Finite Group Databasebeta

← #30650504

Subgroups — #30650504

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#22550692C211yesmaximal, characteristic, complemented
2#22550689C211yesmaximal, characteristic, complemented
3#22550688C211yesmaximal, characteristic, complemented
4#22550685C211yesmaximal, characteristic, complemented
5#22550269C211yesmaximal, characteristic, complemented
6#22550150C211yesmaximal, characteristic, complemented
7#22550142C211yesmaximal, characteristic, complemented
8#22550140C211yesmaximal, characteristic, complemented
9#22550061C211yesmaximal, characteristic, complemented
10#22550057C211yesmaximal, characteristic, complemented
11#22550052C211yesmaximal, characteristic, complemented
12#22550051C211yesmaximal, characteristic, complemented
13#22549997C211yesmaximal, characteristic, complemented
14#22549996C211yesmaximal, characteristic, complemented
15#473×C127C211yesmaximal, characteristic, complemented
16#16861257C2211yescharacteristic
17#16861257C2211yescharacteristic, complemented
18#16861256C2211yescharacteristic
19#16861255C2211yescharacteristic
20#16861242—21no—
21#16861242C2211yescharacteristic
22#16861242C2211yescharacteristic, complemented
23#16861241—21no—
24#16861241—21no—
25#16861241—21no—
26#16861241—21no—
27#16861240C2211yescharacteristic
28#16861240C2211yescharacteristic, complemented
29#16861238—21no—
30#16861237—21no—
31#16861237C2211yescharacteristic
32#16861237C2211yescharacteristic
33#16861236—21no—
34#16861232—21no—
35#16861232—21no—
36#16861231—21no—
37#16861231C2211yescharacteristic
38#16861230—21no—
39#16861229C2211yescharacteristic
40C2×#14523238C2211yescharacteristic, complemented
41Q16×D127—21no—
42Q16×D127—21no—
43C2×#14523237—21no—
44C2×#14523237C2211yescharacteristic
45C2×#14523235C2211yescharacteristic
46SD16×D127—21no—
47#16861217C2211yescharacteristic
48#16861217C2211yescharacteristic, complemented
49#16861215C2211yescharacteristic, complemented
50#16861207C2211yescharacteristic, complemented
51#16861198C2211yescharacteristic
52#16861144C2211yescharacteristic, complemented
53#16861136C2211yescharacteristic
54#16861135C2211yescharacteristic
55#16861135C2211yescharacteristic, complemented
56#16861135C2211yescharacteristic, complemented
57#16861127C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic
59#16861119C2211yescharacteristic, 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#14523249—21no—
72#14523248—21no—
73#14523248—21no—
74#14523248—41no—
75#14523248—41no—
76#14523248C2311yescharacteristic
77#14523248C2311yescharacteristic
78#14523248C2311yescharacteristic
79#14523247—21no—
80#14523247—41no—
81C2×Dic254—21no—
82C2×Dic254C2311yescharacteristic
83Q8×D127—21no—
84Q8×D127—21no—
85Q8×D127—21no—
86Q8×D127—21no—
87Q8×D127—21no—
88C2×D508—21no—
89C2×D508C2311yescharacteristic
90D4×D127—21no—
91D4×D127—41no—
92Dic508—21no—
93Dic508—21no—
94Dic508D411yescharacteristic
95#14523238—21no—
96#14523238—21no—
97#14523238—21no—
98#14523238—21no—
99#14523238D411yescharacteristic
100#14523237—21no—
101#14523237—21no—
102#14523237—21no—
103#14523237—21no—
104#14523237—21no—
105#14523237—21no—
106#14523237D411yescharacteristic
107#14523237D411yescharacteristic
108#14523236D411yescharacteristic
109#14523235—21no—
110#14523235—21no—
111#14523235—21no—
112#14523235—21no—
113#14523235D411yescharacteristic
114D1016—21no—
115D1016—21no—
116D1016D411yescharacteristic
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—
127(C2×D127)⋊C4—41no—
128C4×Dic127C2311yescharacteristic
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—41no—
146Dic254—21no—
147Dic254—21no—
148Dic254—41no—
149Dic254—41no—
150Dic254—41no—
151Dic254C2×D411yescharacteristic
152Dic254C2×D411yescharacteristic
153Dic254C2×D411yescharacteristic
154D508—21no—
155D508—21no—
156D508—41no—
157D508—41no—
158D508—41no—
159D508C2×D411yescharacteristic
160D508C2×D411yescharacteristic
161D508C2×D411yescharacteristic
162C22⋊D127—21no—
163C22⋊D127—21no—
164C22⋊D127—21no—
165C22⋊D127—41no—
166C22⋊D127—41no—
167C2×Dic127—21no—
168C2×Dic127—21no—
169C2×Dic127—41no—
170C4×D127—21no—
171C4×D127—21no—
172C4×D127—21no—
173C4×D127—41no—
174C4×D127—41no—
175C4×D127—41no—
176C4×D127—41no—
177C4×D127—41no—
178C4×D127—41no—
179C127⋊C8—21no—
180C127⋊C8C2×D411yescharacteristic
181C127⋊C8C2×D411yescharacteristic
182Q8×C127—21no—
183Q8×C127—21no—
184Q8×C127—21no—
185Q8×C127—21no—
186Q8×C127C2×D411yescharacteristic
187D4×C127—21no—
188D4×C127C2×D411yescharacteristic
189C2×C4×C127—21no—
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×C127H(5,𝔽2)11yescharacteristic
200C2×D127—21no—
201C2×D127—21no—
202C2×D127—21no—
203C2×D127—41no—
204C2×D127—81no—
205C2×D127—161no—
206C4×C127—21no—
207C4×C127—41no—
208C4×C127—41no—
209C4×C127C22×D411yescharacteristic
210C4×C127C22×D411yescharacteristic
211Dic127—21no—
212Dic127—21no—
213Dic127—21no—
214Dic127—41no—
215Dic127—41no—
216C2×C127—21no—
217C2×C127—41no—
218C2×C127D4211yescharacteristic
219D127—41no—
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#473D12711yescharacteristic, complemented
232#471—1271no—
233#471—1271no—
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—2541no—
249#137—2541no—
250#137—2541no—
251#137—2541no—
252#137—2541no—
253#137—2541no—
254#137C2×D12711yescharacteristic
255#137C2×D12711yescharacteristic
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—1271no—
262#135—2541no—
263#135—2541no—
264#135—2541no—
265#135—2541no—
266C2×Q16—1271no—
267C2×Q16—2541no—
268C2×Q16—2541no—
269C2×Q16C2×D12711yescharacteristic, complemented
270C2×SD16—1271no—
271C2×SD16—1271no—
272C2×SD16—2541no—
273C2×SD16—2541no—
274#131—1271no—
275#131—1271no—
276#131—1271no—
277#131C2×D12711yescharacteristic
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#101C2×D12711yescharacteristic, complemented
288#101C2×D12711yescharacteristic, complemented
289#100—1271no—
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—2541no—
308C2×Q8—2541no—
309C2×Q8—2541no—
310C2×Q8—2541no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8C22×D12711yescharacteristic
314C2×Q8C22×D12711yescharacteristic
315C2×D4—1271no—
316C2×D4—5081no—
317C2×D4—2541no—
318C2×D4—2541no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—1271no—
324Q16—1271no—
325Q16—2541no—
326Q16—2541no—
327Q16—2541no—
328Q16—2541no—
329Q16—2541no—
330Q16—2541no—
331SD16—21no—
332SD16—21no—
333SD16—1271no—
334SD16—1271no—
335SD16—1271no—
336SD16—1271no—
337SD16—2541no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16—2541no—
346SD16—2541no—
347D8—1271no—
348D8—1271no—
349D8—2541no—
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—5081no—
369C23—2541no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—1271no—
375Q8—1271no—
376Q8—1271no—
377Q8—5081no—
378Q8—5081no—
379Q8—5081no—
380Q8—2541no—
381Q8—2541no—
382Q8D4×D12711yescharacteristic
383D4—21no—
384D4—1271no—
385D4—1271no—
386D4—1271no—
387D4—5081no—
388D4—5081no—
389D4—5081no—
390D4—5081no—
391D4—5081no—
392D4—2541no—
393D4—2541no—
394D4—2541no—
395D4—2541no—
396D4—2541no—
397D4D4×D12711yescharacteristic
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—21no—
417C8—1271no—
418C8—1271no—
419C8—2541no—
420C8D4×D12711yescharacteristic
421C8D4×D12711yescharacteristic
422C22—21no—
423C22—20321no—
424C22—10161no—
425C22—5081no—
426C22—2541no—
427C22—2541no—
428C22—2541no—
429C22#1686125511yescharacteristic
430C4—21no—
431C4—41no—
432C4—41no—
433C4—5081no—
434C4—5081no—
435C4—2541no—
436C4—2541no—
437C4—2541no—
438C4C2×D4×D12711yescharacteristic
439C4C2×D4×D12711yescharacteristic
440C2—21no—
441C2—41no—
442C2—10161no—
443C2—5081no—
444C2—5081no—
445C2—5081no—
446C2#2255051011yescharacteristic
subgroupG/H
centerC2#22550510
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#473×C127C2
socleC2×C127D42
hypercenter#473D127
hypocenterC127#3289
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
1. #22550692C2semidirect
2. #22550689C2semidirect
3. #22550688C2semidirect
4. #22550685C2semidirect
5. #22550269C2semidirect
6. #22550150C2semidirect
7. #22550142C2semidirect
8. #22550140C2semidirect
9. #22550061C2semidirect
10. #22550057C2semidirect
11. #22550052C2semidirect
12. #22550051C2semidirect
13. #22549997C2semidirect
14. #22549996C2semidirect
15. #473×C127C2semidirect
17. #16861257C22semidirect
22. #16861242C22semidirect
28. #16861240C22semidirect
40. C2×#14523238C22semidirect
48. #16861217C22semidirect
49. #16861215C22semidirect
50. #16861207C22semidirect
52. #16861144C22semidirect
55. #16861135C22semidirect
56. #16861135C22semidirect
57. #16861127C22semidirect
59. #16861119C22semidirect
62. C2×Q16×C127C22semidirect
64. #108×C127C22semidirect
65. #101×C127C22semidirect
66. #101×C127C22semidirect
224. C127#3289semidirect
231. #473D127semidirect
269. C2×Q16C2×D127semidirect
281. #108C2×D127semidirect
287. #101C2×D127semidirect
288. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506922
#225506892
#225506882
#225506852
#225502692
#225501502
#225501422
#225501402
#225500612
#225500572
#225500522
#225500512
#225499972
#225499962
#473×C1272
#3289127