Finite Group Databasebeta

← #30637701

Subgroups — #30637701

Order 16256.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#22550457C211yesmaximal, characteristic
2#22550017C212yesmaximal, complemented
3#22549796C211yesmaximal, characteristic, complemented
4#22549611C212yesmaximal, complemented
5#324×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—21no—
7C22×Dic254C2211yescharacteristic
8C2×#14523237—22no—
9#16861189—22no—
10#16861184—22no—
11#16861183—21no—
12#16861183C2212yes—
13C2×Dic127⋊C4—21no—
14#16861134—22no—
15C127⋊#95—21no—
16C127⋊#95C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19#14523248—42no—
20#14523248—42no—
21C2×Dic254—21no—
22C2×Dic254—22no—
23C2×Dic254—42no—
24C2×D508—21no—
25C2×D508—21no—
26C2×D508D412yes—
27C2×C22⋊D127—22no—
28C22×Dic127—21no—
29C22×Dic127—21no—
30C2×C4×D127—22no—
31#14523237—44no—
32C4⋊Dic127—21no—
33C4⋊Dic127D412yes—
34Dic127⋊C4—42no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—22no—
37C22×C4×C127D411yescharacteristic
38C22×C4×C127D411yescharacteristic
39C22×C4×C127C2311yescharacteristic
40C2×C8×C127—22no—
41C4⋊C4×C127—22no—
42C22×D127—22no—
43Dic254—24no—
44Dic254—42no—
45Dic254—42no—
46D508—42no—
47D508—42no—
48D508—81no—
49C22⋊D127—42no—
50C22⋊D127—42no—
51C2×Dic127—22no—
52C2×Dic127—22no—
53C2×Dic127—22no—
54C2×Dic127—41no—
55C2×Dic127—41no—
56C2×Dic127—41no—
57C2×Dic127—41no—
58C4×D127—44no—
59C23×C127C2×D411yescharacteristic
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127SD1612yescomplemented
66C2×C4×C127C2×D412yes—
67C8×C127—42no—
68C22×C127—21no—
69C22×C127—21no—
70C22×C127#12011yescharacteristic
71C22×C127C2×SD1611yescharacteristic
72C22×C127AGL(1,ℤ/8)11yescharacteristic
73C2×D127—42no—
74C2×D127—42no—
75C2×D127—42no—
76C4×C127—22no—
77C4×C127—22no—
78C4×C127—41no—
79Dic127—42no—
80Dic127—42no—
81Dic127—42no—
82C2×C127—21no—
83C2×C127—21no—
84C2×C127#45011yescharacteristic
85C2×C127#45611yescharacteristic
86C2×C127B(4,𝔽2)11yescharacteristic
87D127—82no—
88#1604—1271nomaximal, Sylow
89C127#160411yescharacteristic, complemented, Sylow
90#537—1271no—
91#448—1272no—
92#395—1271no—
93#327—1272no—
94#324D12711yescharacteristic, complemented
95C2×#41—2541no—
96C22×Q8—1271no—
97C2×SD16—2542no—
98C22⋊Q8—2542no—
99#121—1272no—
100#121—2542no—
101#121—2541no—
102C2×C4⋊C4—2541no—
103C2×C4⋊C4C2×D12711yescharacteristic
104#103—2542no—
105C22⋊C8C2×D12712yes—
106#95—1271no—
107#95—2541no—
108#41—5082no—
109#41—5082no—
110C2×Q8—5082no—
111C2×Q8—2541no—
112C2×Q8—2542no—
113C2×D4—1272no—
114C2×D4—2541no—
115C2×D4—2541no—
116C2×D4—2542no—
117C22×C4—2542no—
118C22×C4—2541no—
119C22×C4—2541no—
120C22×C4D50811yescharacteristic
121C22×C4D50811yescharacteristic
122C22×C4C22×D12711yescharacteristic
123SD16—5084no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—1272no—
127C4⋊C4—5082no—
128C4⋊C4—2541no—
129C22⋊C4—2542no—
130C22⋊C4—2542no—
131C23—2542no—
132C23C2×D50811yescharacteristic
133Q8—5082no—
134Q8—5082no—
135Q8—2544no—
136D4—10161no—
137D4—5082no—
138D4—5082no—
139D4—5082no—
140D4—5082no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—5084no—
147C2×C4—5081no—
148C2×C4—5081no—
149C2×C4—5081no—
150C2×C4—5081no—
151C2×C4—2542no—
152C2×C4—2542no—
153C2×C4—2542no—
154C2×C4#1452323712yescomplemented
155C2×C4D4×D12712yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—5082no—
160C22—5082no—
161C22—5082no—
162C22#1686117911yescharacteristic
163C22C2×#1452323711yescharacteristic
164C22#1686123711yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—5082no—
169C4—5082no—
170C4—5082no—
171C2—21no—
172C2—21no—
173C2—10162no—
174C2#2255002911yescharacteristic
175C2#2255006111yescharacteristic
176C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#324D127
hypocenterC127#1604
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #22550017C2semidirect
3. #22549796C2semidirect
4. #22549611C2semidirect
5. #324×C127C2semidirect
65. C2×C4×C127SD16semidirect
89. C127#1604semidirect
94. #324D127semidirect
154. C2×C4#14523237semidirect

Maximal subgroups

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

subgroupindex
#225504572
#225500172
#225497962
#225496112
#324×C1272
#1604127