Finite Group Databasebeta

← #30650339

Subgroups — #30650339

Order 16256.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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×#16861236C212yesmaximal, complemented
2#22550519C212yesmaximal, complemented
3#22550119C212yesmaximal, complemented
4#22550088C212yesmaximal, complemented
5#22550037C212yesmaximal, complemented
6#22550008C212yesmaximal, complemented
7C4×D1016C211yesmaximal, characteristic, complemented
8#22549955C211yesmaximal, characteristic, complemented
9#495×C127C211yesmaximal, characteristic, complemented
10C2×#14523249C2212yes—
11C2×D4×D127C2212yescomplemented
12#16861236—28no—
13C2×#14523235C2212yescomplemented
14C2×D1016C2211yescharacteristic
15C2×#14523233C2212yes—
16C2×#14523231C2212yescomplemented
17#16861191—22no—
18#16861185C2212yescomplemented
19#16861182—22no—
20#16861180C2212yes—
21#16861179—22no—
22C4×D508C2211yescharacteristic
23C4×D508C2212yescomplemented
24C22⋊C4×D127—22no—
25#16861143C2211yescharacteristic, complemented
26C4⋊(C127⋊C8)C2211yescharacteristic, complemented
27#16861133C2212yescomplemented
28#16861131C2212yescomplemented
29#16861130C2212yescomplemented
30(C2×D127)⋊C8C2212yescomplemented
31C2×Q16×C127C2211yescharacteristic, complemented
32C2×SD16×C127C2212yescomplemented
33C2×D8×C127C2211yescharacteristic
34#124×C127C2212yescomplemented
35C4×C8×C127C2211yescharacteristic, complemented
36C23×D127—22no—
37#14523249—24no—
38#14523249—42no—
39C2×D508—22no—
40C2×D508D411yescharacteristic
41C2×D508C2312yes—
42C2×C22⋊D127—22no—
43C2×C22⋊D127—22no—
44D4×D127—24no—
45D4×D127—42no—
46C2×C4×D127—22no—
47C2×C4×D127—22no—
48C2×C4×D127C2312yes—
49#14523235—24no—
50D1016D414yescomplemented
51#14523233—24no—
52#14523231—24no—
53C2×C127⋊C8C2312yes—
54C4⋊Dic127D411yescharacteristic
55C4⋊Dic127C2312yes—
56Dic127⋊C4—22no—
57C22⋊Dic127—22no—
58(C2×D127)⋊C4—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4D412yes—
61C2×Q8×C127C2312yes—
62C2×D4×C127C2312yes—
63Q16×C127—41no—
64SD16×C127—24no—
65D8×C127—41no—
66C2×C8×C127C2311yescharacteristic
67C2×C8×C127C2311yescharacteristic
68C22⋊C4×C127—22no—
69C22⋊C4×C127—22no—
70C42×C127C2311yescharacteristic
71C22×D127—22no—
72C22×D127—44no—
73C22×D127—44no—
74C22×D127—82no—
75C22×D127C2×D412yes—
76D508—22no—
77D508—41no—
78D508—44no—
79D508C2×D414yes—
80C22⋊D127—44no—
81C22⋊D127—82no—
82C2×Dic127—22no—
83C2×Dic127C2×D412yes—
84C4×D127—22no—
85C4×D127—22no—
86C4×D127—44no—
87C4×D127—82no—
88C127⋊C8—22no—
89C23×C127—22no—
90Q8×C127—22no—
91Q8×C127—22no—
92D4×C127—22no—
93D4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2411yescharacteristic
98C8×C127—21no—
99C8×C127C2×D412yes—
100C22×C127—42no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127C22×D411yescharacteristic
104C2×D127—22no—
105C2×D127—24no—
106C2×D127—22no—
107C2×D127—22no—
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—82no—
119C2×C127D4211yescharacteristic
120C2×C127#57511yescharacteristic
121C2×C127#57611yescharacteristic
122D127—42no—
123D127—44no—
124#3273—1271nomaximal, Sylow
125C127#327311yescharacteristic, complemented, Sylow
126C2×AGL(1,ℤ/8)—1272no—
127#546—1272no—
128#495D12711yescharacteristic, complemented
129#468—1272no—
130#463—1272no—
131#452—1272no—
132#447—1272no—
133C4×D8—1271no—
134#436—1271no—
135C2×#41—1272no—
136C22×D4—1272no—
137AGL(1,ℤ/8)—2548no—
138C2×Q16C2×D12711yescharacteristic, complemented
139C2×SD16—1272no—
140C2×SD16C2×D12712yescomplemented
141C2×D8—1272no—
142C2×D8—1271no—
143C2×D8C2×D12711yescharacteristic
144C2×M16—1272no—
145#124C2×D12712yescomplemented
146#123—2542no—
147C22⋊Q8—1272no—
148#121—1272no—
149#121—2542no—
150#120—2542no—
151C4×D4—1271no—
152C4×D4—1272no—
153C2×C22⋊C4—2542no—
154#107—1271no—
155C4⋊C8—1271no—
156#103—1272no—
157#102—1272no—
158#102—1272no—
159C22⋊C8—1272no—
160C4×C8C2×D12711yescharacteristic, complemented
161C24—2542no—
162#41—5082no—
163#41—2544no—
164C2×Q8C22×D12712yes—
165C2×D4—1271no—
166C2×D4—1272no—
167C2×D4—5082no—
168C2×D4—2542no—
169C2×D4—2542no—
170C2×D4—2544no—
171C2×D4—2542no—
172C2×D4C22×D12712yes—
173C22×C4—1272no—
174C22×C4—2542no—
175C22×C4—2542no—
176Q16—41no—
177SD16—24no—
178SD16—2544no—
179D8—41no—
180D8—1274no—
181D8—2544no—
182M16—2544no—
183C2×C8—1272no—
184C2×C8C22×D12711yescharacteristic
185C2×C8C22×D12711yescharacteristic
186C4⋊C4—1271no—
187C4⋊C4—1272no—
188C4⋊C4—2542no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—1272no—
192C22⋊C4—2542no—
193C22⋊C4—2542no—
194C22⋊C4—2542no—
195C42C22×D12711yescharacteristic
196C23—22no—
197C23—1272no—
198C23—10162no—
199C23—5084no—
200C23—5084no—
201C23—2542no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—22no—
206D4—1274no—
207D4—10162no—
208D4—5081no—
209D4—5084no—
210D4—5084no—
211D4—2542no—
212C2×C4—22no—
213C2×C4—1272no—
214C2×C4—10162no—
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—
224C8D4×D12712yes—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—10168no—
229C22—5082no—
230C22—5082no—
231C22—2542no—
232C22—2542no—
233C22—2544no—
234C22—2542no—
235C22C2×D4×D12711yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—5082no—
240C4—2542no—
241C4C2×D4×D12711yescharacteristic
242C4#1686125411yescharacteristic
243C2—82no—
244C2—5082no—
245C2—5084no—
246C2#2255051111yescharacteristic
247C2#2255067311yescharacteristic
248C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#495×C127C2
socleC22×C127C22×D4
hypercenter#495D127
hypocenterC127#3273
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. #22550519C2semidirect
3. #22550119C2semidirect
4. #22550088C2semidirect
5. #22550037C2semidirect
6. #22550008C2semidirect
7. C4×D1016C2semidirect
8. #22549955C2semidirect
9. #495×C127C2semidirect
11. C2×D4×D127C22semidirect
13. C2×#14523235C22semidirect
16. C2×#14523231C22semidirect
18. #16861185C22semidirect
23. C4×D508C22semidirect
25. #16861143C22semidirect
26. C4⋊(C127⋊C8)C22semidirect
27. #16861133C22semidirect
28. #16861131C22semidirect
29. #16861130C22semidirect
30. (C2×D127)⋊C8C22semidirect
31. C2×Q16×C127C22semidirect
32. C2×SD16×C127C22semidirect
34. #124×C127C22semidirect
35. C4×C8×C127C22semidirect
50. D1016D4semidirect
125. C127#3273semidirect
128. #495D127semidirect
138. C2×Q16C2×D127semidirect
140. C2×SD16C2×D127semidirect
145. #124C2×D127semidirect
160. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
#225505192
#225501192
#225500882
#225500372
#225500082
C4×D10162
#225499552
#495×C1272
#3273127