Finite Group Databasebeta

← #30637735

Subgroups — #30637735

Order 16256.

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

normal subgroups only

This group has 254 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#22550190C212yesmaximal, complemented
3#22549806C211yesmaximal, 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
12C2×Dic127⋊C4—21no—
13#16861143—22no—
14#16861131—22no—
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—
32Dic127⋊C4—42no—
33(C2×D127)⋊C4—22no—
34(C2×D127)⋊C4—22no—
35C22×C4×C127D411yescharacteristic
36C22×C4×C127D411yescharacteristic
37C22×C4×C127C2311yescharacteristic
38C2×C8×C127—22no—
39C4⋊C4×C127—22no—
40C22×D127—22no—
41Dic254—42no—
42D508—42no—
43D508—42no—
44D508—81no—
45C22⋊D127—42no—
46C22⋊D127—42no—
47C2×Dic127—22no—
48C2×Dic127—22no—
49C2×Dic127—41no—
50C2×Dic127—41no—
51C2×Dic127—41no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C4×D127—44no—
55C23×C127C2×D411yescharacteristic
56C2×C4×C127—21no—
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127—41no—
60C2×C4×C127—41no—
61C2×C4×C127D812yes—
62C2×C4×C127C2×D412yes—
63C8×C127—42no—
64C22×C127—21no—
65C22×C127—21no—
66C22×C127#12011yescharacteristic
67C22×C127C2×D811yescharacteristic
68C22×C127AGL(1,ℤ/8)11yescharacteristic
69C2×D127—42no—
70C2×D127—42no—
71C2×D127—42no—
72C4×C127—22no—
73C4×C127—22no—
74C4×C127—41no—
75Dic127—42no—
76Dic127—42no—
77Dic127—81no—
78C2×C127—21no—
79C2×C127—21no—
80C2×C127#44711yescharacteristic
81C2×C127#45611yescharacteristic
82C2×C127#45811yescharacteristic
83D127—82no—
84#1610—1271nomaximal, Sylow
85C127#161011yescharacteristic, complemented, Sylow
86#537—1271no—
87#480—1272no—
88#398—1271no—
89#327—1272no—
90#324D12711yescharacteristic, complemented
91C2×#41—2541no—
92C22⋊Q8—2542no—
93#121—1272no—
94#121—2542no—
95#121—2541no—
96C2×C4⋊C4—1271no—
97C2×C4⋊C4—2541no—
98C2×C4⋊C4C2×D12711yescharacteristic
99#107—2542no—
100#102—2542no—
101C22⋊C8C2×D12712yes—
102#95—1271no—
103#95—2541no—
104#41—5082no—
105#41—5082no—
106C2×Q8—2541no—
107C2×D4—1272no—
108C2×D4—2541no—
109C2×D4—2541no—
110C2×D4—2542no—
111C22×C4—2542no—
112C22×C4—2541no—
113C22×C4—2541no—
114C22×C4D50811yescharacteristic
115C22×C4D50811yescharacteristic
116C22×C4C22×D12711yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—1272no—
120C4⋊C4—5082no—
121C4⋊C4—2541no—
122C4⋊C4—2542no—
123C22⋊C4—2542no—
124C22⋊C4—2542no—
125C23—2542no—
126C23C2×D50811yescharacteristic
127Q8—5082no—
128D4—10161no—
129D4—5082no—
130D4—5082no—
131D4—5082no—
132D4—5082no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—5084no—
139C2×C4—5081no—
140C2×C4—5081no—
141C2×C4—5081no—
142C2×C4—5081no—
143C2×C4—5081no—
144C2×C4—2542no—
145C2×C4—2542no—
146C2×C4D101612yes—
147C2×C4D4×D12712yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—5082no—
152C22—5082no—
153C22—5082no—
154C22#1686117911yescharacteristic
155C22C2×D101611yescharacteristic
156C22#1686123711yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—10161no—
161C4—5082no—
162C4—5082no—
163C2—21no—
164C2—21no—
165C2—10162no—
166C2#2255001211yescharacteristic
167C2#2255006111yescharacteristic
168C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#324D127
hypocenterC127#1610
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. #22550190C2semidirect
3. #22549806C2semidirect
4. #22549611C2semidirect
5. #324×C127C2semidirect
85. C127#1610semidirect
90. #324D127semidirect

Maximal subgroups

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

subgroupindex
#225504572
#225501902
#225498062
#225496112
#324×C1272
#1610127