Finite Group Databasebeta

← #30648410

Subgroups — #30648410

Order 16256.

504 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861237C211yesmaximal, characteristic, complemented
2#22550454C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550129C212yesmaximal, complemented
5#22550122C212yesmaximal, complemented
6#22550112C212yesmaximal, complemented
7#22550105C212yesmaximal, complemented
8#22549918C211yesmaximal, characteristic, complemented
9#22549873C211yesmaximal, characteristic, complemented
10C2×#16861131C211yesmaximal, characteristic, complemented
11C2×#131×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508C2211yescharacteristic
14#16861237—24no—
15C2×#14523237C2212yes—
16C2×D1016C2212yes—
17#16861205—21no—
18#16861198—21no—
19#16861188C2212yescomplemented
20#16861185—21no—
21#16861183C2212yes—
22#16861183C2212yes—
23#16861183C2212yescomplemented
24#16861181—22no—
25#16861180—21no—
26#16861177—22no—
27#16861168C2211yescharacteristic
28C2×C4⋊Dic127C2211yescharacteristic
29#16861143C2212yescomplemented
30#16861141C2212yescomplemented
31#16861134C2211yescharacteristic, complemented
32#16861134C2211yescharacteristic, complemented
33#16861131C2212yes—
34#16861131C2212yes—
35#16861131C2211yescharacteristic, complemented
36#16861131C2211yescharacteristic, complemented
37C2×#41×C127C2211yescharacteristic
38#131×C127—24no—
39C2×M16×C127C2211yescharacteristic
40C2×M16×C127C2212yescomplemented
41C22×C8×C127C2211yescharacteristic
42C22×C8×C127C2212yescomplemented
43C23×D127—21no—
44#14523248—22no—
45#14523248—22no—
46C2×Dic254C2311yescharacteristic
47C2×D508—22no—
48C2×D508—21no—
49C2×D508—22no—
50C2×D508C2311yescharacteristic
51C2×D508C2312yes—
52C2×C22⋊D127—21no—
53C2×C22⋊D127—22no—
54C2×C22⋊D127—22no—
55C2×C22⋊D127—22no—
56C22×Dic127—21no—
57C2×C4×D127—21no—
58#14523237—24no—
59D1016—24no—
60C4⋊Dic127—21no—
61C4⋊Dic127C2312yes—
62C4⋊Dic127C2311yescharacteristic
63C4⋊Dic127C2311yescharacteristic
64Dic127⋊C4—22no—
65C22⋊Dic127—22no—
66C22⋊Dic127—21no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70C4×Dic127—21no—
71#41×C127—22no—
72#41×C127—22no—
73C2×Q8×C127D411yescharacteristic
74C2×D4×C127D411yescharacteristic
75C2×D4×C127D412yes—
76C22×C4×C127C2312yes—
77C22×C4×C127C2311yescharacteristic
78M16×C127—24no—
79M16×C127D414yescomplemented
80C2×C8×C127—22no—
81C2×C8×C127—24no—
82C2×C8×C127C2311yescharacteristic
83C2×C8×C127C2311yescharacteristic
84C2×C8×C127C2312yes—
85C22×D127—21no—
86C22×D127—22no—
87C22×D127—42no—
88C22×D127—81no—
89C22×D127—81no—
90Dic254—21no—
91Dic254—21no—
92D508—21no—
93D508—22no—
94D508—22no—
95D508—21no—
96D508—42no—
97C22⋊D127—42no—
98C22⋊D127—82no—
99C22⋊D127—82no—
100C22⋊D127—82no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—22no—
104C2×Dic127—21no—
105C2×Dic127—81no—
106C4×D127—42no—
107C23×C127C2×D412yes—
108C23×C127C2×D411yescharacteristic
109Q8×C127—22no—
110D4×C127—24no—
111D4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127C2×D412yes—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127#4112yes—
118C2×C4×C127C2411yescharacteristic
119C8×C127—21no—
120C8×C127—21no—
121C8×C127C2×D414yes—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—22no—
127C22×C127#12112yes—
128C22×C127C22×D411yescharacteristic
129C2×D127—41no—
130C2×D127—42no—
131C2×D127—41no—
132C2×D127—42no—
133C2×D127—42no—
134C2×D127—41no—
135C2×D127—84no—
136C4×C127—22no—
137C4×C127#12112yes—
138C4×C127C22×D411yescharacteristic
139C4×C127C2×#4111yescharacteristic
140Dic127—41no—
141Dic127—42no—
142Dic127—41no—
143Dic127—41no—
144C2×C127—22no—
145C2×C127—42no—
146C2×C127C2×#12111yescharacteristic
147C2×C127#57611yescharacteristic
148C2×C127#57711yescharacteristic
149D127—81no—
150D127—82no—
151#3062—1271nomaximal, Sylow
152C127#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—1271no—
154C2×#131D12711yescharacteristic, complemented
155#537—1271no—
156#535—1271no—
157#469—1272no—
158#468—1272no—
159#466—1272no—
160#465—1272no—
161#428—1271no—
162#416—1271no—
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—24no—
171C2×M16C2×D12711yescharacteristic
172C2×M16C2×D12712yescomplemented
173C22×C8C2×D12711yescharacteristic
174C22×C8C2×D12712yescomplemented
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#117—1271no—
186C2×C4⋊C4—1271no—
187#107—1272no—
188#106—1272no—
189#103—1271no—
190#103—1271no—
191#102—1272no—
192#102—1272no—
193#102—1271no—
194#102—1271no—
195C24—2541no—
196#41—22no—
197#41—22no—
198#41—2542no—
199#41—2542no—
200C2×Q8—1271no—
201C2×Q8D50811yescharacteristic
202C2×D4—1271no—
203C2×D4—1272no—
204C2×D4—2542no—
205C2×D4—2542no—
206C2×D4—2542no—
207C2×D4—2541no—
208C2×D4—2542no—
209C2×D4—2541no—
210C2×D4—2542no—
211C2×D4D50812yes—
212C2×D4D50811yescharacteristic
213C22×C4—2541no—
214C22×C4—2541no—
215C22×C4C22×D12712yes—
216C22×C4C22×D12711yescharacteristic
217SD16—2544no—
218D8—2544no—
219M16—24no—
220M16C22⋊D12714yescomplemented
221C2×C8—22no—
222C2×C8—24no—
223C2×C8C22×D12711yescharacteristic
224C2×C8C22×D12711yescharacteristic
225C2×C8C22×D12712yes—
226C4⋊C4—1272no—
227C4⋊C4—1271no—
228C4⋊C4—1271no—
229C4⋊C4—2542no—
230C4⋊C4—2541no—
231C22⋊C4—2541no—
232C22⋊C4—2542no—
233C22⋊C4—2542no—
234C22⋊C4—2542no—
235C22⋊C4—2542no—
236C42—2541no—
237C23—10161no—
238C23—10161no—
239C23—5082no—
240C23—2541no—
241C23—2542no—
242C23C2×D50812yes—
243C23C2×D50811yescharacteristic
244Q8—22no—
245Q8—2541no—
246Q8—2541no—
247D4—24no—
248D4—22no—
249D4—10162no—
250D4—10162no—
251D4—10162no—
252D4—5082no—
253D4—5082no—
254D4—2541no—
255D4—2542no—
256D4—2542no—
257D4—2541no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—10161no—
261C2×C4—5082no—
262C2×C4—2541no—
263C2×C4—2541no—
264C2×C4—2542no—
265C2×C4—2541no—
266C2×C4C2×C22⋊D12712yes—
267C2×C4C2×D50812yes—
268C2×C4C2×D50811yescharacteristic
269C2×C4#1452324812yes—
270C2×C4C23×D12711yescharacteristic
271C8—21no—
272C8—21no—
273C8C2×C22⋊D12714yes—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—10164no—
280C22—5081no—
281C22—5082no—
282C22—5081no—
283C22—5082no—
284C22—5082no—
285C22—5081no—
286C22#1686118312yes—
287C22C22×D50811yescharacteristic
288C4—22no—
289C4—5081no—
290C4—5082no—
291C4—5081no—
292C4—5081no—
293C4#1686118312yes—
294C4C22×C22⋊D12711yescharacteristic
295C4C2×#1452324811yescharacteristic
296C2—22no—
297C2—42no—
298C2—10162no—
299C2—10161no—
300C2C2×#1686118311yescharacteristic
301C2#2255068111yescharacteristic
302C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#131×C127C2
socleC22×C127C22×D4
hypercenterC2×#131D127
hypocenterC127#3062
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. #22550454C2semidirect
3. #22550440C2semidirect
4. #22550129C2semidirect
5. #22550122C2semidirect
6. #22550112C2semidirect
7. #22550105C2semidirect
8. #22549918C2semidirect
9. #22549873C2semidirect
10. C2×#16861131C2semidirect
11. C2×#131×C127C2semidirect
19. #16861188C22semidirect
23. #16861183C22semidirect
29. #16861143C22semidirect
30. #16861141C22semidirect
31. #16861134C22semidirect
32. #16861134C22semidirect
35. #16861131C22semidirect
36. #16861131C22semidirect
40. C2×M16×C127C22semidirect
42. C22×C8×C127C22semidirect
79. M16×C127D4semidirect
152. C127#3062semidirect
154. C2×#131D127semidirect
172. C2×M16C2×D127semidirect
174. C22×C8C2×D127semidirect
220. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
#225504542
#225504402
#225501292
#225501222
#225501122
#225501052
#225499182
#225498732
C2×#168611312
C2×#131×C1272
#3062127