Finite Group Databasebeta

← #30650502

Subgroups — #30650502

Order 16256.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550693C211yesmaximal, characteristic, complemented
2#22550692C211yesmaximal, characteristic, complemented
3#22550686C211yesmaximal, characteristic, complemented
4#22550682C211yesmaximal, characteristic, complemented
5#22550267C211yesmaximal, characteristic, complemented
6#22550152C211yesmaximal, characteristic, complemented
7#22550150C211yesmaximal, characteristic, complemented
8#22550138C211yesmaximal, characteristic, complemented
9#22550061C211yesmaximal, characteristic, complemented
10#22550057C211yesmaximal, characteristic, complemented
11#22550056C211yesmaximal, characteristic, complemented
12#22550047C211yesmaximal, characteristic, complemented
13#22549997C211yesmaximal, characteristic, complemented
14#22549996C211yesmaximal, characteristic, complemented
15#471×C127C211yesmaximal, characteristic, complemented
16#16861257C2211yescharacteristic
17#16861256C2211yescharacteristic
18#16861256C2211yescharacteristic, complemented
19#16861254C2211yescharacteristic
20#16861242—21no—
21#16861242—21no—
22#16861242C2211yescharacteristic
23#16861242C2211yescharacteristic
24#16861242C2211yescharacteristic
25#16861242C2211yescharacteristic, complemented
26#16861241—21no—
27#16861240C2211yescharacteristic
28#16861238—21no—
29#16861238—21no—
30#16861238—21no—
31#16861236—21no—
32#16861235C2211yescharacteristic
33#16861233—21no—
34#16861231—21no—
35#16861231C2211yescharacteristic
36#16861230—21no—
37#16861230—21no—
38#16861229C2211yescharacteristic
39#16861228—21no—
40C2×Dic508—21no—
41C2×Dic508C2211yescharacteristic, complemented
42C2×#14523238C2211yescharacteristic, complemented
43Q16×D127—21no—
44C2×#14523236C2211yescharacteristic
45SD16×D127—21no—
46SD16×D127—21no—
47#16861217C2211yescharacteristic
48#16861217C2211yescharacteristic, complemented
49#16861215C2211yescharacteristic
50#16861207C2211yescharacteristic, complemented
51#16861196C2211yescharacteristic
52#16861144C2211yescharacteristic, complemented
53#16861136C2211yescharacteristic, complemented
54#16861135C2211yescharacteristic
55#16861135C2211yescharacteristic
56#16861135C2211yescharacteristic, complemented
57#16861127C2211yescharacteristic, complemented
58#16861127C2211yescharacteristic, complemented
59#16861119C2211yescharacteristic, complemented
60#137×C127C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C127C2211yescharacteristic
62C2×SD16×C127C2211yescharacteristic
63#131×C127C2211yescharacteristic, complemented
64#108×C127C2211yescharacteristic, complemented
65#101×C127C2211yescharacteristic, complemented
66#100×C127C2211yescharacteristic
67#14523249—21no—
68#14523249—21no—
69#14523248—21no—
70#14523248—21no—
71#14523248—41no—
72#14523248C2311yescharacteristic
73#14523248C2311yescharacteristic
74#14523248C2311yescharacteristic
75#14523247—21no—
76#14523247—21no—
77#14523247—21no—
78#14523247—21no—
79#14523247—41no—
80#14523247—41no—
81C2×Dic254—21no—
82C2×Dic254—21no—
83C2×Dic254C2311yescharacteristic
84C2×Dic254C2311yescharacteristic
85Q8×D127—21no—
86Q8×D127—21no—
87Q8×D127—21no—
88Q8×D127—21no—
89C2×C22⋊D127—41no—
90D4×D127—21no—
91D4×D127—21no—
92Dic508—21no—
93Dic508—21no—
94Dic508—21no—
95Dic508—21no—
96Dic508—21no—
97Dic508—21no—
98Dic508D411yescharacteristic
99#14523238—21no—
100#14523238—21no—
101#14523238—21no—
102#14523238D411yescharacteristic
103#14523237—21no—
104#14523237—21no—
105#14523237—21no—
106#14523237—21no—
107#14523237D411yescharacteristic
108#14523237D411yescharacteristic
109#14523236—21no—
110#14523236—21no—
111#14523236—21no—
112#14523236D411yescharacteristic
113#14523235—21no—
114#14523235D411yescharacteristic
115D1016D411yescharacteristic
116#14523233—21no—
117#14523233D411yescharacteristic
118#14523232C2311yescharacteristic
119#14523231—21no—
120#14523231—21no—
121#14523231—21no—
122C2×C127⋊C8C2311yescharacteristic
123C8×D127—21no—
124C8×D127—21no—
125C8×D127—21no—
126Dic127⋊C4—41no—
127C22⋊Dic127—41no—
128C4×Dic127C2311yescharacteristic
129#41×C127C2311yescharacteristic
130C2×Q8×C127C2311yescharacteristic
131C2×D4×C127C2311yescharacteristic
132Q16×C127—21no—
133SD16×C127—21no—
134SD16×C127—21no—
135SD16×C127—21no—
136SD16×C127—21no—
137D8×C127—21no—
138M16×C127—21no—
139M16×C127C2311yescharacteristic
140M16×C127C2311yescharacteristic
141M16×C127C2311yescharacteristic
142C2×C8×C127—21no—
143C2×C8×C127C2311yescharacteristic
144C22×D127—41no—
145Dic254—21no—
146Dic254—21no—
147Dic254—21no—
148Dic254—21no—
149Dic254—41no—
150Dic254—41no—
151Dic254—41no—
152Dic254C2×D411yescharacteristic
153Dic254C2×D411yescharacteristic
154Dic254C2×D411yescharacteristic
155D508—41no—
156D508C2×D411yescharacteristic
157D508C2×D411yescharacteristic
158D508C2×D411yescharacteristic
159C22⋊D127—21no—
160C22⋊D127—21no—
161C22⋊D127—21no—
162C22⋊D127—41no—
163C22⋊D127—41no—
164C22⋊D127—41no—
165C22⋊D127—41no—
166C2×Dic127—21no—
167C2×Dic127—21no—
168C2×Dic127—21no—
169C2×Dic127—41no—
170C2×Dic127—41no—
171C2×Dic127—41no—
172C4×D127—21no—
173C4×D127—21no—
174C4×D127—21no—
175C4×D127—41no—
176C4×D127—41no—
177C4×D127—41no—
178C4×D127—41no—
179C127⋊C8—21no—
180C127⋊C8C2×D411yescharacteristic
181C127⋊C8C2×D411yescharacteristic
182C23×C127—21no—
183Q8×C127—21no—
184Q8×C127—21no—
185Q8×C127C2×D411yescharacteristic
186D4×C127—21no—
187D4×C127—21no—
188D4×C127—21no—
189D4×C127C2×D411yescharacteristic
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127C2411yescharacteristic
193C8×C127—21no—
194C8×C127—21no—
195C8×C127—21no—
196C8×C127C2×D411yescharacteristic
197C8×C127C2×D411yescharacteristic
198C22×C127—21no—
199C22×C127—41no—
200C22×C127—81no—
201C22×C127H(5,𝔽2)11yescharacteristic
202C2×D127—21no—
203C2×D127—21no—
204C2×D127—21no—
205C2×D127—161no—
206C4×C127—21no—
207C4×C127—41no—
208C4×C127C22×D411yescharacteristic
209C4×C127C22×D411yescharacteristic
210Dic127—21no—
211Dic127—21no—
212Dic127—21no—
213Dic127—41no—
214Dic127—41no—
215Dic127—41no—
216C2×C127—21no—
217C2×C127—41no—
218C2×C127—81no—
219C2×C127D4211yescharacteristic
220D127—41no—
221D127—41no—
222D127—41no—
223#3289—1271nomaximal, Sylow
224C127#328911yescharacteristic, complemented, Sylow
225#578—1271no—
226#578—1271no—
227#577—1271no—
228#577—1271no—
229#497—1271no—
230#473—1271no—
231#473—1271no—
232#471—1271no—
233#471D12711yescharacteristic, complemented
234#456—1271no—
235#456—1271no—
236#455—1271no—
237#454—1271no—
238#444—1271no—
239#444—1271no—
2402−1+4—1271no—
2412−1+4—1271no—
2422−1+4—1271no—
243H(5,𝔽2)—1271no—
244#137—1271no—
245#137—1271no—
246#137—1271no—
247#137—1271no—
248#137—1271no—
249#137—2541no—
250#137—2541no—
251#137—2541no—
252#137—2541no—
253#137—2541no—
254#137—2541no—
255#137C2×D12711yescharacteristic, complemented
256AGL(1,ℤ/8)—1271no—
257AGL(1,ℤ/8)—2541no—
258AGL(1,ℤ/8)—2541no—
259AGL(1,ℤ/8)C2×D12711yescharacteristic
260#135—1271no—
261#135—1271no—
262#135—2541no—
263#135—2541no—
264#135—2541no—
265#135—2541no—
266C2×Q16—1271no—
267C2×Q16—1271no—
268C2×Q16—2541no—
269C2×Q16—2541no—
270C2×SD16—1271no—
271C2×SD16—2541no—
272C2×SD16—2541no—
273C2×SD16C2×D12711yescharacteristic
274#131—1271no—
275#131—1271no—
276#131—1271no—
277#131C2×D12711yescharacteristic, complemented
278C4⋊Q8—1271no—
279#124—1271no—
280#108—1271no—
281#108C2×D12711yescharacteristic, complemented
282#104—1271no—
283#104—1271no—
284#104—1271no—
285#104—1271no—
286#101—1271no—
287#101—1271no—
288#101C2×D12711yescharacteristic, complemented
289#100C2×D12711yescharacteristic
290#97—1271no—
291#41—1271no—
292#41—1271no—
293#41—1271no—
294#41—5081no—
295#41—5081no—
296#41—5081no—
297#41—2541no—
298#41—2541no—
299#41—2541no—
300#41—2541no—
301#41—2541no—
302#41—2541no—
303#41—2541no—
304#41—2541no—
305#41C22×D12711yescharacteristic
306C2×Q8—1271no—
307C2×Q8—1271no—
308C2×Q8—2541no—
309C2×Q8—2541no—
310C2×Q8—2541no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8—2541no—
314C2×Q8C22×D12711yescharacteristic
315C2×D4—5081no—
316C2×D4—2541no—
317C2×D4—2541no—
318C2×D4C22×D12711yescharacteristic
319Q16—21no—
320Q16—1271no—
321Q16—1271no—
322Q16—2541no—
323Q16—2541no—
324Q16—2541no—
325Q16—2541no—
326Q16—2541no—
327Q16—2541no—
328Q16—2541no—
329Q16—2541no—
330Q16—2541no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—1271no—
336SD16—1271no—
337SD16—1271no—
338SD16—1271no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16—2541no—
346SD16—2541no—
347D8—21no—
348D8—1271no—
349D8—1271no—
350D8—2541no—
351M16—21no—
352M16—1271no—
353M16—2541no—
354M16—2541no—
355M16—2541no—
356M16C22×D12711yescharacteristic
357M16C22×D12711yescharacteristic
358M16C22×D12711yescharacteristic
359C2×C8—21no—
360C2×C8—1271no—
361C2×C8—2541no—
362C2×C8—2541no—
363C2×C8—2541no—
364C2×C8C22×D12711yescharacteristic
365C4⋊C4—5081no—
366C22⋊C4—5081no—
367C42—1271no—
368C23—21no—
369C23—5081no—
370Q8—21no—
371Q8—21no—
372Q8—1271no—
373Q8—1271no—
374Q8—1271no—
375Q8—5081no—
376Q8—5081no—
377Q8—5081no—
378Q8—2541no—
379Q8—2541no—
380Q8—2541no—
381Q8—2541no—
382Q8D4×D12711yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—1271no—
387D4—1271no—
388D4—1271no—
389D4—5081no—
390D4—5081no—
391D4—5081no—
392D4—5081no—
393D4—5081no—
394D4—2541no—
395D4—2541no—
396D4—2541no—
397D4D4×D12711yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—5081no—
401C2×C4—5081no—
402C2×C4—5081no—
403C2×C4—5081no—
404C2×C4—5081no—
405C2×C4—5081no—
406C2×C4—5081no—
407C2×C4—2541no—
408C2×C4—2541no—
409C2×C4—2541no—
410C2×C4—2541no—
411C2×C4—2541no—
412C2×C4—2541no—
413C2×C4C23×D12711yescharacteristic
414C8—21no—
415C8—21no—
416C8—21no—
417C8—1271no—
418C8—1271no—
419C8—2541no—
420C8D4×D12711yescharacteristic
421C8D4×D12711yescharacteristic
422C22—21no—
423C22—41no—
424C22—81no—
425C22—20321no—
426C22—2541no—
427C22—2541no—
428C22—2541no—
429C22#1686125511yescharacteristic
430C4—21no—
431C4—41no—
432C4—5081no—
433C4—5081no—
434C4—5081no—
435C4—2541no—
436C4—2541no—
437C4—2541no—
438C4C2×D4×D12711yescharacteristic
439C4C2×D4×D12711yescharacteristic
440C2—21no—
441C2—41no—
442C2—81no—
443C2—5081no—
444C2—5081no—
445C2—5081no—
446C2#2255051011yescharacteristic
subgroupG/H
centerC2#22550510
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#471×C127C2
socleC2×C127D42
hypercenter#471D127
hypocenterC127#3289
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #22550693C2semidirect
2. #22550692C2semidirect
3. #22550686C2semidirect
4. #22550682C2semidirect
5. #22550267C2semidirect
6. #22550152C2semidirect
7. #22550150C2semidirect
8. #22550138C2semidirect
9. #22550061C2semidirect
10. #22550057C2semidirect
11. #22550056C2semidirect
12. #22550047C2semidirect
13. #22549997C2semidirect
14. #22549996C2semidirect
15. #471×C127C2semidirect
18. #16861256C22semidirect
25. #16861242C22semidirect
41. C2×Dic508C22semidirect
42. C2×#14523238C22semidirect
48. #16861217C22semidirect
50. #16861207C22semidirect
52. #16861144C22semidirect
53. #16861136C22semidirect
56. #16861135C22semidirect
57. #16861127C22semidirect
58. #16861127C22semidirect
59. #16861119C22semidirect
60. #137×C127C22semidirect
63. #131×C127C22semidirect
64. #108×C127C22semidirect
65. #101×C127C22semidirect
224. C127#3289semidirect
233. #471D127semidirect
255. #137C2×D127semidirect
277. #131C2×D127semidirect
281. #108C2×D127semidirect
288. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506932
#225506922
#225506862
#225506822
#225502672
#225501522
#225501502
#225501382
#225500612
#225500572
#225500562
#225500472
#225499972
#225499962
#471×C1272
#3289127