Finite Group Databasebeta

← #30650250

Subgroups — #30650250

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×#16861233C211yesmaximal, characteristic, complemented
2C2×#16861230C211yesmaximal, characteristic, complemented
3#22550530C211yesmaximal, characteristic, complemented
4#22550519C211yesmaximal, characteristic, complemented
5#22550271C211yesmaximal, characteristic, complemented
6#22550133C211yesmaximal, characteristic, complemented
7#22550104C211yesmaximal, characteristic, complemented
8#22550082C211yesmaximal, characteristic, complemented
9#22550041C211yesmaximal, characteristic, complemented
10#22550035C211yesmaximal, characteristic, complemented
11#22550028C211yesmaximal, characteristic, complemented
12#22550018C211yesmaximal, characteristic, complemented
13#22549986C211yesmaximal, characteristic, complemented
14#22549946C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×#14523247C2211yescharacteristic, complemented
19C2×D4×D127C2211yescharacteristic
20#16861233—24no—
21#16861230—24no—
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523238C2211yescharacteristic, complemented
24C2×#14523237C2211yescharacteristic
25C2×#14523237C2211yescharacteristic
26C2×#14523236C2211yescharacteristic
27C2×#14523236C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523231C2211yescharacteristic
30C2×C8×D127C2211yescharacteristic, complemented
31#16861209C2211yescharacteristic, complemented
32#16861193—21no—
33#16861193—21no—
34#16861191—21no—
35#16861189C2211yescharacteristic
36#16861189C2211yescharacteristic, complemented
37#16861185C2211yescharacteristic
38#16861183—21no—
39#16861182—21no—
40#16861180C2211yescharacteristic
41#16861179—21no—
42C4×D508C2211yescharacteristic
43C4×D508C2211yescharacteristic
44C2×C4⋊Dic127—21no—
45C22⋊C4×D127—21no—
46#16861140C2211yescharacteristic, complemented
47#16861134C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861120C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×D8×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic
57C4×D4×C127C2211yescharacteristic, complemented
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic
61C23×D127—21no—
62#14523249—22no—
63#14523249—41no—
64#14523247—22no—
65#14523247—22no—
66#14523247—41no—
67#14523247—41no—
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70C2×D508—21no—
71C2×D508D411yescharacteristic
72C2×D508C2311yescharacteristic
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77D4×D127—22no—
78D4×D127—41no—
79C22×Dic127—21no—
80C22×Dic127—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#14523237—22no—
89#14523236—22no—
90#14523236—41no—
91#14523233D414yescomplemented
92#14523231—22no—
93C2×C127⋊C8C2311yescharacteristic
94C2×C127⋊C8C2311yescharacteristic
95C8×D127—22no—
96C4⋊Dic127—21no—
97C4⋊Dic127—21no—
98C4⋊Dic127—21no—
99C4⋊Dic127—21no—
100C4⋊Dic127D411yescharacteristic
101C4⋊Dic127C2311yescharacteristic
102Dic127⋊C4—21no—
103C22⋊Dic127—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×D4×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127—21no—
114SD16×C127—22no—
115D8×C127—22no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C4⋊C4×C127C2311yescharacteristic
119C22⋊C4×C127—21no—
120C22⋊C4×C127—21no—
121C22⋊C4×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—42no—
125C22×D127—42no—
126C22×D127—81no—
127C22×D127C2×D411yescharacteristic
128C22×D127C2×D411yescharacteristic
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254—21no—
133D508—21no—
134D508—41no—
135D508—42no—
136D508C2×D412yes—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—81no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—42no—
145C2×Dic127—42no—
146C2×Dic127C2×D411yescharacteristic
147C2×Dic127C2×D411yescharacteristic
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—21no—
151C4×D127—21no—
152C4×D127—42no—
153C4×D127—81no—
154C127⋊C8—21no—
155C127⋊C8C2×D412yes—
156C23×C127—21no—
157C23×C127—21no—
158Q8×C127—21no—
159Q8×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127—21no—
163D4×C127C2×D412yes—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—42no—
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×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127C22×D411yescharacteristic
179C2×D127—21no—
180C2×D127—21no—
181C2×D127—22no—
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C2×D127—21no—
186C2×D127—41no—
187C2×D127—41no—
188C2×D127—84no—
189C4×C127—21no—
190C4×C127—21no—
191C4×C127—41no—
192C4×C127—41no—
193C4×C127C22×D411yescharacteristic
194C4×C127H(5,𝔽2)11yescharacteristic
195Dic127—21no—
196Dic127—21no—
197Dic127—41no—
198Dic127—41no—
199Dic127—41no—
200C2×C127—42no—
201C2×C127—81no—
202C2×C127D4211yescharacteristic
203C2×C127#57511yescharacteristic
204C2×C127#57711yescharacteristic
205D127—41no—
206D127—41no—
207D127—42no—
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#463D12711yescharacteristic, complemented
218#461—1271no—
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×SD16—1271no—
237C2×SD16C2×D12711yescharacteristic, complemented
238C2×D8—1271no—
239C2×D8C2×D12711yescharacteristic
240C2×M16—1271no—
241C22×C8—1271no—
242C4⋊Q8—1271no—
243#124C2×D12711yescharacteristic
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, complemented
257C2×C4⋊C4—2541no—
258C2×C22⋊C4—2541no—
259#106—1271no—
260C4⋊C8C2×D12711yescharacteristic, complemented
261#103—1271no—
262#103—1271no—
263#103C2×D12711yescharacteristic, complemented
264#102—1271no—
265#102—1271no—
266#102C2×D12711yescharacteristic
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×Q8—1271no—
279C2×Q8C22×D12711yescharacteristic
280C2×D4—1271no—
281C2×D4—1271no—
282C2×D4—5081no—
283C2×D4—2541no—
284C2×D4—2541no—
285C2×D4—2542no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4C22×D12711yescharacteristic
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—5081no—
302SD16—2542no—
303SD16—2542no—
304SD16—2542no—
305D8—22no—
306D8—1274no—
307M16—2542no—
308C2×C8—1271no—
309C2×C8—1271no—
310C2×C8—2542no—
311C2×C8C22×D12711yescharacteristic
312C2×C8C22×D12711yescharacteristic
313C4⋊C4—1271no—
314C4⋊C4—1271no—
315C4⋊C4—2541no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4—2541no—
320C4⋊C4C22×D12711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—1271no—
325C22⋊C4—1271no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C22⋊C4—2541no—
330C22⋊C4—2541no—
331C42C22×D12711yescharacteristic
332C23—21no—
333C23—21no—
334C23—1271no—
335C23—1271no—
336C23—10161no—
337C23—5082no—
338C23—5082no—
339C23—2541no—
340Q8—21no—
341Q8—21no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345Q8—2541no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—1272no—
350D4—10161no—
351D4—5081no—
352D4—5082no—
353D4—5082no—
354D4—5082no—
355D4—5082no—
356D4—2541no—
357D4D4×D12712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—1271no—
362C2×C4—1271no—
363C2×C4—10161no—
364C2×C4—5082no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—2541no—
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—41no—
385C22—41no—
386C22—41no—
387C22—10164no—
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—5081no—
403C4—5081no—
404C4—5081no—
405C4—2541no—
406C4—2541no—
407C4C2×D4×D12711yescharacteristic
408C4#1686125511yescharacteristic
409C2—42no—
410C2—81no—
411C2—5081no—
412C2—5082no—
413C2—5081no—
414C2#2255051011yescharacteristic
415C2#2255067211yescharacteristic
416C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
hypercenter#463D127
hypocenterC127#3267
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. C2×#16861230C2semidirect
3. #22550530C2semidirect
4. #22550519C2semidirect
5. #22550271C2semidirect
6. #22550133C2semidirect
7. #22550104C2semidirect
8. #22550082C2semidirect
9. #22550041C2semidirect
10. #22550035C2semidirect
11. #22550028C2semidirect
12. #22550018C2semidirect
13. #22549986C2semidirect
14. #22549946C2semidirect
15. #463×C127C2semidirect
18. C2×#14523247C22semidirect
22. C2×#14523238C22semidirect
23. C2×#14523238C22semidirect
30. C2×C8×D127C22semidirect
31. #16861209C22semidirect
36. #16861189C22semidirect
46. #16861140C22semidirect
47. #16861134C22semidirect
48. #16861134C22semidirect
49. #16861130C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861120C22semidirect
54. C2×SD16×C127C22semidirect
57. C4×D4×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
91. #14523233D4semidirect
209. C127#3267semidirect
217. #463D127semidirect
237. C2×SD16C2×D127semidirect
256. C4×D4C2×D127semidirect
260. C4⋊C8C2×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×#168612332
C2×#168612302
#225505302
#225505192
#225502712
#225501332
#225501042
#225500822
#225500412
#225500352
#225500282
#225500182
#225499862
#225499462
#463×C1272
#3267127