Finite Group Databasebeta

← #30650261

Subgroups — #30650261

Order 16256.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2C2×#16861230C211yesmaximal, characteristic, complemented
3#22550529C211yesmaximal, characteristic, complemented
4#22550521C211yesmaximal, characteristic, complemented
5#22550268C211yesmaximal, characteristic, complemented
6#22550133C211yesmaximal, characteristic, complemented
7#22550108C211yesmaximal, characteristic, complemented
8#22550092C211yesmaximal, characteristic, complemented
9#22550041C211yesmaximal, characteristic, complemented
10#22550035C211yesmaximal, characteristic, complemented
11#22550025C211yesmaximal, characteristic, complemented
12#22550024C211yesmaximal, characteristic, complemented
13#22549986C211yesmaximal, characteristic, complemented
14#22549946C211yesmaximal, characteristic, complemented
15#461×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523249C2211yescharacteristic, complemented
18C2×#14523247C2211yescharacteristic
19C2×D4×D127C2211yescharacteristic
20#16861236—24no—
21#16861230—24no—
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523238C2211yescharacteristic, complemented
24C2×#14523237C2211yescharacteristic, complemented
25C2×#14523235C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×D1016C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523231C2211yescharacteristic
30C2×C8×D127C2211yescharacteristic, complemented
31#16861197C2211yescharacteristic
32#16861194—21no—
33#16861194—21no—
34#16861193—21no—
35#16861189C2211yescharacteristic
36#16861185C2211yescharacteristic
37#16861185C2211yescharacteristic, complemented
38#16861184—21no—
39#16861184C2211yescharacteristic
40#16861183—21no—
41#16861179—21no—
42C4×D508C2211yescharacteristic
43C4×D508C2211yescharacteristic, complemented
44C4⋊C4×D127—21no—
45C2×(C2×D127)⋊C4—21no—
46#16861140C2211yescharacteristic, complemented
47#16861134C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic
49#16861130C2211yescharacteristic
50#16861129C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861120C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56C4⋊Q8×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#103×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523249—22no—
63#14523249—22no—
64#14523249—41no—
65#14523249—41no—
66#14523247—22no—
67#14523247—41no—
68C2×Dic254C2311yescharacteristic
69C2×D508—21no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508D411yescharacteristic
73C2×D508C2311yescharacteristic
74C2×D508C2311yescharacteristic
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77D4×D127—22no—
78D4×D127—41no—
79C22×Dic127—21no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127C2311yescharacteristic
84C2×C4×D127C2311yescharacteristic
85#14523238—22no—
86#14523238—41no—
87#14523237—22no—
88#14523235—22no—
89#14523235—41no—
90D1016—22no—
91#14523233D414yescomplemented
92#14523231—22no—
93C2×C127⋊C8C2311yescharacteristic
94C2×C127⋊C8C2311yescharacteristic
95C8×D127—22no—
96C4⋊Dic127—21no—
97C4⋊Dic127D411yescharacteristic
98C4⋊Dic127C2311yescharacteristic
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4D411yescharacteristic
109(C2×D127)⋊C4D411yescharacteristic
110C2×Q8×C127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127—21no—
114SD16×C127—22no—
115SD16×C127—22no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C4⋊C4×C127—21no—
119C4⋊C4×C127—21no—
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—81no—
128C22×D127C2×D411yescharacteristic
129C22×D127C2×D411yescharacteristic
130Dic254—21no—
131Dic254—21no—
132D508—21no—
133D508—21no—
134D508—21no—
135D508—41no—
136D508—42no—
137D508—42no—
138D508—81no—
139D508C2×D412yes—
140C22⋊D127—42no—
141C22⋊D127—42no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—42no—
145C2×Dic127C2×D411yescharacteristic
146C2×Dic127C2×D411yescharacteristic
147C4×D127—21no—
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—21no—
151C4×D127—42no—
152C4×D127—42no—
153C4×D127—42no—
154C127⋊C8—21no—
155C127⋊C8C2×D412yes—
156C23×C127—21no—
157Q8×C127—21no—
158Q8×C127—21no—
159Q8×C127—21no—
160Q8×C127—21no—
161D4×C127—21no—
162D4×C127C2×D412yes—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—81no—
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127—21no—
172C22×C127—22no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127C22×D411yescharacteristic
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—22no—
179C2×D127—21no—
180C2×D127—21no—
181C2×D127—21no—
182C2×D127—21no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C2×D127—84no—
189C4×C127—21no—
190C4×C127—21no—
191C4×C127—41no—
192C4×C127—41no—
193C4×C127—41no—
194C4×C127C22×D411yescharacteristic
195C4×C127H(5,𝔽2)11yescharacteristic
196Dic127—21no—
197Dic127—21no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—42no—
201C2×C127D4211yescharacteristic
202C2×C127#57511yescharacteristic
203C2×C127#57711yescharacteristic
204D127—41no—
205D127—41no—
206D127—42no—
207D127—81no—
208#3267—1271nomaximal, Sylow
209C127#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1271no—
211C2×#135—1271no—
212#547—1271no—
213#546—1271no—
214#497—1271no—
215#470—1271no—
216#465—1271no—
217#463—1271no—
218#461D12711yescharacteristic, complemented
219#452—1271no—
220#452—1271no—
221#450—1271no—
222#449—1271no—
223#442—1271no—
224#435—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C2×#41—1271no—
228C22×D4—1271no—
229AGL(1,ℤ/8)—2544no—
230#135—2544no—
231C2×Q16—1271no—
232C2×Q16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16C2×D12711yescharacteristic
237C2×SD16C2×D12711yescharacteristic
238C2×D8—1271no—
239C2×D8—1271no—
240C2×M16—1271no—
241C22×C8—1271no—
242C4⋊Q8C2×D12711yescharacteristic, complemented
243#124—1271no—
244#123—2541no—
245#123—2541no—
246#123—2541no—
247C22⋊Q8—1271no—
248C22⋊Q8—1271no—
249C22⋊Q8—1271no—
250#121—1271no—
251#121—2541no—
252#121—2541no—
253#120—2541no—
254C4×D4—1271no—
255C4×D4—1271no—
256C4×D4C2×D12711yescharacteristic
257C2×C4⋊C4—2541no—
258C2×C22⋊C4—2541no—
259#106—1271no—
260C4⋊C8C2×D12711yescharacteristic, complemented
261#103—1271no—
262#103C2×D12711yescharacteristic, complemented
263#103C2×D12711yescharacteristic, complemented
264#102—1271no—
265#102—1271no—
266#102—1271no—
267C22⋊C8—1271no—
268C22⋊C8—1271no—
269#97—1271no—
270C24—2541no—
271#41—5081no—
272#41—5081no—
273#41—5081no—
274#41—2542no—
275#41—2542no—
276#41—2542no—
277C2×Q8—1271no—
278C2×Q8C22×D12711yescharacteristic
279C2×Q8C22×D12711yescharacteristic
280C2×D4—1271no—
281C2×D4—1271no—
282C2×D4—1271no—
283C2×D4—5081no—
284C2×D4—2541no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2542no—
290C2×D4C22×D12711yescharacteristic
291C22×C4—21no—
292C22×C4—1271no—
293C22×C4—1271no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4—2541no—
298Q16—5081no—
299Q16—2542no—
300SD16—22no—
301SD16—22no—
302SD16—5081no—
303SD16—2542no—
304SD16—2542no—
305D8—1274no—
306D8—2542no—
307M16—2542no—
308C2×C8—1271no—
309C2×C8—1271no—
310C2×C8—2542no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—1271no—
316C4⋊C4—1271no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4—2541no—
320C4⋊C4C22×D12711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—1271no—
323C22⋊C4—1271no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C22⋊C4—2541no—
331C42C22×D12711yescharacteristic
332C23—21no—
333C23—1271no—
334C23—1271no—
335C23—10161no—
336C23—5082no—
337C23—5082no—
338C23—2541no—
339C23—2541no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2541no—
345Q8—2541no—
346D4—21no—
347D4—1272no—
348D4—10161no—
349D4—5081no—
350D4—5082no—
351D4—5082no—
352D4—5082no—
353D4—5082no—
354D4—2541no—
355D4—2541no—
356D4—2541no—
357D4D4×D12712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—81no—
362C2×C4—1271no—
363C2×C4—1271no—
364C2×C4—5082no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—2541no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4D4×D12711yescharacteristic
375C2×C4D4×D12711yescharacteristic
376C2×C4C23×D12711yescharacteristic
377C8—21no—
378C8—21no—
379C8—1272no—
380C8—2541no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—10164no—
385C22—5081no—
386C22—5081no—
387C22—5081no—
388C22—5081no—
389C22—5081no—
390C22—2541no—
391C22—2541no—
392C22—2541no—
393C22—2541no—
394C22—2542no—
395C22—2541no—
396C22—2541no—
397C22C2×D4×D12711yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—5081no—
404C4—5081no—
405C4—2541no—
406C4—2541no—
407C4C2×D4×D12711yescharacteristic
408C4#1686125511yescharacteristic
409C2—42no—
410C2—10161no—
411C2—5081no—
412C2—5082no—
413C2—5081no—
414C2#2255051011yescharacteristic
415C2#2255067511yescharacteristic
416C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#461×C127C2
socleC22×C127C22×D4
hypercenter#461D127
hypocenterC127#3267
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×#16861230C2semidirect
3. #22550529C2semidirect
4. #22550521C2semidirect
5. #22550268C2semidirect
6. #22550133C2semidirect
7. #22550108C2semidirect
8. #22550092C2semidirect
9. #22550041C2semidirect
10. #22550035C2semidirect
11. #22550025C2semidirect
12. #22550024C2semidirect
13. #22549986C2semidirect
14. #22549946C2semidirect
15. #461×C127C2semidirect
17. C2×#14523249C22semidirect
22. C2×#14523238C22semidirect
23. C2×#14523238C22semidirect
24. C2×#14523237C22semidirect
30. C2×C8×D127C22semidirect
37. #16861185C22semidirect
43. C4×D508C22semidirect
46. #16861140C22semidirect
47. #16861134C22semidirect
50. #16861129C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861120C22semidirect
56. C4⋊Q8×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
60. #103×C127C22semidirect
91. #14523233D4semidirect
209. C127#3267semidirect
218. #461D127semidirect
242. C4⋊Q8C2×D127semidirect
260. C4⋊C8C2×D127semidirect
262. #103C2×D127semidirect
263. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×#168612302
#225505292
#225505212
#225502682
#225501332
#225501082
#225500922
#225500412
#225500352
#225500252
#225500242
#225499862
#225499462
#461×C1272
#3267127