Finite Group Databasebeta

← #30638061

Subgroups — #30638061

Order 16256.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550630C211yesmaximal, characteristic
2#22550282C211yesmaximal, characteristic, complemented
3#22550238C211yesmaximal, characteristic
4#22549618C211yesmaximal, characteristic, complemented
5#22549617C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic
7#491×C127C211yesmaximal, characteristic, complemented
8#16861208—21no—
9#16861204—21no—
10#16861204—21no—
11#16861201C2211yescharacteristic
12#16861197C2211yescharacteristic
13#16861194—21no—
14#16861189—21no—
15#16861187—21no—
16#16861142—21no—
17#16861140—21no—
18#16861132—21no—
19#16861129—21no—
20#16861120C2211yescharacteristic
21#16861120C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23#125×C127C2211yescharacteristic
24#103×C127—21no—
25#103×C127—21no—
26#97×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254D411yescharacteristic
29C2×D508D411yescharacteristic
30C2×C4×D127—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44C4×Dic127—21no—
45C2×Q8×C127—21no—
46C2×C8×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127D411yescharacteristic
52C4⋊C4×C127D411yescharacteristic
53C42×C127C2311yescharacteristic
54C22×D127—21no—
55Dic254—41no—
56Dic254—81no—
57D508—41no—
58C2×Dic127—21no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C4×D127—81no—
62C127⋊C8—22no—
63C127⋊C8—41no—
64Q8×C127—42no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—41no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—41no—
72C22×C127#12011yescharacteristic
73C2×D127—41no—
74C2×D127—41no—
75C2×D127—41no—
76C4×C127—21no—
77C4×C127—21no—
78C4×C127—21no—
79C4×C127—41no—
80C4×C127—41no—
81C4×C127—81no—
82Dic127—41no—
83Dic127—41no—
84Dic127—41no—
85C2×C127#45411yescharacteristic
86C2×C127#45511yescharacteristic
87C2×C127#45611yescharacteristic
88D127—81no—
89#1664—1271nomaximal, Sylow
90C127#166411yescharacteristic, complemented, Sylow
91#563—1271no—
92#501—1271no—
93#491D12711yescharacteristic, complemented
94#489—1271no—
95#330—1271no—
96#330—1271no—
97#329—1271no—
98C4⋊Q8—2541no—
99C4⋊Q8C2×D12711yescharacteristic
100#126—2541no—
101#126—2541no—
102#125—1271no—
103#125C2×D12711yescharacteristic
104#124—1271no—
105#123—2541no—
106C22⋊Q8—2541no—
107C22⋊Q8—2541no—
108#107—2541no—
109#106—2541no—
110#103—21no—
111#103—21no—
112#103—2541no—
113#102—2541no—
114#97—1271no—
115#97—1271no—
116#97C2×D12711yescharacteristic
117C2×Q8—21no—
118C2×Q8—1271no—
119C2×Q8—2541no—
120C2×D4—1271no—
121C22×C4—2541no—
122C2×C8—21no—
123C2×C8—2541no—
124C2×C8—2541no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—1271no—
130C4⋊C4—1271no—
131C4⋊C4—2541no—
132C4⋊C4—2541no—
133C4⋊C4—2541no—
134C4⋊C4—2541no—
135C4⋊C4C22⋊D12711yescharacteristic
136C4⋊C4C22⋊D12711yescharacteristic
137C22⋊C4—2541no—
138C22⋊C4—2541no—
139C22⋊C4—2541no—
140C22⋊C4—2541no—
141C22⋊C4—2541no—
142C42—2541no—
143C42C22×D12711yescharacteristic
144C23—2541no—
145Q8—42no—
146Q8—10161no—
147Q8—5081no—
148D4—5081no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—10161no—
153C2×C4—2541no—
154C2×C4—2541no—
155C2×C4—2541no—
156C2×C4D4×D12711yescharacteristic
157C2×C4D4×D12711yescharacteristic
158C2×C4C2×C22⋊D12711yescharacteristic
159C8—41no—
160C8—5081no—
161C8—2542no—
162C22—5081no—
163C22—5081no—
164C22—5081no—
165C22#1686117711yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—5081no—
173C4—5081no—
174C4—5081no—
175C2—10161no—
176C2#2255004811yescharacteristic
177C2#2255005211yescharacteristic
178C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#491×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenter#491D127
hypocenterC127#1664
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #22550282C2semidirect
4. #22549618C2semidirect
5. #22549617C2semidirect
7. #491×C127C2semidirect
90. C127#1664semidirect
93. #491D127semidirect

Maximal subgroups

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

subgroupindex
#225506302
#225502822
#225502382
#225496182
#225496172
#225496152
#491×C1272
#1664127