Finite Group Databasebeta

← #30637767

Subgroups — #30637767

Order 16256.

246 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550192C212yesmaximal, complemented
2#22549815C211yesmaximal, characteristic, complemented
3#22549610C212yesmaximal, complemented
4#22549602C211yesmaximal, characteristic, complemented
5#537×C127C211yesmaximal, characteristic
6C2×C4⋊Dic127C2211yescharacteristic
7#16861140—22no—
8#16861130—22no—
9C22⋊(C127⋊C8)C2212yes—
10C127⋊#95—21no—
11C127⋊#95—21no—
12C127⋊#95C2211yescharacteristic
13C2×#41×C127—21no—
14C22⋊Q8×C127—22no—
15#121×C127—22no—
16#121×C127—21no—
17#121×C127C2212yes—
18C2×C4⋊C4×C127C2211yescharacteristic
19C22×Dic127—21no—
20C22×Dic127—21no—
21C2×C127⋊C8—22no—
22C4⋊Dic127—22no—
23#41×C127—42no—
24#41×C127—42no—
25C2×Q8×C127—21no—
26C2×D4×C127—21no—
27C2×D4×C127—22no—
28C2×D4×C127—21no—
29C2×D4×C127D412yes—
30C22×C4×C127—22no—
31C22×C4×C127D411yescharacteristic
32C22×C4×C127D411yescharacteristic
33C22×C4×C127C2311yescharacteristic
34C4⋊C4×C127—21no—
35C4⋊C4×C127—22no—
36C4⋊C4×C127D412yes—
37C22⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C2×Dic127—41no—
40C2×Dic127—41no—
41C2×Dic127—42no—
42C2×Dic127—41no—
43C2×Dic127—41no—
44C127⋊C8—42no—
45C23×C127—22no—
46C23×C127C2×D411yescharacteristic
47Q8×C127—42no—
48D4×C127—42no—
49D4×C127—42no—
50D4×C127—42no—
51D4×C127—42no—
52D4×C127—81no—
53C2×C4×C127—21no—
54C2×C4×C127—21no—
55C2×C4×C127—21no—
56C2×C4×C127—22no—
57C2×C4×C127—41no—
58C2×C4×C127—44no—
59C2×C4×C127—41no—
60C2×C4×C127SD1612yes—
61C2×C4×C127C2×D412yes—
62C22×C127—21no—
63C22×C127—21no—
64C22×C127—42no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127#12011yescharacteristic
68C22×C127C2×SD1611yescharacteristic
69C22×C127AGL(1,ℤ/8)11yescharacteristic
70C4×C127—22no—
71C4×C127—22no—
72C4×C127—42no—
73C4×C127—41no—
74Dic127—81no—
75Dic127—81no—
76C2×C127—21no—
77C2×C127—21no—
78C2×C127—82no—
79C2×C127#45011yescharacteristic
80C2×C127#45411yescharacteristic
81C2×C127#45811yescharacteristic
82#1616—1271nomaximal, Sylow
83C127#161611yescharacteristic, complemented, Sylow
84#537D12711yescharacteristic
85#481—1272no—
86#400—1271no—
87#327—1272no—
88#324—1271no—
89C2×#41—21no—
90C22⋊Q8—22no—
91#121—22no—
92#121—21no—
93#121C2×D12712yes—
94C2×C4⋊C4—1271no—
95C2×C4⋊C4C2×D12711yescharacteristic
96#106—2542no—
97#102—2542no—
98C22⋊C8—1272no—
99#95—1271no—
100#95—2541no—
101#95—2541no—
102#41—42no—
103#41—42no—
104C2×Q8—21no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4C22⋊D12712yes—
109C22×C4—22no—
110C22×C4—2541no—
111C22×C4—2541no—
112C22×C4C22⋊D12711yescharacteristic
113C22×C4C22⋊D12711yescharacteristic
114C22×C4C22×D12711yescharacteristic
115C2×C8—2542no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—2542no—
119C4⋊C4C22⋊D12712yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C23—22no—
123C23C2×C22⋊D12711yescharacteristic
124Q8—42no—
125D4—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—81no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—44no—
136C2×C4—41no—
137C2×C4—5081no—
138C2×C4—5081no—
139C2×C4—5082no—
140C2×C4—5081no—
141C2×C4—5081no—
142C2×C4#1452323612yes—
143C2×C4C2×C22⋊D12712yes—
144C8—5082no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22#1686117811yescharacteristic
151C22C2×#1452323611yescharacteristic
152C22#1686123511yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—41no—
157C4—10161no—
158C4—10161no—
159C2—21no—
160C2—21no—
161C2—82no—
162C2#2255002711yescharacteristic
163C2#2255004911yescharacteristic
164C2#2255006611yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#537×C127C2
socleC22×C127#120
hypercenter#537D127
hypocenterC127#1616
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #22550192C2semidirect
2. #22549815C2semidirect
3. #22549610C2semidirect
4. #22549602C2semidirect
83. C127#1616semidirect

Maximal subgroups

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

subgroupindex
#225501922
#225498152
#225496102
#225496022
#537×C1272
#1616127