Finite Group Databasebeta

← #30648877

Subgroups — #30648877

Order 16256.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550423C211yesmaximal, characteristic, complemented
2C4×#14523247C211yesmaximal, characteristic, complemented
3#22550202C211yesmaximal, characteristic, complemented
4#22550198C211yesmaximal, characteristic, complemented
5#22550160C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550075C211yesmaximal, characteristic, complemented
8#22550069C211yesmaximal, characteristic, complemented
9#22550022C211yesmaximal, characteristic, complemented
10#22550018C211yesmaximal, characteristic, complemented
11#22549986C211yesmaximal, characteristic, complemented
12#22549978C211yesmaximal, characteristic, complemented
13#22549976C211yesmaximal, characteristic, complemented
14#22549931C211yesmaximal, characteristic, complemented
15#442×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—21no—
17C2×#14523247C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21#16861208C2211yescharacteristic
22#16861205C2211yescharacteristic
23#16861195—21no—
24#16861184C2211yescharacteristic
25#16861182—21no—
26#16861180C2211yescharacteristic
27C4×Dic254C2211yescharacteristic
28#16861174C2211yescharacteristic
29C4×D508C2211yescharacteristic
30D4×Dic127C2211yescharacteristic
31C4×C22⋊D127—21no—
32C4×C22⋊D127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167C2211yescharacteristic
35#16861165—21no—
36C2×C4×Dic127—21no—
37C42×D127C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861132C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861120C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C4×D4×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic
59#14523248—44no—
60#14523247—22no—
61#14523247—22no—
62C2×Dic254—21no—
63C2×Dic254C2311yescharacteristic
64C2×D508—21no—
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C22×Dic127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73C2×C4×D127D411yescharacteristic
74C2×C4×D127D411yescharacteristic
75C2×C4×D127C2311yescharacteristic
76#14523237—22no—
77#14523236—22no—
78#14523233—22no—
79C2×C127⋊C8C2311yescharacteristic
80C2×C127⋊C8C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91C4×Dic127—21no—
92C4×Dic127—21no—
93C4×Dic127—21no—
94C4×Dic127D411yescharacteristic
95C4×Dic127D411yescharacteristic
96C4×Dic127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127—21no—
100C22×C4×C127—21no—
101D8×C127—22no—
102C2×C8×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C22⋊C4×C127—21no—
108C42×C127C2311yescharacteristic
109C22×D127—21no—
110C22×D127C2×D411yescharacteristic
111Dic254—21no—
112Dic254—42no—
113Dic254#4112yescomplemented
114D508—21no—
115D508—21no—
116D508—42no—
117C22⋊D127—24no—
118C22⋊D127—42no—
119C22⋊D127—42no—
120C22⋊D127—42no—
121C2×Dic127—24no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—42no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—21no—
130C4×D127—21no—
131C4×D127—22no—
132C4×D127—41no—
133C4×D127—42no—
134C127⋊C8—21no—
135C127⋊C8—21no—
136C23×C127—21no—
137C23×C127—21no—
138D4×C127—21no—
139D4×C127—21no—
140D4×C127—21no—
141D4×C127#4112yescomplemented
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—42no—
145C2×C4×C127—42no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2411yescharacteristic
149C8×C127—21no—
150C8×C127—21no—
151C22×C127—22no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127C22×D411yescharacteristic
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127C2×#4111yescharacteristic
169C4×C127C2×#4111yescharacteristic
170Dic127—21no—
171Dic127—22no—
172Dic127—21no—
173Dic127—41no—
174Dic127—41no—
175Dic127AGL(1,ℤ/8)12yes—
176C2×C127—42no—
177C2×C127—81no—
178C2×C127#52511yescharacteristic
179C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C127#57511yescharacteristic
181D127—41no—
182D127—81no—
183#3115—1271nomaximal, Sylow
184C127#311511yescharacteristic, complemented, Sylow
185#532—1271no—
186C4×#41—1271no—
187#482—1271no—
188#482—1271no—
189#475—1271no—
190#474—1271no—
191#460—1271no—
192#459—1271no—
193#449—1271no—
194#449—1271no—
195#442—1271no—
196#442D12711yescharacteristic, complemented
197#440—1271no—
198#440—1271no—
199#432—1271no—
200C2×#41—1271no—
201C2×#41—2541no—
202C2×SD16—1271no—
203C2×SD16—1271no—
204C2×D8—1271no—
205C2×D8C2×D12711yescharacteristic
206C4⋊Q8—1271no—
207#127—1271no—
208#124—2541no—
209#121—1271no—
210#121—1271no—
211#121—2541no—
212C4×Q8—1271no—
213C4×Q8—1271no—
214C4×D4—1271no—
215C4×D4—1271no—
216C4×D4—2541no—
217C4×D4—2541no—
218C4×D4—2541no—
219C4×D4C2×D12711yescharacteristic
220C4×D4C2×D12711yescharacteristic
221#117—1271no—
222#117—2541no—
223C2×C42—1271no—
224C2×C42—2541no—
225#107—1271no—
226#107—1271no—
227#106—1271no—
228#106C2×D12711yescharacteristic
229C4⋊C8—1271no—
230C4⋊C8—1271no—
231#103—1271no—
232#103—1271no—
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237#102C2×D12711yescharacteristic
238#102C2×D12711yescharacteristic
239C22⋊C8—1271no—
240C22⋊C8—1271no—
241#97—1271no—
242#97C2×D12711yescharacteristic
243#41—5084no—
244#41—2542no—
245#41—2542no—
246C2×Q8—1271no—
247C2×Q8—2541no—
248C2×D4—1271no—
249C2×D4—2541no—
250C2×D4—2541no—
251C2×D4—2541no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4C22×D12711yescharacteristic
255C2×D4C22×D12711yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—1271no—
259C22×C4—1271no—
260C22×C4—1271no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4—2541no—
264SD16—2542no—
265SD16—2542no—
266D8—22no—
267D8—2542no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8C22×D12711yescharacteristic
271C2×C8C22×D12711yescharacteristic
272C4⋊C4—1271no—
273C4⋊C4—1271no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4C22×D12711yescharacteristic
278C4⋊C4C22×D12711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C22⋊C4—2541no—
286C42—1271no—
287C42—1271no—
288C42—1271no—
289C42—2541no—
290C42—2541no—
291C42—2541no—
292C42C22×D12711yescharacteristic
293C23—21no—
294C23—21no—
295C23—1271no—
296C23—2541no—
297Q8—1272no—
298Q8—5082no—
299Q8—2541no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—5082no—
304D4—5082no—
305D4—5082no—
306D4—5082no—
307D4—2544no—
308D4—2541no—
309D4—2541no—
310D4#1452324812yescomplemented
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—1271no—
316C2×C4—1271no—
317C2×C4—1271no—
318C2×C4—5082no—
319C2×C4—5081no—
320C2×C4—5082no—
321C2×C4—2542no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4—2544no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4D4×D12711yescharacteristic
329C2×C4D4×D12711yescharacteristic
330C2×C4C23×D12711yescharacteristic
331C8—21no—
332C8—21no—
333C8—2541no—
334C8—2541no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—5081no—
342C22—5081no—
343C22—5081no—
344C22—2541no—
345C22—2541no—
346C22—2541no—
347C22C2×D4×D12711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—1272no—
353C4—5081no—
354C4—5081no—
355C4—2541no—
356C4—2542no—
357C4—2541no—
358C4#41×D12711yescharacteristic
359C4C2×#1452324811yescharacteristic
360C2—42no—
361C2—81no—
362C2—10161no—
363C2—5081no—
364C2#2255038511yescharacteristic
365C2AGL(1,ℤ/8)×D12711yescharacteristic
366C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#442×C127C2
socleC22×C127C22×D4
hypercenter#442D127
hypocenterC127#3115
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #22550423C2semidirect
2. C4×#14523247C2semidirect
3. #22550202C2semidirect
4. #22550198C2semidirect
5. #22550160C2semidirect
6. #22550154C2semidirect
7. #22550075C2semidirect
8. #22550069C2semidirect
9. #22550022C2semidirect
10. #22550018C2semidirect
11. #22549986C2semidirect
12. #22549978C2semidirect
13. #22549976C2semidirect
14. #22549931C2semidirect
15. #442×C127C2semidirect
113. Dic254#41semidirect
141. D4×C127#41semidirect
184. C127#3115semidirect
196. #442D127semidirect
310. D4#14523248semidirect

Maximal subgroups

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

subgroupindex
#225504232
C4×#145232472
#225502022
#225501982
#225501602
#225501542
#225500752
#225500692
#225500222
#225500182
#225499862
#225499782
#225499762
#225499312
#442×C1272
#3115127