Finite Group Databasebeta

← #30649211

Subgroups — #30649211

Order 16256.

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

normal subgroups only

This group has 596 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
2C2×#16861231C211yesmaximal, characteristic, complemented
3#22550443C212yesmaximal, complemented
4#22550264C214yesmaximal, complemented
5#22550259C212yesmaximal, complemented
6#22550251C212yesmaximal, complemented
7#22549831C211yesmaximal, characteristic, complemented
8C22×D8×C127C211yesmaximal, characteristic, complemented
9C2×#14523248C2212yes—
10#16861235—28no—
11#16861231—24no—
12C2×Dic508C2211yescharacteristic, complemented
13C2×#14523237C2212yescomplemented
14C2×#14523236C2214yescomplemented
15C2×D1016C2211yescharacteristic, complemented
16C2×#14523233C2214yescomplemented
17C2×#14523232C2211yescharacteristic
18#16861205C2214yescomplemented
19#16861196C2214yescomplemented
20#16861180—24no—
21#16861178—24no—
22#16861168C2211yescharacteristic
23#16861119C2212yescomplemented
24C8×Dic127C2212yescomplemented
25C22×D4×C127C2212yes—
26C2×D8×C127—24no—
27C2×D8×C127C2212yes—
28C2×D8×C127C2212yes—
29C22×C8×C127C2211yescharacteristic
30#14523248—24no—
31#14523248—24no—
32C2×Dic254C2312yes—
33C2×D508C2312yes—
34C2×C22⋊D127—28no—
35C2×C22⋊D127—22no—
36C2×C4×D127—22no—
37Dic508—22no—
38#14523237—24no—
39#14523236—28no—
40D1016—22no—
41#14523233—28no—
42#14523232D414yescomplemented
43C2×C127⋊C8C2312yes—
44C4⋊Dic127D412yes—
45C22⋊Dic127—28no—
46C22⋊Dic127D412yes—
47C4×Dic127C2312yes—
48C24×C127—22no—
49C2×D4×C127—24no—
50C2×D4×C127—24no—
51C2×D4×C127—22no—
52C2×D4×C127C2314yes—
53C22×C4×C127C2311yescharacteristic
54D8×C127—24no—
55D8×C127—24no—
56C2×C8×C127D414yescomplemented
57C2×C8×C127C2311yescharacteristic
58C2×C8×C127C2311yescharacteristic
59C22×D127—22no—
60Dic254—22no—
61Dic254—22no—
62D508—22no—
63D508—22no—
64C22⋊D127—44no—
65C22⋊D127—88no—
66C2×Dic127—22no—
67C2×Dic127C2×D414yes—
68C4×D127—44no—
69C127⋊C8C2×D414yes—
70C23×C127—24no—
71C23×C127—44no—
72C23×C127—84no—
73C23×C127C2×D411yescharacteristic
74D4×C127—24no—
75D4×C127—24no—
76D4×C127—44no—
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D412yes—
79C2×C4×C127C2×D412yes—
80C2×C4×C127C2411yescharacteristic
81C8×C127C2×D412yes—
82C8×C127C2×D412yes—
83C22×C127—22no—
84C22×C127—44no—
85C22×C127—48no—
86C22×C127—88no—
87C22×C127#12712yes—
88C22×C127C22×D411yescharacteristic
89C2×D127—42no—
90C2×D127—44no—
91C4×C127#12712yes—
92C4×C127C22×D411yescharacteristic
93C4×C127C22×D411yescharacteristic
94Dic127—24no—
95Dic127—42no—
96C2×C127—22no—
97C2×C127—84no—
98C2×C127C2×#12711yescharacteristic
99C2×C127#57612yes—
100D127—82no—
101#3155—1271nomaximal, Sylow
102C127#315511yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1272no—
104C2×#135—1271no—
105C22×D8D12711yescharacteristic, complemented
106#535—1272no—
107#496—1274no—
108#495—1272no—
109#493—1272no—
110#405—1271no—
111C2×#41—1272no—
112C22×D4C2×D12712yes—
113AGL(1,ℤ/8)—2548no—
114#135—2544no—
115C2×Q16—1271no—
116C2×SD16—1274no—
117C2×SD16—1272no—
118C2×D8—24no—
119C2×D8—1274no—
120C2×D8—1271no—
121C2×D8C2×D12712yes—
122C2×D8C2×D12712yes—
123C2×M16—1271no—
124C22×C8C2×D12711yescharacteristic
125#127—1274no—
126#124—1274no—
127#121—2544no—
128#120—2544no—
129#117—1271no—
130#97—1272no—
131C4×C8—1272no—
132C24—22no—
133#41—2544no—
134#41—2544no—
135C2×Q8—1272no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—1272no—
140C2×D4—2548no—
141C2×D4—2542no—
142C2×D4C22×D12714yes—
143C22×C4—2542no—
144C22×C4C22×D12711yescharacteristic
145Q16—2542no—
146SD16—2548no—
147SD16—2544no—
148D8—24no—
149D8—24no—
150D8—2548no—
151D8—2542no—
152M16—1274no—
153C2×C8—1272no—
154C2×C8C22⋊D12714yescomplemented
155C2×C8C22×D12711yescharacteristic
156C2×C8C22×D12711yescharacteristic
157C4⋊C4—1272no—
158C22⋊C4—1272no—
159C22⋊C4—2548no—
160C42—1272no—
161C23—24no—
162C23—44no—
163C23—84no—
164C23—2542no—
165C23D4×D12711yescharacteristic
166Q8—2542no—
167Q8—2542no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—10168no—
172D4—5084no—
173D4—2542no—
174D4—2542no—
175C2×C4—1274no—
176C2×C4—5084no—
177C2×C4—2542no—
178C2×C4D4×D12711yescharacteristic
179C2×C4D4×D12712yes—
180C2×C4C2×C22⋊D12712yes—
181C2×C4C23×D12711yescharacteristic
182C8—1274no—
183C8C2×C22⋊D12712yes—
184C8C2×C22⋊D12712yes—
185C22—22no—
186C22—44no—
187C22—48no—
188C22—88no—
189C22—5082no—
190C22—5084no—
191C22#1686120512yes—
192C22C2×D4×D12711yescharacteristic
193C4—5082no—
194C4—2544no—
195C4#1686120512yes—
196C4C2×D4×D12711yescharacteristic
197C4C22×C22⋊D12711yescharacteristic
198C2—22no—
199C2—84no—
200C2—10162no—
201C2C2×#1686120511yescharacteristic
202C2#2255067712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×D8×C127C2
socleC22×C127C22×D4
hypercenterC22×D8D127
hypocenterC127#3155
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. C2×#16861231C2semidirect
3. #22550443C2semidirect
4. #22550264C2semidirect
5. #22550259C2semidirect
6. #22550251C2semidirect
7. #22549831C2semidirect
8. C22×D8×C127C2semidirect
12. C2×Dic508C22semidirect
13. C2×#14523237C22semidirect
14. C2×#14523236C22semidirect
15. C2×D1016C22semidirect
16. C2×#14523233C22semidirect
18. #16861205C22semidirect
19. #16861196C22semidirect
23. #16861119C22semidirect
24. C8×Dic127C22semidirect
42. #14523232D4semidirect
56. C2×C8×C127D4semidirect
102. C127#3155semidirect
105. C22×D8D127semidirect
154. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
C2×#168612312
#225504432
#225502642
#225502592
#225502512
#225498312
C22×D8×C1272
#3155127