Finite Group Databasebeta

← #30650349

Subgroups — #30650349

Order 16256.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861233C212yesmaximal, complemented
2#22550546C211yesmaximal, characteristic, complemented
3#22550511C211yesmaximal, characteristic, complemented
4#22550126C212yesmaximal, complemented
5#22550073C211yesmaximal, characteristic, complemented
6#22550068C211yesmaximal, characteristic, complemented
7#22550018C212yesmaximal, complemented
8#22550008C212yesmaximal, complemented
9C4×#14523237C211yesmaximal, characteristic, complemented
10#22549955C211yesmaximal, characteristic, complemented
11#493×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yescomplemented
13C2×D4×D127—21no—
14C2×D4×D127C2212yes—
15#16861233—28no—
16C2×#14523237C2211yescharacteristic
17C2×#14523236C2212yescomplemented
18C2×#14523233C2212yes—
19C2×#14523231C2212yescomplemented
20#16861205—21no—
21#16861181—22no—
22#16861180—21no—
23#16861180C2212yes—
24#16861180C2212yescomplemented
25#16861177—22no—
26C4×Dic254C2211yescharacteristic, complemented
27C4×D508C2211yescharacteristic
28C4×D508C2211yescharacteristic
29D4×Dic127—21no—
30#16861141C2211yescharacteristic, complemented
31C4⋊(C127⋊C8)C2211yescharacteristic, complemented
32#16861134C2211yescharacteristic, complemented
33#16861131C2211yescharacteristic
34#16861130C2212yescomplemented
35#16861130C2212yescomplemented
36(C2×D127)⋊C8C2212yescomplemented
37C2×D8×C127C2212yescomplemented
38C2×D8×C127C2212yescomplemented
39#127×C127C2211yescharacteristic
40#127×C127C2211yescharacteristic, complemented
41C4×C8×C127C2211yescharacteristic, complemented
42C23×D127—22no—
43#14523247—24no—
44#14523247—42no—
45C2×Dic254C2311yescharacteristic
46C2×D508D411yescharacteristic
47C2×D508C2311yescharacteristic
48C2×C22⋊D127—22no—
49C2×C22⋊D127—22no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—22no—
52D4×D127—24no—
53D4×D127—42no—
54D4×D127—44no—
55C22×Dic127—22no—
56C2×C4×D127—21no—
57C2×C4×D127C2312yes—
58#14523237D414yescomplemented
59#14523236—24no—
60#14523233—24no—
61#14523231—24no—
62C2×C127⋊C8C2312yes—
63C4⋊Dic127D411yescharacteristic
64C4⋊Dic127C2311yescharacteristic
65C4⋊Dic127C2311yescharacteristic
66Dic127⋊C4—21no—
67C22⋊Dic127—22no—
68C22⋊Dic127—22no—
69(C2×D127)⋊C4—21no—
70(C2×D127)⋊C4D412yes—
71C4×Dic127—21no—
72C2×D4×C127—21no—
73C2×D4×C127—21no—
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127C2312yes—
77C2×D4×C127C2312yes—
78D8×C127—24no—
79D8×C127—42no—
80C2×C8×C127C2311yescharacteristic
81C2×C8×C127C2311yescharacteristic
82C42×C127C2311yescharacteristic
83C22×D127—21no—
84C22×D127—44no—
85C22×D127—44no—
86C22×D127—82no—
87C22×D127C2×D412yes—
88Dic254—21no—
89Dic254C2×D412yescomplemented
90D508—22no—
91D508—21no—
92D508C2×D412yes—
93C22⋊D127—44no—
94C22⋊D127—44no—
95C22⋊D127—44no—
96C22⋊D127—82no—
97C2×Dic127—21no—
98C2×Dic127—21no—
99C2×Dic127—21no—
100C2×Dic127—44no—
101C2×Dic127C2×D412yes—
102C4×D127—22no—
103C4×D127—22no—
104C4×D127—42no—
105C127⋊C8—22no—
106C23×C127—22no—
107C23×C127—22no—
108D4×C127—22no—
109D4×C127—22no—
110D4×C127—22no—
111D4×C127—22no—
112D4×C127—42no—
113D4×C127—42no—
114D4×C127—81no—
115D4×C127—81no—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2411yescharacteristic
119C8×C127—21no—
120C8×C127C2×D412yescomplemented
121C22×C127—42no—
122C22×C127—42no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127—42no—
126C22×C127—42no—
127C22×C127C22×D411yescharacteristic
128C2×D127—22no—
129C2×D127—22no—
130C2×D127—22no—
131C2×D127—22no—
132C2×D127—41no—
133C2×D127—41no—
134C2×D127—88no—
135C4×C127—21no—
136C4×C127AGL(1,ℤ/8)12yescomplemented
137C4×C127C22×D411yescharacteristic
138C4×C127H(5,𝔽2)11yescharacteristic
139Dic127—22no—
140Dic127—22no—
141Dic127—41no—
142Dic127—41no—
143C2×C127—82no—
144C2×C127—82no—
145C2×C127D4211yescharacteristic
146C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C127#57611yescharacteristic
148D127—42no—
149D127—42no—
150#3274—1271nomaximal, Sylow
151C127#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1272no—
153#550—1271no—
154D42—1271no—
155#493D12711yescharacteristic, complemented
156#469—1272no—
157#460—1271no—
158#459—1271no—
159#449—1272no—
160#447—1272no—
161C4×SD16—1271no—
162#436—1271no—
163C2×#41—1272no—
164C22×D4—1272no—
165C22×D4—2541no—
166AGL(1,ℤ/8)—2548no—
167C2×SD16—1272no—
168C2×SD16—1271no—
169C2×D8—1272no—
170C2×D8C2×D12712yescomplemented
171C2×D8C2×D12712yescomplemented
172C2×M16—1272no—
173#127—2541no—
174#127C2×D12711yescharacteristic
175#127C2×D12711yescharacteristic, complemented
176#121—1272no—
177#121—1272no—
178#121—2542no—
179#121—2541no—
180#120—2542no—
181C4×Q8—1271no—
182C4×D4—1271no—
183C4×D4—1271no—
184C4×D4—2541no—
185#106—1271no—
186C4⋊C8—1271no—
187#103—1271no—
188#102—1272no—
189#102—1272no—
190#102—1271no—
191C22⋊C8—1272no—
192C4×C8C2×D12711yescharacteristic, complemented
193C24—2542no—
194#41—5082no—
195#41—2544no—
196C2×Q8—1271no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—1271no—
202C2×D4—1271no—
203C2×D4—5084no—
204C2×D4—5082no—
205C2×D4—2542no—
206C2×D4—2544no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4—2542no—
210C2×D4C22×D12712yes—
211C2×D4C22×D12712yes—
212C22×C4—1272no—
213C22×C4—2542no—
214C22×C4—2541no—
215SD16—1274no—
216SD16—2544no—
217D8—24no—
218D8—42no—
219D8—2544no—
220M16—2544no—
221C2×C8—1272no—
222C2×C8C22×D12711yescharacteristic
223C2×C8C22×D12711yescharacteristic
224C4⋊C4—1271no—
225C4⋊C4—1271no—
226C4⋊C4—1271no—
227C4⋊C4—2541no—
228C22⋊C4—1272no—
229C22⋊C4—2542no—
230C22⋊C4—2542no—
231C22⋊C4—2541no—
232C42—2541no—
233C42C22×D12711yescharacteristic
234C23—22no—
235C23—22no—
236C23—1272no—
237C23—10162no—
238C23—5084no—
239C23—5084no—
240C23—2541no—
241Q8—1272no—
242Q8—2541no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—1272no—
252D4—10162no—
253D4—5084no—
254D4—5084no—
255D4—5084no—
256D4—2542no—
257D4—2541no—
258C2×C4—1272no—
259C2×C4—5082no—
260C2×C4—5084no—
261C2×C4—2541no—
262C2×C4—2541no—
263C2×C4—2541no—
264C2×C4—2542no—
265C2×C4—2542no—
266C2×C4D4×D12711yescharacteristic
267C2×C4D4×D12711yescharacteristic
268C2×C4C23×D12711yescharacteristic
269C8—21no—
270C8—2542no—
271C8D4×D12712yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—10168no—
279C22—5081no—
280C22—5081no—
281C22—2542no—
282C22—2542no—
283C22—2542no—
284C22—2542no—
285C22C2×D4×D12711yescharacteristic
286C4—21no—
287C4—5081no—
288C4—5081no—
289C4—2542no—
290C4—2542no—
291C4#1686123312yescomplemented
292C4C2×D4×D12711yescharacteristic
293C4#1686125411yescharacteristic
294C2—82no—
295C2—82no—
296C2—5082no—
297C2—5082no—
298C2#2255051111yescharacteristic
299C2C2×#1686123311yescharacteristic
300C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#493×C127C2
socleC22×C127C22×D4
hypercenter#493D127
hypocenterC127#3274
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. #22550546C2semidirect
3. #22550511C2semidirect
4. #22550126C2semidirect
5. #22550073C2semidirect
6. #22550068C2semidirect
7. #22550018C2semidirect
8. #22550008C2semidirect
9. C4×#14523237C2semidirect
10. #22549955C2semidirect
11. #493×C127C2semidirect
12. C2×#14523247C22semidirect
17. C2×#14523236C22semidirect
19. C2×#14523231C22semidirect
24. #16861180C22semidirect
26. C4×Dic254C22semidirect
30. #16861141C22semidirect
31. C4⋊(C127⋊C8)C22semidirect
32. #16861134C22semidirect
34. #16861130C22semidirect
35. #16861130C22semidirect
36. (C2×D127)⋊C8C22semidirect
37. C2×D8×C127C22semidirect
38. C2×D8×C127C22semidirect
40. #127×C127C22semidirect
41. C4×C8×C127C22semidirect
58. #14523237D4semidirect
89. Dic254C2×D4semidirect
120. C8×C127C2×D4semidirect
136. C4×C127AGL(1,ℤ/8)semidirect
151. C127#3274semidirect
155. #493D127semidirect
170. C2×D8C2×D127semidirect
171. C2×D8C2×D127semidirect
175. #127C2×D127semidirect
192. C4×C8C2×D127semidirect
271. C8D4×D127semidirect
291. C4#16861233semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
#225505462
#225505112
#225501262
#225500732
#225500682
#225500182
#225500082
C4×#145232372
#225499552
#493×C1272
#3274127