Finite Group Databasebeta

← #30650167

Subgroups — #30650167

Order 16256.

370 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550480C211yesmaximal, characteristic, complemented
2#22550477C211yesmaximal, characteristic, complemented
3#22550178C211yesmaximal, characteristic, complemented
4#22550177C211yesmaximal, characteristic, complemented
5#22550176C211yesmaximal, characteristic, complemented
6#22550174C211yesmaximal, characteristic, complemented
7#22550128C211yesmaximal, characteristic, complemented
8#22550126C211yesmaximal, characteristic, complemented
9#22550111C211yesmaximal, characteristic, complemented
10#22550109C211yesmaximal, characteristic, complemented
11#22550069C211yesmaximal, characteristic, complemented
12#22550069C211yesmaximal, characteristic, complemented
13#22550068C211yesmaximal, characteristic, complemented
14#22549901C211yesmaximal, characteristic, complemented
15#459×C127C211yesmaximal, characteristic, complemented
16C2×#14523233C2211yescharacteristic
17C2×#14523233C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20#16861205C2211yescharacteristic
21#16861201C2211yescharacteristic
22#16861200C2211yescharacteristic
23#16861192—21no—
24#16861191—21no—
25#16861190—21no—
26#16861190—21no—
27#16861184C2211yescharacteristic
28#16861183—21no—
29#16861182—21no—
30#16861181—21no—
31#16861181—21no—
32#16861180C2211yescharacteristic
33#16861180C2211yescharacteristic
34#16861180C2211yescharacteristic
35C4×D508C2211yescharacteristic
36D4×Dic127C2211yescharacteristic
37D4×Dic127C2211yescharacteristic
38#16861166C2211yescharacteristic
39#16861143C2211yescharacteristic
40#16861142C2211yescharacteristic
41#16861141C2211yescharacteristic
42#16861140C2211yescharacteristic
43C4⋊(C127⋊C8)C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45Dic127⋊C8C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic
50#16861130C2211yescharacteristic
51#16861130C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C2×D8×C127C2211yescharacteristic
54#127×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C2×D508—21no—
60C2×D508C2311yescharacteristic
61C2×C22⋊D127—21no—
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C22×Dic127—21no—
67C22×Dic127—21no—
68C2×C4×D127—21no—
69C2×C4×D127C2311yescharacteristic
70#14523233—41no—
71#14523233—41no—
72#14523231—41no—
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C8×D127—41no—
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4D411yescharacteristic
84Dic127⋊C4D411yescharacteristic
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4D411yescharacteristic
93(C2×D127)⋊C4D411yescharacteristic
94C4×Dic127C2311yescharacteristic
95C2×D4×C127—21no—
96C2×D4×C127—21no—
97C2×D4×C127C2311yescharacteristic
98C2×D4×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C22×C4×C127—21no—
101D8×C127—41no—
102D8×C127—41no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127C2×D411yescharacteristic
110D508—21no—
111D508—21no—
112D508—81no—
113C22⋊D127—81no—
114C22⋊D127—81no—
115C22⋊D127—81no—
116C22⋊D127—81no—
117C22⋊D127—81no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—81no—
122C2×Dic127—81no—
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C4×D127—21no—
127C4×D127—21no—
128C4×D127—81no—
129C127⋊C8—21no—
130C127⋊C8—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—21no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—21no—
139D4×C127—21no—
140D4×C127—81no—
141D4×C127—81no—
142C2×C4×C127—21no—
143C2×C4×C127—81no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2411yescharacteristic
147C8×C127—21no—
148C8×C127—21no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127C22×D411yescharacteristic
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—21no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127H(5,𝔽2)11yescharacteristic
169C4×C1272−1+411yescharacteristic
170Dic127—21no—
171Dic127—21no—
172Dic127—21no—
173Dic127—41no—
174Dic127—41no—
175Dic127—41no—
176C2×C127—81no—
177C2×C127—81no—
178C2×C127—81no—
179C2×C127#53711yescharacteristic
180C2×C127#57611yescharacteristic
181C2×C127#57611yescharacteristic
182D127—41no—
183D127—81no—
184#3259—1271nomaximal, Sylow
185C127#325911yescharacteristic, complemented, Sylow
186#540—1271no—
187#540—1271no—
188#478—1271no—
189#478—1271no—
190#478—1271no—
191#478—1271no—
192#469—1271no—
193#469—1271no—
194#466—1271no—
195#466—1271no—
196#459—1271no—
197#459—1271no—
198#459—1271no—
199#459D12711yescharacteristic, complemented
200#424—1271no—
201C2×D8—1271no—
202C2×D8—1271no—
203C2×D8C2×D12711yescharacteristic
204C2×D8C2×D12711yescharacteristic
205C2×M16—1271no—
206C22×C8—1271no—
207#127—1271no—
208#127C2×D12711yescharacteristic
209#125—1271no—
210#125—1271no—
211#123—2541no—
212#123—2541no—
213#123—2541no—
214#123—2541no—
215#121—1271no—
216#121—1271no—
217#121—1271no—
218#121—1271no—
219#121—2541no—
220#121—2541no—
221#121—2541no—
222#121—2541no—
223C4×D4—1271no—
224C4×D4—1271no—
225C4×D4—1271no—
226C4×D4C2×D12711yescharacteristic
227#117—1271no—
228#107—1271no—
229#107—1271no—
230#106—1271no—
231#106—1271no—
232C4⋊C8—1271no—
233C4⋊C8—1271no—
234C4⋊C8—1271no—
235C4⋊C8C2×D12711yescharacteristic
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241#102—1271no—
242#102C2×D12711yescharacteristic
243#102C2×D12711yescharacteristic
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—1271no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4—2541no—
250C2×D4—2541no—
251C2×D4—2541no—
252C2×D4—2541no—
253C2×D4C22×D12711yescharacteristic
254C2×D4C22×D12711yescharacteristic
255C2×D4C22×D12711yescharacteristic
256C22×C4—21no—
257C22×C4—1271no—
258C22×C4—2541no—
259C22×C4—2541no—
260C22×C4—2541no—
261D8—41no—
262D8—41no—
263D8—5081no—
264D8—5081no—
265M16—5081no—
266C2×C8—1271no—
267C2×C8—1271no—
268C2×C8—5081no—
269C2×C8C22×D12711yescharacteristic
270C2×C8C22×D12711yescharacteristic
271C4⋊C4—1271no—
272C4⋊C4—1271no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4C22×D12711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—1271no—
283C22⋊C4—1271no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C42—1271no—
292C42C22×D12711yescharacteristic
293C23—21no—
294C23—21no—
295C23—21no—
296C23—1271no—
297C23—2541no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—81no—
305D4—81no—
306D4—10161no—
307D4—10161no—
308D4—10161no—
309D4—10161no—
310D4—10161no—
311D4—10161no—
312D4—2541no—
313D4—2541no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—1271no—
317C2×C4—1271no—
318C2×C4—1271no—
319C2×C4—10161no—
320C2×C4—10161no—
321C2×C4—10161no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4D4×D12711yescharacteristic
328C2×C4D4×D12711yescharacteristic
329C2×C4C23×D12711yescharacteristic
330C8—21no—
331C8—21no—
332C8—2541no—
333C8—2541no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—5081no—
344C22—5081no—
345C22—5081no—
346C22—2541no—
347C22—2541no—
348C22—2541no—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—5081no—
354C4—5081no—
355C4—5081no—
356C4—2541no—
357C4—2541no—
358C4—2541no—
359C4#1686125411yescharacteristic
360C4#1686125611yescharacteristic
361C2—81no—
362C2—81no—
363C2—81no—
364C2—10161no—
365C2—5081no—
366C2#2255045511yescharacteristic
367C2#2255067711yescharacteristic
368C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3259
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22550480C2semidirect
2. #22550477C2semidirect
3. #22550178C2semidirect
4. #22550177C2semidirect
5. #22550176C2semidirect
6. #22550174C2semidirect
7. #22550128C2semidirect
8. #22550126C2semidirect
9. #22550111C2semidirect
10. #22550109C2semidirect
11. #22550069C2semidirect
12. #22550069C2semidirect
13. #22550068C2semidirect
14. #22549901C2semidirect
15. #459×C127C2semidirect
185. C127#3259semidirect
199. #459D127semidirect

Maximal subgroups

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

subgroupindex
#225504802
#225504772
#225501782
#225501772
#225501762
#225501742
#225501282
#225501262
#225501112
#225501092
#225500692
#225500682
#225499012
#459×C1272
#3259127