Finite Group Databasebeta

← #30651813

Subgroups — #30651813

Order 16256.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550612C211yesmaximal, characteristic, complemented
2#22550609C211yesmaximal, characteristic, complemented
3#22550264C211yesmaximal, characteristic, complemented
4#22550263C211yesmaximal, characteristic, complemented
5#22550259C211yesmaximal, characteristic, complemented
6#22550099C211yesmaximal, characteristic, complemented
7#22550097C211yesmaximal, characteristic, complemented
8#22550090C211yesmaximal, characteristic, complemented
9#22550089C211yesmaximal, characteristic, complemented
10#22550069C211yesmaximal, characteristic, complemented
11#22550005C211yesmaximal, characteristic, complemented
12#22549976C211yesmaximal, characteristic, complemented
13#22549973C211yesmaximal, characteristic, complemented
14#22549956C211yesmaximal, characteristic, complemented
15#459×C127C211yesmaximal, characteristic, complemented
16C2×Dic508C2211yescharacteristic
17C2×#14523237C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523236C2211yescharacteristic
21C2×#14523233C2211yescharacteristic
22C2×#14523233C2211yescharacteristic
23#16861205C2211yescharacteristic
24#16861197C2211yescharacteristic
25#16861196—21no—
26#16861196C2211yescharacteristic
27#16861196C2211yescharacteristic
28#16861196C2211yescharacteristic
29#16861195—21no—
30#16861192—21no—
31#16861191—21no—
32#16861190—21no—
33#16861190—21no—
34C4×Dic254C2211yescharacteristic
35C4×Dic254C2211yescharacteristic
36#16861174C2211yescharacteristic
37D4×Dic127—21no—
38D4×Dic127C2211yescharacteristic
39D4×Dic127C2211yescharacteristic
40#16861171C2211yescharacteristic
41C4×C22⋊D127—21no—
42#16861142C2211yescharacteristic
43Dic127⋊C8C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45#16861134C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861120C2211yescharacteristic
50#16861119C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C2×D8×C127C2211yescharacteristic
54#127×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C2×Dic254D411yescharacteristic
60C2×Dic254C2311yescharacteristic
61C2×Dic254C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C22×Dic127—21no—
66C22×Dic127—21no—
67C2×C4×D127—21no—
68Dic508—41no—
69#14523237—41no—
70#14523236—22no—
71#14523236—22no—
72#14523236—41no—
73#14523233—22no—
74#14523233—41no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C4⋊Dic127D411yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4D411yescharacteristic
82Dic127⋊C4D411yescharacteristic
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97C4×Dic127C2311yescharacteristic
98C2×D4×C127—21no—
99C2×D4×C127—21no—
100C2×D4×C127C2311yescharacteristic
101C2×D4×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104D8×C127—41no—
105D8×C127—41no—
106C2×C8×C127C2311yescharacteristic
107C2×C8×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112Dic254—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—21no—
116Dic254—41no—
117D508—21no—
118D508—21no—
119C22⋊D127—42no—
120C22⋊D127—81no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—81no—
125C2×Dic127—81no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C4×D127—81no—
131C127⋊C8—21no—
132C127⋊C8#4112yes—
133C23×C127—21no—
134C23×C127—21no—
135C23×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—21no—
139D4×C127—21no—
140D4×C127—21no—
141D4×C127—21no—
142D4×C127—42no—
143D4×C127—81no—
144C2×C4×C127—21no—
145C2×C4×C127—81no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2411yescharacteristic
149C8×C127—21no—
150C8×C127—21no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127C22×D411yescharacteristic
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—41no—
167C4×C127C2×#4111yescharacteristic
168C4×C127H(5,𝔽2)11yescharacteristic
169Dic127—21no—
170Dic127—21no—
171Dic127—21no—
172Dic127—21no—
173Dic127—41no—
174Dic127—41no—
175Dic127—41no—
176C2×C127—81no—
177C2×C127—81no—
178C2×C127—81no—
179C2×C127#55011yescharacteristic
180C2×C127#57511yescharacteristic
181C2×C127#57611yescharacteristic
182D127—81no—
183#3403—1271nomaximal, Sylow
184C127#340311yescharacteristic, complemented, Sylow
185#559—1271no—
186#559—1271no—
187#496—1271no—
188#496—1271no—
189#495—1271no—
190#464—1271no—
191#464—1271no—
192#463—1271no—
193#463—1271no—
194#459—1271no—
195#459D12711yescharacteristic, complemented
196#446—1271no—
197#440—1271no—
198#440—1271no—
199C4×D8—1271no—
200C2×Q16—1271no—
201C2×SD16—1271no—
202C2×SD16—1271no—
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C2×D8—1271no—
207C2×D8C2×D12711yescharacteristic
208C2×D8C2×D12711yescharacteristic
209#127—1271no—
210#127C2×D12711yescharacteristic
211#124—1271no—
212#124—1271no—
213#124—1271no—
214#124—1271no—
215#124—2541no—
216#124—2541no—
217#123—2541no—
218#123—2541no—
219#123—2541no—
220#123—2541no—
221C4×Q8—1271no—
222C4×Q8—1271no—
223C4×Q8—1271no—
224C4×D4—1271no—
225C4×D4—1271no—
226C4×D4—1271no—
227C4×D4—2541no—
228C4×D4—2541no—
229C4×D4C2×D12711yescharacteristic
230#107—1271no—
231C4⋊C8—1271no—
232C4⋊C8—1271no—
233C4⋊C8C2×D12711yescharacteristic
234#103—1271no—
235#103—1271no—
236#102—1271no—
237#102—1271no—
238#102C2×D12711yescharacteristic
239#102C2×D12711yescharacteristic
240#97—1271no—
241#97—1271no—
242C4×C8—1271no—
243C2×Q8—1271no—
244C2×Q8—1271no—
245C2×Q8—1271no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—1271no—
249C2×D4—2541no—
250C2×D4—2541no—
251C2×D4C22×D12711yescharacteristic
252C2×D4C22×D12711yescharacteristic
253C2×D4C22×D12711yescharacteristic
254C22×C4—21no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4—2541no—
258Q16—5081no—
259SD16—5081no—
260SD16—5081no—
261SD16—2542no—
262SD16—2542no—
263D8—41no—
264D8—41no—
265D8—5081no—
266D8—2542no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8C22×D12711yescharacteristic
270C2×C8C22×D12711yescharacteristic
271C4⋊C4—1271no—
272C4⋊C4—1271no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4C22×D12711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C42—1271no—
291C42—1271no—
292C42—2541no—
293C42—2541no—
294C42C22×D12711yescharacteristic
295C23—21no—
296C23—21no—
297C23—21no—
298C23—2541no—
299Q8—5081no—
300Q8—2541no—
301Q8—2541no—
302Q8—2541no—
303Q8—2541no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—21no—
310D4—42no—
311D4—81no—
312D4—10161no—
313D4—5082no—
314D4—2541no—
315D4—2541no—
316C2×C4—21no—
317C2×C4—81no—
318C2×C4—1271no—
319C2×C4—1271no—
320C2×C4—1271no—
321C2×C4—1271no—
322C2×C4—10161no—
323C2×C4—10161no—
324C2×C4—10161no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4D4×D12711yescharacteristic
329C2×C4D4×D12711yescharacteristic
330C2×C4C23×D12711yescharacteristic
331C8—21no—
332C8—21no—
333C8—1272no—
334C8—2541no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—5081no—
345C22—5081no—
346C22—5081no—
347C22C2×D4×D12711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—5081no—
352C4—5081no—
353C4—5081no—
354C4—2541no—
355C4—2541no—
356C4—2541no—
357C4—2541no—
358C4#41×D12711yescharacteristic
359C4#1686125411yescharacteristic
360C2—81no—
361C2—81no—
362C2—81no—
363C2—10161no—
364C2#2255054411yescharacteristic
365C2#2255067211yescharacteristic
366C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3403
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. #22550612C2semidirect
2. #22550609C2semidirect
3. #22550264C2semidirect
4. #22550263C2semidirect
5. #22550259C2semidirect
6. #22550099C2semidirect
7. #22550097C2semidirect
8. #22550090C2semidirect
9. #22550089C2semidirect
10. #22550069C2semidirect
11. #22550005C2semidirect
12. #22549976C2semidirect
13. #22549973C2semidirect
14. #22549956C2semidirect
15. #459×C127C2semidirect
184. C127#3403semidirect
195. #459D127semidirect

Maximal subgroups

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

subgroupindex
#225506122
#225506092
#225502642
#225502632
#225502592
#225500992
#225500972
#225500902
#225500892
#225500692
#225500052
#225499762
#225499732
#225499562
#459×C1272
#3403127