Finite Group Databasebeta

← #30651320

Subgroups — #30651320

Order 16256.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550590C211yesmaximal, characteristic, complemented
2#22550526C211yesmaximal, characteristic, complemented
3#22550273C211yesmaximal, characteristic, complemented
4#22550198C211yesmaximal, characteristic, complemented
5#22550197C211yesmaximal, characteristic, complemented
6#22550193C211yesmaximal, characteristic, complemented
7#22550192C211yesmaximal, characteristic, complemented
8#22550164C211yesmaximal, characteristic, complemented
9#22550154C211yesmaximal, characteristic, complemented
10#22550126C211yesmaximal, characteristic, complemented
11D8×Dic127C211yesmaximal, characteristic, complemented
12#22549952C211yesmaximal, characteristic, complemented
13#22549918C211yesmaximal, characteristic, complemented
14#22549915C211yesmaximal, characteristic, complemented
15#469×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×#14523231C2211yescharacteristic
18#16861208C2211yescharacteristic
19#16861208C2211yescharacteristic
20#16861196—21no—
21#16861195—21no—
22#16861191—21no—
23#16861190—21no—
24#16861186—21no—
25#16861181—21no—
26#16861180C2211yescharacteristic
27#16861180C2211yescharacteristic
28D4×Dic127C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30C4×C22⋊D127C2211yescharacteristic
31#16861168C2211yescharacteristic
32#16861167C2211yescharacteristic
33#16861165—21no—
34C2×C4⋊Dic127C2211yescharacteristic
35C4⋊C4×D127C2211yescharacteristic
36#16861143C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861141C2211yescharacteristic
41#16861140C2211yescharacteristic
42#16861140C2211yescharacteristic
43#16861140C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C2×M16×C127C2211yescharacteristic
54#121×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#106×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127D411yescharacteristic
65C22×Dic127—21no—
66C22×Dic127—21no—
67C22×Dic127—21no—
68C2×C4×D127—21no—
69C2×C4×D127C2311yescharacteristic
70#14523231—22no—
71C2×C127⋊C8C2311yescharacteristic
72C2×C127⋊C8C2311yescharacteristic
73C4⋊Dic127—21no—
74C4⋊Dic127C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4D411yescharacteristic
82C22⋊Dic127—21no—
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127D411yescharacteristic
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4D411yescharacteristic
90C4×Dic127—21no—
91C4×Dic127—21no—
92C4×Dic127C2311yescharacteristic
93C2×D4×C127—21no—
94C2×D4×C127—21no—
95C2×D4×C127C2311yescharacteristic
96C2×D4×C127C2311yescharacteristic
97C22×C4×C127C2311yescharacteristic
98D8×C127—22no—
99M16×C127—22no—
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22×D127C2×D411yescharacteristic
107Dic254—42no—
108Dic254—81no—
109C22⋊D127—22no—
110C22⋊D127—42no—
111C22⋊D127—81no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—42no—
117C2×Dic127—42no—
118C2×Dic127—42no—
119C2×Dic127C2×D411yescharacteristic
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C4×D127—21no—
123C4×D127—21no—
124C4×D127—42no—
125C127⋊C8—21no—
126C127⋊C8—21no—
127C23×C127—21no—
128C23×C127—21no—
129C23×C127C2×D411yescharacteristic
130D4×C127—21no—
131D4×C127—21no—
132D4×C127—21no—
133D4×C127—21no—
134D4×C127—42no—
135D4×C127—81no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2411yescharacteristic
142C8×C127—21no—
143C8×C127#4112yescomplemented
144C22×C127—21no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127C22×D411yescharacteristic
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—21no—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127C2×#4111yescharacteristic
161C4×C1272−1+411yescharacteristic
162Dic127—21no—
163Dic127—21no—
164Dic127—41no—
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168Dic127AGL(1,ℤ/8)12yes—
169C2×C127—41no—
170C2×C127—81no—
171C2×C127—81no—
172C2×C127#54711yescharacteristic
173C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C127#57711yescharacteristic
175D127—41no—
176#3354—1271nomaximal, Sylow
177C127#335411yescharacteristic, complemented, Sylow
178#555—1271no—
179#547—1271no—
180#498—1271no—
181#482—1271no—
182#482—1271no—
183#481—1271no—
184#481—1271no—
185#476—1271no—
186#474—1271no—
187#469—1271no—
188#469D12711yescharacteristic, complemented
189C4×D8—1271no—
190#436—1271no—
191#428—1271no—
192#428—1271no—
193C2×#41—2541no—
194C2×D8C2×D12711yescharacteristic
195C2×M16—1271no—
196C2×M16C2×D12711yescharacteristic
197C4⋊Q8—1271no—
198C4⋊Q8—1271no—
199#124—2541no—
200#124—2541no—
201#123—2541no—
202#123—2541no—
203C22⋊Q8—2541no—
204#121—1271no—
205#121—1271no—
206#121—2541no—
207#121C2×D12711yescharacteristic
208#121C2×D12711yescharacteristic
209C4×D4—1271no—
210C4×D4—1271no—
211C4×D4—1271no—
212#117—1271no—
213#117—1271no—
214#117—2541no—
215C2×C4⋊C4—1271no—
216C2×C4⋊C4—1271no—
217#107—1271no—
218#107—1271no—
219#107—1271no—
220#106—1271no—
221#106—1271no—
222#106—1271no—
223#106—1271no—
224#106—1271no—
225#106C2×D12711yescharacteristic
226C4⋊C8—1271no—
227#102—1271no—
228#102—1271no—
229#102—1271no—
230#102—1271no—
231#102C2×D12711yescharacteristic
232#102C2×D12711yescharacteristic
233C22⋊C8—1271no—
234C22⋊C8—1271no—
235C4×C8—1271no—
236#41—5084no—
237C2×Q8—2541no—
238C2×Q8—2541no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—1271no—
242C2×D4—2541no—
243C2×D4—2541no—
244C2×D4C22×D12711yescharacteristic
245C2×D4C22×D12711yescharacteristic
246C22×C4—1271no—
247C22×C4—2541no—
248C22×C4—2541no—
249C22×C4—2541no—
250C22×C4—2541no—
251C22×C4C22×D12711yescharacteristic
252D8—22no—
253M16—22no—
254M16—2542no—
255C2×C8—1271no—
256C2×C8—1271no—
257C2×C8C22×D12711yescharacteristic
258C2×C8C22×D12711yescharacteristic
259C4⋊C4—1271no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—1271no—
263C4⋊C4—1271no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4C22×D12711yescharacteristic
269C4⋊C4C22×D12711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—1271no—
273C22⋊C4—1271no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C42—1271no—
281C42—2541no—
282C42—2541no—
283C23—21no—
284C23—21no—
285C23—1271no—
286C23D4×D12711yescharacteristic
287Q8—10161no—
288Q8—5082no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—42no—
294D4—81no—
295D4—10161no—
296D4—5082no—
297D4—2542no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—1271no—
303C2×C4—1271no—
304C2×C4—1271no—
305C2×C4—5082no—
306C2×C4—5082no—
307C2×C4—5082no—
308C2×C4—5082no—
309C2×C4—2541no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4D4×D12711yescharacteristic
316C2×C4C23×D12711yescharacteristic
317C8—21no—
318C8—2541no—
319C8—2541no—
320C8#1452324712yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—2541no—
331C22—2541no—
332C22—2541no—
333C22C2×D4×D12711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—1272no—
338C4—5081no—
339C4—5081no—
340C4—5081no—
341C4—5081no—
342C4—2541no—
343C4—2541no—
344C4C2×#1452324711yescharacteristic
345C4#1686125611yescharacteristic
346C2—41no—
347C2—81no—
348C2—81no—
349C2—5081no—
350C2#2255052611yescharacteristic
351C2AGL(1,ℤ/8)×D12711yescharacteristic
352C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#469×C127C2
socleC22×C127C22×D4
hypercenter#469D127
hypocenterC127#3354
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #22550590C2semidirect
2. #22550526C2semidirect
3. #22550273C2semidirect
4. #22550198C2semidirect
5. #22550197C2semidirect
6. #22550193C2semidirect
7. #22550192C2semidirect
8. #22550164C2semidirect
9. #22550154C2semidirect
10. #22550126C2semidirect
11. D8×Dic127C2semidirect
12. #22549952C2semidirect
13. #22549918C2semidirect
14. #22549915C2semidirect
15. #469×C127C2semidirect
143. C8×C127#41semidirect
177. C127#3354semidirect
188. #469D127semidirect
320. C8#14523247semidirect

Maximal subgroups

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

subgroupindex
#225505902
#225505262
#225502732
#225501982
#225501972
#225501932
#225501922
#225501642
#225501542
#225501262
D8×Dic1272
#225499522
#225499182
#225499152
#469×C1272
#3354127