Finite Group Databasebeta

← #30648445

Subgroups — #30648445

Order 16256.

452 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550675C211yesmaximal, characteristic, complemented
2#22550672C211yesmaximal, characteristic, complemented
3C2×#16861230C211yesmaximal, characteristic, complemented
4#131×D127C211yesmaximal, characteristic, complemented
5#22550150C211yesmaximal, characteristic, complemented
6#22550149C211yesmaximal, characteristic, complemented
7#22550145C211yesmaximal, characteristic, complemented
8#22550144C211yesmaximal, characteristic, complemented
9#22550140C211yesmaximal, characteristic, complemented
10#22550139C211yesmaximal, characteristic, complemented
11#22550136C211yesmaximal, characteristic, complemented
12#108×D127C211yesmaximal, characteristic, complemented
13#22549859C211yesmaximal, characteristic, complemented
14#22549857C211yesmaximal, characteristic, complemented
15#471×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18#41×D127C2211yescharacteristic
19#16861242C2211yescharacteristic
20#16861241—21no—
21#16861240C2211yescharacteristic
22#16861239C2211yescharacteristic
23#16861238—21no—
24#16861237C2211yescharacteristic
25#16861236—21no—
26#16861235C2211yescharacteristic
27#16861234C2211yescharacteristic
28#16861233—21no—
29#16861232—21no—
30#16861230—21no—
31#16861230—21no—
32#16861230—21no—
33#16861230—21no—
34#16861230—21no—
35#16861230—21no—
36#16861228—21no—
37C2×#14523238C2211yescharacteristic, complemented
38C2×#14523237C2211yescharacteristic, complemented
39C2×#14523233C2211yescharacteristic, complemented
40#16861217—21no—
41#16861217C2211yescharacteristic
42#16861216C2211yescharacteristic
43#16861215—21no—
44#16861215C2211yescharacteristic
45M16×D127—21no—
46M16×D127C2211yescharacteristic
47M16×D127C2211yescharacteristic
48M16×D127C2211yescharacteristic
49C2×C8×D127—21no—
50C2×C8×D127C2211yescharacteristic
51#16861145C2211yescharacteristic, complemented
52#16861144C2211yescharacteristic, complemented
53#16861144C2211yescharacteristic, complemented
54#16861128C2211yescharacteristic, complemented
55#16861127C2211yescharacteristic, complemented
56#16861127C2211yescharacteristic, complemented
57#16861126C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic, complemented
59#16861125C2211yescharacteristic, complemented
60#137×C127C2211yescharacteristic
61AGL(1,ℤ/8)×C127C2211yescharacteristic
62C2×SD16×C127C2211yescharacteristic, complemented
63#131×C127C2211yescharacteristic
64#108×C127C2211yescharacteristic, complemented
65#101×C127C2211yescharacteristic, complemented
66#100×C127C2211yescharacteristic, complemented
67#14523249—21no—
68#14523249—21no—
69#14523249—21no—
70#14523249—21no—
71#14523249D411yescharacteristic
72#14523248—21no—
73#14523248C2311yescharacteristic
74#14523247—21no—
75#14523247—21no—
76#14523247—21no—
77#14523247—21no—
78#14523247—21no—
79#14523247D411yescharacteristic
80C2×Dic254C2311yescharacteristic
81Q8×D127D411yescharacteristic
82C2×D508—21no—
83C2×D508C2311yescharacteristic
84C2×C22⋊D127—21no—
85D4×D127—21no—
86D4×D127D411yescharacteristic
87C22×Dic127—21no—
88C2×C4×D127—21no—
89C2×C4×D127—21no—
90C2×C4×D127C2311yescharacteristic
91Dic508—21no—
92#14523238—21no—
93#14523238—21no—
94#14523238—21no—
95#14523238—21no—
96#14523237—21no—
97#14523237—21no—
98#14523237—21no—
99#14523237—21no—
100#14523236—21no—
101#14523236—21no—
102#14523235—21no—
103#14523235—21no—
104D1016—21no—
105#14523233—21no—
106#14523233—21no—
107#14523233—21no—
108#14523233—21no—
109#14523232—21no—
110#14523232C2311yescharacteristic
111#14523232C2311yescharacteristic
112#14523232C2311yescharacteristic
113#14523231—21no—
114#14523231—21no—
115#14523231—21no—
116#14523231—21no—
117#14523231—21no—
118C2×C127⋊C8—21no—
119C2×C127⋊C8C2311yescharacteristic
120C8×D127—21no—
121C8×D127—21no—
122C8×D127—21no—
123C8×D127—21no—
124C8×D127—21no—
125C8×D127D411yescharacteristic
126C8×D127D411yescharacteristic
127C8×D127D411yescharacteristic
128C8×D127D411yescharacteristic
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—21no—
145C22×D127—21no—
146C22×D127#4111yescharacteristic
147Dic254—21no—
148Dic254—21no—
149Dic254—21no—
150Dic254C2×D411yescharacteristic
151D508—21no—
152D508—21no—
153D508—21no—
154D508—41no—
155D508—41no—
156D508C2×D411yescharacteristic
157C22⋊D127—21no—
158C22⋊D127—21no—
159C22⋊D127—21no—
160C22⋊D127—41no—
161C22⋊D127—41no—
162C2×Dic127—21no—
163C2×Dic127—21no—
164C2×Dic127—41no—
165C2×Dic127—41no—
166C2×Dic127#4111yescharacteristic
167C4×D127—21no—
168C4×D127—21no—
169C4×D127—21no—
170C4×D127—41no—
171C4×D127—41no—
172C4×D127C2×D411yescharacteristic
173C4×D127C2×D411yescharacteristic
174C4×D127C2×D411yescharacteristic
175C4×D127C2×D411yescharacteristic
176C127⋊C8—21no—
177C127⋊C8—21no—
178C127⋊C8—21no—
179C127⋊C8C2×D411yescharacteristic
180C127⋊C8C2×D411yescharacteristic
181C23×C127—21no—
182Q8×C127—21no—
183Q8×C127—21no—
184Q8×C127C2×D411yescharacteristic
185D4×C127—21no—
186D4×C127—21no—
187D4×C127—21no—
188D4×C127C2×D411yescharacteristic
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127C2411yescharacteristic
192C8×C127—21no—
193C8×C127—21no—
194C8×C127—21no—
195C8×C127C2×D411yescharacteristic
196C8×C127C2×D411yescharacteristic
197C22×C127—21no—
198C22×C127—41no—
199C22×C127—81no—
200C22×C127C2×#4111yescharacteristic
201C2×D127—21no—
202C2×D127—41no—
203C2×D127—42no—
204C2×D127—41no—
205C2×D127—81no—
206C2×D127#12111yescharacteristic
207C2×D127#12111yescharacteristic
208C4×C127—21no—
209C4×C127—41no—
210C4×C127C22×D411yescharacteristic
211C4×C127C22×D411yescharacteristic
212Dic127—21no—
213Dic127—41no—
214Dic127#12111yescharacteristic
215Dic127#12111yescharacteristic
216C2×C127—21no—
217C2×C127—41no—
218C2×C127—81no—
219C2×C127C2×#12111yescharacteristic
220D127—21no—
221D127—21no—
222D127—41no—
223D127—81no—
224#3067—1271nomaximal, Sylow
225C127#306711yescharacteristic, complemented, Sylow
226#575—1271no—
227#575—1271no—
228C2×#135—1271no—
229C2×#131—1271no—
230#473—1271no—
231#473—1271no—
232#472—1271no—
233#472—1271no—
234#471—1271no—
235#471—1271no—
236#471—1271no—
237#471D12711yescharacteristic, complemented
238C2×#108—1271no—
239#413—1271no—
240#413—1271no—
241C2×#41—1271no—
242C2×#41—1271no—
243C2×#41—1271no—
244#137—1271no—
245#137—1271no—
246#137—1271no—
247#137—2541no—
248#137—2541no—
249#137C2×D12711yescharacteristic
250AGL(1,ℤ/8)—1271no—
251AGL(1,ℤ/8)—1271no—
252AGL(1,ℤ/8)—1271no—
253AGL(1,ℤ/8)—2541no—
254AGL(1,ℤ/8)—2541no—
255AGL(1,ℤ/8)C2×D12711yescharacteristic
256#135—2541no—
257#135—2541no—
258#135—2541no—
259#135—2541no—
260#135—2541no—
261#135—2541no—
262#135—2541no—
263#135—2541no—
264C2×Q16—1271no—
265C2×SD16—1271no—
266C2×SD16C2×D12711yescharacteristic, complemented
267C2×D8—1271no—
268#131—1271no—
269#131—1271no—
270#131—1271no—
271#131—2541no—
272#131—2541no—
273#131C2×D12711yescharacteristic
274C2×M16—1271no—
275C2×M16—1271no—
276C2×M16—1271no—
277C2×M16—2541no—
278C22×C8—1271no—
279C22×C8—2541no—
280#108—1271no—
281#108—1271no—
282#108—1271no—
283#108C2×D12711yescharacteristic, complemented
284#101—1271no—
285#101—1271no—
286#101—1271no—
287#101C2×D12711yescharacteristic, complemented
288#100—1271no—
289#100—1271no—
290#100—1271no—
291#100C2×D12711yescharacteristic, complemented
292#41—1271no—
293#41—1271no—
294#41—1271no—
295#41—2541no—
296#41—2541no—
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×Q8C22×D12711yescharacteristic
309C2×D4—1271no—
310C2×D4—1271no—
311C2×D4—2541no—
312C2×D4—2541no—
313C2×D4—2541no—
314C2×D4C22×D12711yescharacteristic
315C22×C4—1271no—
316C22×C4—2541no—
317C22×C4—2541no—
318C22×C4—2541no—
319Q16—21no—
320Q16—2541no—
321Q16—2541no—
322Q16—2541no—
323Q16—2541no—
324Q16—2541no—
325SD16—21no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—2541no—
330SD16—2541no—
331SD16—2541no—
332SD16—2541no—
333SD16—2541no—
334SD16—2541no—
335SD16—2541no—
336SD16—2541no—
337D8—21no—
338D8—2541no—
339D8—2541no—
340D8—2541no—
341D8—2541no—
342D8—2541no—
343M16—21no—
344M16—1271no—
345M16—1271no—
346M16—1271no—
347M16—2541no—
348M16—2541no—
349M16—2541no—
350M16—2541no—
351M16—2541no—
352M16—2541no—
353M16C22×D12711yescharacteristic
354M16C22×D12711yescharacteristic
355M16C22×D12711yescharacteristic
356C2×C8—21no—
357C2×C8—1271no—
358C2×C8—1271no—
359C2×C8—1271no—
360C2×C8—1271no—
361C2×C8—1271no—
362C2×C8—2541no—
363C2×C8—2541no—
364C2×C8—2541no—
365C2×C8—2541no—
366C2×C8—2541no—
367C2×C8—2541no—
368C2×C8C22×D12711yescharacteristic
369C23—21no—
370C23—1271no—
371C23—2541no—
372C23—2541no—
373Q8—21no—
374Q8—21no—
375Q8—1271no—
376Q8—2541no—
377Q8—2541no—
378Q8—2541no—
379Q8D4×D12711yescharacteristic
380D4—21no—
381D4—21no—
382D4—21no—
383D4—1271no—
384D4—5081no—
385D4—5081no—
386D4—5081no—
387D4—5081no—
388D4—2541no—
389D4—2541no—
390D4—2541no—
391D4—2541no—
392D4—2541no—
393D4—2541no—
394D4D4×D12711yescharacteristic
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—1271no—
398C2×C4—1271no—
399C2×C4—1271no—
400C2×C4—1271no—
401C2×C4—1271no—
402C2×C4—5081no—
403C2×C4—5081no—
404C2×C4—5081no—
405C2×C4—5081no—
406C2×C4—2541no—
407C2×C4—2541no—
408C2×C4—2541no—
409C2×C4—2541no—
410C2×C4—2541no—
411C2×C4C23×D12711yescharacteristic
412C8—21no—
413C8—21no—
414C8—21no—
415C8—1271no—
416C8—1271no—
417C8—2541no—
418C8—2541no—
419C8—2541no—
420C8D4×D12711yescharacteristic
421C8D4×D12711yescharacteristic
422C22—21no—
423C22—41no—
424C22—81no—
425C22—1271no—
426C22—1271no—
427C22—10161no—
428C22—5081no—
429C22—5082no—
430C22—5081no—
431C22—2541no—
432C22#41×D12711yescharacteristic
433C4—21no—
434C4—41no—
435C4—1271no—
436C4—1271no—
437C4—5081no—
438C4—2541no—
439C4C2×D4×D12711yescharacteristic
440C4C2×D4×D12711yescharacteristic
441C2—21no—
442C2—41no—
443C2—81no—
444C2—10161no—
445C2—5081no—
446C2—2541no—
447C2—2541no—
448C2#121×D12711yescharacteristic
subgroupG/H
centerC2#121×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#471×C127C2
socleC2×C127C2×#121
hypercenter#471D127
hypocenterC127#3067
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. #22550675C2semidirect
2. #22550672C2semidirect
3. C2×#16861230C2semidirect
4. #131×D127C2semidirect
5. #22550150C2semidirect
6. #22550149C2semidirect
7. #22550145C2semidirect
8. #22550144C2semidirect
9. #22550140C2semidirect
10. #22550139C2semidirect
11. #22550136C2semidirect
12. #108×D127C2semidirect
13. #22549859C2semidirect
14. #22549857C2semidirect
15. #471×C127C2semidirect
37. C2×#14523238C22semidirect
38. C2×#14523237C22semidirect
39. C2×#14523233C22semidirect
51. #16861145C22semidirect
52. #16861144C22semidirect
53. #16861144C22semidirect
54. #16861128C22semidirect
55. #16861127C22semidirect
56. #16861127C22semidirect
57. #16861126C22semidirect
58. #16861125C22semidirect
59. #16861125C22semidirect
62. C2×SD16×C127C22semidirect
64. #108×C127C22semidirect
65. #101×C127C22semidirect
66. #100×C127C22semidirect
225. C127#3067semidirect
237. #471D127semidirect
266. C2×SD16C2×D127semidirect
283. #108C2×D127semidirect
287. #101C2×D127semidirect
291. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506752
#225506722
C2×#168612302
#131×D1272
#225501502
#225501492
#225501452
#225501442
#225501402
#225501392
#225501362
#108×D1272
#225498592
#225498572
#471×C1272
#3067127