Finite Group Databasebeta

← #30650520

Subgroups — #30650520

Order 16256.

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

normal subgroups only

This group has 460 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#22550685C212yesmaximal, complemented
2#22550684C212yesmaximal, complemented
3#22550250C211yesmaximal, characteristic, complemented
4#22550142C212yesmaximal, complemented
5#22550141C212yesmaximal, complemented
6#22550058C211yesmaximal, characteristic, complemented
7#22550045C212yesmaximal, complemented
8#22549990C212yesmaximal, complemented
9#456×C127C211yesmaximal, characteristic, complemented
10#16861257C2211yescharacteristic
11#16861255C2212yescomplemented
12#16861241—22no—
13#16861240—22no—
14#16861240C2212yescomplemented
15#16861237C2212yes—
16#16861236—22no—
17#16861234—22no—
18#16861234C2212yes—
19#16861232—22no—
20#16861229—22no—
21#16861229C2212yes—
22C2×#14523235—22no—
23C2×#14523235C2212yescomplemented
24C2×#14523235C2212yescomplemented
25SD16×D127—22no—
26#16861217C2212yescomplemented
27#16861216C2212yescomplemented
28#16861206C2211yescharacteristic
29#16861144C2212yescomplemented
30#16861137C2212yescomplemented
31#16861128C2211yescharacteristic, complemented
32#16861125C2212yescomplemented
33C4×C127⋊C8C2211yescharacteristic, complemented
342−1+4×C127C2211yescharacteristic
35#137×C127C2212yescomplemented
36C4⋊Q8×C127C2211yescharacteristic, complemented
37#104×C127C2212yescomplemented
38#101×C127C2211yescharacteristic, complemented
39#14523249—22no—
40#14523249—22no—
41#14523248—22no—
42#14523248—41no—
43#14523248C2312yes—
44Q8×D127—22no—
45C2×D508—22no—
46C2×D508—41no—
47C2×D508C2312yes—
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#14523235—22no—
59#14523235—22no—
60#14523235D412yescomplemented
61D1016—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×Q8×C127C2311yescharacteristic
76Q16×C127—22no—
77SD16×C127—22no—
78M16×C127C2312yes—
79C4⋊C4×C127—41no—
80C42×C127C2311yescharacteristic
81C22×D127—22no—
82C22×D127—42no—
83Dic254—41no—
84Dic254C2×D412yes—
85D508—22no—
86D508—22no—
87D508—42no—
88D508—41no—
89D508—82no—
90D508C2×D412yes—
91C22⋊D127—22no—
92C22⋊D127—42no—
93C4×D127—22no—
94C4×D127—42no—
95C4×D127—42no—
96C127⋊C8—22no—
97C127⋊C8C2×D412yes—
98C127⋊C8C2×D412yes—
99Q8×C127—22no—
100Q8×C127—22no—
101Q8×C127—41no—
102Q8×C127C2×D412yes—
103D4×C127—22no—
104D4×C127—41no—
105D4×C127C2×D412yes—
106C2×C4×C127—21no—
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×C127H(5,𝔽2)11yescharacteristic
115C2×D127—22no—
116C2×D127—42no—
117C2×D127—82no—
118C2×D127—162no—
119C4×C127—22no—
120C4×C127—22no—
121C4×C127—41no—
122C4×C127—41no—
123C4×C127C22×D412yes—
124Dic127—22no—
125C2×C127—21no—
126C2×C127—42no—
127C2×C127D4211yescharacteristic
128D127—42no—
129D127—82no—
130#3291—1271nomaximal, Sylow
131C127#329111yescharacteristic, complemented, Sylow
132#577—1272no—
133#577—1272no—
134#492—1271no—
135#471—1272no—
136#471—1272no—
137#456—1271no—
138#456D12711yescharacteristic, complemented
139#453—1272no—
140#443—1272no—
1412−1+4—1271no—
1422−1+4C2×D12711yescharacteristic
143H(5,𝔽2)—1272no—
144#137—1272no—
145#137—2542no—
146#137—2542no—
147#137C2×D12712yescomplemented
148AGL(1,ℤ/8)—1272no—
149AGL(1,ℤ/8)—1272no—
150AGL(1,ℤ/8)—2542no—
151AGL(1,ℤ/8)—2542no—
152#135—1272no—
153#135—2542no—
154#135—2542no—
155C2×SD16—1272no—
156C2×SD16—1272no—
157C2×SD16—2542no—
158C2×SD16—2542no—
159#131—1272no—
160#131—1272no—
161C4⋊Q8C2×D12711yescharacteristic, complemented
162#127—1271no—
163#108—1272no—
164#104—1272no—
165#104C2×D12712yescomplemented
166#101—1271no—
167#101C2×D12711yescharacteristic, complemented
168#100—1272no—
169C4×C8—1271no—
170#41—22no—
171#41—41no—
172#41—1272no—
173#41—5081no—
174#41—2542no—
175#41—2542no—
176#41—2542no—
177#41C22×D12712yes—
178C2×Q8—22no—
179C2×Q8—2542no—
180C2×Q8C22×D12711yescharacteristic
181C2×Q8C22×D12711yescharacteristic
182C2×D4—1272no—
183C2×D4—5081no—
184C2×D4—5082no—
185C2×D4—2542no—
186C2×D4—2542no—
187Q16—22no—
188Q16—1272no—
189Q16—2542no—
190SD16—22no—
191SD16—1272no—
192SD16—1272no—
193SD16—2542no—
194SD16—2542no—
195SD16—2542no—
196SD16—2542no—
197SD16—2542no—
198SD16—2542no—
199SD16—2542no—
200D8—1272no—
201D8—2542no—
202D8—2542no—
203M16—1272no—
204M16—2542no—
205M16—2542no—
206M16C22×D12712yes—
207C2×C8—1272no—
208C2×C8—2542no—
209C2×C8—2542no—
210C4⋊C4—41no—
211C42C22×D12711yescharacteristic
212C23—5082no—
213C23—2542no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—1272no—
218Q8—5081no—
219Q8D4×D12712yes—
220D4—22no—
221D4—41no—
222D4—1272no—
223D4—10162no—
224D4—5082no—
225D4—5081no—
226D4—5082no—
227D4—2542no—
228D4—2542no—
229D4—2542no—
230D4D4×D12712yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—5082no—
237C2×C4—5082no—
238C2×C4—2542no—
239C2×C4C23×D12711yescharacteristic
240C8—22no—
241C8—1272no—
242C8—1272no—
243C8—2542no—
244C22—22no—
245C22—20322no—
246C22—10162no—
247C22—5082no—
248C22—2542no—
249C22#1686125411yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—2542no—
255C4C2×D4×D12712yes—
256C2—21no—
257C2—42no—
258C2—10162no—
259C2—5082no—
260C2#2255050811yescharacteristic
subgroupG/H
centerC2#22550508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#456×C127C2
socleC2×C127D42
hypercenter#456D127
hypocenterC127#3291
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #22550685C2semidirect
2. #22550684C2semidirect
3. #22550250C2semidirect
4. #22550142C2semidirect
5. #22550141C2semidirect
6. #22550058C2semidirect
7. #22550045C2semidirect
8. #22549990C2semidirect
9. #456×C127C2semidirect
11. #16861255C22semidirect
14. #16861240C22semidirect
23. C2×#14523235C22semidirect
24. C2×#14523235C22semidirect
26. #16861217C22semidirect
27. #16861216C22semidirect
29. #16861144C22semidirect
30. #16861137C22semidirect
31. #16861128C22semidirect
32. #16861125C22semidirect
33. C4×C127⋊C8C22semidirect
35. #137×C127C22semidirect
36. C4⋊Q8×C127C22semidirect
37. #104×C127C22semidirect
38. #101×C127C22semidirect
51. #14523238D4semidirect
53. #14523236D4semidirect
60. #14523235D4semidirect
63. #14523233D4semidirect
131. C127#3291semidirect
138. #456D127semidirect
147. #137C2×D127semidirect
161. C4⋊Q8C2×D127semidirect
165. #104C2×D127semidirect
167. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506852
#225506842
#225502502
#225501422
#225501412
#225500582
#225500452
#225499902
#456×C1272
#3291127