Finite Group Databasebeta

← #30648206

Subgroups — #30648206

Order 16256.

584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861235C212yesmaximal, complemented
2C4×#14523248C211yesmaximal, characteristic, complemented
3#22550073C214yesmaximal, complemented
4#22550068C214yesmaximal, complemented
5#22549900C211yesmaximal, characteristic, complemented
6#22549881C212yesmaximal, complemented
7C2×#127×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—28no—
10C2×#14523236C2214yescomplemented
11C2×#14523233C2214yescomplemented
12C2×#14523232C2212yes—
13C4×Dic254C2212yescomplemented
14C4×D508C2212yescomplemented
15C4×C22⋊D127—22no—
16#16861168C2211yescharacteristic
17#16861166—21no—
18C42×D127—21no—
19C4⋊(C127⋊C8)C2214yescomplemented
20#16861130C2218yescomplemented
21C22×D4×C127—22no—
22C22×D4×C127C2212yes—
23#127×C127—21no—
24#127×C127—21no—
25#127×C127C2214yes—
26C2×C42×C127C2211yescharacteristic
27#14523248—22no—
28#14523248D414yescomplemented
29C2×Dic254C2311yescharacteristic
30C2×D508C2311yescharacteristic
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33C2×C4×D127—22no—
34#14523236—28no—
35#14523233—28no—
36#14523232—24no—
37C2×C127⋊C8C2314yes—
38C4⋊Dic127C2312yes—
39Dic127⋊C4—22no—
40C22⋊Dic127—21no—
41(C2×D127)⋊C4—22no—
42C4×Dic127—21no—
43C4×Dic127—22no—
44C24×C127—22no—
45C2×D4×C127—24no—
46C2×D4×C127—22no—
47C2×D4×C127—24no—
48C2×D4×C127—24no—
49C2×D4×C127—24no—
50C2×D4×C127—48no—
51C2×D4×C127C2314yes—
52C22×C4×C127D412yes—
53C22×C4×C127C2311yescharacteristic
54C42×C127D411yescharacteristic
55C42×C127D411yescharacteristic
56C42×C127C2312yes—
57C22×D127—21no—
58Dic254—21no—
59Dic254C2×D412yescomplemented
60D508—21no—
61D508C2×D412yescomplemented
62C22⋊D127—24no—
63C2×Dic127—21no—
64C2×Dic127—22no—
65C4×D127—24no—
66C4×D127—44no—
67C127⋊C8—24no—
68C23×C127—24no—
69C23×C127—48no—
70C23×C127—44no—
71C23×C127C2×D411yescharacteristic
72D4×C127—24no—
73D4×C127—24no—
74D4×C127—48no—
75D4×C127—44no—
76D4×C127—48no—
77C2×C4×C127—24no—
78C2×C4×C127C2×D412yes—
79C2×C4×C127C2×D412yes—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D412yes—
82C2×C4×C127#4112yes—
83C2×C4×C127C2411yescharacteristic
84C22×C127—22no—
85C22×C127—44no—
86C22×C127—48no—
87C22×C127—88no—
88C22×C127#12112yes—
89C22×C127C22×D411yescharacteristic
90C2×D127—22no—
91C2×D127—42no—
92C4×C127—22no—
93C4×C127#12112yes—
94C4×C127AGL(1,ℤ/8)14yescomplemented
95C4×C127C22×D411yescharacteristic
96C4×C127C2×#4111yescharacteristic
97Dic127—22no—
98Dic127—42no—
99C2×C127—22no—
100C2×C127—84no—
101C2×C127C2×#12111yescharacteristic
102C2×C127C2×AGL(1,ℤ/8)12yes—
103D127—42no—
104#3038—1271nomaximal, Sylow
105C127#303811yescharacteristic, complemented, Sylow
106C2×AGL(1,ℤ/8)—1272no—
107C2×#127D12711yescharacteristic, complemented
108C4×#41—1271no—
109#460—1274no—
110#459—1274no—
111#423—1271no—
112#418—1272no—
113C2×#41—1271no—
114C22×D4—22no—
115C22×D4C2×D12712yes—
116AGL(1,ℤ/8)—2548no—
117C2×SD16—1274no—
118C2×D8—1274no—
119C2×M16—1272no—
120#127—21no—
121#127—21no—
122#127C2×D12714yes—
123C4×Q8—1272no—
124C4×D4—1272no—
125C4×D4—2542no—
126#117—1271no—
127#117—2541no—
128C2×C42—2541no—
129C2×C42C2×D12711yescharacteristic
130C4⋊C8—1274no—
131#102—1278no—
132C24—22no—
133#41—1274no—
134#41—2542no—
135C2×Q8—1271no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—48no—
142C2×D4—1271no—
143C2×D4—2541no—
144C2×D4C22×D12714yes—
145C22×C4—2542no—
146C22×C4—2541no—
147C22×C4C22⋊D12712yes—
148C22×C4C22×D12711yescharacteristic
149SD16—2548no—
150D8—2548no—
151M16—2544no—
152C2×C8—1274no—
153C4⋊C4—1272no—
154C4⋊C4—2542no—
155C22⋊C4—2541no—
156C22⋊C4—2542no—
157C42—2541no—
158C42—2542no—
159C42C22⋊D12711yescharacteristic
160C42C22⋊D12711yescharacteristic
161C42C22×D12712yes—
162C23—24no—
163C23—48no—
164C23—44no—
165C23—2541no—
166C23C2×C22⋊D12711yescharacteristic
167Q8—1272no—
168Q8—2541no—
169D4—24no—
170D4—24no—
171D4—48no—
172D4—48no—
173D4—44no—
174D4—1272no—
175D4—2544no—
176D4—2541no—
177C2×C4—24no—
178C2×C4—5084no—
179C2×C4—2542no—
180C2×C4—2541no—
181C2×C4—2544no—
182C2×C4D4×D12712yes—
183C2×C4C2×C22⋊D12712yes—
184C2×C4C2×C22⋊D12712yes—
185C2×C4C2×C22⋊D12711yescharacteristic
186C2×C4#1452324712yes—
187C2×C4C23×D12711yescharacteristic
188C8—2544no—
189C22—22no—
190C22—44no—
191C22—48no—
192C22—88no—
193C22—5082no—
194C22—2542no—
195C22#1686118012yes—
196C22C22×C22⋊D12711yescharacteristic
197C4—22no—
198C4—5082no—
199C4—2542no—
200C4#1686118012yes—
201C4#1686123514yescomplemented
202C4C2×D4×D12711yescharacteristic
203C4C2×#1452324711yescharacteristic
204C2—22no—
205C2—84no—
206C2—5082no—
207C2C2×#1686118011yescharacteristic
208C2C2×#1686123512yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#127×C127C2
socleC22×C127C22×D4
hypercenterC2×#127D127
hypocenterC127#3038
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. C4×#14523248C2semidirect
3. #22550073C2semidirect
4. #22550068C2semidirect
5. #22549900C2semidirect
6. #22549881C2semidirect
7. C2×#127×C127C2semidirect
10. C2×#14523236C22semidirect
11. C2×#14523233C22semidirect
13. C4×Dic254C22semidirect
14. C4×D508C22semidirect
19. C4⋊(C127⋊C8)C22semidirect
20. #16861130C22semidirect
28. #14523248D4semidirect
59. Dic254C2×D4semidirect
61. D508C2×D4semidirect
94. C4×C127AGL(1,ℤ/8)semidirect
105. C127#3038semidirect
107. C2×#127D127semidirect
201. C4#16861235semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C4×#145232482
#225500732
#225500682
#225499002
#225498812
C2×#127×C1272
#3038127