Finite Group Databasebeta

← #30640678

Subgroups — #30640678

Order 16256.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550676C211yesmaximal, characteristic, complemented
2#22550314C212yesmaximal, complemented
3#22550129C211yesmaximal, characteristic, complemented
4#22549687C212yesmaximal, complemented
5#349×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—21no—
7#16861242—22no—
8#16861237—22no—
9#16861231—21no—
10#16861231C2212yes—
11C2×D1016C2211yescharacteristic, complemented
12#16861183—21no—
13#16861154—22no—
14D2032—22no—
15#16861141C2211yescharacteristic, complemented
16#16861131—21no—
17C2×M16×C127C2211yescharacteristic
18M32×C127C2212yescomplemented
19#14523248—22no—
20#14523248—22no—
21#14523248—41no—
22#14523248—41no—
23C2×Dic254—21no—
24C2×D508—21no—
25C2×D508—21no—
26C2×C22⋊D127—41no—
27C2×C22⋊D127—41no—
28C2×C4×D127—41no—
29Dic508—21no—
30Dic508D412yescomplemented
31#14523237—22no—
32#14523237—22no—
33D1016—21no—
34D1016—22no—
35D1016D412yescomplemented
36C4⋊Dic127—21no—
37(C2×D127)⋊C4—41no—
38C22×C4×C127D411yescharacteristic
39M16×C127—22no—
40C2×C8×C127D411yescharacteristic
41C2×C8×C127C2311yescharacteristic
42C16×C127—22no—
43C22×D127—41no—
44C22×D127—41no—
45Dic254—22no—
46Dic254—41no—
47D508—22no—
48D508—41no—
49D508—41no—
50D508—41no—
51C22⋊D127—42no—
52C22⋊D127—42no—
53C22⋊D127—82no—
54C2×Dic127—41no—
55C2×Dic127—41no—
56C4×D127—42no—
57C4×D127—42no—
58C23×C127D811yescharacteristic
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127D811yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C8×C127—21no—
64C8×C127C2×D412yes—
65C22×C127—21no—
66C22×C127—41no—
67C22×C127C2×D811yescharacteristic
68C2×D127—42no—
69C2×D127—82no—
70C2×D127—81no—
71C2×D127—81no—
72C2×D127—81no—
73C4×C127—21no—
74C4×C127#12011yescharacteristic
75C4×C127AGL(1,ℤ/8)11yescharacteristic
76Dic127—42no—
77Dic127—81no—
78C2×C127—21no—
79C2×C127—41no—
80C2×C127#44711yescharacteristic
81D127—82no—
82D127—161no—
83#2190—1271nomaximal, Sylow
84C127#219011yescharacteristic, complemented, Sylow
85#575—1271no—
86#509—1272no—
87#469—1271no—
88#360—1272no—
89#349D12711yescharacteristic, complemented
90C2×#41—2541no—
91#137—2542no—
92AGL(1,ℤ/8)—2542no—
93#135—1272no—
94#135—2541no—
95C2×D8—1271no—
96C2×M16C2×D12711yescharacteristic
97#121—2541no—
98SD32—2542no—
99D16—2542no—
100M32C2×D12712yescomplemented
101#106—1271no—
102#102—2541no—
103#41—5081no—
104#41—5081no—
105#41—2542no—
106#41—2542no—
107C2×Q8—2541no—
108C2×D4—5081no—
109C2×D4—5081no—
110C2×D4—2541no—
111C2×D4—2541no—
112C22×C4—5081no—
113C22×C4D50811yescharacteristic
114Q16—1272no—
115Q16—2541no—
116SD16—2542no—
117SD16—2542no—
118D8—1272no—
119D8—2541no—
120D8—2542no—
121M16—22no—
122C2×C8D50811yescharacteristic
123C2×C8C22×D12711yescharacteristic
124C4⋊C4—2541no—
125C22⋊C4—5081no—
126C16—22no—
127C23—5081no—
128C23—5081no—
129C23D101611yescharacteristic
130Q8—5081no—
131Q8—2542no—
132D4—10162no—
133D4—5081no—
134D4—5082no—
135D4—5082no—
136D4—5081no—
137D4—5081no—
138D4—2542no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—5082no—
142C2×C4—5081no—
143C2×C4—5081no—
144C2×C4—5082no—
145C2×C4D101611yescharacteristic
146C2×C4C2×D50811yescharacteristic
147C8—21no—
148C8D4×D12712yes—
149C22—21no—
150C22—41no—
151C22—10161no—
152C22—10161no—
153C22—10162no—
154C22—10161no—
155C22—5082no—
156C22C2×D101611yescharacteristic
157C4—21no—
158C4—10161no—
159C4—5082no—
160C4#1686117911yescharacteristic
161C4#1686123711yescharacteristic
162C2—21no—
163C2—41no—
164C2—20321no—
165C2—10162no—
166C2#2255001211yescharacteristic
subgroupG/H
centerC2#22550012
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#349×C127C2
socleC2×C127#447
hypercenter#349D127
hypocenterC127#2190
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
1. #22550676C2semidirect
2. #22550314C2semidirect
3. #22550129C2semidirect
4. #22549687C2semidirect
5. #349×C127C2semidirect
11. C2×D1016C22semidirect
15. #16861141C22semidirect
18. M32×C127C22semidirect
30. Dic508D4semidirect
35. D1016D4semidirect
84. C127#2190semidirect
89. #349D127semidirect
100. M32C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225503142
#225501292
#225496872
#349×C1272
#2190127