Finite Group Databasebeta

← #30677752

Subgroups — #30677752

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#22554123C211yesmaximal, characteristic, complemented
2#22554120C211yesmaximal, characteristic, complemented
3#22554119C211yesmaximal, characteristic, complemented
4#22553683C211yesmaximal, characteristic, complemented
5#22553577C211yesmaximal, characteristic, complemented
6#22553575C211yesmaximal, characteristic, complemented
7#22553573C211yesmaximal, characteristic, complemented
8#22553572C211yesmaximal, characteristic, complemented
9#22553495C211yesmaximal, characteristic, complemented
10#22553493C211yesmaximal, characteristic, complemented
11#22553480C211yesmaximal, characteristic, complemented
12#22553478C211yesmaximal, characteristic, complemented
13#22553429C211yesmaximal, characteristic, complemented
14#22553427C211yesmaximal, characteristic, complemented
15#577×C131C211yesmaximal, characteristic, complemented
16#16862293C2211yescharacteristic
17#16862292C2211yescharacteristic, complemented
18#16862278—21no—
19#16862277—21no—
20#16862277C2211yescharacteristic, complemented
21#16862276—21no—
22#16862276C2211yescharacteristic, complemented
23#16862276C2211yescharacteristic, complemented
24#16862272—21no—
25#16862272C2211yescharacteristic
26#16862271—21no—
27#16862271C2211yescharacteristic
28#16862271C2211yescharacteristic
29#16862270—21no—
30#16862267—21no—
31#16862266—21no—
32#16862266—21no—
33#16862266C2211yescharacteristic
34C2×#14523502C2211yescharacteristic, complemented
35C2×#14523501—21no—
36C2×#14523501C2211yescharacteristic, complemented
37C2×#14523500—21no—
38C2×#14523500C2211yescharacteristic, complemented
39SD16×D131—21no—
40#16862253C2211yescharacteristic, complemented
41#16862253C2211yescharacteristic, complemented
42#16862252C2211yescharacteristic, complemented
43#16862244C2211yescharacteristic, complemented
44#16862242C2211yescharacteristic
45#16862182C2211yescharacteristic, complemented
46#16862181C2211yescharacteristic, complemented
47#16862173C2211yescharacteristic, complemented
48#16862173C2211yescharacteristic, complemented
49#16862173C2211yescharacteristic, complemented
50#16862172C2211yescharacteristic, complemented
51#16862165C2211yescharacteristic, complemented
52#16862164C2211yescharacteristic, complemented
53#16862163C2211yescharacteristic, complemented
54#16862162C2211yescharacteristic, complemented
55C8×Dic131C2211yescharacteristic, complemented
562−1+4×C131C2211yescharacteristic
57H(5,𝔽2)×C131C2211yescharacteristic, complemented
58#137×C131—21no—
59#137×C131C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C131—21no—
61AGL(1,ℤ/8)×C131C2211yescharacteristic
62#135×C131—21no—
63#135×C131C2211yescharacteristic
64C2×SD16×C131—21no—
65C2×SD16×C131C2211yescharacteristic, complemented
66#131×C131C2211yescharacteristic, complemented
67#14523514—21no—
68#14523513—21no—
69#14523513—21no—
70#14523513C2311yescharacteristic
71#14523512—21no—
72#14523512—41no—
73C2×Dic262—21no—
74C2×Dic262C2311yescharacteristic
75Q8×D131—21no—
76C2×D524—21no—
77C2×D524C2311yescharacteristic
78C2×C22⋊D131—41no—
79D4×D131—21no—
80D4×D131—41no—
81#14523503—21no—
82#14523503—21no—
83#14523503—21no—
84#14523503D411yescharacteristic, complemented
85#14523502—21no—
86#14523502—21no—
87#14523501—21no—
88#14523501—21no—
89#14523501—21no—
90#14523501—21no—
91#14523501—21no—
92#14523501D411yescharacteristic, complemented
93#14523500—21no—
94#14523500—21no—
95#14523500—21no—
96#14523500—21no—
97#14523500—21no—
98#14523500D411yescharacteristic, complemented
99#14523498—21no—
100#14523498—21no—
101#14523498—21no—
102#14523498D411yescharacteristic, complemented
103#14523497—21no—
104#14523497—21no—
105#14523497C2311yescharacteristic
106#14523497C2311yescharacteristic
107#14523497C2311yescharacteristic
108#14523496—21no—
109C2×C131⋊C8—21no—
110C2×C131⋊C8—21no—
111C2×C131⋊C8C2311yescharacteristic
112C8×D131—21no—
113Dic131⋊C4—41no—
114C4×Dic131C2311yescharacteristic
115#41×C131—21no—
116#41×C131—21no—
117#41×C131—21no—
118#41×C131—21no—
119#41×C131—41no—
120#41×C131C2311yescharacteristic
121#41×C131C2311yescharacteristic
122#41×C131C2311yescharacteristic
123C2×Q8×C131—21no—
124C2×Q8×C131—21no—
125C2×Q8×C131C2311yescharacteristic
126C2×D4×C131—21no—
127C2×D4×C131—21no—
128C2×D4×C131—41no—
129C2×D4×C131C2311yescharacteristic
130Q16×C131—21no—
131Q16×C131D411yescharacteristic, complemented
132SD16×C131—21no—
133SD16×C131—21no—
134SD16×C131—21no—
135SD16×C131—21no—
136SD16×C131D411yescharacteristic, complemented
137SD16×C131D411yescharacteristic, complemented
138D8×C131—21no—
139D8×C131D411yescharacteristic, complemented
140M16×C131—21no—
141M16×C131C2311yescharacteristic
142C2×C8×C131—21no—
143C2×C8×C131C2311yescharacteristic
144C22×D131—21no—
145C22×D131—41no—
146Dic262—21no—
147Dic262—21no—
148Dic262—41no—
149Dic262C2×D411yescharacteristic
150D524—21no—
151D524—21no—
152D524—41no—
153D524C2×D411yescharacteristic
154C22⋊D131—21no—
155C22⋊D131—41no—
156C22⋊D131—41no—
157C22⋊D131—82no—
158C2×Dic131—21no—
159C2×Dic131—21no—
160C2×Dic131—41no—
161C4×D131—21no—
162C4×D131—41no—
163C4×D131—41no—
164C131⋊C8—21no—
165C131⋊C8—21no—
166C131⋊C8—21no—
167C131⋊C8C2×D411yescharacteristic
168C131⋊C8C2×D411yescharacteristic
169C23×C131—21no—
170C23×C131—41no—
171Q8×C131—21no—
172Q8×C131—21no—
173Q8×C131—41no—
174Q8×C131C2×D411yescharacteristic
175Q8×C131C2×D411yescharacteristic
176Q8×C131C2×D411yescharacteristic
177D4×C131—21no—
178D4×C131—21no—
179D4×C131—21no—
180D4×C131—21no—
181D4×C131—21no—
182D4×C131—41no—
183D4×C131—41no—
184D4×C131—41no—
185D4×C131C2×D411yescharacteristic
186D4×C131C2×D411yescharacteristic
187D4×C131C2×D411yescharacteristic
188C2×C4×C131—21no—
189C2×C4×C131—21no—
190C2×C4×C131—21no—
191C2×C4×C131—21no—
192C2×C4×C131—41no—
193C2×C4×C131—41no—
194C2×C4×C131—41no—
195C2×C4×C131C2411yescharacteristic
196C8×C131—21no—
197C8×C131C2×D411yescharacteristic
198C8×C131C2×D411yescharacteristic
199C22×C131—21no—
200C22×C131—21no—
201C22×C131—21no—
202C22×C131—41no—
203C22×C131—81no—
204C22×C131—161no—
205C22×C131H(5,𝔽2)11yescharacteristic
206C2×D131—21no—
207C2×D131—41no—
208C2×D131—81no—
209C2×D131—161no—
210C4×C131—21no—
211C4×C131—21no—
212C4×C131—21no—
213C4×C131—41no—
214C4×C131C22×D411yescharacteristic
215C4×C131C22×D411yescharacteristic
216Dic131—22no—
217Dic131—21no—
218Dic131—41no—
219C2×C131—21no—
220C2×C131—41no—
221C2×C131—41no—
222C2×C131—41no—
223C2×C131—81no—
224C2×C131D4211yescharacteristic
225D131—41no—
226D131—81no—
227C131#329111yescharacteristic, complemented, Sylow
228#3291—1311nomaximal, Sylow
229#577—1311no—
230#577—1311no—
231#577—1311no—
232#577D13111yescharacteristic, complemented
233#492—1311no—
234#471—1311no—
235#471—1311no—
236#471—1311no—
237#471—1311no—
238#456—1311no—
239#456—1311no—
240#453—1311no—
241#453—1311no—
242#443—1311no—
243#443—1311no—
2442−1+4—1311no—
2452−1+4C2×D13111yescharacteristic
246H(5,𝔽2)—1311no—
247H(5,𝔽2)C2×D13111yescharacteristic, complemented
248#137—21no—
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)—21no—
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—21no—
265#135—2621no—
266#135—2621no—
267#135—2621no—
268#135—1311no—
269#135C2×D13111yescharacteristic
270C2×SD16—21no—
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#108—1311no—
286#104—1311no—
287#104—1311no—
288#104—1311no—
289#104—1311no—
290#101—1311no—
291#101—1311no—
292#100—1311no—
293#100—1311no—
294C4×C8—1311no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—2621no—
301#41—2621no—
302#41—2621no—
303#41—2621no—
304#41—5241no—
305#41—1311no—
306#41C22×D13111yescharacteristic
307#41C22×D13111yescharacteristic
308#41C22×D13111yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—2621no—
312C2×Q8—2621no—
313C2×Q8—1311no—
314C2×Q8C22×D13111yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—2621no—
319C2×D4—2621no—
320C2×D4—5241no—
321C2×D4—5241no—
322C2×D4—1311no—
323C2×D4C22×D13111yescharacteristic
324Q16—21no—
325Q16—2621no—
326Q16—2621no—
327Q16—2621no—
328Q16—1311no—
329Q16C22⋊D13111yescharacteristic, complemented
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—
348SD16C22⋊D13111yescharacteristic, complemented
349SD16C22⋊D13111yescharacteristic, complemented
350D8—21no—
351D8—2621no—
352D8—2621no—
353D8—2621no—
354D8—1311no—
355D8C22⋊D13111yescharacteristic, complemented
356M16—21no—
357M16—2621no—
358M16—2621no—
359M16—2621no—
360M16—1311no—
361M16—1311no—
362M16—1311no—
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—41no—
374C23—2621no—
375C23—5241no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—2621no—
380Q8—2621no—
381Q8—5241no—
382Q8—1311no—
383Q8D4×D13111yescharacteristic
384Q8C2×C22⋊D13111yescharacteristic
385Q8C2×C22⋊D13111yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—2621no—
395D4—2621no—
396D4—2621no—
397D4—5241no—
398D4—5241no—
399D4—5241no—
400D4—10482no—
401D4—1311no—
402D4D4×D13111yescharacteristic
403D4C2×C22⋊D13111yescharacteristic
404D4C2×C22⋊D13111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—2621no—
413C2×C4—2621no—
414C2×C4—2621no—
415C2×C4—5241no—
416C2×C4—5241no—
417C2×C4—5241no—
418C2×C4C23×D13111yescharacteristic
419C8—21no—
420C8—2621no—
421C8—2621no—
422C8—2621no—
423C8—1311no—
424C8—1311no—
425C8C2×C22⋊D13111yescharacteristic
426C8C2×C22⋊D13111yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—2621no—
432C22—81no—
433C22—5241no—
434C22—161no—
435C22—10481no—
436C22—20961no—
437C22#1686229111yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—2622no—
443C4—2621no—
444C4—5241no—
445C4C2×D4×D13111yescharacteristic
446C4C22×C22⋊D13111yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—5241no—
453C2—10481no—
454C2D4×C22⋊D13111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#577×C131C2
socleC2×C131D42
hypercenter#577D131
hypocenterC131#3291
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #22554123C2semidirect
2. #22554120C2semidirect
3. #22554119C2semidirect
4. #22553683C2semidirect
5. #22553577C2semidirect
6. #22553575C2semidirect
7. #22553573C2semidirect
8. #22553572C2semidirect
9. #22553495C2semidirect
10. #22553493C2semidirect
11. #22553480C2semidirect
12. #22553478C2semidirect
13. #22553429C2semidirect
14. #22553427C2semidirect
15. #577×C131C2semidirect
17. #16862292C22semidirect
20. #16862277C22semidirect
22. #16862276C22semidirect
23. #16862276C22semidirect
34. C2×#14523502C22semidirect
36. C2×#14523501C22semidirect
38. C2×#14523500C22semidirect
40. #16862253C22semidirect
41. #16862253C22semidirect
42. #16862252C22semidirect
43. #16862244C22semidirect
45. #16862182C22semidirect
46. #16862181C22semidirect
47. #16862173C22semidirect
48. #16862173C22semidirect
49. #16862173C22semidirect
50. #16862172C22semidirect
51. #16862165C22semidirect
52. #16862164C22semidirect
53. #16862163C22semidirect
54. #16862162C22semidirect
55. C8×Dic131C22semidirect
57. H(5,𝔽2)×C131C22semidirect
59. #137×C131C22semidirect
65. C2×SD16×C131C22semidirect
66. #131×C131C22semidirect
84. #14523503D4semidirect
92. #14523501D4semidirect
98. #14523500D4semidirect
102. #14523498D4semidirect
131. Q16×C131D4semidirect
136. SD16×C131D4semidirect
137. SD16×C131D4semidirect
139. D8×C131D4semidirect
227. C131#3291semidirect
232. #577D131semidirect
247. H(5,𝔽2)C2×D131semidirect
255. #137C2×D131semidirect
277. C2×SD16C2×D131semidirect
281. #131C2×D131semidirect
329. Q16C22⋊D131semidirect
348. SD16C22⋊D131semidirect
349. SD16C22⋊D131semidirect
355. D8C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
#225541232
#225541202
#225541192
#225536832
#225535772
#225535752
#225535732
#225535722
#225534952
#225534932
#225534802
#225534782
#225534292
#225534272
#577×C1312
#3291131