Finite Group Databasebeta

← #30638213

Subgroups — #30638213

Order 16256.

184 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550252C211yesmaximal, characteristic
2#22550077C211yesmaximal, characteristic, complemented
3#22550073C211yesmaximal, characteristic, complemented
4#22550003C211yesmaximal, characteristic, complemented
5#22549626C211yesmaximal, characteristic, complemented
6#22549619C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic, complemented
8C2×#14523237—21no—
9C2×#14523236—21no—
10C2×#14523233—21no—
11C2×#14523233—21no—
12#16861206C2211yescharacteristic
13C4×Dic254C2211yescharacteristic
14C4⋊(C127⋊C8)C2211yescharacteristic
15Dic127⋊C8—21no—
16#16861131—21no—
17#16861130—21no—
18#16861119—21no—
19C4×C127⋊C8C2211yescharacteristic
20#127×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22C4⋊C8×C127C2211yescharacteristic
23C2×Dic254D411yescharacteristic
24C2×D508—21no—
25C2×D508—21no—
26C2×D508—21no—
27#14523237—42no—
28#14523236—42no—
29#14523233—42no—
30#14523233—42no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8D411yescharacteristic
33C2×C127⋊C8D411yescharacteristic
34C4⋊Dic127D411yescharacteristic
35Dic127⋊C4—21no—
36C4×Dic127—21no—
37C2×D4×C127—21no—
38C2×D4×C127—21no—
39C2×D4×C127—21no—
40C2×C8×C127—21no—
41C2×C8×C127—21no—
42C42×C127C2311yescharacteristic
43C22×D127—41no—
44Dic254—21no—
45Dic254—21no—
46D508—41no—
47D508—41no—
48D508—41no—
49D508—41no—
50D508—81no—
51C2×Dic127—21no—
52C2×Dic127—21no—
53C127⋊C8—21no—
54C127⋊C8—41no—
55C127⋊C8SD1612yescomplemented
56C23×C127—41no—
57D4×C127—41no—
58D4×C127—41no—
59D4×C127—41no—
60D4×C127—41no—
61D4×C127—81no—
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127#4111yescharacteristic
65C8×C127—41no—
66C8×C127—41no—
67C22×C127—81no—
68C22×C127—81no—
69C22×C127—81no—
70C22×C127#12111yescharacteristic
71C2×D127—81no—
72C2×D127—81no—
73C2×D127—81no—
74C4×C127—21no—
75C4×C127C2×SD1611yescharacteristic
76C4×C127AGL(1,ℤ/8)11yescharacteristic
77C4×C127AGL(1,ℤ/8)11yescharacteristic
78C4×C127AGL(1,ℤ/8)11yescharacteristic
79Dic127—41no—
80Dic127—41no—
81C2×C127—161no—
82C2×C127#46011yescharacteristic
83C2×C127#46911yescharacteristic
84C2×C127#47211yescharacteristic
85D127—161no—
86#1688—1271nomaximal, Sylow
87C127#168811yescharacteristic, complemented, Sylow
88#493—1271no—
89#460—1271no—
90#460—1271no—
91#446—1271no—
92#334—1271no—
93#331—1271no—
94#331D12711yescharacteristic, complemented
95C2×SD16—2541no—
96C2×SD16—2541no—
97C2×D8—2541no—
98C2×D8—2541no—
99#127—1271no—
100#127C2×D12711yescharacteristic
101C4×Q8—1271no—
102C4⋊C8—1271no—
103C4⋊C8—2541no—
104C4⋊C8C2×D12711yescharacteristic
105C4⋊C8C2×D12711yescharacteristic
106#102—2541no—
107#102—2541no—
108#97—2541no—
109C4×C8—1271no—
110C2×Q8—1271no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—2541no—
115C2×D4—2541no—
116C2×D4—2541no—
117SD16—5082no—
118SD16—5082no—
119D8—5082no—
120D8—5082no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—1271no—
124C2×C8—1271no—
125C2×C8—2541no—
126C4⋊C4—1271no—
127C4⋊C4—2541no—
128C42—2541no—
129C42C22×D12711yescharacteristic
130C23—41no—
131C23—5081no—
132Q8—2541no—
133Q8—2541no—
134D4—41no—
135D4—41no—
136D4—41no—
137D4—41no—
138D4—81no—
139D4—10161no—
140D4—5081no—
141D4—5081no—
142D4—5081no—
143D4—5081no—
144C2×C4—2541no—
145C2×C4—2541no—
146C2×C4D4×D12711yescharacteristic
147C2×C4D4×D12711yescharacteristic
148C2×C4#1452324811yescharacteristic
149C8—41no—
150C8—41no—
151C8—1272no—
152C8—5081no—
153C8—2541no—
154C22—81no—
155C22—81no—
156C22—81no—
157C22—10161no—
158C22—10161no—
159C22—10161no—
160C22#1686118211yescharacteristic
161C4—21no—
162C4—5081no—
163C4—5081no—
164C4SD16×D12711yescharacteristic
165C4#1686123311yescharacteristic
166C4#1686123311yescharacteristic
167C4#1686123611yescharacteristic
168C2—161no—
169C2—20321no—
170C2#2255007511yescharacteristic
171C2#2255012811yescharacteristic
172C2#2255014511yescharacteristic
subgroupG/H
centerC22#16861182
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#121
prime radical#22550252C2
hypercenter#331D127
hypocenterC127#1688
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22550077C2semidirect
3. #22550073C2semidirect
4. #22550003C2semidirect
5. #22549626C2semidirect
6. #22549619C2semidirect
7. #331×C127C2semidirect
55. C127⋊C8SD16semidirect
87. C127#1688semidirect
94. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225502522
#225500772
#225500732
#225500032
#225496262
#225496192
#331×C1272
#1688127