Finite Group Databasebeta

← #30648413

Subgroups — #30648413

Order 16256.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16861234C211yesmaximal, characteristic, complemented
2C2×#16861215C211yesmaximal, characteristic, complemented
3#22550454C211yesmaximal, characteristic, complemented
4#22550445C211yesmaximal, characteristic, complemented
5#22550128C212yesmaximal, complemented
6#22550125C212yesmaximal, complemented
7#22550111C212yesmaximal, complemented
8#22550108C212yesmaximal, complemented
9#22549919C211yesmaximal, characteristic, complemented
10C2×#16861131C211yesmaximal, characteristic, complemented
11#416×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508C2211yescharacteristic
14#16861234—24no—
15C2×#14523235C2212yes—
16C2×#14523233C2212yes—
17#16861215—24no—
18C2×#14523232C2211yescharacteristic
19C2×#14523231C2212yescomplemented
20C2×C8×D127C2212yescomplemented
21#16861206—21no—
22#16861197—21no—
23#16861188—21no—
24#16861185C2212yescomplemented
25#16861184C2212yes—
26#16861184C2212yes—
27#16861183—21no—
28#16861181—22no—
29#16861180C2212yescomplemented
30#16861179—22no—
31C2×C4⋊Dic127C2211yescharacteristic
32#16861142C2212yescomplemented
33#16861140C2212yescomplemented
34#16861131C2212yes—
35#16861131C2212yes—
36C2×#41×C127C2211yescharacteristic
37C22×C8×C127C2211yescharacteristic
38#117×C127C2211yescharacteristic
39#103×C127C2211yescharacteristic, complemented
40#103×C127C2211yescharacteristic, complemented
41#102×C127C2211yescharacteristic, complemented
42#102×C127C2211yescharacteristic, complemented
43C23×D127—21no—
44#14523248—22no—
45#14523248—22no—
46C2×Dic254D411yescharacteristic
47C2×D508—22no—
48C2×D508—22no—
49C2×D508—22no—
50C2×D508—21no—
51C2×D508—22no—
52C2×D508D411yescharacteristic
53C2×D508C2312yes—
54C2×C22⋊D127—22no—
55C2×C22⋊D127D412yes—
56C22×Dic127—21no—
57C2×C4×D127C2312yes—
58#14523235—24no—
59#14523233—24no—
60#14523232D414yescomplemented
61#14523231—24no—
62C2×C127⋊C8C2312yes—
63C8×D127—24no—
64C4⋊Dic127—21no—
65C4⋊Dic127C2312yes—
66Dic127⋊C4—22no—
67C22⋊Dic127—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—22no—
71#41×C127—22no—
72#41×C127—22no—
73C2×Q8×C127C2311yescharacteristic
74C2×D4×C127—21no—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78C2×C8×C127—22no—
79C2×C8×C127C2311yescharacteristic
80C2×C8×C127C2311yescharacteristic
81C4⋊C4×C127C2311yescharacteristic
82C4⋊C4×C127C2311yescharacteristic
83C22⋊C4×C127—21no—
84C42×C127—21no—
85C22×D127—22no—
86C22×D127—42no—
87C22×D127—81no—
88C22×D127—81no—
89C22×D127C2×D412yes—
90Dic254—22no—
91D508—22no—
92D508—22no—
93D508—22no—
94D508—42no—
95D508—82no—
96D508—82no—
97C22⋊D127—24no—
98C22⋊D127—82no—
99C2×Dic127—22no—
100C2×Dic127—81no—
101C2×Dic127C2×D412yes—
102C4×D127—22no—
103C4×D127—22no—
104C127⋊C8C2×D414yes—
105C23×C127—21no—
106C23×C127C2×D411yescharacteristic
107Q8×C127—21no—
108Q8×C127—21no—
109D4×C127—21no—
110D4×C127—21no—
111D4×C127—42no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—42no—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D412yes—
118C2×C4×C127#4112yes—
119C2×C4×C127C2411yescharacteristic
120C8×C127—21no—
121C8×C127—21no—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127#12112yes—
128C22×C127C22×D411yescharacteristic
129C2×D127—22no—
130C2×D127—22no—
131C2×D127—22no—
132C2×D127—42no—
133C2×D127—42no—
134C2×D127—42no—
135C2×D127—84no—
136C4×C127—41no—
137C4×C127—41no—
138C4×C127—41no—
139C4×C127#12112yes—
140C4×C127C22×D411yescharacteristic
141C4×C127C2×#4111yescharacteristic
142Dic127—22no—
143Dic127—42no—
144C2×C127—22no—
145C2×C127—81no—
146C2×C127C2×#12111yescharacteristic
147C2×C127#57611yescharacteristic
148C2×C127#57711yescharacteristic
149D127—42no—
150D127—82no—
151#3062—1271nomaximal, Sylow
152C127#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—1271no—
154C2×#131—1271no—
155#537—1271no—
156#535—1271no—
157#469—1272no—
158#468—1272no—
159#466—1272no—
160#465—1272no—
161#428—1271no—
162#416D12711yescharacteristic, complemented
163C2×#102—1271no—
164C2×#41—1271no—
165C2×#41C2×D12711yescharacteristic
166C22×D4—1271no—
167AGL(1,ℤ/8)—2544no—
168C2×SD16—1272no—
169C2×D8—1272no—
170#131—2544no—
171C2×M16—1271no—
172C2×M16—1272no—
173C22×C8—1272no—
174C22×C8C2×D12711yescharacteristic
175#127—2541no—
176#124—2541no—
177C22⋊Q8—1272no—
178C22⋊Q8—2541no—
179#121—1272no—
180#121—1272no—
181#121—1272no—
182#121—2541no—
183#121—2542no—
184#120—2542no—
185#117C2×D12711yescharacteristic
186C2×C4⋊C4—1271no—
187#107—1272no—
188#106—1272no—
189#103C2×D12711yescharacteristic, complemented
190#103C2×D12711yescharacteristic, complemented
191#102—1272no—
192#102—1272no—
193#102C2×D12711yescharacteristic, complemented
194#102C2×D12711yescharacteristic, complemented
195C24—2541no—
196#41—22no—
197#41—22no—
198#41—2542no—
199#41—2542no—
200C2×Q8—1271no—
201C2×Q8C22×D12711yescharacteristic
202C2×D4—21no—
203C2×D4—1272no—
204C2×D4—1271no—
205C2×D4—1272no—
206C2×D4—2542no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4—2541no—
210C2×D4—2542no—
211C2×D4—2542no—
212C2×D4C22×D12711yescharacteristic
213C22×C4—21no—
214C22×C4—1272no—
215C22×C4—2541no—
216C22×C4C22×D12711yescharacteristic
217SD16—2544no—
218D8—2544no—
219M16—1274no—
220M16—2544no—
221C2×C8—22no—
222C2×C8—1272no—
223C2×C8—2544no—
224C2×C8C22×D12711yescharacteristic
225C2×C8C22×D12711yescharacteristic
226C4⋊C4—1272no—
227C4⋊C4—2542no—
228C4⋊C4—2541no—
229C4⋊C4C22×D12711yescharacteristic
230C4⋊C4C22×D12711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—2542no—
233C22⋊C4—2542no—
234C22⋊C4—2542no—
235C22⋊C4—2542no—
236C42—21no—
237C23—21no—
238C23—1272no—
239C23—10161no—
240C23—10161no—
241C23—5082no—
242C23—2542no—
243C23D4×D12711yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—2542no—
247D4—21no—
248D4—21no—
249D4—42no—
250D4—10162no—
251D4—10162no—
252D4—10162no—
253D4—5082no—
254D4—2544no—
255D4—2542no—
256D4—2542no—
257D4—2542no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—1272no—
263C2×C4—10161no—
264C2×C4—2542no—
265C2×C4—2542no—
266C2×C4—2542no—
267C2×C4D4×D12712yes—
268C2×C4D4×D12711yescharacteristic
269C2×C4#1452324812yes—
270C2×C4C23×D12711yescharacteristic
271C8—21no—
272C8—21no—
273C8—1274no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—10164no—
280C22—5082no—
281C22—5082no—
282C22—5082no—
283C22—2542no—
284C22—2542no—
285C22—2542no—
286C22#1686118212yes—
287C22C2×D4×D12711yescharacteristic
288C4—41no—
289C4—41no—
290C4—41no—
291C4—5082no—
292C4—2542no—
293C4#1686118212yes—
294C4C2×D4×D12711yescharacteristic
295C4C2×#1452324811yescharacteristic
296C2—22no—
297C2—81no—
298C2—10162no—
299C2—5082no—
300C2C2×#1686118211yescharacteristic
301C2#2255067811yescharacteristic
302C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
hypercenter#416D127
hypocenterC127#3062
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×#16861215C2semidirect
3. #22550454C2semidirect
4. #22550445C2semidirect
5. #22550128C2semidirect
6. #22550125C2semidirect
7. #22550111C2semidirect
8. #22550108C2semidirect
9. #22549919C2semidirect
10. C2×#16861131C2semidirect
11. #416×C127C2semidirect
19. C2×#14523231C22semidirect
20. C2×C8×D127C22semidirect
24. #16861185C22semidirect
29. #16861180C22semidirect
32. #16861142C22semidirect
33. #16861140C22semidirect
39. #103×C127C22semidirect
40. #103×C127C22semidirect
41. #102×C127C22semidirect
42. #102×C127C22semidirect
60. #14523232D4semidirect
152. C127#3062semidirect
162. #416D127semidirect
189. #103C2×D127semidirect
190. #103C2×D127semidirect
193. #102C2×D127semidirect
194. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168612152
#225504542
#225504452
#225501282
#225501252
#225501112
#225501082
#225499192
C2×#168611312
#416×C1272
#3062127