Finite Group Databasebeta

← #30637769

Subgroups — #30637769

Order 16256.

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

normal subgroups only

This group has 246 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#22550195C212yesmaximal, complemented
3#22549815C211yesmaximal, characteristic, complemented
4#22549611C212yesmaximal, complemented
5#324×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—21no—
7#16861189—22no—
8#16861184—22no—
9#16861183—21no—
10#16861183C2212yes—
11C2×C4⋊Dic127C2211yescharacteristic
12#16861141—22no—
13#16861131—22no—
14C127⋊#95—21no—
15C127⋊#95—21no—
16C127⋊#95C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19#14523248—42no—
20#14523248—42no—
21C2×Dic254—21no—
22C2×D508—21no—
23C2×D508—21no—
24C2×D508D412yes—
25C2×C22⋊D127—22no—
26C22×Dic127—21no—
27C22×Dic127—21no—
28C2×C4×D127—22no—
29C4⋊Dic127—21no—
30C4⋊Dic127—22no—
31C4⋊Dic127D412yes—
32(C2×D127)⋊C4—22no—
33(C2×D127)⋊C4—22no—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37C2×C8×C127—22no—
38C4⋊C4×C127—22no—
39C22×D127—22no—
40Dic254—42no—
41D508—42no—
42D508—42no—
43D508—81no—
44C22⋊D127—42no—
45C22⋊D127—42no—
46C2×Dic127—22no—
47C2×Dic127—41no—
48C2×Dic127—41no—
49C2×Dic127—42no—
50C2×Dic127—41no—
51C2×Dic127—41no—
52C4×D127—44no—
53C23×C127C2×D411yescharacteristic
54C2×C4×C127—21no—
55C2×C4×C127—21no—
56C2×C4×C127—21no—
57C2×C4×C127—41no—
58C2×C4×C127—41no—
59C2×C4×C127SD1612yes—
60C2×C4×C127C2×D412yes—
61C8×C127—42no—
62C22×C127—21no—
63C22×C127—21no—
64C22×C127#12011yescharacteristic
65C22×C127C2×SD1611yescharacteristic
66C22×C127AGL(1,ℤ/8)11yescharacteristic
67C2×D127—42no—
68C2×D127—42no—
69C2×D127—42no—
70C4×C127—22no—
71C4×C127—22no—
72C4×C127—41no—
73Dic127—42no—
74Dic127—81no—
75Dic127—81no—
76C2×C127—21no—
77C2×C127—21no—
78C2×C127#45011yescharacteristic
79C2×C127#45411yescharacteristic
80C2×C127#45811yescharacteristic
81D127—82no—
82#1616—1271nomaximal, Sylow
83C127#161611yescharacteristic, complemented, Sylow
84#537—1271no—
85#481—1272no—
86#400—1271no—
87#327—1272no—
88#324D12711yescharacteristic, complemented
89C2×#41—2541no—
90C22⋊Q8—2542no—
91#121—1272no—
92#121—2542no—
93#121—2541no—
94C2×C4⋊C4—1271no—
95C2×C4⋊C4C2×D12711yescharacteristic
96#106—2542no—
97#102—2542no—
98C22⋊C8C2×D12712yes—
99#95—1271no—
100#95—2541no—
101#95—2541no—
102#41—5082no—
103#41—5082no—
104C2×Q8—2541no—
105C2×D4—1272no—
106C2×D4—2541no—
107C2×D4—2541no—
108C2×D4—2542no—
109C22×C4—2542no—
110C22×C4—2541no—
111C22×C4—2541no—
112C22×C4D50811yescharacteristic
113C22×C4D50811yescharacteristic
114C22×C4C22×D12711yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—1272no—
118C4⋊C4—2541no—
119C4⋊C4—2542no—
120C22⋊C4—2542no—
121C22⋊C4—2542no—
122C23—2542no—
123C23C2×D50811yescharacteristic
124Q8—5082no—
125D4—10161no—
126D4—5082no—
127D4—5082no—
128D4—5082no—
129D4—5082no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—5084no—
136C2×C4—5081no—
137C2×C4—5081no—
138C2×C4—5082no—
139C2×C4—5081no—
140C2×C4—5081no—
141C2×C4—2542no—
142C2×C4#1452323712yes—
143C2×C4D4×D12712yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—5082no—
148C22—5082no—
149C22—5082no—
150C22#1686117911yescharacteristic
151C22C2×#1452323711yescharacteristic
152C22#1686123711yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—10161no—
157C4—10161no—
158C4—5082no—
159C2—21no—
160C2—21no—
161C2—10162no—
162C2#2255002911yescharacteristic
163C2#2255005111yescharacteristic
164C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#324D127
hypocenterC127#1616
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. #22550195C2semidirect
3. #22549815C2semidirect
4. #22549611C2semidirect
5. #324×C127C2semidirect
83. C127#1616semidirect
88. #324D127semidirect

Maximal subgroups

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

subgroupindex
#225504572
#225501952
#225498152
#225496112
#324×C1272
#1616127