22553574C211yesmaximal, characteristic, complemented
9#22553497C211yesmaximal, characteristic, complemented
10#22553493C211yesmaximal, characteristic, complemented
11#22553482C211yesmaximal, characteristic, complemented
12#22553478C211yesmaximal, characteristic, complemented
13#22553428C211yesmaximal, characteristic, complemented
14#22553427C211yesmaximal, characteristic, complemented
15#471×C131C211yesmaximal, characteristic, complemented
16#16862294C2211yescharacteristic
17#16862293C2211yescharacteristic
18#16862292C2211yescharacteristic, complemented
19#16862291C2211yescharacteristic, complemented
20#16862279—21no—
21#16862279C2211yescharacteristic, complemented
22#16862279C2211yescharacteristic, complemented
23#16862278—21no—
24#16862277C2211yescharacteristic, complemented
25#16862275—21no—
26#16862275—21no—
27#16862274—21no—
28#16862274C2211yescharacteristic
29#16862274C2211yescharacteristic
30#16862273—21no—
31#16862273—21no—
32#16862272C2211yescharacteristic
33#16862270—21no—
34#16862269—21no—
35#16862268—21no—
36#16862268C2211yescharacteristic
37#16862267—21no—
38#16862266C2211yescharacteristic
39#16862265—21no—
40C2×#14523502—21no—
41C2×#14523502C2211yescharacteristic, complemented
42C2×#14523501C2211yescharacteristic, complemented
43C2×#14523500C2211yescharacteristic, complemented
44SD16×D131—21no—
45SD16×D131—21no—
46SD16×D131—21no—
47#16862254C2211yescharacteristic, complemented
48#16862254C2211yescharacteristic, complemented
49#16862252C2211yescharacteristic, complemented
50#16862244C2211yescharacteristic, complemented
51#16862242C2211yescharacteristic
52#16862181C2211yescharacteristic, complemented
53#16862173C2211yescharacteristic, complemented
54#16862172C2211yescharacteristic, complemented
55#16862172C2211yescharacteristic, complemented
56#16862172C2211yescharacteristic, complemented
57#16862164C2211yescharacteristic, complemented
58#16862162C2211yescharacteristic, complemented
59C8×Dic131C2211yescharacteristic, complemented
60#137×C131C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C131C2211yescharacteristic
62C2×SD16×C131C2211yescharacteristic, complemented
63#131×C131C2211yescharacteristic, complemented
64#108×C131C2211yescharacteristic, complemented
65#101×C131C2211yescharacteristic, complemented
66#100×C131C2211yescharacteristic, complemented
67#14523514—21no—
68#14523514—21no—
69#14523514—21no—
70#14523513—21no—
71#14523513—21no—
72#14523513—41no—
73#14523513C2311yescharacteristic
74#14523513C2311yescharacteristic
75#14523513C2311yescharacteristic
76#14523512—21no—
77#14523512—21no—
78#14523512—21no—
79#14523512—41no—
80C2×Dic262—21no—
81C2×Dic262C2311yescharacteristic
82Q8×D131—21no—
83Q8×D131—21no—
84Q8×D131—21no—
85C2×D524—21no—
86C2×D524C2311yescharacteristic
87C2×C22⋊D131—41no—
88C2×C22⋊D131—41no—
89D4×D131—21no—
90D4×D131—21no—
91D4×D131—21no—
92D4×D131—41no—
93Dic524—21no—
94Dic524—21no—
95Dic524D411yescharacteristic, complemented
96#14523503—21no—
97#14523503D411yescharacteristic, complemented
98#14523502—21no—
99#14523502—21no—
100#14523502—21no—
101#14523502—21no—
102#14523502—21no—
103#14523502—21no—
104#14523502D411yescharacteristic, complemented
105#14523502D411yescharacteristic, complemented
106#14523501—21no—
107#14523501—21no—
108#14523501—21no—
109#14523501D411yescharacteristic, complemented
110#14523500—21no—
111#14523500—21no—
112#14523500—21no—
113#14523500D411yescharacteristic, complemented
114D1048—21no—
115D1048—21no—
116D1048D411yescharacteristic, complemented
117#14523498—21no—
118#14523498D411yescharacteristic, complemented
119#14523497C2311yescharacteristic
120#14523496—21no—
121#14523496—21no—
122#14523496—21no—
123C2×C131⋊C8C2311yescharacteristic
124C8×D131—21no—
125C8×D131—21no—
126C8×D131—21no—
127Dic131⋊C4—41no—
128C4×Dic131C2311yescharacteristic
129#41×C131C2311yescharacteristic
130C2×Q8×C131C2311yescharacteristic
131C2×D4×C131C2311yescharacteristic
132Q16×C131—21no—
133SD16×C131—21no—
134SD16×C131—21no—
135SD16×C131—21no—
136SD16×C131—21no—
137D8×C131—21no—
138M16×C131—21no—
139M16×C131C2311yescharacteristic
140M16×C131C2311yescharacteristic
141M16×C131C2311yescharacteristic
142C2×C8×C131—21no—
143C2×C8×C131C2311yescharacteristic
144C22×D131—21no—
145C22×D131—41no—
146C22×D131—41no—
147Dic262—21no—
148Dic262—21no—
149Dic262—41no—
150Dic262—41no—
151Dic262C2×D411yescharacteristic
152Dic262C2×D411yescharacteristic
153Dic262C2×D411yescharacteristic
154D524—21no—
155D524—21no—
156D524—41no—
157D524—41no—
158D524C2×D411yescharacteristic
159D524C2×D411yescharacteristic
160D524C2×D411yescharacteristic
161C22⋊D131—21no—
162C22⋊D131—21no—
163C22⋊D131—21no—
164C22⋊D131—41no—
165C22⋊D131—41no—
166C22⋊D131—41no—
167C22⋊D131—41no—
168C22⋊D131—82no—
169C2×Dic131—21no—
170C2×Dic131—21no—
171C2×Dic131—41no—
172C2×Dic131—41no—
173C4×D131—21no—
174C4×D131—21no—
175C4×D131—21no—
176C4×D131—41no—
177C4×D131—41no—
178C4×D131—41no—
179C4×D131—41no—
180C131⋊C8—21no—
181C131⋊C8C2×D411yescharacteristic
182C131⋊C8C2×D411yescharacteristic
183C23×C131—21no—
184Q8×C131—21no—
185Q8×C131—21no—
186Q8×C131C2×D411yescharacteristic
187D4×C131—21no—
188D4×C131—21no—
189D4×C131—21no—
190D4×C131C2×D411yescharacteristic
191C2×C4×C131—21no—
192C2×C4×C131—21no—
193C2×C4×C131C2411yescharacteristic
194C8×C131—21no—
195C8×C131—21no—
196C8×C131—21no—
197C8×C131C2×D411yescharacteristic
198C8×C131C2×D411yescharacteristic
199C22×C131—21no—
200C22×C131—41no—
201C22×C131—81no—
202C22×C131H(5,𝔽2)11yescharacteristic
203C2×D131—21no—
204C2×D131—21no—
205C2×D131—21no—
206C2×D131—41no—
207C2×D131—81no—
208C2×D131—161no—
209C2×D131—161no—
210C4×C131—21no—
211C4×C131—41no—
212C4×C131C22×D411yescharacteristic
213C4×C131C22×D411yescharacteristic
214Dic131—22no—
215Dic131—21no—
216Dic131—21no—
217Dic131—21no—
218Dic131—41no—
219C2×C131—21no—
220C2×C131—41no—
221C2×C131—81no—
222C2×C131D4211yescharacteristic
223D131—41no—
224D131—41no—
225D131—41no—
226D131—81no—
227C131#329111yescharacteristic, complemented, Sylow
228#3291—1311nomaximal, Sylow
229#577—1311no—
230#577—1311no—
231#577—1311no—
232#577—1311no—
233#492—1311no—
234#471—1311no—
235#471—1311no—
236#471—1311no—
237#471D13111yescharacteristic, complemented
238#456—1311no—
239#456—1311no—
240#453—1311no—
241#453—1311no—
242#443—1311no—
243#443—1311no—
2442−1+4—1311no—
2452−1+4—1311no—
246H(5,𝔽2)—1311no—
247H(5,𝔽2)—1311no—
248#137—2621no—
249#137—2621no—
250#137—2621no—
251#137—2621no—
252#137—1311no—
253#137—1311no—
254#137—1311no—
255#137C2×D13111yescharacteristic, complemented
256AGL(1,ℤ/8)—2621no—
257AGL(1,ℤ/8)—2621no—
258AGL(1,ℤ/8)—2621no—
259AGL(1,ℤ/8)—2621no—
260AGL(1,ℤ/8)—1311no—
261AGL(1,ℤ/8)—1311no—
262AGL(1,ℤ/8)—1311no—
263AGL(1,ℤ/8)C2×D13111yescharacteristic
264#135—2621no—
265#135—2621no—
266#135—2621no—
267#135—2621no—
268#135—1311no—
269#135—1311no—
270C2×SD16—2621no—
271C2×SD16—2621no—
272C2×SD16—2621no—
273C2×SD16—2621no—
274C2×SD16—1311no—
275C2×SD16—1311no—
276C2×SD16—1311no—
277C2×SD16C2×D13111yescharacteristic, complemented
278#131—1311no—
279#131—1311no—
280#131—1311no—
281#131C2×D13111yescharacteristic, complemented
282C4⋊Q8—1311no—
283#127—1311no—
284#108—1311no—
285#108C2×D13111yescharacteristic, complemented
286#104—1311no—
287#104—1311no—
288#104—1311no—
289#104—1311no—
290#101—1311no—
291#101C2×D13111yescharacteristic, complemented
292#100—1311no—
293#100C2×D13111yescharacteristic, complemented
294C4×C8—1311no—
295#41—2621no—
296#41—2621no—
297#41—2621no—
298#41—2621no—
299#41—2621no—
300#41—2621no—
301#41—2621no—
302#41—2621no—
303#41—5241no—
304#41—5241no—
305#41—1311no—
306#41—1311no—
307#41—1311no—
308#41C22×D13111yescharacteristic
309C2×Q8—2621no—
310C2×Q8—2621no—
311C2×Q8—2621no—
312C2×Q8—2621no—
313C2×Q8—1311no—
314C2×Q8C22×D13111yescharacteristic
315C2×D4—2621no—
316C2×D4—2621no—
317C2×D4—2621no—
318C2×D4—2621no—
319C2×D4—5241no—
320C2×D4—5241no—
321C2×D4—5241no—
322C2×D4—1311no—
323C2×D4C22×D13111yescharacteristic
324Q16—21no—
325Q16—2621no—
326Q16—2621no—
327Q16—2621no—
328Q16—1311no—
329Q16—1311no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—2621no—
335SD16—2621no—
336SD16—2621no—
337SD16—2621no—
338SD16—2621no—
339SD16—2621no—
340SD16—2621no—
341SD16—2621no—
342SD16—2621no—
343SD16—2621no—
344SD16—2621no—
345SD16—2621no—
346SD16—1311no—
347SD16—1311no—
348SD16—1311no—
349SD16—1311no—
350D8—21no—
351D8—2621no—
352D8—2621no—
353D8—2621no—
354D8—1311no—
355D8—1311no—
356M16—21no—
357M16—2621no—
358M16—2621no—
359M16—2621no—
360M16—1311no—
361M16C22×D13111yescharacteristic
362M16C22×D13111yescharacteristic
363M16C22×D13111yescharacteristic
364C2×C8—21no—
365C2×C8—2621no—
366C2×C8—2621no—
367C2×C8—2621no—
368C2×C8—1311no—
369C2×C8C22×D13111yescharacteristic
370C4⋊C4—5241no—
371C42—1311no—
372C23—21no—
373C23—2621no—
374C23—5241no—
375C23—5241no—
376Q8—21no—
377Q8—21no—
378Q8—2621no—
379Q8—2621no—
380Q8—5241no—
381Q8—5241no—
382Q8—1311no—
383Q8—1311no—
384Q8—1311no—
385Q8D4×D13111yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—2621no—
390D4—2621no—
391D4—2621no—
392D4—2621no—
393D4—2621no—
394D4—5241no—
395D4—5241no—
396D4—5241no—
397D4—5241no—
398D4—5241no—
399D4—5241no—
400D4—10482no—
401D4—1311no—
402D4—1311no—
403D4—1311no—
404D4D4×D13111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—2621no—
408C2×C4—2621no—
409C2×C4—2621no—
410C2×C4—2621no—
411C2×C4—2621no—
412C2×C4—5241no—
413C2×C4—5241no—
414C2×C4—5241no—
415C2×C4—5241no—
416C2×C4—5241no—
417C2×C4—5241no—
418C2×C4C23×D13111yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—2621no—
423C8—1311no—
424C8—1311no—
425C8D4×D13111yescharacteristic
426C8D4×D13111yescharacteristic
427C22—21no—
428C22—41no—
429C22—2621no—
430C22—2621no—
431C22—2621no—
432C22—81no—
433C22—5241no—
434C22—10481no—
435C22—20961no—
436C22—20961no—
437C22#1686229211yescharacteristic
438C4—21no—
439C4—41no—
440C4—2622no—
441C4—2621no—
442C4—2621no—
443C4—2621no—
444C4—5241no—
445C4C2×D4×D13111yescharacteristic
446C4C2×D4×D13111yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—5241no—
451C2—5241no—
452C2—5241no—
453C2—10481no—
454C2#2255394611yescharacteristic
subgroupG/H
centerC2#22553946
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#471×C131C2
socleC2×C131D42
hypercenter#471D131
hypocenterC131#3291
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #22554124C2semidirect
2. #22554122C2semidirect
3. #22554121C2semidirect
4. #22554118C2semidirect
5. #22553683C2semidirect
6. #22553578C2semidirect
7. #22553576C2semidirect
8. #22553574C2semidirect
9. #22553497C2semidirect
10. #22553493C2semidirect
11. #22553482C2semidirect
12. #22553478C2semidirect
13. #22553428C2semidirect
14. #22553427C2semidirect
15. #471×C131C2semidirect
18. #16862292C22semidirect
19. #16862291C22semidirect
21. #16862279C22semidirect
22. #16862279C22semidirect
24. #16862277C22semidirect
41. C2×#14523502C22semidirect
42. C2×#14523501C22semidirect
43. C2×#14523500C22semidirect
47. #16862254C22semidirect
48. #16862254C22semidirect
49. #16862252C22semidirect
50. #16862244C22semidirect
52. #16862181C22semidirect
53. #16862173C22semidirect
54. #16862172C22semidirect
55. #16862172C22semidirect
56. #16862172C22semidirect
57. #16862164C22semidirect
58. #16862162C22semidirect
59. C8×Dic131C22semidirect
60. #137×C131C22semidirect
62. C2×SD16×C131C22semidirect
63. #131×C131C22semidirect
64. #108×C131C22semidirect
65. #101×C131C22semidirect
66. #100×C131C22semidirect
95. Dic524D4semidirect
97. #14523503D4semidirect
104. #14523502D4semidirect
105. #14523502D4semidirect
109. #14523501D4semidirect
113. #14523500D4semidirect
116. D1048D4semidirect
118. #14523498D4semidirect
227. C131#3291semidirect
237. #471D131semidirect
255. #137C2×D131semidirect
277. C2×SD16C2×D131semidirect
281. #131C2×D131semidirect
285. #108C2×D131semidirect
291. #101C2×D131semidirect
293. #100C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541242
#225541222
#225541212
#225541182
#225536832
#225535782
#225535762
#225535742
#225534972
#225534932
#225534822
#225534782
#225534282
#225534272
#471×C1312
#3291131