Finite Group Databasebeta

← #30651432

Subgroups — #30651432

Order 16256.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550585C211yesmaximal, characteristic, complemented
2#22550546C211yesmaximal, characteristic, complemented
3#22550240C211yesmaximal, characteristic, complemented
4#22550239C211yesmaximal, characteristic, complemented
5#22550221C211yesmaximal, characteristic, complemented
6#22550177C211yesmaximal, characteristic, complemented
7#22550174C211yesmaximal, characteristic, complemented
8#22550164C211yesmaximal, characteristic, complemented
9#22550069C211yesmaximal, characteristic, complemented
10#22550069C211yesmaximal, characteristic, complemented
11#22550005C211yesmaximal, characteristic, complemented
12#22549988C211yesmaximal, characteristic, complemented
13#22549986C211yesmaximal, characteristic, complemented
14D8×Dic127C211yesmaximal, characteristic, complemented
15#459×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×#14523233C2211yescharacteristic
18#16861208C2211yescharacteristic
19#16861205C2211yescharacteristic
20#16861205C2211yescharacteristic
21#16861201C2211yescharacteristic
22#16861200C2211yescharacteristic
23#16861188—21no—
24#16861186—21no—
25#16861186—21no—
26#16861181—21no—
27#16861181—21no—
28#16861180—21no—
29C4×Dic254C2211yescharacteristic
30C4×D508C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32D4×Dic127C2211yescharacteristic
33D4×Dic127C2211yescharacteristic
34D4×Dic127C2211yescharacteristic
35C2×Dic127⋊C4—21no—
36#16861143C2211yescharacteristic
37#16861141C2211yescharacteristic
38#16861140C2211yescharacteristic
39Dic127⋊C8C2211yescharacteristic
40Dic127⋊C8C2211yescharacteristic
41#16861131C2211yescharacteristic
42#16861131C2211yescharacteristic
43#16861130C2211yescharacteristic
44#16861130C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49#16861120C2211yescharacteristic
50#16861119C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C2×D8×C127C2211yescharacteristic
54#127×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—44no—
60C2×Dic254—21no—
61C2×Dic254D411yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C22×Dic127—21no—
68C22×Dic127—21no—
69C22×Dic127—21no—
70C22×Dic127—21no—
71C2×C4×D127—21no—
72#14523233—22no—
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C4⋊Dic127—21no—
76C4⋊Dic127D411yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4D411yescharacteristic
85Dic127⋊C4D411yescharacteristic
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90(C2×D127)⋊C4—21no—
91C4×Dic127C2311yescharacteristic
92C4×Dic127C2311yescharacteristic
93C2×D4×C127—21no—
94C2×D4×C127—21no—
95C2×D4×C127C2311yescharacteristic
96C2×D4×C127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C22×C4×C127—21no—
99D8×C127—22no—
100D8×C127—22no—
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C22⋊C4×C127—21no—
105C42×C127C2311yescharacteristic
106C22×D127—21no—
107Dic254—22no—
108Dic254—81no—
109D508—21no—
110D508—21no—
111C22⋊D127—42no—
112C22⋊D127—42no—
113C22⋊D127—81no—
114C22⋊D127—81no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—42no—
119C2×Dic127—42no—
120C2×Dic127—42no—
121C2×Dic127—81no—
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C4×D127—42no—
127C127⋊C8—21no—
128C127⋊C8—21no—
129C23×C127—21no—
130C23×C127—21no—
131C23×C127—21no—
132D4×C127—21no—
133D4×C127—21no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—42no—
138D4×C127—81no—
139D4×C127#4112yescomplemented
140C2×C4×C127—21no—
141C2×C4×C127—81no—
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2411yescharacteristic
145C8×C127—21no—
146C8×C127—21no—
147C22×C127—22no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127C22×D411yescharacteristic
157C2×D127—41no—
158C2×D127—41no—
159C2×D127—41no—
160C4×C127—21no—
161C4×C127—21no—
162C4×C127—41no—
163C4×C127C2×#4111yescharacteristic
164C4×C1272−1+411yescharacteristic
165Dic127—21no—
166Dic127—21no—
167Dic127—21no—
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171Dic127AGL(1,ℤ/8)12yes—
172C2×C127—42no—
173C2×C127—81no—
174C2×C127—81no—
175C2×C127#54811yescharacteristic
176C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C127#57611yescharacteristic
178D127—81no—
179#3366—1271nomaximal, Sylow
180C127#336611yescharacteristic, complemented, Sylow
181#554—1271no—
182#550—1271no—
183#490—1271no—
184#490—1271no—
185#486—1271no—
186#478—1271no—
187#478—1271no—
188#476—1271no—
189#459—1271no—
190#459—1271no—
191#459D12711yescharacteristic, complemented
192#446—1271no—
193#442—1271no—
194#442—1271no—
195C4×D8—1271no—
196C2×#41—2541no—
197C2×D8—1271no—
198C2×D8C2×D12711yescharacteristic
199C2×D8C2×D12711yescharacteristic
200C4⋊Q8—1271no—
201#127—1271no—
202#127—1271no—
203#127C2×D12711yescharacteristic
204#125—1271no—
205#125—1271no—
206C22⋊Q8—2541no—
207C22⋊Q8—2541no—
208C22⋊Q8—2541no—
209#121—2541no—
210#121—2541no—
211#121—2541no—
212C4×Q8—1271no—
213C4×D4—1271no—
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4—1271no—
217C4×D4—1271no—
218C4×D4C2×D12711yescharacteristic
219C2×C4⋊C4—2541no—
220#107—1271no—
221#106—1271no—
222#106—1271no—
223C4⋊C8—1271no—
224C4⋊C8—1271no—
225C4⋊C8C2×D12711yescharacteristic
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102—1271no—
230#102—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102C2×D12711yescharacteristic
235#102C2×D12711yescharacteristic
236#97—1271no—
237#97—1271no—
238C4×C8—1271no—
239#41—5084no—
240C2×Q8—1271no—
241C2×Q8—2541no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—1271no—
245C2×D4—2541no—
246C2×D4—2541no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4C22×D12711yescharacteristic
250C2×D4C22×D12711yescharacteristic
251C2×D4C22×D12711yescharacteristic
252C22×C4—21no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4—2541no—
256C22×C4—2541no—
257C22×C4—2541no—
258D8—22no—
259D8—22no—
260D8—2542no—
261C2×C8—1271no—
262C2×C8—1271no—
263C2×C8C22×D12711yescharacteristic
264C2×C8C22×D12711yescharacteristic
265C4⋊C4—1271no—
266C4⋊C4—1271no—
267C4⋊C4—1271no—
268C4⋊C4—1271no—
269C4⋊C4—1271no—
270C4⋊C4—1271no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C42—1271no—
284C42—1271no—
285C42C22×D12711yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—2541no—
290Q8—10161no—
291Q8—2542no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—81no—
299D4—10161no—
300D4—10161no—
301D4—5082no—
302D4—5082no—
303D4—2541no—
304D4—2541no—
305D4#1452324712yescomplemented
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—1271no—
309C2×C4—1271no—
310C2×C4—1271no—
311C2×C4—1271no—
312C2×C4—10161no—
313C2×C4—5082no—
314C2×C4—5082no—
315C2×C4—5082no—
316C2×C4—5082no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4D4×D12711yescharacteristic
321C2×C4D4×D12711yescharacteristic
322C2×C4C23×D12711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2541no—
326C8—2541no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—5081no—
337C22—5081no—
338C22—5081no—
339C22C2×D4×D12711yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—1272no—
344C4—5081no—
345C4—5081no—
346C4—5081no—
347C4—2541no—
348C4—2541no—
349C4—2541no—
350C4C2×#1452324711yescharacteristic
351C4#1686125611yescharacteristic
352C2—42no—
353C2—81no—
354C2—81no—
355C2—10161no—
356C2#2255053211yescharacteristic
357C2AGL(1,ℤ/8)×D12711yescharacteristic
358C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3366
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. #22550585C2semidirect
2. #22550546C2semidirect
3. #22550240C2semidirect
4. #22550239C2semidirect
5. #22550221C2semidirect
6. #22550177C2semidirect
7. #22550174C2semidirect
8. #22550164C2semidirect
9. #22550069C2semidirect
10. #22550069C2semidirect
11. #22550005C2semidirect
12. #22549988C2semidirect
13. #22549986C2semidirect
14. D8×Dic127C2semidirect
15. #459×C127C2semidirect
139. D4×C127#41semidirect
180. C127#3366semidirect
191. #459D127semidirect
305. D4#14523247semidirect

Maximal subgroups

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

subgroupindex
#225505852
#225505462
#225502402
#225502392
#225502212
#225501772
#225501742
#225501642
#225500692
#225500052
#225499882
#225499862
D8×Dic1272
#459×C1272
#3366127