Finite Group Databasebeta

← #30637901

Subgroups — #30637901

Order 16256.

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

normal subgroups only

This group has 194 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#22550599C211yesmaximal, characteristic
2#22550226C211yesmaximal, characteristic, complemented
3#22550163C211yesmaximal, characteristic
4#22549621C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic
7#477×C127C211yesmaximal, characteristic, complemented
8#16861209C2211yescharacteristic
9#16861204—22no—
10#16861197—21no—
11#16861185—21no—
12C4×Dic254—21no—
13C4×D508C2211yescharacteristic
14#16861167—21no—
15#16861140—21no—
16C4⋊(C127⋊C8)C2211yescharacteristic
17#16861133—22no—
18#16861129—21no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×Q8×C127C2211yescharacteristic
22#107×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25C2×Dic254—21no—
26C2×D508D411yescharacteristic
27C2×C4×D127—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127D412yes—
32C4⋊Dic127D411yescharacteristic
33Dic127⋊C4—22no—
34(C2×D127)⋊C4—22no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—21no—
37C4×Dic127—22no—
38C2×Q8×C127—21no—
39C2×Q8×C127D411yescharacteristic
40C2×C8×C127—21no—
41C4⋊C4×C127—21no—
42C4⋊C4×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127D411yescharacteristic
45C42×C127—21no—
46C42×C127C2311yescharacteristic
47C22×D127—21no—
48Dic254—41no—
49Dic254—41no—
50D508—21no—
51D508—21no—
52C2×Dic127—21no—
53C2×Dic127—22no—
54C4×D127—41no—
55C4×D127—41no—
56C127⋊C8—41no—
57C127⋊C8—41no—
58Q8×C127—21no—
59Q8×C127—21no—
60Q8×C127—42no—
61C2×C4×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—41no—
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C8×C127—41no—
68C22×C127#12011yescharacteristic
69C2×D127—41no—
70C2×D127—41no—
71C2×D127—41no—
72C4×C127—21no—
73C4×C127—41no—
74C4×C127—41no—
75C4×C127—81no—
76C4×C127#13511yescharacteristic
77C4×C127#13511yescharacteristic
78C4×C127#13711yescharacteristic
79C4×C127#13711yescharacteristic
80Dic127—41no—
81Dic127—42no—
82C2×C127#45211yescharacteristic
83C2×C127#45211yescharacteristic
84C2×C127#45411yescharacteristic
85D127—81no—
86#1638—1271nomaximal, Sylow
87C127#163811yescharacteristic, complemented, Sylow
88#556—1271no—
89#487—1271no—
90#477D12711yescharacteristic, complemented
91#476—1271no—
92#332—1271no—
93Q8⋊C8—1271no—
94D4⋊C8—1271no—
95C4⋊Q8—1271no—
96C4⋊Q8C2×D12711yescharacteristic
97#126—2542no—
98#124—2541no—
99C22⋊Q8—2541no—
100C4×Q8—2541no—
101C4×Q8C2×D12711yescharacteristic
102C4×D4—1271no—
103#117—2541no—
104#107—21no—
105#106—2541no—
106C4⋊C8—1271no—
107C4⋊C8C2×D12711yescharacteristic
108#103—21no—
109#103—2542no—
110#102—2541no—
111C4×C8—1271no—
112C2×Q8—21no—
113C2×Q8—2541no—
114C2×Q8C22⋊D12711yescharacteristic
115C2×D4—1271no—
116C22×C4—2541no—
117C2×C8—21no—
118C2×C8—2541no—
119C2×C8—2541no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—1272no—
124C4⋊C4—1271no—
125C4⋊C4—2542no—
126C4⋊C4—2541no—
127C4⋊C4C22⋊D12711yescharacteristic
128C22⋊C4—2542no—
129C22⋊C4—2542no—
130C22⋊C4—2541no—
131C42—21no—
132C42—2542no—
133C42C22×D12711yescharacteristic
134C23—2541no—
135Q8—21no—
136Q8—21no—
137Q8—42no—
138Q8—5081no—
139Q8—5081no—
140D4—2541no—
141D4—2541no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—5081no—
146C2×C4—5081no—
147C2×C4—2541no—
148C2×C4—2542no—
149C2×C4D4×D12711yescharacteristic
150C2×C4D4×D12711yescharacteristic
151C2×C4C2×C22⋊D12711yescharacteristic
152C8—41no—
153C8—5081no—
154C8—5081no—
155C22—5081no—
156C22—5081no—
157C22—5081no—
158C22#1686117711yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—81no—
163C4—5081no—
164C4—5082no—
165C4#1686122911yescharacteristic
166C4#1686123211yescharacteristic
167C4#1686124011yescharacteristic
168C4#1686124111yescharacteristic
169C2—10161no—
170C2#2255003611yescharacteristic
171C2#2255003711yescharacteristic
172C2#2255004711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#477×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenter#477D127
hypocenterC127#1638
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #22550226C2semidirect
4. #22549621C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
7. #477×C127C2semidirect
87. C127#1638semidirect
90. #477D127semidirect

Maximal subgroups

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

subgroupindex
#225505992
#225502262
#225501632
#225496212
Q8⋊(C127⋊C8)2
#225496032
#477×C1272
#1638127