Finite Group Databasebeta

← #30650524

Subgroups — #30650524

Order 16256.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550688C212yesmaximal, complemented
2#22550683C212yesmaximal, complemented
3#22550249C211yesmaximal, characteristic, complemented
4#22550140C212yesmaximal, complemented
5#22550139C212yesmaximal, complemented
6#22550060C212yesmaximal, complemented
7#22550045C212yesmaximal, complemented
8#22549994C211yesmaximal, characteristic, complemented
9#443×C127C211yesmaximal, characteristic, complemented
10#16861256C2212yes—
11#16861255C2212yescomplemented
12#16861242—22no—
13#16861242C2212yescomplemented
14#16861241—22no—
15#16861239C2212yescomplemented
16#16861237—22no—
17#16861237C2212yes—
18#16861234C2212yes—
19#16861233—22no—
20#16861231—22no—
21#16861231C2211yescharacteristic
22#16861230—22no—
23C2×#14523237—22no—
24C2×#14523237C2212yescomplemented
25C2×#14523237C2212yescomplemented
26SD16×D127—22no—
27#16861215C2212yescomplemented
28#16861209C2211yescharacteristic, complemented
29#16861206C2211yescharacteristic
30#16861145C2211yescharacteristic, complemented
31#16861137C2212yescomplemented
32#16861127C2212yescomplemented
33#16861125C2212yescomplemented
34#135×C127C2211yescharacteristic
35#131×C127C2212yescomplemented
36#108×C127C2211yescharacteristic, complemented
37#104×C127C2212yescomplemented
38C4×C8×C127C2211yescharacteristic, complemented
39#14523249—22no—
40#14523248—22no—
41#14523248—22no—
42#14523248C2312yes—
43#14523247—22no—
44#14523247—42no—
45C2×Dic254—22no—
46C2×Dic254C2312yes—
47Q8×D127—22no—
48C2×D508—22no—
49C2×D508—41no—
50C2×D508C2312yes—
51D4×D127—22no—
52D4×D127—42no—
53Dic508—22no—
54Dic508D411yescharacteristic, complemented
55#14523238—22no—
56#14523237—22no—
57#14523237—22no—
58#14523237—22no—
59#14523237—22no—
60#14523237—22no—
61#14523237—22no—
62#14523237D412yescomplemented
63#14523236—22no—
64#14523235—22no—
65D1016—22no—
66D1016D411yescharacteristic, complemented
67#14523233—22no—
68#14523232C2312yes—
69#14523231—22no—
70C8×D127—22no—
71C4⋊Dic127—41no—
72#41×C127C2312yes—
73Q16×C127D411yescharacteristic, complemented
74SD16×C127D412yescomplemented
75D8×C127D411yescharacteristic, complemented
76M16×C127—22no—
77M16×C127C2312yes—
78C2×C8×C127—22no—
79C2×C8×C127C2311yescharacteristic
80C2×C8×C127C2311yescharacteristic
81C42×C127C2311yescharacteristic
82C22×D127—22no—
83C22×D127—42no—
84Dic254—22no—
85Dic254—22no—
86Dic254—42no—
87Dic254C2×D412yes—
88D508—22no—
89D508—22no—
90D508—42no—
91D508—82no—
92D508C2×D412yes—
93C22⋊D127—22no—
94C22⋊D127—42no—
95C22⋊D127—42no—
96C2×Dic127—22no—
97C2×Dic127—42no—
98C4×D127—22no—
99C4×D127—42no—
100C4×D127—42no—
101C127⋊C8—22no—
102Q8×C127C2×D412yes—
103D4×C127—22no—
104D4×C127C2×D412yes—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127C2411yescharacteristic
108C8×C127—22no—
109C8×C127C2×D411yescharacteristic
110C8×C127C2×D411yescharacteristic
111C8×C127C2×D411yescharacteristic
112C8×C127C2×D411yescharacteristic
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×C127C22×D411yescharacteristic
122C4×C127C22×D411yescharacteristic
123Dic127—22no—
124Dic127—42no—
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—1272no—
138#453—1272no—
139#443—1271no—
140#443D12711yescharacteristic, complemented
1412−1+4—1272no—
142H(5,𝔽2)—1272no—
143#137—1272no—
144#137—1272no—
145#137—2542no—
146#137—2542no—
147AGL(1,ℤ/8)—1272no—
148AGL(1,ℤ/8)—1272no—
149AGL(1,ℤ/8)—2542no—
150AGL(1,ℤ/8)—2542no—
151#135—1271no—
152#135—2542no—
153#135—2542no—
154#135C2×D12711yescharacteristic
155C2×SD16—1272no—
156C2×SD16—1272no—
157C2×SD16—2542no—
158C2×SD16—2542no—
159#131—1272no—
160#131C2×D12712yescomplemented
161C4⋊Q8—1271no—
162#127—1271no—
163#108—1271no—
164#108C2×D12711yescharacteristic, complemented
165#104—1272no—
166#104C2×D12712yescomplemented
167#101—1272no—
168#100—1272no—
169C4×C8C2×D12711yescharacteristic, complemented
170#41—1272no—
171#41—5082no—
172#41—2542no—
173#41—2542no—
174#41—2542no—
175#41—2542no—
176#41C22×D12712yes—
177C2×Q8—1272no—
178C2×Q8—2542no—
179C2×Q8—2542no—
180C2×D4—1272no—
181C2×D4—5081no—
182C2×D4—5082no—
183C2×D4—2542no—
184C2×D4—2542no—
185Q16—1271no—
186Q16—2542no—
187Q16—2542no—
188Q16D50811yescharacteristic, complemented
189SD16—1272no—
190SD16—2542no—
191SD16—2542no—
192SD16—2542no—
193SD16—2542no—
194SD16—2542no—
195SD16—2542no—
196SD16—2542no—
197SD16—2542no—
198SD16D50812yescomplemented
199D8—1271no—
200D8—2542no—
201D8—2542no—
202D8D50811yescharacteristic, complemented
203M16—22no—
204M16—1272no—
205M16—2542no—
206M16C22×D12712yes—
207C2×C8—22no—
208C2×C8—2542no—
209C2×C8C22×D12711yescharacteristic
210C2×C8C22×D12711yescharacteristic
211C4⋊C4—5081no—
212C42C22×D12711yescharacteristic
213C23—5082no—
214C23—2542no—
215Q8—1272no—
216Q8—5082no—
217Q8—2542no—
218Q8—2542no—
219Q8C2×D50812yes—
220D4—22no—
221D4—1272no—
222D4—10162no—
223D4—5082no—
224D4—5082no—
225D4—5082no—
226D4—2542no—
227D4—2542no—
228D4—2542no—
229D4C2×D50812yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—5082no—
233C2×C4—5082no—
234C2×C4—5082no—
235C2×C4—2542no—
236C2×C4—2542no—
237C2×C4C23×D12711yescharacteristic
238C8—22no—
239C8—2542no—
240C8D4×D12711yescharacteristic
241C8D4×D12711yescharacteristic
242C8C2×D50811yescharacteristic
243C8C2×D50811yescharacteristic
244C22—22no—
245C22—20322no—
246C22—10162no—
247C22—5082no—
248C22—2542no—
249C22#1686125511yescharacteristic
250C4—22no—
251C4—22no—
252C4—5082no—
253C4—2542no—
254C4C2×D4×D12711yescharacteristic
255C4C22×D50811yescharacteristic
256C2—21no—
257C2—42no—
258C2—10162no—
259C2—5082no—
260C2D4×D50811yescharacteristic
subgroupG/H
centerC2D4×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#443×C127C2
socleC2×C127D42
hypercenter#443D127
hypocenterC127#3291
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #22550688C2semidirect
2. #22550683C2semidirect
3. #22550249C2semidirect
4. #22550140C2semidirect
5. #22550139C2semidirect
6. #22550060C2semidirect
7. #22550045C2semidirect
8. #22549994C2semidirect
9. #443×C127C2semidirect
11. #16861255C22semidirect
13. #16861242C22semidirect
15. #16861239C22semidirect
24. C2×#14523237C22semidirect
25. C2×#14523237C22semidirect
27. #16861215C22semidirect
28. #16861209C22semidirect
30. #16861145C22semidirect
31. #16861137C22semidirect
32. #16861127C22semidirect
33. #16861125C22semidirect
35. #131×C127C22semidirect
36. #108×C127C22semidirect
37. #104×C127C22semidirect
38. C4×C8×C127C22semidirect
54. Dic508D4semidirect
62. #14523237D4semidirect
66. D1016D4semidirect
73. Q16×C127D4semidirect
74. SD16×C127D4semidirect
75. D8×C127D4semidirect
131. C127#3291semidirect
140. #443D127semidirect
160. #131C2×D127semidirect
164. #108C2×D127semidirect
166. #104C2×D127semidirect
169. C4×C8C2×D127semidirect
188. Q16D508semidirect
198. SD16D508semidirect
202. D8D508semidirect

Maximal subgroups

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

subgroupindex
#225506882
#225506832
#225502492
#225501402
#225501392
#225500602
#225500452
#225499942
#443×C1272
#3291127