Finite Group Databasebeta

← #30647956

Subgroups — #30647956

Order 16256.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D508C211yesmaximal, characteristic, complemented
2C2×#16861234C212yesmaximal, complemented
3#22550026C214yesmaximal, complemented
4#22550009C214yesmaximal, complemented
5#22549880C212yesmaximal, complemented
6#22549840C211yesmaximal, characteristic, complemented
7#525×C127C211yesmaximal, characteristic, complemented
8C24×D127—21no—
9C22×D508—28no—
10C22×D508—22no—
11C22×D508—21no—
12C22×D508C2214yes—
13C22×D508C2212yes—
14#16861234—28no—
15C2×#14523235C2214yescomplemented
16C2×#14523233C2214yescomplemented
17C2×#14523232C2212yescomplemented
18#16861129C2218yescomplemented
19C22⋊(C127⋊C8)C2214yescomplemented
20C2×#41×C127C2211yescharacteristic, complemented
21C23×C4×C127C2211yescharacteristic
22#123×C127—21no—
23C22⋊Q8×C127C2212yescomplemented
24#121×C127C2212yescomplemented
25#120×C127—21no—
26C4×D4×C127—22no—
27#117×C127C2211yescharacteristic, complemented
28C23×D127—28no—
29C23×D127—22no—
30C23×D127—24no—
31C23×D127—48no—
32C2×D508—28no—
33C2×D508—216no—
34C2×D508—216no—
35C2×D508—24no—
36C2×D508—24no—
37C2×D508—22no—
38C2×D508D418yescomplemented
39C2×D508C2314yes—
40#14523235—28no—
41#14523233—28no—
42#14523232—24no—
43C2×C127⋊C8C2314yescomplemented
44C24×C127D411yescharacteristic
45#41×C127—24no—
46C2×Q8×C127C2311yescharacteristic, complemented
47C2×D4×C127—22no—
48C2×D4×C127—21no—
49C2×D4×C127C2311yescharacteristic, complemented
50C22×C4×C127—24no—
51C22×C4×C127—21no—
52C22×C4×C127D412yes—
53C22×C4×C127D411yescharacteristic
54C22×C4×C127C2312yes—
55C22×C4×C127C2311yescharacteristic
56C4⋊C4×C127—22no—
57C4⋊C4×C127C2312yescomplemented
58C22⋊C4×C127—21no—
59C22⋊C4×C127—22no—
60C22⋊C4×C127—22no—
61C42×C127—21no—
62C22×D127—216no—
63C22×D127—28no—
64C22×D127—24no—
65C22×D127—432no—
66C22×D127—48no—
67C22×D127—48no—
68C22×D127—48no—
69D508—216no—
70D508—28no—
71D508—24no—
72D508C2×D418yes—
73C127⋊C8—24no—
74C23×C127—22no—
75C23×C127—24no—
76C23×C127—21no—
77C23×C127C2×D411yescharacteristic
78C23×C127C2×D412yes—
79Q8×C127—22no—
80D4×C127—22no—
81D4×C127—42no—
82D4×C127—44no—
83C2×C4×C127—22no—
84C2×C4×C127—24no—
85C2×C4×C127—24no—
86C2×C4×C127—22no—
87C2×C4×C127—21no—
88C2×C4×C127—42no—
89C2×C4×C127C2×D414yes—
90C2×C4×C127C2×D412yes—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2411yescharacteristic
93C22×C127—28no—
94C22×C127—22no—
95C22×C127—22no—
96C22×C127—22no—
97C22×C127—41no—
98C22×C127—42no—
99C22×C127#12012yes—
100C22×C127AGL(1,ℤ/8)14yescomplemented
101C22×C127C22×D411yescharacteristic
102C2×D127—28no—
103C2×D127—416no—
104C2×D127—432no—
105C2×D127—48no—
106C4×C127—22no—
107C4×C127—41no—
108C4×C127—42no—
109C4×C127#12012yes—
110C4×C127C22×D412yes—
111C2×C127—22no—
112C2×C127—24no—
113C2×C127—81no—
114C2×C127C2×#12011yescharacteristic
115C2×C127C2×AGL(1,ℤ/8)12yes—
116D127—48no—
117#3003—1271nomaximal, Sylow
118C127#300311yescharacteristic, complemented, Sylow
119C23×D4—1271no—
120C2×AGL(1,ℤ/8)—1272no—
121#525D12711yescharacteristic, complemented
122#450—1274no—
123#447—1274no—
124#418—1272no—
125#407—1271no—
126C25—2541no—
127C2×#41C2×D12711yescharacteristic, complemented
128C22×D4—1272no—
129C22×D4—1274no—
130C22×D4—2548no—
131C22×D4—2542no—
132C22×D4—2541no—
133C23×C4C2×D12711yescharacteristic
134AGL(1,ℤ/8)—2548no—
135C2×SD16—1274no—
136C2×D8—1274no—
137C2×M16—1272no—
138#123—21no—
139C22⋊Q8C2×D12712yescomplemented
140#121C2×D12712yescomplemented
141#120—21no—
142C4×D4—22no—
143#117C2×D12711yescharacteristic, complemented
144#102—1278no—
145C22⋊C8—1274no—
146C24—5088no—
147C24—2548no—
148C24—2542no—
149C24—2544no—
150C24C22⋊D12711yescharacteristic
151#41—24no—
152C2×Q8C22×D12711yescharacteristic, complemented
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—1278no—
156C2×D4—1274no—
157C2×D4—25416no—
158C2×D4—2548no—
159C2×D4—25416no—
160C2×D4—2544no—
161C2×D4—2544no—
162C2×D4—2542no—
163C2×D4C22×D12711yescharacteristic, complemented
164C22×C4—24no—
165C22×C4—21no—
166C22×C4C22⋊D12712yes—
167C22×C4C22⋊D12711yescharacteristic
168C22×C4C22×D12712yes—
169C22×C4C22×D12711yescharacteristic
170SD16—2548no—
171D8—2548no—
172M16—2544no—
173C2×C8—1274no—
174C4⋊C4—22no—
175C4⋊C4C22×D12712yescomplemented
176C22⋊C4—21no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C42—21no—
180C23—24no—
181C23—22no—
182C23—21no—
183C23—50832no—
184C23—5088no—
185C23—5088no—
186C23—5088no—
187C23—25416no—
188C23—2544no—
189C23—2548no—
190C23C2×C22⋊D12712yes—
191C23C2×C22⋊D12711yescharacteristic
192Q8—22no—
193D4—22no—
194D4—44no—
195D4—42no—
196D4—1278no—
197D4—25416no—
198D4—2544no—
199D4—2548no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4D4×D12714yes—
207C2×C4C2×C22⋊D12712yes—
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4C23×D12711yescharacteristic
210C8—2544no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—41no—
216C22—42no—
217C22—50832no—
218C22—50816no—
219C22—5088no—
220C22—2548no—
221C22#1686117712yes—
222C22#1686123414yescomplemented
223C22C22×C22⋊D12711yescharacteristic
224C4—22no—
225C4—41no—
226C4—42no—
227C4#1686117712yes—
228C4C2×D4×D12712yes—
229C2—24no—
230C2—22no—
231C2—81no—
232C2—5088no—
233C2C2×#1686117711yescharacteristic
234C2C2×#1686123412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#525×C127C2
socleC22×C127C22×D4
hypercenter#525D127
hypocenterC127#3003
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C23×D508C2semidirect
2. C2×#16861234C2semidirect
3. #22550026C2semidirect
4. #22550009C2semidirect
5. #22549880C2semidirect
6. #22549840C2semidirect
7. #525×C127C2semidirect
15. C2×#14523235C22semidirect
16. C2×#14523233C22semidirect
17. C2×#14523232C22semidirect
18. #16861129C22semidirect
19. C22⋊(C127⋊C8)C22semidirect
20. C2×#41×C127C22semidirect
23. C22⋊Q8×C127C22semidirect
24. #121×C127C22semidirect
27. #117×C127C22semidirect
38. C2×D508D4semidirect
43. C2×C127⋊C8C23semidirect
46. C2×Q8×C127C23semidirect
49. C2×D4×C127C23semidirect
57. C4⋊C4×C127C23semidirect
100. C22×C127AGL(1,ℤ/8)semidirect
118. C127#3003semidirect
121. #525D127semidirect
127. C2×#41C2×D127semidirect
139. C22⋊Q8C2×D127semidirect
140. #121C2×D127semidirect
143. #117C2×D127semidirect
152. C2×Q8C22×D127semidirect
163. C2×D4C22×D127semidirect
175. C4⋊C4C22×D127semidirect
222. C22#16861234semidirect

Maximal subgroups

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

subgroupindex
C23×D5082
C2×#168612342
#225500262
#225500092
#225498802
#225498402
#525×C1272
#3003127