Finite Group Databasebeta

← #30647580

Subgroups — #30647580

Order 16256.

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

normal subgroups only

This group has 528 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×#14523233C211yesmaximal, characteristic
2#22550350C211yesmaximal, characteristic, complemented
3#22549986C214yesmaximal, complemented
4C4×#14523233C214yesmaximal, complemented
5#22549912C211yesmaximal, characteristic, complemented
6#22549881C211yesmaximal, characteristic, complemented
7#22549880C211yesmaximal, characteristic, complemented
8#22549832C211yesmaximal, characteristic, complemented
9#518×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C2×#14523233C418yescomplemented
12C2×#14523233C2212yes—
13C2×#14523233C2212yes—
14C2×#14523232C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16C4×D508C2214yescomplemented
17#16861171—22no—
18#16861168C2211yescharacteristic
19C22⋊C4×D127—22no—
20#16861142C2212yescomplemented
21#16861140C2212yescomplemented
22#16861130C2214yescomplemented
23#16861129C2214yescomplemented
24#16861120C2212yescomplemented
25C4×C127⋊C8C2212yescomplemented
26C22×D4×C127C2211yescharacteristic
27C4×D4×C127—22no—
28C4×D4×C127C2214yescomplemented
29#117×C127C2211yescharacteristic
30#117×C127C2211yescharacteristic
31C2×C22⋊C4×C127—22no—
32C23×D127—21no—
33C2×D508—21no—
34C2×D508—22no—
35C2×D508C2×C414yescomplemented
36C2×D508C2312yes—
37C2×C4×D127—24no—
38#14523233C2×C418yescomplemented
39#14523233C2×C418yescomplemented
40#14523232—22no—
41C2×C127⋊C8C2×C414yescomplemented
42C2×C127⋊C8C2312yes—
43C2×C127⋊C8C2311yescharacteristic
44C2×C127⋊C8C2311yescharacteristic
45C4⋊Dic127C2312yes—
46C22⋊Dic127—21no—
47(C2×D127)⋊C4—24no—
48C4×Dic127—21no—
49C24×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—22no—
52C2×D4×C127C2×C414yescomplemented
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—
63C22×D127—22no—
64C22×D127—24no—
65C22×D127—82no—
66D508—24no—
67D508—22no—
68D508C22×C414yes—
69C2×Dic127—22no—
70C4×D127—84no—
71C127⋊C8—22no—
72C127⋊C8C22×C412yes—
73C127⋊C8C22×C412yes—
74C23×C127—22no—
75C23×C127—24no—
76C23×C127—82no—
77C23×C127C2×D411yescharacteristic
78D4×C127—24no—
79D4×C127—22no—
80D4×C127C22×C414yes—
81C2×C4×C127—22no—
82C2×C4×C127—84no—
83C2×C4×C127C22×C412yes—
84C2×C4×C127C2×D414yes—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4112yes—
87C2×C4×C127C2411yescharacteristic
88C22×C127—22no—
89C22×C127—24no—
90C22×C127—44no—
91C22×C127—84no—
92C22×C127C4×D412yes—
93C22×C127C22×D411yescharacteristic
94C2×D127—24no—
95C2×D127—44no—
96C2×D127—84no—
97C4×C127—24no—
98C4×C127—42no—
99C4×C127C4×D412yes—
100C4×C127C23×C411yescharacteristic
101C4×C127C2×#4111yescharacteristic
102Dic127—42no—
103C2×C127—22no—
104C2×C127—44no—
105C2×C127C2×C4×D411yescharacteristic
106C2×C127#57612yes—
107D127—44no—
108#2948—1271nomaximal, Sylow
109C127#294811yescharacteristic, complemented, Sylow
110C22×D8—1271no—
111#518—1271no—
112#518D12711yescharacteristic, complemented
113#442—1274no—
114C4×D8—1274no—
115#427—1271no—
116#418—1271no—
117#418—1271no—
118#405—1271no—
119C22×D4—1271no—
120C22×D4C2×D12711yescharacteristic
121C2×D8—1278no—
122C2×D8—1272no—
123C2×D8—1272no—
124C2×M16—1271no—
125C22×C8—1271no—
126C4×D4—22no—
127C4×D4—1274no—
128C4×D4—2542no—
129C4×D4C2×D12714yescomplemented
130#117—1271no—
131#117C2×D12711yescharacteristic
132#117C2×D12711yescharacteristic
133C2×C22⋊C4—22no—
134C2×C22⋊C4—2542no—
135#107—1272no—
136#106—1272no—
137#102—1274no—
138#102—1274no—
139#97—1272no—
140C4×C8—1272no—
141C24—21no—
142C24—2541no—
143C2×D4—21no—
144C2×D4—22no—
145C2×D4—1274no—
146C2×D4—1272no—
147C2×D4—2541no—
148C2×D4—2542no—
149C2×D4C4×D12714yescomplemented
150C2×D4C22×D12712yes—
151C22×C4—24no—
152C22×C4—2544no—
153C22×C4C22×D12711yescharacteristic
154D8—1278no—
155D8—1278no—
156M16—2542no—
157C2×C8—1274no—
158C2×C8—1271no—
159C2×C8—1271no—
160C2×C8—1272no—
161C4⋊C4—1272no—
162C4⋊C4C22⋊D12712yes—
163C4⋊C4C22×D12712yes—
164C22⋊C4—21no—
165C22⋊C4—24no—
166C22⋊C4—2541no—
167C22⋊C4—2544no—
168C22⋊C4C22⋊D12712yes—
169C42—21no—
170C42—2541no—
171C42C22×D12712yes—
172C23—22no—
173C23—24no—
174C23—82no—
175C23—10162no—
176C23—2544no—
177C23—2542no—
178C23C2×C22⋊D12711yescharacteristic
179D4—22no—
180D4—24no—
181D4—1274no—
182D4—2542no—
183D4—2544no—
184D4C2×C4×D12714yes—
185C2×C4—22no—
186C2×C4—84no—
187C2×C4—10164no—
188C2×C4—2542no—
189C2×C4C2×C4×D12712yes—
190C2×C4C2×C22⋊D12714yes—
191C2×C4C2×C22⋊D12711yescharacteristic
192C2×C4#1452324812yes—
193C2×C4C23×D12711yescharacteristic
194C8—1272no—
195C8—1272no—
196C8—2542no—
197C22—22no—
198C22—24no—
199C22—44no—
200C22—84no—
201C22—10164no—
202C22—5084no—
203C22—2544no—
204C22C4×C22⋊D12712yes—
205C22C22×C22⋊D12711yescharacteristic
206C4—24no—
207C4—42no—
208C4—5082no—
209C4C4×C22⋊D12712yes—
210C4C22×C4×D12711yescharacteristic
211C4C2×#1452324811yescharacteristic
212C2—22no—
213C2—44no—
214C2—5084no—
215C2C2×C4×C22⋊D12711yescharacteristic
216C2#2255067912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
Fitting#518×C127C2
socleC22×C127C22×D4
prime radicalC22×#14523233C2
cosocleC2×C4×C127C24
hypercenter#518D127
hypocenterC127#2948
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. #22550350C2semidirect
3. #22549986C2semidirect
4. C4×#14523233C2semidirect
5. #22549912C2semidirect
6. #22549881C2semidirect
7. #22549880C2semidirect
8. #22549832C2semidirect
9. #518×C127C2semidirect
11. C2×#14523233C4semidirect
16. C4×D508C22semidirect
20. #16861142C22semidirect
21. #16861140C22semidirect
22. #16861130C22semidirect
23. #16861129C22semidirect
24. #16861120C22semidirect
25. C4×C127⋊C8C22semidirect
28. C4×D4×C127C22semidirect
35. C2×D508C2×C4semidirect
38. #14523233C2×C4semidirect
39. #14523233C2×C4semidirect
41. C2×C127⋊C8C2×C4semidirect
52. C2×D4×C127C2×C4semidirect
109. C127#2948semidirect
112. #518D127semidirect
129. C4×D4C2×D127semidirect
149. C2×D4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
#225503502
#225499862
C4×#145232332
#225499122
#225498812
#225498802
#225498322
#518×C1272
#2948127