Finite Group Databasebeta

← #30650585

Subgroups — #30650585

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550576C211yesmaximal, characteristic, complemented
2#22550521C211yesmaximal, characteristic, complemented
3#22550231C211yesmaximal, characteristic, complemented
4#22550213C211yesmaximal, characteristic, complemented
5#22550203C211yesmaximal, characteristic, complemented
6#22550176C211yesmaximal, characteristic, complemented
7#22550111C211yesmaximal, characteristic, complemented
8#22550104C211yesmaximal, characteristic, complemented
9#22550091C211yesmaximal, characteristic, complemented
10#22550018C211yesmaximal, characteristic, complemented
11#22550008C211yesmaximal, characteristic, complemented
12C8×D508C211yesmaximal, characteristic, complemented
13#22549869C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15C4×D8×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18C2×#14523236C2211yescharacteristic, complemented
19C2×#14523233C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic, complemented
22#16861202—21no—
23#16861201C2211yescharacteristic, complemented
24#16861197C2211yescharacteristic
25#16861193—21no—
26#16861192—21no—
27#16861189C2211yescharacteristic
28#16861184C2211yescharacteristic
29#16861183—21no—
30#16861182—21no—
31#16861180C2211yescharacteristic
32#16861180C2211yescharacteristic, complemented
33#16861179—21no—
34C4×D508C2211yescharacteristic
35C4×C22⋊D127—21no—
36#16861167C2211yescharacteristic, complemented
37C4⋊C4×D127C2211yescharacteristic
38C2×(C2×D127)⋊C4—21no—
39#16861142C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41C4⋊(C127⋊C8)C2211yescharacteristic, complemented
42#16861134C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861132C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861129C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52C2×D8×C127C2211yescharacteristic
53C4×D4×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic, complemented
55#107×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic, complemented
58C4×C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523247—22no—
61#14523247—41no—
62C2×Dic254C2311yescharacteristic
63C2×D508—21no—
64C2×D508D411yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70D4×D127—22no—
71D4×D127—41no—
72C22×Dic127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2311yescharacteristic
75C2×C4×D127C2311yescharacteristic
76#14523236—41no—
77#14523233—41no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—41no—
81C8×D127—41no—
82C4⋊Dic127—21no—
83C4⋊Dic127D411yescharacteristic
84C4⋊Dic127C2311yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4D411yescharacteristic
97(C2×D127)⋊C4D411yescharacteristic
98C4×Dic127—21no—
99C2×D4×C127C2311yescharacteristic
100C2×D4×C127C2311yescharacteristic
101C22×C4×C127—21no—
102C22×C4×C127—21no—
103D8×C127—22no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C22⋊C4×C127—21no—
109C22⋊C4×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112C22×D127—42no—
113C22×D127—42no—
114C22×D127—81no—
115C22×D127C2×D411yescharacteristic
116C22×D127C2×D411yescharacteristic
117Dic254—21no—
118Dic254—21no—
119D508—22no—
120D508—21no—
121D508—21no—
122D508—81no—
123C22⋊D127—42no—
124C22⋊D127—42no—
125C22⋊D127—42no—
126C22⋊D127—81no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—42no—
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C4×D127—21no—
134C4×D127—21no—
135C4×D127—21no—
136C4×D127—21no—
137C4×D127—81no—
138C127⋊C8—21no—
139C127⋊C8—21no—
140C23×C127—21no—
141C23×C127—21no—
142D4×C127—21no—
143D4×C127—21no—
144D4×C127—21no—
145D4×C127#4112yes—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—81no—
149C2×C4×C127—81no—
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2411yescharacteristic
153C8×C127—21no—
154C8×C127—21no—
155C22×C127—22no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127C22×D411yescharacteristic
162C2×D127—21no—
163C2×D127—21no—
164C2×D127—21no—
165C2×D127—21no—
166C2×D127—21no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—84no—
171C2×D127#13512yes—
172C4×C127—21no—
173C4×C127—21no—
174C4×C127—41no—
175C4×C127—41no—
176C4×C127C2×#4111yescharacteristic
177C4×C127H(5,𝔽2)11yescharacteristic
178Dic127—21no—
179Dic127—21no—
180Dic127—41no—
181Dic127—41no—
182Dic127—41no—
183C2×C127—42no—
184C2×C127—81no—
185C2×C127#54611yescharacteristic
186C2×C127C2×#13511yescharacteristic
187C2×C127#57611yescharacteristic
188D127—22no—
189D127—41no—
190D127—81no—
191#3297—1271nomaximal, Sylow
192C127#329711yescharacteristic, complemented, Sylow
193#553—1271no—
194#546—1271no—
195#488—1271no—
196#484—1271no—
197#482—1271no—
198#478—1271no—
199#466—1271no—
200#465—1271no—
201#463—1271no—
202#449—1271no—
203#447—1271no—
204C4×D8D12711yescharacteristic, complemented
205C8×D4—1271no—
206#416—1271no—
207C2×#102—1271no—
208C2×#41—1271no—
209C22×D4—1271no—
210C2×SD16—1271no—
211C2×D8—1271no—
212C2×D8C2×D12711yescharacteristic
213C22×C8—1271no—
214C22×C8—1271no—
215#126—2541no—
216#125—1271no—
217#124—1271no—
218#123—2541no—
219#123—2541no—
220C22⋊Q8—1271no—
221#121—1271no—
222#121—1271no—
223#121—1271no—
224#121—2541no—
225#121—2541no—
226#120—2541no—
227C4×D4—1271no—
228C4×D4—2541no—
229C4×D4C2×D12711yescharacteristic
230C4×D4C2×D12711yescharacteristic, complemented
231#117—1271no—
232C2×C4⋊C4—1271no—
233C2×C22⋊C4—2541no—
234#107—1271no—
235#107C2×D12711yescharacteristic, complemented
236#106—1271no—
237C4⋊C8—1271no—
238#103—1271no—
239#103—1271no—
240#103—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102C2×D12711yescharacteristic
247#102C2×D12711yescharacteristic, complemented
248C22⋊C8—1271no—
249C22⋊C8—1271no—
250C4×C8C2×D12711yescharacteristic, complemented
251C24—2541no—
252#41—5081no—
253#41—2542no—
254C2×Q8—1271no—
255C2×D4—1271no—
256C2×D4—1271no—
257C2×D4—5081no—
258C2×D4—2541no—
259C2×D4—2541no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2542no—
264C2×D4C22×D12711yescharacteristic
265C2×D4C22×D12711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1271no—
269C22×C4—1271no—
270C22×C4—2541no—
271C22×C4—2541no—
272SD16—5081no—
273D8—22no—
274D8—5081no—
275C2×C8—1271no—
276C2×C8—1271no—
277C2×C8—5081no—
278C2×C8—5081no—
279C2×C8C22×D12711yescharacteristic
280C2×C8C22×D12711yescharacteristic
281C4⋊C4—1271no—
282C4⋊C4—1271no—
283C4⋊C4—1271no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
288C4⋊C4C22×D12711yescharacteristic
289C4⋊C4C22×D12711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—1271no—
293C22⋊C4—1271no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C22⋊C4—2541no—
300C22⋊C4—2541no—
301C42—2541no—
302C42C22×D12711yescharacteristic
303C23—21no—
304C23—21no—
305C23—1271no—
306C23—1271no—
307C23—10161no—
308C23—5082no—
309C23—5082no—
310C23—2541no—
311Q8—2541no—
312Q8—2541no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—10161no—
317D4—10161no—
318D4—5082no—
319D4—5082no—
320D4—5082no—
321D4—2542no—
322D4—2541no—
323D4—2541no—
324D4#1452324812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—1271no—
330C2×C4—1271no—
331C2×C4—10161no—
332C2×C4—5082no—
333C2×C4—2541no—
334C2×C4—2541no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4D4×D12711yescharacteristic
341C2×C4D4×D12711yescharacteristic
342C2×C4C23×D12711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2541no—
346C8—2541no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—1272no—
354C22—10164no—
355C22—5081no—
356C22—5081no—
357C22—5081no—
358C22—2541no—
359C22—2541no—
360C22—2541no—
361C22—2541no—
362C22—2541no—
363C22C2×D4×D12711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—5081no—
369C4—5081no—
370C4—5081no—
371C4—2541no—
372C4—2541no—
373C4C2×#1452324811yescharacteristic
374C4#1686125511yescharacteristic
375C2—42no—
376C2—81no—
377C2—10161no—
378C2—5081no—
379C2—2542no—
380C2#2255052211yescharacteristic
381C2#135×D12711yescharacteristic
382C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×D8×C127C2
socleC22×C127C22×D4
hypercenterC4×D8D127
hypocenterC127#3297
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #22550576C2semidirect
2. #22550521C2semidirect
3. #22550231C2semidirect
4. #22550213C2semidirect
5. #22550203C2semidirect
6. #22550176C2semidirect
7. #22550111C2semidirect
8. #22550104C2semidirect
9. #22550091C2semidirect
10. #22550018C2semidirect
11. #22550008C2semidirect
12. C8×D508C2semidirect
13. #22549869C2semidirect
14. #102×D127C2semidirect
15. C4×D8×C127C2semidirect
18. C2×#14523236C22semidirect
21. C2×C8×D127C22semidirect
23. #16861201C22semidirect
32. #16861180C22semidirect
36. #16861167C22semidirect
39. #16861142C22semidirect
40. #16861140C22semidirect
41. C4⋊(C127⋊C8)C22semidirect
42. #16861134C22semidirect
43. #16861134C22semidirect
44. #16861132C22semidirect
51. (C2×D127)⋊C8C22semidirect
54. C4×D4×C127C22semidirect
55. #107×C127C22semidirect
57. #102×C127C22semidirect
58. C4×C8×C127C22semidirect
192. C127#3297semidirect
204. C4×D8D127semidirect
230. C4×D4C2×D127semidirect
235. #107C2×D127semidirect
247. #102C2×D127semidirect
250. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505762
#225505212
#225502312
#225502132
#225502032
#225501762
#225501112
#225501042
#225500912
#225500182
#225500082
C8×D5082
#225498692
#102×D1272
C4×D8×C1272
#3297127