Finite Group Databasebeta

← #30650651

Subgroups — #30650651

Order 16256.

398 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#22550590C211yesmaximal, characteristic, complemented
2#22550525C211yesmaximal, characteristic, complemented
3#22550225C211yesmaximal, characteristic, complemented
4#22550218C211yesmaximal, characteristic, complemented
5#22550216C211yesmaximal, characteristic, complemented
6#22550164C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550102C211yesmaximal, characteristic, complemented
9#22550079C211yesmaximal, characteristic, complemented
10#22550018C211yesmaximal, characteristic, complemented
11D8×Dic127C211yesmaximal, characteristic, complemented
12(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
13#22549872C211yesmaximal, characteristic, complemented
14#22549869C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20#16861208C2211yescharacteristic
21#16861207C2211yescharacteristic
22#16861196—21no—
23#16861195—21no—
24#16861192—21no—
25#16861190—21no—
26#16861188C2211yescharacteristic
27#16861185C2211yescharacteristic
28#16861181—21no—
29#16861181—21no—
30#16861180C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32D4×Dic127C2211yescharacteristic
33C4×C22⋊D127C2211yescharacteristic
34#16861168C2211yescharacteristic
35#16861167C2211yescharacteristic
36#16861165—21no—
37C2×Dic127⋊C4—21no—
38#16861143C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861133C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×#41×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×D8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—41no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127D411yescharacteristic
66C22×Dic127—21no—
67C22×Dic127—21no—
68C22×Dic127—21no—
69C2×C4×D127C2311yescharacteristic
70#14523236—41no—
71C2×C127⋊C8—41no—
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C8×D127—41no—
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4D411yescharacteristic
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127D411yescharacteristic
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4D411yescharacteristic
93C4×Dic127—21no—
94C4×Dic127—21no—
95C4×Dic127C2311yescharacteristic
96#41×C127—22no—
97#41×C127—41no—
98C2×Q8×C127C2311yescharacteristic
99C2×D4×C127—21no—
100C2×D4×C127—21no—
101C2×D4×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104C22×C4×C127C2311yescharacteristic
105SD16×C127—41no—
106D8×C127—22no—
107C2×C8×C127C2311yescharacteristic
108C2×C8×C127C2311yescharacteristic
109C4⋊C4×C127C2311yescharacteristic
110C22⋊C4×C127—21no—
111C22×D127C2×D411yescharacteristic
112Dic254—21no—
113Dic254—21no—
114Dic254—81no—
115C22⋊D127—22no—
116C22⋊D127—42no—
117C22⋊D127—81no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—42no—
123C2×Dic127—42no—
124C2×Dic127—42no—
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C4×D127—21no—
129C4×D127—21no—
130C127⋊C8—21no—
131C127⋊C8—21no—
132C23×C127—21no—
133C23×C127—21no—
134C23×C127C2×D411yescharacteristic
135Q8×C127—21no—
136Q8×C127—21no—
137D4×C127—21no—
138D4×C127—21no—
139D4×C127—21no—
140D4×C127—42no—
141D4×C127—81no—
142D4×C127#4112yes—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—42no—
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—21no—
153C22×C127—22no—
154C22×C127—21no—
155C22×C127—21no—
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—
165C4×C127—21no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127C2×#4111yescharacteristic
169C4×C127H(5,𝔽2)11yescharacteristic
170Dic127—21no—
171Dic127—21no—
172Dic127—41no—
173Dic127—41no—
174Dic127—41no—
175Dic127—41no—
176Dic127#13512yes—
177C2×C127—41no—
178C2×C127—42no—
179C2×C127—81no—
180C2×C127#54611yescharacteristic
181C2×C127C2×#13511yescharacteristic
182C2×C127#57711yescharacteristic
183D127—41no—
184#3303—1271nomaximal, Sylow
185C127#330311yescharacteristic, complemented, Sylow
186#555—1271no—
187#547—1271no—
188#487—1271no—
189#485—1271no—
190#485—1271no—
191#476—1271no—
192#465—1271no—
193#465—1271no—
194#461—1271no—
195#449—1271no—
196#449D12711yescharacteristic, complemented
197C4×D8—1271no—
198C8×D4—1271no—
199#416—1271no—
200#416—1271no—
201C2×#41—1271no—
202C2×#41C2×D12711yescharacteristic
203C2×SD16—1271no—
204C2×SD16C2×D12711yescharacteristic
205C2×D8C2×D12711yescharacteristic
206C22×C8—1271no—
207C22×C8—1271no—
208C4⋊Q8—1271no—
209C4⋊Q8—1271no—
210#124—2541no—
211#124—2541no—
212#123—2541no—
213#123—2541no—
214C22⋊Q8—1271no—
215C22⋊Q8—1271no—
216#121—1271no—
217#121—2541no—
218#121—2541no—
219#121C2×D12711yescharacteristic
220C4×D4—1271no—
221C4×D4—1271no—
222C4×D4—1271no—
223#117—1271no—
224#117—1271no—
225#117—2541no—
226C2×C4⋊C4—2541no—
227#107—1271no—
228#106—1271no—
229#106—1271no—
230C4⋊C8—1271no—
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#103C2×D12711yescharacteristic
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102C2×D12711yescharacteristic
241C22⋊C8—1271no—
242C22⋊C8C2×D12711yescharacteristic
243C4×C8—1271no—
244#41—22no—
245#41—41no—
246#41—5081no—
247#41—2542no—
248C2×Q8—1271no—
249C2×Q8—2541no—
250C2×Q8C22×D12711yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—1271no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4C22×D12711yescharacteristic
257C2×D4C22×D12711yescharacteristic
258C22×C4—21no—
259C22×C4—1271no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4C22×D12711yescharacteristic
264SD16—41no—
265SD16—5081no—
266D8—22no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8—5081no—
270C2×C8—5081no—
271C2×C8C22×D12711yescharacteristic
272C2×C8C22×D12711yescharacteristic
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—1271no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4C22×D12711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—1271no—
285C22⋊C4—1271no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C42—1271no—
294C42—2541no—
295C42—2541no—
296C23—21no—
297C23—21no—
298C23—1271no—
299C23D4×D12711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—10161no—
303Q8—2541no—
304Q8—2541no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—10161no—
311D4—5082no—
312D4—2542no—
313D4#1452324712yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—1271no—
320C2×C4—1271no—
321C2×C4—1271no—
322C2×C4—5082no—
323C2×C4—5082no—
324C2×C4—5082no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4D4×D12711yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—21no—
334C8—21no—
335C8—2541no—
336C8—2541no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—2541no—
347C22—2541no—
348C22—2541no—
349C22C2×D4×D12711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—1272no—
354C4—5081no—
355C4—5081no—
356C4—5081no—
357C4—5081no—
358C4—2541no—
359C4—2541no—
360C4C2×#1452324711yescharacteristic
361C4#1686125411yescharacteristic
362C2—41no—
363C2—42no—
364C2—81no—
365C2—5081no—
366C2#2255051311yescharacteristic
367C2#135×D12711yescharacteristic
368C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3303
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #22550590C2semidirect
2. #22550525C2semidirect
3. #22550225C2semidirect
4. #22550218C2semidirect
5. #22550216C2semidirect
6. #22550164C2semidirect
7. #22550103C2semidirect
8. #22550102C2semidirect
9. #22550079C2semidirect
10. #22550018C2semidirect
11. D8×Dic127C2semidirect
12. (C22⋊D127)⋊C8C2semidirect
13. #22549872C2semidirect
14. #22549869C2semidirect
15. #449×C127C2semidirect
185. C127#3303semidirect
196. #449D127semidirect

Maximal subgroups

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

subgroupindex
#225505902
#225505252
#225502252
#225502182
#225502162
#225501642
#225501032
#225501022
#225500792
#225500182
D8×Dic1272
(C22⋊D127)⋊C82
#225498722
#225498692
#449×C1272
#3303127