Finite Group Databasebeta

← #30648404

Subgroups — #30648404

Order 16256.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861217C211yesmaximal, characteristic, complemented
2#22550454C211yesmaximal, characteristic, complemented
3#22550443C211yesmaximal, characteristic, complemented
4#22550126C212yesmaximal, complemented
5#22550119C212yesmaximal, complemented
6#22550109C212yesmaximal, complemented
7#22550102C212yesmaximal, complemented
8#22549918C211yesmaximal, characteristic, complemented
9#22549872C211yesmaximal, characteristic, complemented
10C2×#16861130C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861217—24no—
14C2×#14523231C2212yescomplemented
15C22×C127⋊C8C2211yescharacteristic
16C2×C8×D127C2212yescomplemented
17#16861205—21no—
18#16861196—21no—
19#16861188—21no—
20#16861185C2212yescomplemented
21#16861183—21no—
22#16861181—22no—
23#16861180C2212yes—
24#16861180C2212yes—
25#16861180C2212yescomplemented
26#16861178—22no—
27#16861168C2211yescharacteristic
28C2×C4⋊Dic127C2211yescharacteristic
29#16861143C2212yescomplemented
30#16861141C2212yescomplemented
31#16861133C2211yescharacteristic, complemented
32#16861133C2211yescharacteristic, complemented
33#16861130C2212yes—
34#16861130C2212yes—
35#16861130C2211yescharacteristic, complemented
36#16861130C2211yescharacteristic, complemented
37C2×#41×C127C2211yescharacteristic
38C22×D4×C127C2211yescharacteristic
39AGL(1,ℤ/8)×C127—24no—
40C2×SD16×C127C2212yes—
41C2×D8×C127C2212yes—
42C2×M16×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254D411yescharacteristic
46C2×D508D411yescharacteristic
47C2×C22⋊D127—22no—
48C2×C22⋊D127—22no—
49C2×C22⋊D127—22no—
50C2×C22⋊D127D412yes—
51C22×Dic127—21no—
52C2×C4×D127C2312yes—
53#14523231—24no—
54C2×C127⋊C8—22no—
55C2×C127⋊C8C2311yescharacteristic
56C2×C127⋊C8C2311yescharacteristic
57C8×D127—24no—
58C4⋊Dic127—21no—
59C4⋊Dic127C2312yes—
60C4⋊Dic127C2311yescharacteristic
61C4⋊Dic127C2311yescharacteristic
62Dic127⋊C4—22no—
63C22⋊Dic127—22no—
64C22⋊Dic127—22no—
65C22⋊Dic127—22no—
66C22⋊Dic127—21no—
67(C2×D127)⋊C4—22no—
68C4×Dic127—21no—
69C24×C127—21no—
70#41×C127—22no—
71#41×C127—22no—
72C2×Q8×C127C2311yescharacteristic
73C2×D4×C127—21no—
74C2×D4×C127—22no—
75C2×D4×C127—21no—
76C2×D4×C127—22no—
77C2×D4×C127C2311yescharacteristic
78C2×D4×C127C2312yes—
79C22×C4×C127—21no—
80C22×C4×C127C2311yescharacteristic
81SD16×C127—24no—
82D8×C127—24no—
83M16×C127D414yescomplemented
84C2×C8×C127C2312yes—
85C22×D127C2×D412yes—
86Dic254—22no—
87D508—22no—
88C22⋊D127—24no—
89C22⋊D127—82no—
90C22⋊D127—82no—
91C22⋊D127—82no—
92C2×Dic127—21no—
93C2×Dic127—21no—
94C2×Dic127—22no—
95C2×Dic127—81no—
96C2×Dic127C2×D412yes—
97C4×D127—22no—
98C4×D127—22no—
99C127⋊C8—21no—
100C127⋊C8—21no—
101C23×C127—21no—
102C23×C127—22no—
103C23×C127—42no—
104C23×C127—81no—
105C23×C127—81no—
106C23×C127C2×D411yescharacteristic
107Q8×C127—21no—
108Q8×C127—21no—
109D4×C127—21no—
110D4×C127—22no—
111D4×C127—22no—
112D4×C127—21no—
113D4×C127—42no—
114D4×C127—42no—
115C2×C4×C127—21no—
116C2×C4×C127—42no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D412yes—
119C2×C4×C127#4112yes—
120C2×C4×C127C2411yescharacteristic
121C8×C127C2×D414yes—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—41no—
125C22×C127—42no—
126C22×C127—42no—
127C22×C127—41no—
128C22×C127—41no—
129C22×C127—42no—
130C22×C127—84no—
131C22×C127#12112yes—
132C22×C127C22×D411yescharacteristic
133C2×D127—22no—
134C2×D127—22no—
135C2×D127—22no—
136C4×C127—41no—
137C4×C127#12112yes—
138C4×C127C22×D411yescharacteristic
139C4×C127C2×#4111yescharacteristic
140Dic127—22no—
141Dic127—42no—
142Dic127—41no—
143Dic127—41no—
144C2×C127—22no—
145C2×C127—81no—
146C2×C127—82no—
147C2×C127C2×#12111yescharacteristic
148C2×C127#57611yescharacteristic
149C2×C127#57711yescharacteristic
150D127—42no—
151#3062—1271nomaximal, Sylow
152C127#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
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#416—1271no—
163C2×#102—1271no—
164C2×#41—1271no—
165C2×#41C2×D12711yescharacteristic
166C22×D4C2×D12711yescharacteristic
167AGL(1,ℤ/8)—24no—
168C2×SD16C2×D12712yes—
169C2×D8C2×D12712yes—
170#131—2544no—
171C2×M16—1272no—
172C2×M16C2×D12711yescharacteristic
173C22×C8—1271no—
174C22×C8—1272no—
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—21no—
196#41—22no—
197#41—22no—
198#41—2542no—
199#41—2542no—
200C2×Q8—1271no—
201C2×Q8C22×D12711yescharacteristic
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—1272no—
207C2×D4—1271no—
208C2×D4—2542no—
209C2×D4—2542no—
210C2×D4—2542no—
211C2×D4C22×D12711yescharacteristic
212C2×D4C22×D12712yes—
213C22×C4—21no—
214C22×C4—1272no—
215C22×C4—2541no—
216C22×C4C22×D12711yescharacteristic
217SD16—24no—
218D8—24no—
219M16—2544no—
220M16C22⋊D12714yescomplemented
221C2×C8—1271no—
222C2×C8—1271no—
223C2×C8—2542no—
224C2×C8—2544no—
225C2×C8C22×D12712yes—
226C4⋊C4—1272no—
227C4⋊C4—1271no—
228C4⋊C4—1271no—
229C4⋊C4—2542no—
230C4⋊C4—2541no—
231C22⋊C4—2542no—
232C22⋊C4—2542no—
233C22⋊C4—2542no—
234C22⋊C4—2542no—
235C22⋊C4—2541no—
236C42—2541no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—81no—
241C23—81no—
242C23—1272no—
243C23D4×D12711yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—2542no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—21no—
251D4—42no—
252D4—42no—
253D4—10162no—
254D4—10162no—
255D4—10162no—
256D4—2544no—
257D4—2542no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—1272no—
261C2×C4—10161no—
262C2×C4—2541no—
263C2×C4—2541no—
264C2×C4—2542no—
265C2×C4—2542no—
266C2×C4—2542no—
267C2×C4D4×D12711yescharacteristic
268C2×C4C2×C22⋊D12712yes—
269C2×C4#1452324712yes—
270C2×C4C23×D12711yescharacteristic
271C8—2541no—
272C8—2541no—
273C8C2×C22⋊D12714yes—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—84no—
283C22—2542no—
284C22—2542no—
285C22—2542no—
286C22#1686118012yes—
287C22C2×D4×D12711yescharacteristic
288C4—41no—
289C4—5082no—
290C4—5081no—
291C4—5081no—
292C4—2542no—
293C4#1686118012yes—
294C4C22×C22⋊D12711yescharacteristic
295C4C2×#1452324711yescharacteristic
296C2—22no—
297C2—82no—
298C2—81no—
299C2—5082no—
300C2C2×#1686118011yescharacteristic
301C2#2255068011yescharacteristic
302C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3062
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861217C2semidirect
2. #22550454C2semidirect
3. #22550443C2semidirect
4. #22550126C2semidirect
5. #22550119C2semidirect
6. #22550109C2semidirect
7. #22550102C2semidirect
8. #22549918C2semidirect
9. #22549872C2semidirect
10. C2×#16861130C2semidirect
11. C2×AGL(1,ℤ/8)×C127C2semidirect
14. C2×#14523231C22semidirect
16. C2×C8×D127C22semidirect
20. #16861185C22semidirect
25. #16861180C22semidirect
29. #16861143C22semidirect
30. #16861141C22semidirect
31. #16861133C22semidirect
32. #16861133C22semidirect
35. #16861130C22semidirect
36. #16861130C22semidirect
83. M16×C127D4semidirect
152. C127#3062semidirect
153. C2×AGL(1,ℤ/8)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×#168612172
#225504542
#225504432
#225501262
#225501192
#225501092
#225501022
#225499182
#225498722
C2×#168611302
C2×AGL(1,ℤ/8)×C1272
#3062127