Finite Group Databasebeta

← #30650498

Subgroups — #30650498

Order 16256.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#22550684C212yesmaximal, complemented
2#22550680C212yesmaximal, complemented
3#22550266C211yesmaximal, characteristic, complemented
4#22550144C212yesmaximal, complemented
5#22550142C212yesmaximal, complemented
6#22550053C211yesmaximal, characteristic, complemented
7#22550045C212yesmaximal, complemented
8#22549995C212yesmaximal, complemented
9#454×C127C211yesmaximal, characteristic, complemented
10#16861255C2212yescomplemented
11#16861254C2211yescharacteristic
12#16861240—22no—
13#16861240C2212yescomplemented
14#16861237C2212yes—
15#16861236—22no—
16#16861234—22no—
17#16861234C2212yescomplemented
18#16861233—22no—
19#16861230—22no—
20#16861229—22no—
21#16861229C2212yes—
22C2×#14523235—22no—
23C2×#14523235C2212yescomplemented
24C2×#14523233C2212yescomplemented
25D8×D127—22no—
26#16861217C2212yescomplemented
27#16861216C2212yescomplemented
28#16861206C2211yescharacteristic, complemented
29#16861144C2212yescomplemented
30#16861137C2212yescomplemented
31#16861126C2211yescharacteristic, complemented
32#16861125C2212yescomplemented
33#16861120C2211yescharacteristic, complemented
342−1+4×C127C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C127C2212yescomplemented
36#124×C127C2211yescharacteristic, complemented
37#104×C127C2212yescomplemented
38#101×C127C2211yescharacteristic, complemented
39#14523249—22no—
40#14523248—22no—
41#14523248C2312yes—
42#14523247—22no—
43C2×D508—22no—
44C2×D508—41no—
45C2×D508C2312yes—
46C2×C22⋊D127—41no—
47D4×D127—22no—
48D4×D127—22no—
49D4×D127—42no—
50#14523238—22no—
51#14523238D412yescomplemented
52#14523237—22no—
53#14523236D412yescomplemented
54#14523235—22no—
55#14523235—22no—
56#14523235—22no—
57#14523235—22no—
58#14523235D412yescomplemented
59D1016—22no—
60#14523233—22no—
61#14523233—22no—
62#14523233—22no—
63#14523233D412yescomplemented
64#14523232—22no—
65#14523232C2312yes—
66#14523231—22no—
67C2×C127⋊C8—22no—
68C2×C127⋊C8C2312yes—
69C8×D127—22no—
70#41×C127—22no—
71#41×C127—41no—
72#41×C127C2312yes—
73C2×Q8×C127—22no—
74C2×Q8×C127C2311yescharacteristic
75C2×D4×C127C2311yescharacteristic
76SD16×C127—22no—
77D8×C127—22no—
78M16×C127C2312yes—
79C22⋊C4×C127—41no—
80C42×C127C2311yescharacteristic
81C22×D127—22no—
82C22×D127—41no—
83C22×D127—42no—
84Dic254C2×D412yes—
85D508—22no—
86D508—22no—
87D508—42no—
88D508—82no—
89D508C2×D412yes—
90C22⋊D127—22no—
91C22⋊D127—42no—
92C22⋊D127—42no—
93C2×Dic127—41no—
94C4×D127—22no—
95C4×D127—42no—
96C127⋊C8—22no—
97C127⋊C8C2×D412yes—
98C127⋊C8C2×D412yes—
99C23×C127—21no—
100Q8×C127—22no—
101Q8×C127—41no—
102Q8×C127C2×D412yes—
103D4×C127—22no—
104D4×C127—22no—
105D4×C127—41no—
106D4×C127C2×D412yes—
107C2×C4×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—42no—
111C2×C4×C127C2411yescharacteristic
112C8×C127—22no—
113C22×C127—22no—
114C22×C127—41no—
115C22×C127—81no—
116C22×C127H(5,𝔽2)11yescharacteristic
117C2×D127—22no—
118C2×D127—42no—
119C2×D127—82no—
120C2×D127—161no—
121C2×D127—162no—
122C4×C127—22no—
123C4×C127—41no—
124C4×C127—41no—
125C4×C127C22×D412yes—
126Dic127—22no—
127C2×C127—21no—
128C2×C127—42no—
129C2×C127—81no—
130C2×C127D4211yescharacteristic
131D127—42no—
132D127—82no—
133#3288—1271nomaximal, Sylow
134C127#328811yescharacteristic, complemented, Sylow
135#577—1272no—
136#576—1272no—
137#496—1271no—
138#472—1272no—
139#471—1272no—
140#455—1271no—
141#454D12711yescharacteristic, complemented
142#453—1272no—
143#444—1272no—
1442−1+4C2×D12711yescharacteristic, complemented
145H(5,𝔽2)—1272no—
146H(5,𝔽2)—1271no—
147#137—1272no—
148#137—2542no—
149AGL(1,ℤ/8)—1272no—
150AGL(1,ℤ/8)—1272no—
151AGL(1,ℤ/8)—2542no—
152AGL(1,ℤ/8)—2542no—
153AGL(1,ℤ/8)—2542no—
154AGL(1,ℤ/8)C2×D12712yescomplemented
155#135—1272no—
156#135—2542no—
157#135—2542no—
158C2×SD16—1272no—
159C2×SD16—2542no—
160C2×D8—1272no—
161C2×D8—2542no—
162#131—1272no—
163#131—1272no—
164#127—1271no—
165#124C2×D12711yescharacteristic, complemented
166#108—1272no—
167#104—1272no—
168#104C2×D12712yescomplemented
169#101C2×D12711yescharacteristic, complemented
170#100—1271no—
171#100—1272no—
172#97—1271no—
173#41—22no—
174#41—41no—
175#41—1272no—
176#41—2542no—
177#41—2542no—
178#41—2542no—
179#41C22×D12712yes—
180C2×Q8—22no—
181C2×Q8C22×D12711yescharacteristic
182C2×D4—1272no—
183C2×D4—5081no—
184C2×D4—5082no—
185C2×D4—5081no—
186C2×D4—2542no—
187C2×D4—2542no—
188C2×D4—2542no—
189C2×D4C22×D12711yescharacteristic
190Q16—1272no—
191Q16—2542no—
192SD16—22no—
193SD16—1272no—
194SD16—1272no—
195SD16—2542no—
196SD16—2542no—
197SD16—2542no—
198SD16—2542no—
199SD16—2542no—
200D8—22no—
201D8—1272no—
202D8—2542no—
203D8—2542no—
204D8—2542no—
205D8—2542no—
206M16—1272no—
207M16—2542no—
208M16—2542no—
209M16C22×D12712yes—
210C2×C8—1272no—
211C2×C8—2542no—
212C2×C8—2542no—
213C22⋊C4—41no—
214C42C22×D12711yescharacteristic
215C23—21no—
216C23—5082no—
217C23—5081no—
218C23—2542no—
219Q8—22no—
220Q8—41no—
221Q8—1272no—
222Q8D4×D12712yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—1272no—
227D4—10162no—
228D4—5082no—
229D4—5082no—
230D4—5082no—
231D4—2542no—
232D4—2542no—
233D4—2542no—
234D4D4×D12712yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—5082no—
240C2×C4—5081no—
241C2×C4—2542no—
242C2×C4C23×D12711yescharacteristic
243C8—22no—
244C8—1272no—
245C8—1272no—
246C8—2542no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—20322no—
251C22—20321no—
252C22—10162no—
253C22—5082no—
254C22—2542no—
255C22#1686125411yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—2542no—
260C4C2×D4×D12712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—10162no—
265C2—5082no—
266C2#2255050811yescharacteristic
subgroupG/H
centerC2#22550508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#454×C127C2
socleC2×C127D42
hypercenter#454D127
hypocenterC127#3288
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #22550684C2semidirect
2. #22550680C2semidirect
3. #22550266C2semidirect
4. #22550144C2semidirect
5. #22550142C2semidirect
6. #22550053C2semidirect
7. #22550045C2semidirect
8. #22549995C2semidirect
9. #454×C127C2semidirect
10. #16861255C22semidirect
13. #16861240C22semidirect
17. #16861234C22semidirect
23. C2×#14523235C22semidirect
24. C2×#14523233C22semidirect
26. #16861217C22semidirect
27. #16861216C22semidirect
28. #16861206C22semidirect
29. #16861144C22semidirect
30. #16861137C22semidirect
31. #16861126C22semidirect
32. #16861125C22semidirect
33. #16861120C22semidirect
34. 2−1+4×C127C22semidirect
35. AGL(1,ℤ/8)×C127C22semidirect
36. #124×C127C22semidirect
37. #104×C127C22semidirect
38. #101×C127C22semidirect
51. #14523238D4semidirect
53. #14523236D4semidirect
58. #14523235D4semidirect
63. #14523233D4semidirect
134. C127#3288semidirect
141. #454D127semidirect
144. 2−1+4C2×D127semidirect
154. AGL(1,ℤ/8)C2×D127semidirect
165. #124C2×D127semidirect
168. #104C2×D127semidirect
169. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506842
#225506802
#225502662
#225501442
#225501422
#225500532
#225500452
#225499952
#454×C1272
#3288127