Finite Group Databasebeta

← #30638165

Subgroups — #30638165

Order 16256.

182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550222C211yesmaximal, characteristic, complemented
2#22550081C211yesmaximal, characteristic, complemented
3D4×C127⋊C8C211yesmaximal, characteristic, complemented
4#22549628C211yesmaximal, characteristic, complemented
5#22549623C211yesmaximal, characteristic, complemented
6#22549620C211yesmaximal, characteristic, complemented
7#459×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9C22×C127⋊C8—21no—
10#16861209C2211yescharacteristic
11C4⋊(C127⋊C8)C2211yescharacteristic
12C4⋊(C127⋊C8)C2211yescharacteristic
13#16861132—21no—
14#16861130—21no—
15#16861130—21no—
16C22⋊(C127⋊C8)—21no—
17C4×C127⋊C8C2211yescharacteristic
18C2×D8×C127—21no—
19#127×C127C2211yescharacteristic
20C4×D4×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22#102×C127—21no—
23C2×Dic254—21no—
24#14523236—42no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—42no—
28C2×C127⋊C8D411yescharacteristic
29C2×C127⋊C8D411yescharacteristic
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C2×D4×C127—21no—
33C2×D4×C127—21no—
34C2×D4×C127—21no—
35C2×D4×C127D411yescharacteristic
36C22×C4×C127—21no—
37D8×C127—42no—
38C2×C8×C127—21no—
39C4⋊C4×C127D411yescharacteristic
40C22⋊C4×C127—21no—
41C42×C127C2311yescharacteristic
42Dic254—41no—
43Dic254—41no—
44C2×Dic127—41no—
45C127⋊C8—21no—
46C127⋊C8—21no—
47C127⋊C8—41no—
48C127⋊C8—41no—
49C23×C127—21no—
50C23×C127—41no—
51D4×C127—21no—
52D4×C127—41no—
53D4×C127—41no—
54D4×C127—41no—
55D4×C127—41no—
56D4×C127—81no—
57D4×C127D812yes—
58C2×C4×C127—21no—
59C2×C4×C127—41no—
60C2×C4×C127—41no—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127#4111yescharacteristic
64C8×C127—41no—
65C22×C127—22no—
66C22×C127—41no—
67C22×C127—41no—
68C22×C127—81no—
69C22×C127—81no—
70C22×C127—81no—
71C22×C127#12111yescharacteristic
72C4×C127—21no—
73C4×C127—41no—
74C4×C127C2×D811yescharacteristic
75C4×C127#13511yescharacteristic
76C4×C127#13511yescharacteristic
77C4×C127AGL(1,ℤ/8)11yescharacteristic
78Dic127—81no—
79C2×C127—42no—
80C2×C127—161no—
81C2×C127#46311yescharacteristic
82C2×C127#46611yescharacteristic
83C2×C127#47111yescharacteristic
84#1681—1271nomaximal, Sylow
85C127#168111yescharacteristic, complemented, Sylow
86#486—1271no—
87#461—1271no—
88#459D12711yescharacteristic
89C8×D4—1271no—
90#335—1271no—
91#332—1271no—
92#331—1271no—
93C2×SD16—2541no—
94C2×D8—21no—
95C22×C8—2541no—
96C4⋊Q8—1271no—
97#127C2×D12711yescharacteristic
98C4×D4C2×D12711yescharacteristic
99C4⋊C8—1271no—
100C4⋊C8—1271no—
101C4⋊C8C2×D12711yescharacteristic
102#103—2541no—
103#102—21no—
104#102—2541no—
105#102—2541no—
106C22⋊C8—2541no—
107C4×C8—1271no—
108C2×Q8—2541no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4C22⋊D12711yescharacteristic
113C22×C4—21no—
114SD16—5082no—
115D8—42no—
116C2×C8—21no—
117C2×C8—1271no—
118C2×C8—1271no—
119C2×C8—5082no—
120C2×C8—2541no—
121C2×C8—2541no—
122C4⋊C4—2541no—
123C4⋊C4—2541no—
124C4⋊C4C22⋊D12711yescharacteristic
125C22⋊C4—21no—
126C42C22×D12711yescharacteristic
127C23—21no—
128C23—41no—
129Q8—5081no—
130Q8—5081no—
131D4—21no—
132D4—41no—
133D4—41no—
134D4—41no—
135D4—41no—
136D4—81no—
137D4#1452323312yes—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—5081no—
142C2×C4D4×D12711yescharacteristic
143C2×C4C2×C22⋊D12711yescharacteristic
144C2×C4#1452324711yescharacteristic
145C8—41no—
146C8—5081no—
147C8—5081no—
148C8—2541no—
149C8—2541no—
150C22—22no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22—81no—
155C22—81no—
156C22#1686118111yescharacteristic
157C4—21no—
158C4—41no—
159C4—10161no—
160C4C2×#1452323311yescharacteristic
161C4#1686122811yescharacteristic
162C4#1686122911yescharacteristic
163C4#1686123311yescharacteristic
164C2—42no—
165C2—161no—
166C2#2255008911yescharacteristic
167C2#2255011011yescharacteristic
168C2#2255013711yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radical#459×C127C2
socleC22×C127#121
hypercenter#459D127
hypocenterC127#1681
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22550222C2semidirect
2. #22550081C2semidirect
3. D4×C127⋊C8C2semidirect
4. #22549628C2semidirect
5. #22549623C2semidirect
6. #22549620C2semidirect
85. C127#1681semidirect

Maximal subgroups

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

subgroupindex
#225502222
#225500812
D4×C127⋊C82
#225496282
#225496232
#225496202
#459×C1272
#1681127