Finite Group Databasebeta

← #30638194

Subgroups — #30638194

Order 16256.

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

normal subgroups only

This group has 182 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#22550248C211yesmaximal, characteristic, complemented
2#22550157C211yesmaximal, characteristic, complemented
3#22550068C211yesmaximal, characteristic
4#22549953C211yesmaximal, characteristic, complemented
5#22549628C211yesmaximal, characteristic, complemented
6#22549623C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic, complemented
8C2×#14523236—21no—
9C2×#14523236—21no—
10C2×#14523233—21no—
11C2×#14523231—21no—
12#16861209C2211yescharacteristic
13C4×D508C2211yescharacteristic
14#16861143—21no—
15C4⋊(C127⋊C8)C2211yescharacteristic
16#16861131—21no—
17#16861130—21no—
18(C2×D127)⋊C8—21no—
19C4×C127⋊C8C2211yescharacteristic
20#127×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22C4⋊C8×C127C2211yescharacteristic
23C2×Dic254—21no—
24C2×D508D411yescharacteristic
25C2×C4×D127—21no—
26#14523236—42no—
27#14523236—42no—
28#14523233—42no—
29#14523231—42no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8D411yescharacteristic
32C2×C127⋊C8D411yescharacteristic
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D411yescharacteristic
36(C2×D127)⋊C4—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—21no—
44Dic254—41no—
45Dic254—41no—
46D508—21no—
47D508—21no—
48C2×Dic127—21no—
49C2×Dic127—41no—
50C4×D127—41no—
51C4×D127—41no—
52C127⋊C8—21no—
53C127⋊C8—41no—
54C127⋊C8D812yescomplemented
55C23×C127—41no—
56D4×C127—41no—
57D4×C127—41no—
58D4×C127—41no—
59D4×C127—41no—
60D4×C127—81no—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127#4111yescharacteristic
64C8×C127—41no—
65C8×C127—41no—
66C22×C127—81no—
67C22×C127—81no—
68C22×C127—81no—
69C22×C127#12111yescharacteristic
70C2×D127—41no—
71C2×D127—41no—
72C2×D127—41no—
73C4×C127—21no—
74C4×C127C2×D811yescharacteristic
75C4×C127AGL(1,ℤ/8)11yescharacteristic
76C4×C127AGL(1,ℤ/8)11yescharacteristic
77C4×C127#13711yescharacteristic
78Dic127—41no—
79Dic127—81no—
80C2×C127—161no—
81C2×C127#45911yescharacteristic
82C2×C127#46811yescharacteristic
83C2×C127#47111yescharacteristic
84D127—81no—
85#1685—1271nomaximal, Sylow
86C127#168511yescharacteristic, complemented, Sylow
87#492—1271no—
88#474—1271no—
89#459—1271no—
90#436—1271no—
91#335—1271no—
92#332—1271no—
93#331D12711yescharacteristic, complemented
94C2×SD16—2541no—
95C2×SD16—2541no—
96C2×D8—2541no—
97C2×M16—2541no—
98C4⋊Q8—1271no—
99#127C2×D12711yescharacteristic
100C4×D4—1271no—
101#107—2541no—
102C4⋊C8—1271no—
103C4⋊C8C2×D12711yescharacteristic
104C4⋊C8C2×D12711yescharacteristic
105#102—2541no—
106#102—2541no—
107C22⋊C8—2541no—
108C4×C8—1271no—
109C2×Q8—2541no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—1271no—
114C22×C4—2541no—
115SD16—5082no—
116SD16—5082no—
117D8—5082no—
118M16—5082no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—1271no—
122C2×C8—1271no—
123C2×C8—2541no—
124C4⋊C4—1271no—
125C4⋊C4—2541no—
126C4⋊C4—2541no—
127C22⋊C4—2541no—
128C42C22×D12711yescharacteristic
129C23—41no—
130C23—2541no—
131Q8—5081no—
132Q8—5081no—
133D4—41no—
134D4—41no—
135D4—41no—
136D4—41no—
137D4—81no—
138D4—2541no—
139D4—2541no—
140C2×C4—5081no—
141C2×C4—5081no—
142C2×C4—5081no—
143C2×C4—2541no—
144C2×C4D4×D12711yescharacteristic
145C2×C4D4×D12711yescharacteristic
146C2×C4#1452324811yescharacteristic
147C8—41no—
148C8—41no—
149C8—1272no—
150C8—5081no—
151C8—2541no—
152C22—81no—
153C22—81no—
154C22—81no—
155C22—5081no—
156C22—5081no—
157C22—5081no—
158C22#1686118211yescharacteristic
159C4—21no—
160C4—10161no—
161C4—5081no—
162C4D8×D12711yescharacteristic
163C4#1686123311yescharacteristic
164C4#1686123611yescharacteristic
165C4#1686123811yescharacteristic
166C2—161no—
167C2—10161no—
168C2#2255007011yescharacteristic
169C2#2255012111yescharacteristic
170C2#2255013811yescharacteristic
subgroupG/H
centerC22#16861182
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#121
prime radical#22550068C2
hypercenter#331D127
hypocenterC127#1685
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #22550248C2semidirect
2. #22550157C2semidirect
4. #22549953C2semidirect
5. #22549628C2semidirect
6. #22549623C2semidirect
7. #331×C127C2semidirect
54. C127⋊C8D8semidirect
86. C127#1685semidirect
93. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225502482
#225501572
#225500682
#225499532
#225496282
#225496232
#331×C1272
#1685127