Finite Group Databasebeta

← #30637863

Subgroups — #30637863

Order 16256.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550586C211yesmaximal, characteristic
2#22550255C211yesmaximal, characteristic, complemented
3#22550163C211yesmaximal, characteristic
4#22549624C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic
7#477×C127C211yesmaximal, characteristic, complemented
8C2×#14523238—22no—
9#16861209C2211yescharacteristic
10#16861201—21no—
11#16861187—22no—
12#16861185—21no—
13C4×D508—21no—
14C4×D508C2211yescharacteristic
15C4⋊C4×D127—21no—
16#16861140—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861129—21no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×Q8×C127C2211yescharacteristic
22#107×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25C2×Dic254D412yes—
26C2×D508D411yescharacteristic
27C2×C4×D127—22no—
28C2×C4×D127—21no—
29#14523238—44no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35Dic127⋊C4—22no—
36Dic127⋊C4—22no—
37(C2×D127)⋊C4—22no—
38(C2×D127)⋊C4—21no—
39C2×Q8×C127—21no—
40C2×Q8×C127D411yescharacteristic
41C2×C8×C127—21no—
42C4⋊C4×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C42×C127—21no—
47C42×C127C2311yescharacteristic
48C22×D127—21no—
49Dic254—42no—
50D508—21no—
51D508Q1612yescomplemented
52C2×Dic127—21no—
53C2×Dic127—22no—
54C4×D127—41no—
55C4×D127—41no—
56C4×D127—82no—
57C127⋊C8—22no—
58C127⋊C8—41no—
59Q8×C127—21no—
60Q8×C127—21no—
61Q8×C127—42no—
62C2×C4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127—41no—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C8×C127—41no—
69C22×C127#12011yescharacteristic
70C2×D127—22no—
71C2×D127—41no—
72C2×D127—41no—
73C4×C127—21no—
74C4×C127—41no—
75C4×C127—41no—
76C4×C127—81no—
77C4×C127C2×Q1611yescharacteristic
78C4×C127#13511yescharacteristic
79C4×C127#13711yescharacteristic
80C4×C127#13711yescharacteristic
81Dic127—41no—
82Dic127—42no—
83C2×C127#45111yescharacteristic
84C2×C127#45211yescharacteristic
85C2×C127#45311yescharacteristic
86D127—42no—
87#1632—1271nomaximal, Sylow
88C127#163211yescharacteristic, complemented, Sylow
89#554—1271no—
90#494—1271no—
91#477D12711yescharacteristic, complemented
92#476—1271no—
93#333—1271no—
94Q8⋊C8—1271no—
95D4⋊C8—1271no—
96C2×Q16—2542no—
97C4⋊Q8—1271no—
98C4⋊Q8C2×D12711yescharacteristic
99#125—2541no—
100C22⋊Q8—2542no—
101C22⋊Q8—2541no—
102C4×Q8C2×D12711yescharacteristic
103C4×D4—1271no—
104C4×D4—2541no—
105C2×C4⋊C4—2541no—
106#107—21no—
107#106—2541no—
108C4⋊C8—1271no—
109C4⋊C8C2×D12711yescharacteristic
110#103—21no—
111#102—2541no—
112C4×C8—1271no—
113C2×Q8—21no—
114C2×Q8—1272no—
115C2×Q8C22⋊D12711yescharacteristic
116C2×D4—1271no—
117C22×C4—2542no—
118C22×C4—2541no—
119Q16—5084no—
120C2×C8—21no—
121C2×C8—2541no—
122C2×C8—2541no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—1271no—
127C4⋊C4—2542no—
128C4⋊C4—2542no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4C22⋊D12711yescharacteristic
132C22⋊C4—2542no—
133C22⋊C4—2541no—
134C42—21no—
135C42C22×D12711yescharacteristic
136C23—2541no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—5082no—
141D4—1272no—
142D4—2541no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—10162no—
147C2×C4—5081no—
148C2×C4—5081no—
149C2×C4—2541no—
150C2×C4—2542no—
151C2×C4D4×D12711yescharacteristic
152C2×C4D4×D12711yescharacteristic
153C2×C4C2×C22⋊D12711yescharacteristic
154C8—41no—
155C8—5081no—
156C8—2542no—
157C22—5081no—
158C22—5081no—
159C22—2542no—
160C22#1686117711yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—5081no—
166C4—5082no—
167C4Q16×D12711yescharacteristic
168C4#1686122911yescharacteristic
169C4#1686124011yescharacteristic
170C4#1686124111yescharacteristic
171C2—5082no—
172C2#2255003011yescharacteristic
173C2#2255003611yescharacteristic
174C2#2255004211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#477×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenter#477D127
hypocenterC127#1632
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. #22550255C2semidirect
4. #22549624C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
7. #477×C127C2semidirect
51. D508Q16semidirect
88. C127#1632semidirect
91. #477D127semidirect

Maximal subgroups

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

subgroupindex
#225505862
#225502552
#225501632
#225496242
Q8⋊(C127⋊C8)2
#225496032
#477×C1272
#1632127