Finite Group Databasebeta

← #30650743

Subgroups — #30650743

Order 16256.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550576C211yesmaximal, characteristic, complemented
2#22550546C211yesmaximal, characteristic, complemented
3#22550240C211yesmaximal, characteristic, complemented
4#22550199C211yesmaximal, characteristic, complemented
5#22550188C211yesmaximal, characteristic, complemented
6#22550184C211yesmaximal, characteristic, complemented
7#22550126C211yesmaximal, characteristic, complemented
8#22550109C211yesmaximal, characteristic, complemented
9#22550018C211yesmaximal, characteristic, complemented
10#22550018C211yesmaximal, characteristic, complemented
11#22549976C211yesmaximal, characteristic, complemented
12#22549946C211yesmaximal, characteristic, complemented
13#22549874C211yesmaximal, characteristic, complemented
14#22549869C211yesmaximal, characteristic, complemented
15#459×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523231C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861205—21no—
22#16861202—21no—
23#16861201C2211yescharacteristic
24#16861192—21no—
25#16861182—21no—
26#16861181—21no—
27#16861181—21no—
28#16861180C2211yescharacteristic
29#16861180C2211yescharacteristic
30#16861180C2211yescharacteristic
31#16861180C2211yescharacteristic
32C4×Dic254C2211yescharacteristic
33C4×D508C2211yescharacteristic
34D4×Dic127—21no—
35C4×C22⋊D127—21no—
36#16861167C2211yescharacteristic
37C4⋊C4×D127C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861141C2211yescharacteristic
41#16861134C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861132C2211yescharacteristic
44#16861130C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861120C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C2×D8×C127C2211yescharacteristic
54#127×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—22no—
61#14523247—41no—
62#14523247—41no—
63C2×Dic254C2311yescharacteristic
64C2×D508D411yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C22×Dic127—21no—
72C22×Dic127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2311yescharacteristic
75C2×C4×D127C2311yescharacteristic
76#14523236—22no—
77#14523231—41no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—41no—
81C4⋊Dic127D411yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84C4⋊Dic127C2311yescharacteristic
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4D411yescharacteristic
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127—21no—
97C4×Dic127—21no—
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×D127C2×D411yescharacteristic
112C22×D127C2×D411yescharacteristic
113Dic254—21no—
114Dic254#4112yes—
115D508—41no—
116C22⋊D127—42no—
117C22⋊D127—42no—
118C22⋊D127—42no—
119C22⋊D127—81no—
120C22⋊D127—81no—
121C22⋊D127—81no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—42no—
127C2×Dic127—42no—
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C4×D127—21no—
131C4×D127—21no—
132C4×D127—21no—
133C4×D127—21no—
134C4×D127—81no—
135C127⋊C8—21no—
136C127⋊C8—21no—
137C23×C127—21no—
138C23×C127—21no—
139C23×C127—21no—
140D4×C127—21no—
141D4×C127—21no—
142D4×C127—21no—
143D4×C127—21no—
144D4×C127—21no—
145D4×C127—21no—
146D4×C127—42no—
147D4×C127—81no—
148C2×C4×C127—21no—
149C2×C4×C127—81no—
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2411yescharacteristic
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127C22×D411yescharacteristic
165C2×D127—21no—
166C2×D127—21no—
167C2×D127—21no—
168C2×D127—21no—
169C2×D127—21no—
170C2×D127—21no—
171C4×C127—21no—
172C4×C127—21no—
173C4×C127—41no—
174C4×C127C2×#4111yescharacteristic
175C4×C127H(5,𝔽2)11yescharacteristic
176Dic127—21no—
177Dic127—21no—
178Dic127—22no—
179Dic127—41no—
180Dic127—41no—
181Dic127—41no—
182C2×C127—81no—
183C2×C127—81no—
184C2×C127—81no—
185C2×C127#54611yescharacteristic
186C2×C127#57511yescharacteristic
187C2×C127#57611yescharacteristic
188D127—41no—
189D127—41no—
190#3309—1271nomaximal, Sylow
191C127#330911yescharacteristic, complemented, Sylow
192#553—1271no—
193#550—1271no—
194#490—1271no—
195#482—1271no—
196#480—1271no—
197#479—1271no—
198#469—1271no—
199#466—1271no—
200#459D12711yescharacteristic, complemented
201#449—1271no—
202#449—1271no—
203#440—1271no—
204#435—1271no—
205#417—1271no—
206#416—1271no—
207C2×#41—1271no—
208C2×#41—1271no—
209C2×SD16—1271no—
210C2×D8C2×D12711yescharacteristic
211C2×D8C2×D12711yescharacteristic
212C2×M16—1271no—
213C22×C8—1271no—
214#127—2541no—
215#127C2×D12711yescharacteristic
216#126—2541no—
217#125—1271no—
218#123—2541no—
219#121—1271no—
220#121—1271no—
221#121—1271no—
222#121—1271no—
223#121—2541no—
224#121—2541no—
225#121—2541no—
226C4×Q8—1271no—
227C4×D4—1271no—
228C4×D4—2541no—
229C4×D4—2541no—
230C4×D4C2×D12711yescharacteristic
231#117—1271no—
232C2×C4⋊C4—1271no—
233#107—1271no—
234#107—1271no—
235#106—1271no—
236C4⋊C8C2×D12711yescharacteristic
237#103—1271no—
238#103—1271no—
239#103—1271no—
240#102—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102—1271no—
245#102C2×D12711yescharacteristic
246#102C2×D12711yescharacteristic
247C22⋊C8—1271no—
248C22⋊C8—1271no—
249#97—1271no—
250#41—5081no—
251#41—5081no—
252#41—2542no—
253#41—2542no—
254C2×Q8—1271no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—1271no—
258C2×D4—2541no—
259C2×D4—2541no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4C22×D12711yescharacteristic
265C2×D4C22×D12711yescharacteristic
266C2×D4C22×D12711yescharacteristic
267C22×C4—21no—
268C22×C4—1271no—
269C22×C4—1271no—
270C22×C4—2541no—
271C22×C4—2541no—
272C22×C4—2541no—
273SD16—2542no—
274D8—41no—
275D8—41no—
276M16—5081no—
277C2×C8—1271no—
278C2×C8—1271no—
279C2×C8—5081no—
280C2×C8C22×D12711yescharacteristic
281C2×C8C22×D12711yescharacteristic
282C4⋊C4—1271no—
283C4⋊C4—1271no—
284C4⋊C4—1271no—
285C4⋊C4—1271no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
288C4⋊C4—2541no—
289C4⋊C4—2541no—
290C4⋊C4C22×D12711yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—1271no—
293C22⋊C4—1271no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C42—2541no—
300C42—2541no—
301C42C22×D12711yescharacteristic
302C23—21no—
303C23—21no—
304C23—21no—
305C23—1271no—
306C23—1271no—
307Q8—1272no—
308Q8—2541no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—81no—
317D4—10161no—
318D4—10161no—
319D4—10161no—
320D4—5081no—
321D4—5082no—
322D4—5082no—
323D4—5082no—
324C2×C4—21no—
325C2×C4—81no—
326C2×C4—1271no—
327C2×C4—1271no—
328C2×C4—10161no—
329C2×C4—5082no—
330C2×C4—5082no—
331C2×C4—2541no—
332C2×C4—2541no—
333C2×C4—2541no—
334C2×C4—2541no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4D4×D12711yescharacteristic
340C2×C4D4×D12711yescharacteristic
341C2×C4C23×D12711yescharacteristic
342C8—21no—
343C8—21no—
344C8—2541no—
345C8—2541no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—2541no—
356C22—2541no—
357C22—2541no—
358C22—2541no—
359C22—2541no—
360C22—2541no—
361C22C2×D4×D12711yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—5081no—
366C4—5081no—
367C4—5081no—
368C4—2541no—
369C4—2541no—
370C4—2542no—
371C4#41×D12711yescharacteristic
372C4#1686125411yescharacteristic
373C2—81no—
374C2—81no—
375C2—81no—
376C2—5081no—
377C2—5081no—
378C2#2255052311yescharacteristic
379C2#2255067211yescharacteristic
380C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3309
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. #22550576C2semidirect
2. #22550546C2semidirect
3. #22550240C2semidirect
4. #22550199C2semidirect
5. #22550188C2semidirect
6. #22550184C2semidirect
7. #22550126C2semidirect
8. #22550109C2semidirect
9. #22550018C2semidirect
10. #22550018C2semidirect
11. #22549976C2semidirect
12. #22549946C2semidirect
13. #22549874C2semidirect
14. #22549869C2semidirect
15. #459×C127C2semidirect
191. C127#3309semidirect
200. #459D127semidirect

Maximal subgroups

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

subgroupindex
#225505762
#225505462
#225502402
#225501992
#225501882
#225501842
#225501262
#225501092
#225500182
#225499762
#225499462
#225498742
#225498692
#459×C1272
#3309127