Finite Group Databasebeta

← #30637969

Subgroups — #30637969

Order 16256.

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

normal subgroups only

This group has 194 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#22550601C211yesmaximal, characteristic
2#22550158C211yesmaximal, characteristic
3#22550083C211yesmaximal, characteristic
4#22549622C211yesmaximal, characteristic
5#22549608C211yesmaximal, characteristic
6#22549608C211yesmaximal, characteristic
7#500×C127C211yesmaximal, characteristic
8C2×#14523238—21no—
9#16861207—21no—
10#16861200—21no—
11#16861199—22no—
12Q8×Dic127—21no—
13C4×Dic254C2211yescharacteristic
14C4×Dic254C2211yescharacteristic
15#16861174—21no—
16#16861140—21no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861133—21no—
20#16861132—21no—
21C4⋊Q8×C127C2211yescharacteristic
22C4⋊Q8×C127C2211yescharacteristic
23#107×C127—22no—
24C4×C8×C127C2211yescharacteristic
25C2×Dic254D411yescharacteristic
26C2×Dic254D411yescharacteristic
27#14523238—42no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—22no—
31C4⋊Dic127D411yescharacteristic
32C4⋊Dic127D411yescharacteristic
33Dic127⋊C4—22no—
34Dic127⋊C4—22no—
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37C4×Dic127—22no—
38C4×Dic127—21no—
39C4×Dic127—21no—
40C2×Q8×C127—21no—
41C2×Q8×C127—21no—
42C2×C8×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127D412yes—
47C42×C127C2311yescharacteristic
48Dic254—21no—
49Dic254—21no—
50Dic254—21no—
51Dic254Q1612yescomplemented
52C2×Dic127—21no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C127⋊C8—41no—
57C127⋊C8—41no—
58Q8×C127—41no—
59Q8×C127—41no—
60Q8×C127—41no—
61Q8×C127—41no—
62C2×C4×C127—22no—
63C2×C4×C127—41no—
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C8×C127—22no—
68C22×C127#12011yescharacteristic
69C4×C127—21no—
70C4×C127—42no—
71C4×C127—81no—
72C4×C127C2×Q1611yescharacteristic
73C4×C127#13511yescharacteristic
74C4×C127AGL(1,ℤ/8)11yescharacteristic
75C4×C127#13711yescharacteristic
76Dic127—22no—
77Dic127—41no—
78Dic127—41no—
79Dic127—41no—
80C2×C127#44911yescharacteristic
81C2×C127#45111yescharacteristic
82C2×C127#45611yescharacteristic
83#1648—1271nomaximal, Sylow
84C127#164811yescharacteristic, complemented, Sylow
85#557—1271no—
86#500D12711yescharacteristic
87#475—1271no—
88#462—1271no—
89#332—1271no—
90Q8⋊C8—1271no—
91Q8⋊C8—1271no—
92C2×Q16—2541no—
93C4⋊Q8—2541no—
94C4⋊Q8C2×D12711yescharacteristic
95C4⋊Q8C2×D12711yescharacteristic
96#125—2541no—
97#125—2542no—
98C4×Q8—1271no—
99C4×Q8—1271no—
100C4×Q8—2541no—
101C4×Q8—2541no—
102#107—22no—
103#106—2541no—
104C4⋊C8—1271no—
105C4⋊C8—1271no—
106#103—2541no—
107#103—2541no—
108C4×C8C2×D12711yescharacteristic
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—1271no—
112C2×Q8—1271no—
113Q16—5082no—
114C2×C8—21no—
115C2×C8—2541no—
116C2×C8—2541no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—1271no—
121C4⋊C4—1271no—
122C4⋊C4—2542no—
123C4⋊C4—2542no—
124C4⋊C4—2542no—
125C4⋊C4—2541no—
126C4⋊C4—2541no—
127C4⋊C4C22⋊D12712yes—
128C42—2542no—
129C42—2541no—
130C42—2541no—
131C42C22×D12711yescharacteristic
132Q8—41no—
133Q8—41no—
134Q8—41no—
135Q8—41no—
136Q8—1272no—
137Q8—2541no—
138Q8—2541no—
139Q8—2541no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—2541no—
143C2×C4—2541no—
144C2×C4—2541no—
145C2×C4—2541no—
146C2×C4D4×D12711yescharacteristic
147C2×C4D4×D12711yescharacteristic
148C2×C4C2×C22⋊D12711yescharacteristic
149C8—22no—
150C8—5081no—
151C8—5081no—
152C22#1686117711yescharacteristic
153C4—21no—
154C4—42no—
155C4—81no—
156C4—5081no—
157C4—5081no—
158C4—5081no—
159C4—2542no—
160C4Q16×D12711yescharacteristic
161C4#1686122811yescharacteristic
162C4#1686123311yescharacteristic
163C4#1686124111yescharacteristic
164C2#2255001811yescharacteristic
165C2#2255003011yescharacteristic
166C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#500×C127C2
socle, prime radicalC22×C127#120
hypercenter#500D127
hypocenterC127#1648
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
51. Dic254Q16semidirect
84. C127#1648semidirect

Maximal subgroups

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

subgroupindex
#225506012
#225501582
#225500832
#225496222
#225496082
#500×C1272
#1648127