Finite Group Databasebeta

← #30650715

Subgroups — #30650715

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550535C211yesmaximal, characteristic, complemented
2#22550238C211yesmaximal, characteristic, complemented
3#22550211C211yesmaximal, characteristic, complemented
4#22550189C211yesmaximal, characteristic, complemented
5#22550168C211yesmaximal, characteristic, complemented
6#22550120C211yesmaximal, characteristic, complemented
7#22550110C211yesmaximal, characteristic, complemented
8#22550091C211yesmaximal, characteristic, complemented
9#22550019C211yesmaximal, characteristic, complemented
10#22550014C211yesmaximal, characteristic, complemented
11#22549976C211yesmaximal, characteristic, complemented
12#22549948C211yesmaximal, characteristic, complemented
13#22549875C211yesmaximal, characteristic, complemented
14C2×#16861132C211yesmaximal, characteristic, complemented
15#547×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C2×#14523232C2211yescharacteristic
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#16861142C2211yescharacteristic
33#16861142C2211yescharacteristic
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
47#16861120C2211yescharacteristic
48C2×#41×C127—21no—
49C4⋊Q8×C127C2211yescharacteristic
50#123×C127—21no—
51#123×C127—21no—
52C22⋊Q8×C127—21no—
53#121×C127C2211yescharacteristic
54#121×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56C4×D4×C127C2211yescharacteristic
57C2×C4⋊C4×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—
74#14523232—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—
88#41×C127—44no—
89C2×Q8×C127—21no—
90C2×D4×C127—21no—
91C2×D4×C127—21no—
92C2×D4×C127D411yescharacteristic
93C2×D4×C127C2311yescharacteristic
94C22×C4×C127—21no—
95C22×C4×C127—21no—
96C22×C4×C127—21no—
97C22×C4×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C4⋊C4×C127—21no—
100C4⋊C4×C127—21no—
101C4⋊C4×C127—21no—
102C4⋊C4×C127D411yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C22⋊C4×C127—21no—
108C22⋊C4×C127—21no—
109C22⋊C4×C127D411yescharacteristic
110C22⋊C4×C127D411yescharacteristic
111C42×C127C2311yescharacteristic
112C22×D127—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—21no—
116Dic254—42no—
117Dic254#4112yescomplemented
118D508—21no—
119D508—21no—
120C22⋊D127—42no—
121C22⋊D127—42no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—42no—
126C4×D127—42no—
127C127⋊C8—21no—
128C127⋊C8—21no—
129C127⋊C8—21no—
130C127⋊C8—21no—
131C23×C127—21no—
132C23×C127C2×D411yescharacteristic
133C23×C127C2×D411yescharacteristic
134Q8×C127—42no—
135D4×C127—22no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—42no—
139D4×C127—42no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—42no—
148C2×C4×C127—42no—
149C2×C4×C127—42no—
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2×D411yescharacteristic
152C2×C4×C127C2×D411yescharacteristic
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2411yescharacteristic
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127AGL(1,ℤ/8)12yes—
164C22×C127C22×D411yescharacteristic
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C4×C127—21no—
169C4×C127—21no—
170C4×C127—21no—
171C4×C127—21no—
172C4×C127—41no—
173C4×C127—41no—
174C4×C127—41no—
175C4×C127C2×#4111yescharacteristic
176C4×C127H(5,𝔽2)11yescharacteristic
177Dic127—22no—
178Dic127—41no—
179Dic127—41no—
180C2×C127—22no—
181C2×C127—41no—
182C2×C127—81no—
183C2×C127#54611yescharacteristic
184C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C127#57811yescharacteristic
186D127—81no—
187#3308—1271nomaximal, Sylow
188C127#330811yescharacteristic, complemented, Sylow
189#548—1271no—
190#547D12711yescharacteristic, complemented
191#489—1271no—
192#484—1271no—
193#480—1271no—
194#477—1271no—
195#468—1271no—
196#466—1271no—
197#463—1271no—
198#449—1271no—
199#448—1271no—
200#440—1271no—
201#435—1271no—
202#417—1271no—
203C2×#103—1271no—
204C2×#41—21no—
205C2×#41—1271no—
206C22×Q8—1271no—
207C2×SD16—1271no—
208C2×SD16—1271no—
209C2×D8—1271no—
210C2×M16—1271no—
211C22×C8—1271no—
212C4⋊Q8C2×D12711yescharacteristic
213#124—1271no—
214#124—2541no—
215#123—21no—
216#123—21no—
217C22⋊Q8—21no—
218C22⋊Q8—1271no—
219C22⋊Q8—2541no—
220#121—1271no—
221#121—2541no—
222#121C2×D12711yescharacteristic
223#121C2×D12711yescharacteristic
224C4×Q8—1271no—
225C4×D4—2541no—
226C4×D4C2×D12711yescharacteristic
227C4×D4C2×D12711yescharacteristic
228C2×C4⋊C4C2×D12711yescharacteristic
229C2×C4⋊C4C2×D12711yescharacteristic
230#107—1271no—
231#107—1271no—
232#107—1271no—
233C4⋊C8—1271no—
234#103—1271no—
235#103—1271no—
236#103—1271no—
237#103—1271no—
238#103—1271no—
239#103—1271no—
240#102—1271no—
241#102—1271no—
242#102—1271no—
243#102—1271no—
244C22⋊C8—1271no—
245C22⋊C8—1271no—
246#97—1271no—
247#41—44no—
248#41—5081no—
249#41—2542no—
250C2×Q8—21no—
251C2×Q8—1271no—
252C2×Q8—1271no—
253C2×Q8—5081no—
254C2×Q8—2542no—
255C2×Q8—2541no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—1271no—
259C2×D4—2541no—
260C2×D4—2541no—
261C2×D4C22⋊D12711yescharacteristic
262C2×D4C22×D12711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—2541no—
267C22×C4—2541no—
268C22×C4C22×D12711yescharacteristic
269C22×C4C22×D12711yescharacteristic
270SD16—5081no—
271SD16—2542no—
272D8—5081no—
273M16—5081no—
274C2×C8—1271no—
275C2×C8—1271no—
276C2×C8—1271no—
277C2×C8—1271no—
278C2×C8—5081no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—1271no—
283C4⋊C4—2541no—
284C4⋊C4—2541no—
285C4⋊C4C22⋊D12711yescharacteristic
286C4⋊C4C22×D12711yescharacteristic
287C4⋊C4C22×D12711yescharacteristic
288C4⋊C4C22×D12711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4C22⋊D12711yescharacteristic
297C22⋊C4C22⋊D12711yescharacteristic
298C42—2541no—
299C42C22×D12711yescharacteristic
300C23—21no—
301C23—2541no—
302C23C2×C22⋊D12711yescharacteristic
303C23C2×C22⋊D12711yescharacteristic
304Q8—42no—
305Q8—1272no—
306Q8—5082no—
307Q8—2541no—
308Q8—2541no—
309Q8—2541no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—5082no—
316D4—5082no—
317D4—2541no—
318D4—2541no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—5082no—
330C2×C4—5082no—
331C2×C4—2541no—
332C2×C4—2541no—
333C2×C4—2541no—
334C2×C4C2×C22⋊D12711yescharacteristic
335C2×C4C2×C22⋊D12711yescharacteristic
336C2×C4C2×C22⋊D12711yescharacteristic
337C2×C4C2×C22⋊D12711yescharacteristic
338C2×C4C23×D12711yescharacteristic
339C8—2541no—
340C8—2541no—
341C8—2541no—
342C8—2541no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—5081no—
352C22—5081no—
353C22—5081no—
354C22#1686123512yes—
355C22C22×C22⋊D12711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—5081no—
364C4—5081no—
365C4—2542no—
366C4#41×D12711yescharacteristic
367C4#1686125511yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—10161no—
372C2#2255051411yescharacteristic
373C2C2×#1686123511yescharacteristic
374C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#547×C127C2
socleC22×C127C22×D4
hypercenter#547D127
hypocenterC127#3308
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #22550535C2semidirect
2. #22550238C2semidirect
3. #22550211C2semidirect
4. #22550189C2semidirect
5. #22550168C2semidirect
6. #22550120C2semidirect
7. #22550110C2semidirect
8. #22550091C2semidirect
9. #22550019C2semidirect
10. #22550014C2semidirect
11. #22549976C2semidirect
12. #22549948C2semidirect
13. #22549875C2semidirect
14. C2×#16861132C2semidirect
15. #547×C127C2semidirect
117. Dic254#41semidirect
188. C127#3308semidirect
190. #547D127semidirect

Maximal subgroups

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

subgroupindex
#225505352
#225502382
#225502112
#225501892
#225501682
#225501202
#225501102
#225500912
#225500192
#225500142
#225499762
#225499482
#225498752
C2×#168611322
#547×C1272
#3308127