Finite Group Databasebeta

← #30650340

Subgroups — #30650340

Order 16256.

520 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 129 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×#16861233C212yesmaximal, complemented
2#22550521C212yesmaximal, complemented
3#22550260C211yesmaximal, characteristic, complemented
4#22550123C212yesmaximal, complemented
5#22550091C212yesmaximal, complemented
6#22550039C212yesmaximal, complemented
7#22550011C212yesmaximal, complemented
8#22549955C211yesmaximal, characteristic, complemented
9C4×D8×C127C211yesmaximal, characteristic, complemented
10C2×#14523247C2212yes—
11C2×D4×D127C2212yescomplemented
12#16861233—28no—
13C2×Dic508C2211yescharacteristic, complemented
14C2×#14523237C2212yescomplemented
15C2×#14523236C2212yescomplemented
16C2×D1016C2211yescharacteristic
17C2×#14523233C2212yes—
18C2×#14523231C2212yescomplemented
19#16861197C2212yescomplemented
20#16861193—22no—
21#16861189C2212yescomplemented
22#16861184C2212yes—
23#16861183—22no—
24#16861179—22no—
25C4×D508C2211yescharacteristic
26C2×(C2×D127)⋊C4—22no—
27C4⋊(C127⋊C8)C2211yescharacteristic, complemented
28#16861132C2212yescomplemented
29#16861129C2212yescomplemented
30(C2×D127)⋊C8C2212yescomplemented
31C2×D8×C127C2211yescharacteristic
32C4×D4×C127C2212yescomplemented
33#107×C127C2211yescharacteristic, complemented
34#102×C127C2212yescomplemented
35C4×C8×C127C2211yescharacteristic, complemented
36C23×D127—22no—
37#14523247—24no—
38#14523247—42no—
39C2×Dic254C2312yes—
40C2×D508—22no—
41C2×D508D411yescharacteristic
42C2×D508C2312yes—
43C2×C22⋊D127—22no—
44C2×C22⋊D127—22no—
45D4×D127—24no—
46D4×D127—42no—
47C22×Dic127—22no—
48C2×C4×D127C2312yes—
49Dic508—41no—
50#14523237—24no—
51#14523236—24no—
52D1016—41no—
53#14523233—24no—
54#14523231—24no—
55C2×C127⋊C8C2312yes—
56C4⋊Dic127—22no—
57C4⋊Dic127D411yescharacteristic
58(C2×D127)⋊C4—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4D412yes—
63C2×D4×C127C2312yes—
64C22×C4×C127—22no—
65D8×C127D414yescomplemented
66C2×C8×C127C2311yescharacteristic
67C2×C8×C127C2311yescharacteristic
68C4⋊C4×C127C2312yes—
69C22⋊C4×C127—22no—
70C42×C127C2311yescharacteristic
71C22×D127—22no—
72C22×D127—44no—
73C22×D127—44no—
74C22×D127—82no—
75C22×D127C2×D412yes—
76Dic254—22no—
77Dic254—22no—
78D508—22no—
79D508—22no—
80D508—41no—
81D508—82no—
82C22⋊D127—44no—
83C22⋊D127—44no—
84C2×Dic127—22no—
85C2×Dic127—44no—
86C2×Dic127C2×D412yes—
87C4×D127—22no—
88C4×D127—22no—
89C127⋊C8—22no—
90C23×C127—22no—
91D4×C127—22no—
92D4×C127C2×D414yes—
93C2×C4×C127—22no—
94C2×C4×C127—82no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2411yescharacteristic
98C8×C127—21no—
99C8×C127C2×D412yes—
100C22×C127—24no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127C22×D411yescharacteristic
104C2×D127—22no—
105C2×D127—22no—
106C2×D127—22no—
107C2×D127—42no—
108C2×D127—42no—
109C2×D127—42no—
110C2×D127—88no—
111C4×C127—21no—
112C4×C127—21no—
113C4×C127—42no—
114C4×C127C22×D411yescharacteristic
115C4×C127H(5,𝔽2)11yescharacteristic
116Dic127—22no—
117Dic127—42no—
118C2×C127—44no—
119C2×C127D4211yescharacteristic
120C2×C127#57511yescharacteristic
121C2×C127#57611yescharacteristic
122D127—42no—
123D127—82no—
124#3273—1271nomaximal, Sylow
125C127#327311yescharacteristic, complemented, Sylow
126C2×AGL(1,ℤ/8)—1272no—
127#546—1272no—
128#495—1271no—
129#468—1272no—
130#463—1272no—
131#452—1272no—
132#447—1272no—
133C4×D8D12711yescharacteristic, complemented
134#436—1271no—
135C2×#41—1272no—
136C22×D4—1272no—
137AGL(1,ℤ/8)—2548no—
138C2×Q16—1271no—
139C2×SD16—1272no—
140C2×SD16—1272no—
141C2×D8—1272no—
142C2×D8—1271no—
143C2×D8C2×D12711yescharacteristic
144C2×M16—1272no—
145#124—1272no—
146#123—2542no—
147C22⋊Q8—1272no—
148#121—1272no—
149#121—2542no—
150#120—2542no—
151C4×D4—1271no—
152C4×D4C2×D12712yescomplemented
153C2×C22⋊C4—2542no—
154#107C2×D12711yescharacteristic, complemented
155C4⋊C8—1271no—
156#103—1272no—
157#102—1272no—
158#102C2×D12712yescomplemented
159C22⋊C8—1272no—
160C4×C8C2×D12711yescharacteristic, complemented
161C24—2542no—
162#41—5082no—
163#41—2544no—
164C2×Q8—1272no—
165C2×D4—1271no—
166C2×D4—1272no—
167C2×D4—5082no—
168C2×D4—2542no—
169C2×D4—2542no—
170C2×D4—2542no—
171C2×D4—2544no—
172C2×D4C22×D12712yes—
173C22×C4—22no—
174C22×C4—1272no—
175C22×C4—2542no—
176Q16—5081no—
177SD16—2544no—
178SD16—2544no—
179D8—5081no—
180D8—2544no—
181D8D50814yescomplemented
182M16—2544no—
183C2×C8—1272no—
184C2×C8C22×D12711yescharacteristic
185C2×C8C22×D12711yescharacteristic
186C4⋊C4—1271no—
187C4⋊C4—2542no—
188C4⋊C4C22×D12712yes—
189C22⋊C4—22no—
190C22⋊C4—1272no—
191C22⋊C4—2542no—
192C22⋊C4—2542no—
193C22⋊C4—2542no—
194C22⋊C4—2542no—
195C42C22×D12711yescharacteristic
196C23—22no—
197C23—1272no—
198C23—10162no—
199C23—5084no—
200C23—5084no—
201C23—2542no—
202Q8—2542no—
203Q8—2542no—
204D4—22no—
205D4—10162no—
206D4—5081no—
207D4—5084no—
208D4—5084no—
209D4—2542no—
210D4—2542no—
211D4C2×D50814yes—
212C2×C4—22no—
213C2×C4—82no—
214C2×C4—1272no—
215C2×C4—5084no—
216C2×C4—2542no—
217C2×C4—2542no—
218C2×C4—2542no—
219C2×C4D4×D12711yescharacteristic
220C2×C4D4×D12711yescharacteristic
221C2×C4C23×D12711yescharacteristic
222C8—21no—
223C8—2542no—
224C8C2×D50812yes—
225C22—24no—
226C22—42no—
227C22—42no—
228C22—10168no—
229C22—5082no—
230C22—5082no—
231C22—5082no—
232C22—2542no—
233C22—2542no—
234C22—2542no—
235C22C2×D4×D12711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—5082no—
240C4—2542no—
241C4C22×D50811yescharacteristic
242C4#1686125511yescharacteristic
243C2—44no—
244C2—10162no—
245C2—5082no—
246C2D4×D50811yescharacteristic
247C2#2255067311yescharacteristic
248C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×D8×C127C2
socleC22×C127C22×D4
hypercenterC4×D8D127
hypocenterC127#3273
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. #22550521C2semidirect
3. #22550260C2semidirect
4. #22550123C2semidirect
5. #22550091C2semidirect
6. #22550039C2semidirect
7. #22550011C2semidirect
8. #22549955C2semidirect
9. C4×D8×C127C2semidirect
11. C2×D4×D127C22semidirect
13. C2×Dic508C22semidirect
14. C2×#14523237C22semidirect
15. C2×#14523236C22semidirect
18. C2×#14523231C22semidirect
19. #16861197C22semidirect
21. #16861189C22semidirect
27. C4⋊(C127⋊C8)C22semidirect
28. #16861132C22semidirect
29. #16861129C22semidirect
30. (C2×D127)⋊C8C22semidirect
32. C4×D4×C127C22semidirect
33. #107×C127C22semidirect
34. #102×C127C22semidirect
35. C4×C8×C127C22semidirect
65. D8×C127D4semidirect
125. C127#3273semidirect
133. C4×D8D127semidirect
152. C4×D4C2×D127semidirect
154. #107C2×D127semidirect
158. #102C2×D127semidirect
160. C4×C8C2×D127semidirect
181. D8D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
#225505212
#225502602
#225501232
#225500912
#225500392
#225500112
#225499552
C4×D8×C1272
#3273127