Finite Group Databasebeta

← #30651321

Subgroups — #30651321

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#22550589C211yesmaximal, characteristic, complemented
2#22550527C211yesmaximal, characteristic, complemented
3#22550273C211yesmaximal, characteristic, complemented
4#22550203C211yesmaximal, characteristic, complemented
5#22550200C211yesmaximal, characteristic, complemented
6#22550196C211yesmaximal, characteristic, complemented
7#22550194C211yesmaximal, characteristic, complemented
8#22550165C211yesmaximal, characteristic, complemented
9#22550155C211yesmaximal, characteristic, complemented
10#22550130C211yesmaximal, characteristic, complemented
11#22550129C211yesmaximal, characteristic, complemented
12#22549958C211yesmaximal, characteristic, complemented
13#22549952C211yesmaximal, characteristic, complemented
14#22549917C211yesmaximal, characteristic, complemented
15#428×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—21no—
17C2×D1016C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19#16861208C2211yescharacteristic
20#16861208C2211yescharacteristic
21#16861197—21no—
22#16861195—21no—
23#16861194—21no—
24#16861192—21no—
25#16861187—21no—
26#16861184C2211yescharacteristic
27#16861184C2211yescharacteristic
28#16861183C2211yescharacteristic
29#16861183C2211yescharacteristic
30#16861182—21no—
31#16861171C2211yescharacteristic
32#16861171C2211yescharacteristic
33C4×C22⋊D127C2211yescharacteristic
34#16861167C2211yescharacteristic
35#16861166—21no—
36C4⋊C4×D127C2211yescharacteristic
37#16861142C2211yescharacteristic
38#16861141C2211yescharacteristic
39#16861140C2211yescharacteristic
40#16861140C2211yescharacteristic
41#16861140C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861131C2211yescharacteristic
44#16861131C2211yescharacteristic
45#16861129C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×M16×C127C2211yescharacteristic
53#117×C127C2211yescharacteristic
54C2×C4⋊C4×C127C2211yescharacteristic
55#107×C127C2211yescharacteristic
56#107×C127C2211yescharacteristic
57#106×C127C2211yescharacteristic
58#106×C127C2211yescharacteristic
59#14523248—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×D508—21no—
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127D411yescharacteristic
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127C2311yescharacteristic
73D1016—22no—
74#14523231—22no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4D411yescharacteristic
83C22⋊Dic127D411yescharacteristic
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4D411yescharacteristic
92C4×Dic127—21no—
93C4×Dic127C2311yescharacteristic
94C22×C4×C127—21no—
95C22×C4×C127C2311yescharacteristic
96M16×C127—22no—
97C2×C8×C127C2311yescharacteristic
98C2×C8×C127C2311yescharacteristic
99C4⋊C4×C127—21no—
100C4⋊C4×C127C2311yescharacteristic
101C4⋊C4×C127C2311yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C22⋊C4×C127—21no—
105C42×C127—21no—
106C22×D127—21no—
107C22×D127—21no—
108C22×D127C2×D411yescharacteristic
109Dic254—42no—
110Dic254—81no—
111D508—21no—
112D508—21no—
113D508—21no—
114D508—21no—
115D508—42no—
116D508—81no—
117C22⋊D127—22no—
118C22⋊D127—42no—
119C22⋊D127—81no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C2×Dic127C2×D411yescharacteristic
125C4×D127—21no—
126C4×D127—21no—
127C4×D127—42no—
128C4×D127—42no—
129C4×D127—42no—
130C127⋊C8—21no—
131C127⋊C8—21no—
132C23×C127C2×D411yescharacteristic
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—42no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2411yescharacteristic
142C8×C127—21no—
143C8×C127#4112yescomplemented
144C22×C127—21no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127C22×D411yescharacteristic
148C2×D127—21no—
149C2×D127—21no—
150C2×D127—21no—
151C2×D127—41no—
152C2×D127—41no—
153C2×D127—41no—
154C2×D127—41no—
155C2×D127—41no—
156C2×D127—41no—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127C2×#4111yescharacteristic
163C4×C1272−1+411yescharacteristic
164Dic127—21no—
165Dic127—21no—
166Dic127—41no—
167Dic127—41no—
168Dic127AGL(1,ℤ/8)12yes—
169C2×C127—41no—
170C2×C127#54711yescharacteristic
171C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C127#57711yescharacteristic
173D127—41no—
174D127—81no—
175D127—81no—
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#469—1271no—
189C4×D8—1271no—
190#436—1271no—
191#428—1271no—
192#428D12711yescharacteristic, complemented
193C2×#41—2541no—
194C2×D8—1271no—
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—1271no—
207#121—1271no—
208#121—2541no—
209C4×D4—1271no—
210C4×D4—1271no—
211C4×D4—1271no—
212#117—1271no—
213#117—2541no—
214#117C2×D12711yescharacteristic
215C2×C4⋊C4—1271no—
216C2×C4⋊C4C2×D12711yescharacteristic
217#107—1271no—
218#107C2×D12711yescharacteristic
219#107C2×D12711yescharacteristic
220#106—1271no—
221#106—1271no—
222#106—1271no—
223#106—1271no—
224#106C2×D12711yescharacteristic
225#106C2×D12711yescharacteristic
226C4⋊C8—1271no—
227#102—1271no—
228#102—1271no—
229#102—1271no—
230#102—1271no—
231#102—1271no—
232#102—1271no—
233C22⋊C8—1271no—
234C22⋊C8—1271no—
235C4×C8—1271no—
236#41—5084no—
237C2×Q8—2541no—
238C2×Q8—2541no—
239C2×D4—1271no—
240C2×D4—1271no—
241C2×D4—1271no—
242C2×D4—2541no—
243C2×D4—2541no—
244C2×D4—2541no—
245C2×D4—2541no—
246C22×C4—21no—
247C22×C4—1271no—
248C22×C4—2541no—
249C22×C4—2541no—
250C22×C4—2541no—
251C22×C4C22×D12711yescharacteristic
252D8—2542no—
253M16—22no—
254M16—2542no—
255C2×C8—1271no—
256C2×C8—1271no—
257C2×C8C22×D12711yescharacteristic
258C2×C8C22×D12711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—1271no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4C22×D12711yescharacteristic
267C4⋊C4C22×D12711yescharacteristic
268C4⋊C4C22×D12711yescharacteristic
269C4⋊C4C22×D12711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—1271no—
272C22⋊C4—1271no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C42—21no—
281C42—1271no—
282C42—2541no—
283C23—1271no—
284C23—2541no—
285C23—2541no—
286C23D4×D12711yescharacteristic
287Q8—10161no—
288Q8—5082no—
289D4—10161no—
290D4—10161no—
291D4—5082no—
292D4—5082no—
293D4—2542no—
294D4—2541no—
295D4—2541no—
296D4—2541no—
297D4—2541no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—1271no—
306C2×C4—1271no—
307C2×C4—1271no—
308C2×C4—5082no—
309C2×C4—5082no—
310C2×C4—5082no—
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#1452324812yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—5081no—
325C22—5081no—
326C22—5081no—
327C22—5081no—
328C22—5081no—
329C22—5081no—
330C22—2541no—
331C22—2541no—
332C22—2541no—
333C22C2×D4×D12711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—1272no—
340C4—5081no—
341C4—5081no—
342C4—2541no—
343C4—2541no—
344C4C2×#1452324811yescharacteristic
345C4#1686125711yescharacteristic
346C2—41no—
347C2—10161no—
348C2—10161no—
349C2—5081no—
350C2#2255052711yescharacteristic
351C2AGL(1,ℤ/8)×D12711yescharacteristic
352C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
hypercenter#428D127
hypocenterC127#3354
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #22550589C2semidirect
2. #22550527C2semidirect
3. #22550273C2semidirect
4. #22550203C2semidirect
5. #22550200C2semidirect
6. #22550196C2semidirect
7. #22550194C2semidirect
8. #22550165C2semidirect
9. #22550155C2semidirect
10. #22550130C2semidirect
11. #22550129C2semidirect
12. #22549958C2semidirect
13. #22549952C2semidirect
14. #22549917C2semidirect
15. #428×C127C2semidirect
143. C8×C127#41semidirect
177. C127#3354semidirect
192. #428D127semidirect
320. C8#14523248semidirect

Maximal subgroups

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

subgroupindex
#225505892
#225505272
#225502732
#225502032
#225502002
#225501962
#225501942
#225501652
#225501552
#225501302
#225501292
#225499582
#225499522
#225499172
#428×C1272
#3354127