Finite Group Databasebeta

← #30637867

Subgroups — #30637867

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#22550256C211yesmaximal, characteristic, complemented
3#22550171C211yesmaximal, characteristic, complemented
4#22550167C211yesmaximal, characteristic
5#22549624C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic
7Q8⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×Dic508—22no—
9#16861209C2211yescharacteristic
10#16861209C2211yescharacteristic
11#16861201—21no—
12#16861187—22no—
13#16861185—21no—
14C4×D508—21no—
15C4×D508C2211yescharacteristic
16C4⋊C4×D127—21no—
17#16861142—21no—
18#16861141—21no—
19C4⋊(C127⋊C8)C2211yescharacteristic
20#16861133—21no—
21#16861131—21no—
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25C2×Dic254—21no—
26C2×Dic254D412yes—
27C2×D508D411yescharacteristic
28C2×C4×D127—22no—
29C2×C4×D127—21no—
30Dic508—44no—
31C2×C127⋊C8—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—22no—
38Dic127⋊C4—22no—
39(C2×D127)⋊C4—22no—
40(C2×D127)⋊C4—21no—
41C2×Q8×C127D411yescharacteristic
42C2×C8×C127—21no—
43C2×C8×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127D411yescharacteristic
46C42×C127—21no—
47C42×C127C2311yescharacteristic
48C22×D127—21no—
49Dic254—42no—
50Dic254—42no—
51D508—21no—
52D508Q1612yescomplemented
53C2×Dic127—21no—
54C2×Dic127—22no—
55C2×Dic127—41no—
56C4×D127—41no—
57C4×D127—41no—
58C4×D127—82no—
59C127⋊C8—41no—
60Q8×C127—21no—
61Q8×C127—21no—
62C2×C4×C127—21no—
63C2×C4×C127—21no—
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C8×C127—22no—
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×C127C2×Q1611yescharacteristic
77C4×C127#13511yescharacteristic
78C4×C127#13711yescharacteristic
79C4×C127#13711yescharacteristic
80Dic127—41no—
81Dic127—42no—
82Dic127—81no—
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#477—1271no—
92#476—1271no—
93#333—1271no—
94Q8⋊C8D12711yescharacteristic, complemented
95D4⋊C8—1271no—
96C2×Q16—2542no—
97C4⋊Q8—1271no—
98C4⋊Q8—1271no—
99#125—2541no—
100C22⋊Q8—2542no—
101C22⋊Q8—2541no—
102C4×Q8C2×D12711yescharacteristic
103C4×D4—1271no—
104C4×D4—2541no—
105C2×C4⋊C4—2541no—
106#107—2541no—
107#106—2541no—
108C4⋊C8—1271no—
109C4⋊C8C2×D12711yescharacteristic
110#103—2541no—
111#102—2541no—
112C4×C8C2×D12711yescharacteristic
113C2×Q8—1272no—
114C2×Q8—2541no—
115C2×Q8D50811yescharacteristic
116C2×D4—1271no—
117C22×C4—2542no—
118C22×C4—2541no—
119Q16—5084no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2541no—
123C4⋊C4—21no—
124C4⋊C4—1271no—
125C4⋊C4—2542no—
126C4⋊C4—2542no—
127C4⋊C4—2541no—
128C4⋊C4—2541no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4D50811yescharacteristic
132C22⋊C4—2542no—
133C22⋊C4—2541no—
134C42—21no—
135C42C22×D12711yescharacteristic
136C23—2541no—
137Q8—21no—
138Q8—21no—
139Q8—5082no—
140Q8—5082no—
141D4—1272no—
142D4—2541no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—10162no—
146C2×C4—5081no—
147C2×C4—5081no—
148C2×C4—5081no—
149C2×C4—2541no—
150C2×C4—2542no—
151C2×C4D4×D12711yescharacteristic
152C2×C4D4×D12711yescharacteristic
153C2×C4C2×D50811yescharacteristic
154C8—22no—
155C8—41no—
156C8—5081no—
157C22—5081no—
158C22—5081no—
159C22—2542no—
160C22#1686117911yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—10161no—
165C4—5081no—
166C4—5082no—
167C4Q16×D12711yescharacteristic
168C4#1686123111yescharacteristic
169C4#1686123811yescharacteristic
170C4#1686124211yescharacteristic
171C2—5082no—
172C2#2255003311yescharacteristic
173C2#2255004011yescharacteristic
174C2#2255004511yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingQ8⋊C8×C127C2
socleC22×C127#120
prime radicalC2×D508D4
hypercenterQ8⋊C8D127
hypocenterC127#1632
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #22550256C2semidirect
3. #22550171C2semidirect
5. #22549624C2semidirect
7. Q8⋊C8×C127C2semidirect
52. D508Q16semidirect
88. C127#1632semidirect
94. Q8⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225505862
#225502562
#225501712
#225501672
#225496242
#225496052
Q8⋊C8×C1272
#1632127