Finite Group Databasebeta

← #30650626

Subgroups — #30650626

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#22550535C211yesmaximal, characteristic, complemented
2#22550233C211yesmaximal, characteristic, complemented
3#22550217C211yesmaximal, characteristic, complemented
4#22550194C211yesmaximal, characteristic, complemented
5#22550180C211yesmaximal, characteristic, complemented
6#22550110C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550091C211yesmaximal, characteristic, complemented
9#22550019C211yesmaximal, characteristic, complemented
10#22550014C211yesmaximal, characteristic, complemented
11C4×#14523236C211yesmaximal, characteristic, complemented
12D4×C127⋊C8C211yesmaximal, characteristic, complemented
13#22549870C211yesmaximal, characteristic, complemented
14C2×#16861132C211yesmaximal, characteristic, complemented
15#553×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C22×C127⋊C8C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23#16861197C2211yescharacteristic
24#16861195—21no—
25#16861188C2211yescharacteristic
26#16861187—21no—
27#16861183C2211yescharacteristic
28#16861181—21no—
29C4×Dic254C2211yescharacteristic
30(C22⋊D127)⋊C4—21no—
31#16861142C2211yescharacteristic
32#16861140C2211yescharacteristic
33#16861140C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35#16861132C2211yescharacteristic
36#16861132C2211yescharacteristic
37#16861132C2211yescharacteristic
38#16861132C2211yescharacteristic
39#16861132C2211yescharacteristic
40#16861132C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861129C2211yescharacteristic
43#16861129C2211yescharacteristic
44#16861129C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46C22⋊(C127⋊C8)C2211yescharacteristic
47C4×C127⋊C8C2211yescharacteristic
48#126×C127—21no—
49#125×C127C2211yescharacteristic
50#123×C127—21no—
51#121×C127—21no—
52#121×C127C2211yescharacteristic
53#121×C127C2211yescharacteristic
54C4×D4×C127—21no—
55C4×D4×C127C2211yescharacteristic
56C4×D4×C127C2211yescharacteristic
57#117×C127C2211yescharacteristic
58C2×C4⋊C4×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—41no—
61C2×Dic254—22no—
62C2×Dic254—21no—
63C2×Dic254—41no—
64C2×Dic254C2311yescharacteristic
65C2×Dic254C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C22×Dic127—21no—
70C2×C4×D127—21no—
71#14523236—22no—
72#14523236—41no—
73#14523233—41no—
74C2×C127⋊C8—41no—
75C2×C127⋊C8—41no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83C22⋊Dic127—21no—
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87C4×Dic127—21no—
88C2×D4×C127—21no—
89C2×D4×C127—21no—
90C2×D4×C127D411yescharacteristic
91C2×D4×C127C2311yescharacteristic
92C22×C4×C127—21no—
93C22×C4×C127—21no—
94C22×C4×C127C2311yescharacteristic
95C22×C4×C127C2311yescharacteristic
96C4⋊C4×C127—21no—
97C4⋊C4×C127—21no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127D411yescharacteristic
100C4⋊C4×C127C2311yescharacteristic
101C4⋊C4×C127C2311yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C22⋊C4×C127—21no—
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C22⋊C4×C127D411yescharacteristic
108C22⋊C4×C127D411yescharacteristic
109C42×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112Dic254—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—42no—
116Dic254#4112yes—
117D508—21no—
118D508—21no—
119C22⋊D127—42no—
120C22⋊D127—42no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—42no—
125C4×D127—42no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C127⋊C8—21no—
129C127⋊C8—21no—
130C23×C127—21no—
131C23×C127C2×D411yescharacteristic
132C23×C127C2×D411yescharacteristic
133D4×C127—22no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—42no—
137D4×C127—81no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—81no—
146C2×C4×C127—81no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2411yescharacteristic
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127#13512yes—
161C22×C127C22×D411yescharacteristic
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—41no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127C2×#4111yescharacteristic
173C4×C127H(5,𝔽2)11yescharacteristic
174Dic127—22no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—22no—
178C2×C127—41no—
179C2×C127—81no—
180C2×C127#54611yescharacteristic
181C2×C127C2×#13511yescharacteristic
182C2×C127#57711yescharacteristic
183D127—81no—
184#3301—1271nomaximal, Sylow
185C127#330111yescharacteristic, complemented, Sylow
186#553D12711yescharacteristic, complemented
187#548—1271no—
188#488—1271no—
189#485—1271no—
190#481—1271no—
191#479—1271no—
192#466—1271no—
193#465—1271no—
194#463—1271no—
195#449—1271no—
196#448—1271no—
197C4×SD16—1271no—
198C8×D4—1271no—
199#416—1271no—
200C2×#103—1271no—
201C2×#41—1271no—
202C22×Q8—1271no—
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C22×C8—1271no—
207C22×C8—1271no—
208#126—21no—
209#125C2×D12711yescharacteristic
210#124—1271no—
211#124—2541no—
212#123—21no—
213C22⋊Q8—1271no—
214C22⋊Q8—2541no—
215#121—21no—
216#121—1271no—
217#121—2541no—
218#121C2×D12711yescharacteristic
219#121C2×D12711yescharacteristic
220C4×Q8—1271no—
221C4×D4—21no—
222C4×D4—2541no—
223C4×D4C2×D12711yescharacteristic
224C4×D4C2×D12711yescharacteristic
225#117C2×D12711yescharacteristic
226C2×C4⋊C4C2×D12711yescharacteristic
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#103—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241C22⋊C8—1271no—
242C22⋊C8—1271no—
243C4×C8—1271no—
244#41—5081no—
245#41—2542no—
246C2×Q8—1271no—
247C2×Q8—1271no—
248C2×Q8—5081no—
249C2×Q8—2542no—
250C2×Q8—2541no—
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—21no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4C22×D12711yescharacteristic
263C22×C4C22×D12711yescharacteristic
264SD16—5081no—
265SD16—2542no—
266D8—5081no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—1271no—
271C2×C8—5081no—
272C2×C8—5081no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—1271no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4C22⋊D12711yescharacteristic
280C4⋊C4C22×D12711yescharacteristic
281C4⋊C4C22×D12711yescharacteristic
282C4⋊C4C22×D12711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4C22⋊D12711yescharacteristic
292C22⋊C4C22⋊D12711yescharacteristic
293C42—21no—
294C42—2541no—
295C42C22×D12711yescharacteristic
296C23—21no—
297C23—2541no—
298C23C2×C22⋊D12711yescharacteristic
299C23C2×C22⋊D12711yescharacteristic
300Q8—1272no—
301Q8—5082no—
302Q8—2541no—
303Q8—2541no—
304Q8—2541no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—5082no—
311D4—5082no—
312D4—2541no—
313D4—2541no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—5082no—
324C2×C4—5082no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4C2×C22⋊D12711yescharacteristic
329C2×C4C2×C22⋊D12711yescharacteristic
330C2×C4C2×C22⋊D12711yescharacteristic
331C2×C4C2×C22⋊D12711yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—2541no—
334C8—2541no—
335C8—2541no—
336C8—2541no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—5081no—
346C22—5081no—
347C22—5081no—
348C22#1686122912yes—
349C22C22×C22⋊D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5081no—
358C4—5081no—
359C4—2542no—
360C4#41×D12711yescharacteristic
361C4#1686125511yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—10161no—
366C2#2255051411yescharacteristic
367C2C2×#1686122911yescharacteristic
368C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#553×C127C2
socleC22×C127C22×D4
hypercenter#553D127
hypocenterC127#3301
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #22550535C2semidirect
2. #22550233C2semidirect
3. #22550217C2semidirect
4. #22550194C2semidirect
5. #22550180C2semidirect
6. #22550110C2semidirect
7. #22550103C2semidirect
8. #22550091C2semidirect
9. #22550019C2semidirect
10. #22550014C2semidirect
11. C4×#14523236C2semidirect
12. D4×C127⋊C8C2semidirect
13. #22549870C2semidirect
14. C2×#16861132C2semidirect
15. #553×C127C2semidirect
185. C127#3301semidirect
186. #553D127semidirect

Maximal subgroups

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

subgroupindex
#225505352
#225502332
#225502172
#225501942
#225501802
#225501102
#225501032
#225500912
#225500192
#225500142
C4×#145232362
D4×C127⋊C82
#225498702
C2×#168611322
#553×C1272
#3301127