Finite Group Databasebeta

← #30648465

Subgroups — #30648465

Order 16256.

446 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550675C211yesmaximal, characteristic
2C2×#16861241C211yesmaximal, characteristic, complemented
3#137×D127C211yesmaximal, characteristic, complemented
4#22550641C211yesmaximal, characteristic, complemented
5#22550150C211yesmaximal, characteristic, complemented
6#22550149C211yesmaximal, characteristic, complemented
7#22550148C211yesmaximal, characteristic, complemented
8#22550142C211yesmaximal, characteristic, complemented
9#22550141C211yesmaximal, characteristic, complemented
10#22550140C211yesmaximal, characteristic, complemented
11#22550139C211yesmaximal, characteristic, complemented
12#22549923C211yesmaximal, characteristic, complemented
13#22549859C211yesmaximal, characteristic, complemented
14#101×D127C211yesmaximal, characteristic, complemented
15#473×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17#41×D127C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861242C2211yescharacteristic
20#16861241—21no—
21#16861241—21no—
22#16861241—21no—
23#16861241—21no—
24#16861241—21no—
25#16861241—21no—
26#16861240C2211yescharacteristic
27#16861240C2211yescharacteristic
28#16861239C2211yescharacteristic
29#16861238—21no—
30#16861237C2211yescharacteristic
31#16861236—21no—
32#16861234C2211yescharacteristic
33#16861232—21no—
34#16861230—21no—
35C2×#14523238C2211yescharacteristic, complemented
36Q16×D127—21no—
37C2×#14523237C2211yescharacteristic, complemented
38C2×#14523235C2211yescharacteristic, complemented
39SD16×D127—21no—
40#16861217—21no—
41#16861217C2211yescharacteristic
42#16861216C2211yescharacteristic
43#16861215—21no—
44#16861215C2211yescharacteristic
45C2×#14523231—21no—
46C2×#14523231C2211yescharacteristic
47M16×D127—21no—
48M16×D127C2211yescharacteristic
49M16×D127C2211yescharacteristic
50M16×D127C2211yescharacteristic
51#16861145C2211yescharacteristic, complemented
52#16861144C2211yescharacteristic, complemented
53#16861144C2211yescharacteristic, complemented
54#16861128C2211yescharacteristic, complemented
55#16861128C2211yescharacteristic, complemented
56#16861127C2211yescharacteristic, complemented
57#16861127C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic, complemented
59#16861125C2211yescharacteristic, complemented
60#137×C127C2211yescharacteristic
61#137×C127C2211yescharacteristic
62C2×Q16×C127C2211yescharacteristic, complemented
63#131×C127C2211yescharacteristic
64#108×C127C2211yescharacteristic, complemented
65#101×C127C2211yescharacteristic, complemented
66#101×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#14523247D411yescharacteristic
76C2×Dic254—21no—
77C2×Dic254C2311yescharacteristic
78Q8×D127—21no—
79Q8×D127—21no—
80Q8×D127—21no—
81Q8×D127—21no—
82Q8×D127D411yescharacteristic
83C2×D508—21no—
84C2×D508C2311yescharacteristic
85D4×D127—21no—
86D4×D127D411yescharacteristic
87C2×C4×D127—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#14523238—21no—
97#14523237—21no—
98#14523237—21no—
99#14523237—21no—
100#14523237—21no—
101#14523236—21no—
102#14523235—21no—
103#14523235—21no—
104#14523235—21no—
105#14523235—21no—
106#14523235—21no—
107D1016—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—
118#14523231D411yescharacteristic
119#14523231D411yescharacteristic
120#14523231D411yescharacteristic
121#14523231D411yescharacteristic
122C2×C127⋊C8—21no—
123C2×C127⋊C8C2311yescharacteristic
124C8×D127—21no—
125C8×D127—21no—
126C8×D127—21no—
127C8×D127—21no—
128C8×D127—21no—
129#41×C127C2311yescharacteristic
130C2×Q8×C127C2311yescharacteristic
131C2×Q8×C127C2311yescharacteristic
132Q16×C127—21no—
133Q16×C127—21no—
134Q16×C127—21no—
135Q16×C127—21no—
136SD16×C127—21no—
137SD16×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—
150Dic254—41no—
151Dic254—41no—
152Dic254C2×D411yescharacteristic
153D508—21no—
154D508—21no—
155D508—21no—
156D508—41no—
157D508—41no—
158D508C2×D411yescharacteristic
159C22⋊D127—21no—
160C22⋊D127—21no—
161C22⋊D127—21no—
162C2×Dic127—21no—
163C2×Dic127—21no—
164C2×Dic127#4111yescharacteristic
165C4×D127—21no—
166C4×D127—21no—
167C4×D127—21no—
168C4×D127—41no—
169C4×D127—41no—
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
181Q8×C127—21no—
182Q8×C127—21no—
183Q8×C127—21no—
184Q8×C127—21no—
185Q8×C127C2×D411yescharacteristic
186D4×C127—21no—
187D4×C127C2×D411yescharacteristic
188C2×C4×C127—21no—
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×C127C2×#4111yescharacteristic
199C2×D127—21no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—81no—
203C2×D127—81no—
204C2×D127#12111yescharacteristic
205C2×D127#12111yescharacteristic
206C4×C127—21no—
207C4×C127—41no—
208C4×C127—41no—
209C4×C127C22×D411yescharacteristic
210C4×C127C22×D411yescharacteristic
211Dic127—21no—
212Dic127—41no—
213Dic127#12111yescharacteristic
214Dic127#12111yescharacteristic
215C2×C127—21no—
216C2×C127—41no—
217C2×C127C2×#12111yescharacteristic
218D127—21no—
219D127—21no—
220D127—41no—
221D127—81no—
222#3069—1271nomaximal, Sylow
223C127#306911yescharacteristic, complemented, Sylow
224#575—1271no—
225C2×#137—1271no—
226C2×#137—1271no—
227#568—1271no—
228#473—1271no—
229#473—1271no—
230#473—1271no—
231#473D12711yescharacteristic, complemented
232#471—1271no—
233#471—1271no—
234#471—1271no—
235#471—1271no—
236#430—1271no—
237#413—1271no—
238C2×#101—1271no—
239C2×#41—1271no—
240C2×#41—1271no—
241C22×Q8—1271no—
242#137—1271no—
243#137—1271no—
244#137—1271no—
245#137—1271no—
246#137—2541no—
247#137—2541no—
248#137—2541no—
249#137—2541no—
250#137—2541no—
251#137—2541no—
252#137—2541no—
253#137C2×D12711yescharacteristic
254#137C2×D12711yescharacteristic
255AGL(1,ℤ/8)—1271no—
256AGL(1,ℤ/8)—1271no—
257AGL(1,ℤ/8)—2541no—
258#135—2541no—
259#135—2541no—
260C2×Q16—1271no—
261C2×Q16—2541no—
262C2×Q16C2×D12711yescharacteristic, complemented
263C2×SD16—1271no—
264C2×SD16—1271no—
265C2×SD16—2541no—
266#131—1271no—
267#131—1271no—
268#131—1271no—
269#131—2541no—
270#131—2541no—
271#131C2×D12711yescharacteristic
272C2×M16—1271no—
273C2×M16—1271no—
274C2×M16—1271no—
275C2×M16—1271no—
276C2×M16—2541no—
277C2×M16—2541no—
278#108—1271no—
279#108—1271no—
280#108—1271no—
281#108C2×D12711yescharacteristic, complemented
282#101—1271no—
283#101—1271no—
284#101—1271no—
285#101—1271no—
286#101C2×D12711yescharacteristic, complemented
287#101C2×D12711yescharacteristic, complemented
288#100—1271no—
289#100—1271no—
290#41—1271no—
291#41—1271no—
292#41—1271no—
293#41—2541no—
294#41—2541no—
295#41—2541no—
296#41—2541no—
297#41—2541no—
298#41—2541no—
299#41C22×D12711yescharacteristic
300C2×Q8—1271no—
301C2×Q8—1271no—
302C2×Q8—2541no—
303C2×Q8—2541no—
304C2×Q8—2541no—
305C2×Q8—2541no—
306C2×Q8—2541no—
307C2×Q8C22×D12711yescharacteristic
308C2×Q8C22×D12711yescharacteristic
309C2×D4—1271no—
310C2×D4—1271no—
311C2×D4—2541no—
312C2×D4—2541no—
313C22×C4—1271no—
314C22×C4—2541no—
315C22×C4—2541no—
316C22×C4—2541no—
317Q16—21no—
318Q16—21no—
319Q16—21no—
320Q16—21no—
321Q16—2541no—
322Q16—2541no—
323Q16—2541no—
324Q16—2541no—
325Q16—2541no—
326Q16—2541no—
327SD16—21no—
328SD16—21no—
329SD16—2541no—
330SD16—2541no—
331SD16—2541no—
332SD16—2541no—
333SD16—2541no—
334SD16—2541no—
335SD16—2541no—
336SD16—2541no—
337SD16—2541no—
338SD16—2541no—
339D8—2541no—
340D8—2541no—
341M16—21no—
342M16—1271no—
343M16—1271no—
344M16—1271no—
345M16—1271no—
346M16—1271no—
347M16—1271no—
348M16—1271no—
349M16—2541no—
350M16—2541no—
351M16—2541no—
352M16—2541no—
353M16—2541no—
354M16—2541no—
355M16C22×D12711yescharacteristic
356M16C22×D12711yescharacteristic
357M16C22×D12711yescharacteristic
358C2×C8—21no—
359C2×C8—1271no—
360C2×C8—2541no—
361C2×C8—2541no—
362C2×C8—2541no—
363C2×C8—2541no—
364C2×C8—2541no—
365C2×C8—2541no—
366C2×C8C22×D12711yescharacteristic
367C23—1271no—
368C23—2541no—
369C23—2541no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—1271no—
375Q8—5081no—
376Q8—5081no—
377Q8—2541no—
378Q8—2541no—
379Q8—2541no—
380Q8D4×D12711yescharacteristic
381D4—21no—
382D4—1271no—
383D4—5081no—
384D4—5081no—
385D4—2541no—
386D4—2541no—
387D4—2541no—
388D4—2541no—
389D4—2541no—
390D4—2541no—
391D4D4×D12711yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—1271no—
396C2×C4—1271no—
397C2×C4—1271no—
398C2×C4—1271no—
399C2×C4—1271no—
400C2×C4—5081no—
401C2×C4—5081no—
402C2×C4—5081no—
403C2×C4—5081no—
404C2×C4—2541no—
405C2×C4—2541no—
406C2×C4—2541no—
407C2×C4—2541no—
408C2×C4—2541no—
409C2×C4C23×D12711yescharacteristic
410C8—21no—
411C8—21no—
412C8—21no—
413C8—1271no—
414C8—1271no—
415C8—2541no—
416C8—2541no—
417C8—2541no—
418C8D4×D12711yescharacteristic
419C8D4×D12711yescharacteristic
420C22—21no—
421C22—1271no—
422C22—1271no—
423C22—10161no—
424C22—10161no—
425C22—5081no—
426C22—5081no—
427C22—2541no—
428C22#41×D12711yescharacteristic
429C4—21no—
430C4—41no—
431C4—41no—
432C4—1271no—
433C4—1271no—
434C4—5081no—
435C4—2541no—
436C4C2×D4×D12711yescharacteristic
437C4C2×D4×D12711yescharacteristic
438C2—21no—
439C2—41no—
440C2—10161no—
441C2—5081no—
442C2—2541no—
443C2—2541no—
444C2#121×D12711yescharacteristic
subgroupG/H
centerC2#121×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#473×C127C2
socleC2×C127C2×#121
prime radical#22550675C2
hypercenter#473D127
hypocenterC127#3069
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
2. C2×#16861241C2semidirect
3. #137×D127C2semidirect
4. #22550641C2semidirect
5. #22550150C2semidirect
6. #22550149C2semidirect
7. #22550148C2semidirect
8. #22550142C2semidirect
9. #22550141C2semidirect
10. #22550140C2semidirect
11. #22550139C2semidirect
12. #22549923C2semidirect
13. #22549859C2semidirect
14. #101×D127C2semidirect
15. #473×C127C2semidirect
35. C2×#14523238C22semidirect
37. C2×#14523237C22semidirect
38. C2×#14523235C22semidirect
51. #16861145C22semidirect
52. #16861144C22semidirect
53. #16861144C22semidirect
54. #16861128C22semidirect
55. #16861128C22semidirect
56. #16861127C22semidirect
57. #16861127C22semidirect
58. #16861125C22semidirect
59. #16861125C22semidirect
62. C2×Q16×C127C22semidirect
64. #108×C127C22semidirect
65. #101×C127C22semidirect
66. #101×C127C22semidirect
223. C127#3069semidirect
231. #473D127semidirect
262. C2×Q16C2×D127semidirect
281. #108C2×D127semidirect
286. #101C2×D127semidirect
287. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506752
C2×#168612412
#137×D1272
#225506412
#225501502
#225501492
#225501482
#225501422
#225501412
#225501402
#225501392
#225499232
#225498592
#101×D1272
#473×C1272
#3069127