Finite Group Databasebeta

← #30647587

Subgroups — #30647587

Order 16256.

492 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C22×#14523236C211yesmaximal, characteristic
2#22550353C211yesmaximal, characteristic, complemented
3#22549976C214yesmaximal, complemented
4C4×#14523236C214yesmaximal, complemented
5#22549912C211yesmaximal, characteristic, complemented
6#22549884C211yesmaximal, characteristic, complemented
7#22549881C211yesmaximal, characteristic
8#22549832C211yesmaximal, characteristic, complemented
9#518×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11C2×#14523236C418yescomplemented
12C2×#14523236C2212yes—
13C2×#14523236C2212yes—
14C2×#14523232C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16C4×Dic254C2214yescomplemented
17#16861174—22no—
18#16861168C2211yescharacteristic
19#16861165—22no—
20#16861142C2212yescomplemented
21#16861140C2212yescomplemented
22#16861132C2214yescomplemented
23#16861130C2214yes—
24#16861120C2212yescomplemented
25C4×C127⋊C8C2212yescomplemented
26C22×D4×C127C2211yescharacteristic
27C4×D4×C127—22no—
28C4×D4×C127C2214yes—
29#117×C127C2211yescharacteristic
30#117×C127C2211yescharacteristic
31C2×C22⋊C4×C127—22no—
32C2×Dic254—21no—
33C2×Dic254—22no—
34C2×Dic254C2×C414yescomplemented
35C2×Dic254C2312yes—
36C22×Dic127—21no—
37#14523236C2×C418yescomplemented
38#14523236C2×C418yescomplemented
39#14523232—22no—
40C2×C127⋊C8C2×C414yescomplemented
41C2×C127⋊C8C2312yes—
42C2×C127⋊C8C2311yescharacteristic
43C2×C127⋊C8C2311yescharacteristic
44C4⋊Dic127C2312yes—
45Dic127⋊C4—24no—
46C22⋊Dic127—21no—
47C4×Dic127—24no—
48C4×Dic127—21no—
49C24×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—22no—
52C2×D4×C127C2×C414yes—
53C2×D4×C127C2312yes—
54C22×C4×C127—24no—
55C22×C4×C127C2311yescharacteristic
56C4⋊C4×C127D412yes—
57C4⋊C4×C127C2312yes—
58C22⋊C4×C127—21no—
59C22⋊C4×C127—24no—
60C22⋊C4×C127D412yes—
61C42×C127—21no—
62C42×C127C2312yes—
63Dic254—24no—
64Dic254—22no—
65Dic254C22×C414yes—
66C2×Dic127—24no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C127⋊C8—22no—
70C127⋊C8C22×C412yes—
71C127⋊C8C22×C412yes—
72C23×C127—22no—
73C23×C127—24no—
74C23×C127—82no—
75C23×C127C2×D411yescharacteristic
76D4×C127—24no—
77D4×C127—22no—
78D4×C127C22×C414yes—
79C2×C4×C127—22no—
80C2×C4×C127—84no—
81C2×C4×C127C22×C412yes—
82C2×C4×C127C2×D414yes—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4112yes—
85C2×C4×C127C2411yescharacteristic
86C22×C127—22no—
87C22×C127—24no—
88C22×C127—44no—
89C22×C127—84no—
90C22×C127C4×D412yes—
91C22×C127C22×D411yescharacteristic
92C4×C127—24no—
93C4×C127—42no—
94C4×C127C4×D412yes—
95C4×C127C23×C411yescharacteristic
96C4×C127C2×#4111yescharacteristic
97Dic127—24no—
98Dic127—42no—
99C2×C127—22no—
100C2×C127—44no—
101C2×C127C2×C4×D411yescharacteristic
102C2×C127#57712yes—
103#2949—1271nomaximal, Sylow
104C127#294911yescharacteristic, complemented, Sylow
105C22×SD16—1271no—
106#519—1271no—
107#518D12711yescharacteristic
108#440—1274no—
109C4×SD16—1274no—
110#427—1271no—
111#419—1271no—
112#418—1271no—
113#405—1271no—
114C22×Q8—1271no—
115C22×D4C2×D12711yescharacteristic
116C2×SD16—1278no—
117C2×SD16—1272no—
118C2×SD16—1272no—
119C2×M16—1271no—
120C22×C8—1271no—
121C4×Q8—1274no—
122C4×Q8—2542no—
123C4×D4—22no—
124C4×D4C2×D12714yes—
125#117—1271no—
126#117—2542no—
127#117C2×D12711yescharacteristic
128#117C2×D12711yescharacteristic
129C2×C22⋊C4—22no—
130#107—1272no—
131#106—1272no—
132#103—1274no—
133#102—1274no—
134#97—1272no—
135C4×C8—1272no—
136C24—21no—
137C2×Q8—1274no—
138C2×Q8—1272no—
139C2×Q8—2541no—
140C2×Q8—2542no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4C4×D12714yes—
144C2×D4C22×D12712yes—
145C22×C4—24no—
146C22×C4—2541no—
147C22×C4C22×D12711yescharacteristic
148SD16—1278no—
149SD16—1278no—
150M16—2542no—
151C2×C8—1274no—
152C2×C8—1271no—
153C2×C8—1271no—
154C2×C8—1272no—
155C4⋊C4—1272no—
156C4⋊C4—2544no—
157C4⋊C4C22⋊D12712yes—
158C4⋊C4C22×D12712yes—
159C22⋊C4—21no—
160C22⋊C4—24no—
161C22⋊C4—2541no—
162C22⋊C4C22⋊D12712yes—
163C42—21no—
164C42—2541no—
165C42—2544no—
166C42C22×D12712yes—
167C23—22no—
168C23—24no—
169C23—82no—
170C23C2×C22⋊D12711yescharacteristic
171Q8—1274no—
172Q8—2542no—
173Q8—2544no—
174D4—22no—
175D4—24no—
176D4C2×C4×D12714yes—
177C2×C4—22no—
178C2×C4—84no—
179C2×C4—2542no—
180C2×C4—2542no—
181C2×C4—2544no—
182C2×C4C2×C4×D12712yes—
183C2×C4C2×C22⋊D12714yes—
184C2×C4C2×C22⋊D12711yescharacteristic
185C2×C4#1452324812yes—
186C2×C4C23×D12711yescharacteristic
187C8—1272no—
188C8—1272no—
189C8—2542no—
190C22—22no—
191C22—24no—
192C22—44no—
193C22—84no—
194C22C4×C22⋊D12712yes—
195C22C22×C22⋊D12711yescharacteristic
196C4—24no—
197C4—42no—
198C4—5082no—
199C4—2544no—
200C4C4×C22⋊D12712yes—
201C4C22×C4×D12711yescharacteristic
202C4C2×#1452324811yescharacteristic
203C2—22no—
204C2—44no—
205C2C2×C4×C22⋊D12711yescharacteristic
206C2#2255068712yes—
subgroupG/H
centerC22C22×C22⋊D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
Fitting#518×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×C127C22
cosocleC2×C4×C127C24
hypercenter#518D127
hypocenterC127#2949
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #22550353C2semidirect
3. #22549976C2semidirect
4. C4×#14523236C2semidirect
5. #22549912C2semidirect
6. #22549884C2semidirect
8. #22549832C2semidirect
11. C2×#14523236C4semidirect
16. C4×Dic254C22semidirect
20. #16861142C22semidirect
21. #16861140C22semidirect
22. #16861132C22semidirect
24. #16861120C22semidirect
25. C4×C127⋊C8C22semidirect
34. C2×Dic254C2×C4semidirect
37. #14523236C2×C4semidirect
38. #14523236C2×C4semidirect
40. C2×C127⋊C8C2×C4semidirect
104. C127#2949semidirect

Maximal subgroups

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

subgroupindex
C22×#145232362
#225503532
#225499762
C4×#145232362
#225499122
#225498842
#225498812
#225498322
#518×C1272
#2949127