Finite Group Databasebeta

← #30650506

Subgroups — #30650506

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#22550690C211yesmaximal, characteristic, complemented
3#22550688C211yesmaximal, characteristic, complemented
4#22550683C211yesmaximal, characteristic, complemented
5#22550270C211yesmaximal, characteristic, complemented
6#22550152C211yesmaximal, characteristic, complemented
7#22550151C211yesmaximal, characteristic, complemented
8#22550140C211yesmaximal, characteristic, complemented
9#22550139C211yesmaximal, characteristic, complemented
10#22550060C211yesmaximal, characteristic, complemented
11#22550060C211yesmaximal, characteristic, complemented
12#22550055C211yesmaximal, characteristic, complemented
13#22550050C211yesmaximal, characteristic, complemented
14#22549999C211yesmaximal, characteristic, complemented
15#444×C127C211yesmaximal, characteristic, complemented
16#16861256C2211yescharacteristic
17#16861256C2211yescharacteristic
18#16861256C2211yescharacteristic, complemented
19#16861255C2211yescharacteristic
20#16861242—21no—
21#16861242—21no—
22#16861242—21no—
23#16861242C2211yescharacteristic
24#16861242C2211yescharacteristic
25#16861242C2211yescharacteristic, complemented
26#16861241—21no—
27#16861239C2211yescharacteristic
28#16861239C2211yescharacteristic
29#16861239C2211yescharacteristic, complemented
30#16861238—21no—
31#16861238—21no—
32#16861237—21no—
33#16861237C2211yescharacteristic
34#16861234C2211yescharacteristic
35#16861233—21no—
36#16861231—21no—
37#16861231—21no—
38#16861231C2211yescharacteristic
39#16861230—21no—
40#16861228—21no—
41C2×Dic508—21no—
42C2×Dic508C2211yescharacteristic, complemented
43C2×Dic508C2211yescharacteristic, complemented
44Q16×D127—21no—
45C2×#14523237—21no—
46C2×#14523237C2211yescharacteristic
47C2×#14523237C2211yescharacteristic
48SD16×D127—21no—
49#16861215C2211yescharacteristic
50#16861215C2211yescharacteristic, complemented
51#16861209C2211yescharacteristic, complemented
52#16861197C2211yescharacteristic
53#16861145C2211yescharacteristic, complemented
54#16861137C2211yescharacteristic
55#16861137C2211yescharacteristic, complemented
56#16861127C2211yescharacteristic, complemented
57#16861127C2211yescharacteristic, complemented
58#16861127C2211yescharacteristic, complemented
59#16861125C2211yescharacteristic
60#135×C127C2211yescharacteristic
61#131×C127C2211yescharacteristic
62#131×C127C2211yescharacteristic, complemented
63#108×C127C2211yescharacteristic, complemented
64#104×C127C2211yescharacteristic
65#104×C127C2211yescharacteristic, complemented
66#97×C127C2211yescharacteristic, complemented
67#14523249—21no—
68#14523248—21no—
69#14523248—21no—
70#14523248—21no—
71#14523248—21no—
72#14523248C2311yescharacteristic
73#14523248C2311yescharacteristic
74#14523247—21no—
75#14523247—21no—
76#14523247—21no—
77#14523247—41no—
78#14523247—41no—
79#14523247—41no—
80C2×Dic254—21no—
81C2×Dic254—21no—
82C2×Dic254—21no—
83C2×Dic254C2311yescharacteristic
84C2×Dic254C2311yescharacteristic
85C2×Dic254C2311yescharacteristic
86Q8×D127—21no—
87Q8×D127—21no—
88Q8×D127—21no—
89C2×D508—21no—
90C2×D508C2311yescharacteristic
91D4×D127—21no—
92D4×D127—41no—
93Dic508—21no—
94Dic508—21no—
95Dic508—21no—
96Dic508—21no—
97Dic508—21no—
98Dic508—21no—
99Dic508—21no—
100Dic508D411yescharacteristic
101#14523238—21no—
102#14523238—21no—
103#14523238—21no—
104#14523237—21no—
105#14523237—21no—
106#14523237—21no—
107#14523237—21no—
108#14523237—21no—
109#14523237—21no—
110#14523237—21no—
111#14523237—21no—
112#14523237D411yescharacteristic
113#14523237D411yescharacteristic
114#14523236—21no—
115#14523236—21no—
116#14523236—21no—
117#14523235—21no—
118D1016—21no—
119D1016D411yescharacteristic
120#14523233—21no—
121#14523232C2311yescharacteristic
122#14523232C2311yescharacteristic
123#14523231—21no—
124#14523231—21no—
125C8×D127—21no—
126C8×D127—21no—
127C4⋊Dic127—41no—
128(C2×D127)⋊C4—41no—
129#41×C127C2311yescharacteristic
130#41×C127C2311yescharacteristic
131Q16×C127D411yescharacteristic
132SD16×C127D411yescharacteristic
133SD16×C127D411yescharacteristic
134D8×C127D411yescharacteristic
135M16×C127—21no—
136M16×C127—21no—
137M16×C127C2311yescharacteristic
138M16×C127C2311yescharacteristic
139C2×C8×C127—21no—
140C2×C8×C127—21no—
141C2×C8×C127C2311yescharacteristic
142C2×C8×C127C2311yescharacteristic
143C42×C127C2311yescharacteristic
144C22×D127—21no—
145C22×D127—41no—
146Dic254—21no—
147Dic254—21no—
148Dic254—21no—
149Dic254—21no—
150Dic254—21no—
151Dic254—21no—
152Dic254—41no—
153Dic254—41no—
154Dic254—41no—
155Dic254C2×D411yescharacteristic
156Dic254C2×D411yescharacteristic
157D508—21no—
158D508—21no—
159D508—41no—
160D508C2×D411yescharacteristic
161D508C2×D411yescharacteristic
162C22⋊D127—21no—
163C22⋊D127—21no—
164C22⋊D127—41no—
165C22⋊D127—41no—
166C22⋊D127—41no—
167C22⋊D127—41no—
168C2×Dic127—21no—
169C2×Dic127—21no—
170C2×Dic127—21no—
171C2×Dic127—41no—
172C2×Dic127—41no—
173C2×Dic127—41no—
174C4×D127—21no—
175C4×D127—21no—
176C4×D127—41no—
177C4×D127—41no—
178C4×D127—41no—
179C4×D127—41no—
180C127⋊C8—21no—
181C127⋊C8—21no—
182Q8×C127C2×D411yescharacteristic
183Q8×C127C2×D411yescharacteristic
184D4×C127—21no—
185D4×C127—21no—
186D4×C127C2×D411yescharacteristic
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×C127C2×D411yescharacteristic
195C8×C127C2×D411yescharacteristic
196C8×C127C2×D411yescharacteristic
197C8×C127C2×D411yescharacteristic
198C22×C127—21no—
199C22×C127—21no—
200C22×C127H(5,𝔽2)11yescharacteristic
201C2×D127—21no—
202C2×D127—21no—
203C2×D127—41no—
204C2×D127—81no—
205C2×D127—161no—
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—41no—
209C4×C127C22×D411yescharacteristic
210C4×C127C22×D411yescharacteristic
211Dic127—21no—
212Dic127—21no—
213Dic127—41no—
214Dic127—41no—
215Dic127—41no—
216C2×C127—21no—
217C2×C127—41no—
218C2×C127—41no—
219C2×C127D4211yescharacteristic
220D127—41no—
221D127—41no—
222D127—81no—
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#471—1271no—
234#456—1271no—
235#456—1271no—
236#455—1271no—
237#454—1271no—
238#444—1271no—
239#444D12711yescharacteristic, complemented
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—1271no—
250#137—2541no—
251#137—2541no—
252#137—2541no—
253#137—2541no—
254#137—2541no—
255#137—2541no—
256AGL(1,ℤ/8)—1271no—
257AGL(1,ℤ/8)—1271no—
258AGL(1,ℤ/8)—2541no—
259AGL(1,ℤ/8)—2541no—
260#135—1271no—
261#135—2541no—
262#135—2541no—
263#135—2541no—
264#135—2541no—
265#135C2×D12711yescharacteristic
266C2×Q16—1271no—
267C2×Q16—1271no—
268C2×Q16—2541no—
269C2×Q16—2541no—
270C2×SD16—1271no—
271C2×SD16—1271no—
272C2×SD16—2541no—
273C2×SD16—2541no—
274#131—1271no—
275#131—1271no—
276#131C2×D12711yescharacteristic
277#131C2×D12711yescharacteristic, complemented
278C4⋊Q8—1271no—
279#124—1271no—
280#108—1271no—
281#108C2×D12711yescharacteristic, complemented
282#104—1271no—
283#104—1271no—
284#104C2×D12711yescharacteristic
285#104C2×D12711yescharacteristic, complemented
286#101—1271no—
287#101—1271no—
288#101—1271no—
289#100—1271no—
290#97C2×D12711yescharacteristic, complemented
291#41—1271no—
292#41—1271no—
293#41—5081no—
294#41—5081no—
295#41—5081no—
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#41C22×D12711yescharacteristic
305#41C22×D12711yescharacteristic
306C2×Q8—1271no—
307C2×Q8—1271no—
308C2×Q8—1271no—
309C2×Q8—2541no—
310C2×Q8—2541no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8—2541no—
314C2×Q8—2541no—
315C2×D4—1271no—
316C2×D4—5081no—
317C2×D4—2541no—
318C2×D4—2541no—
319Q16—1271no—
320Q16—2541no—
321Q16—2541no—
322Q16—2541no—
323Q16—2541no—
324Q16—2541no—
325Q16—2541no—
326Q16—2541no—
327Q16—2541no—
328Q16—2541no—
329Q16—2541no—
330Q16D50811yescharacteristic
331SD16—1271no—
332SD16—1271no—
333SD16—2541no—
334SD16—2541no—
335SD16—2541no—
336SD16—2541no—
337SD16—2541no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16D50811yescharacteristic
346SD16D50811yescharacteristic
347D8—1271no—
348D8—2541no—
349D8—2541no—
350D8D50811yescharacteristic
351M16—21no—
352M16—21no—
353M16—1271no—
354M16—1271no—
355M16—2541no—
356M16—2541no—
357M16C22×D12711yescharacteristic
358M16C22×D12711yescharacteristic
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—2541no—
362C2×C8—2541no—
363C2×C8C22×D12711yescharacteristic
364C2×C8C22×D12711yescharacteristic
365C4⋊C4—5081no—
366C22⋊C4—5081no—
367C42C22×D12711yescharacteristic
368C23—5081no—
369C23—2541no—
370Q8—1271no—
371Q8—1271no—
372Q8—5081no—
373Q8—5081no—
374Q8—5081no—
375Q8—2541no—
376Q8—2541no—
377Q8—2541no—
378Q8—2541no—
379Q8—2541no—
380Q8—2541no—
381Q8C2×D50811yescharacteristic
382Q8C2×D50811yescharacteristic
383D4—21no—
384D4—21no—
385D4—1271no—
386D4—1271no—
387D4—5081no—
388D4—5081no—
389D4—5081no—
390D4—5081no—
391D4—5081no—
392D4—2541no—
393D4—2541no—
394D4—2541no—
395D4—2541no—
396D4C2×D50811yescharacteristic
397D4C2×D50811yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—5081no—
402C2×C4—5081no—
403C2×C4—5081no—
404C2×C4—5081no—
405C2×C4—5081no—
406C2×C4—5081no—
407C2×C4—5081no—
408C2×C4—2541no—
409C2×C4—2541no—
410C2×C4—2541no—
411C2×C4—2541no—
412C2×C4—2541no—
413C2×C4C23×D12711yescharacteristic
414C8—21no—
415C8—21no—
416C8—2541no—
417C8—2541no—
418C8D4×D12711yescharacteristic
419C8D4×D12711yescharacteristic
420C8C2×D50811yescharacteristic
421C8C2×D50811yescharacteristic
422C22—21no—
423C22—21no—
424C22—20321no—
425C22—10161no—
426C22—5081no—
427C22—2541no—
428C22—2541no—
429C22#1686125511yescharacteristic
430C4—21no—
431C4—21no—
432C4—41no—
433C4—5081no—
434C4—5081no—
435C4—5081no—
436C4—2541no—
437C4—2541no—
438C4C2×D4×D12711yescharacteristic
439C4C22×D50811yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—10161no—
444C2—5081no—
445C2—5081no—
446C2D4×D50811yescharacteristic
subgroupG/H
centerC2D4×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#444×C127C2
socleC2×C127D42
hypercenter#444D127
hypocenterC127#3289
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #22550693C2semidirect
2. #22550690C2semidirect
3. #22550688C2semidirect
4. #22550683C2semidirect
5. #22550270C2semidirect
6. #22550152C2semidirect
7. #22550151C2semidirect
8. #22550140C2semidirect
9. #22550139C2semidirect
10. #22550060C2semidirect
11. #22550060C2semidirect
12. #22550055C2semidirect
13. #22550050C2semidirect
14. #22549999C2semidirect
15. #444×C127C2semidirect
18. #16861256C22semidirect
25. #16861242C22semidirect
29. #16861239C22semidirect
42. C2×Dic508C22semidirect
43. C2×Dic508C22semidirect
50. #16861215C22semidirect
51. #16861209C22semidirect
53. #16861145C22semidirect
55. #16861137C22semidirect
56. #16861127C22semidirect
57. #16861127C22semidirect
58. #16861127C22semidirect
62. #131×C127C22semidirect
63. #108×C127C22semidirect
65. #104×C127C22semidirect
66. #97×C127C22semidirect
224. C127#3289semidirect
239. #444D127semidirect
277. #131C2×D127semidirect
281. #108C2×D127semidirect
285. #104C2×D127semidirect
290. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506932
#225506902
#225506882
#225506832
#225502702
#225501522
#225501512
#225501402
#225501392
#225500602
#225500552
#225500502
#225499992
#444×C1272
#3289127