Finite Group Databasebeta

← #30637950

Subgroups — #30637950

Order 16256.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×Dic254C211yesmaximal, characteristic
2#22550228C211yesmaximal, characteristic
3#22550083C211yesmaximal, characteristic
4#22549624C211yesmaximal, characteristic
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic
6#22549606C211yesmaximal, characteristic
7#475×C127C211yesmaximal, characteristic
8C2×#14523238—21no—
9#16861209—21no—
10#16861209C2211yescharacteristic
11#16861208—22no—
12#16861208—21no—
13Q8×Dic127—21no—
14C4×Dic254—21no—
15C4×Dic254C2211yescharacteristic
16C4⋊(C127⋊C8)C2211yescharacteristic
17#16861133—21no—
18#16861132—22no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×Q8×C127C2211yescharacteristic
22#106×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#103×C127—21no—
25C2×Dic254—22no—
26C2×Dic254D411yescharacteristic
27C2×Dic254D412yes—
28#14523238—42no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35Dic127⋊C4—22no—
36Dic127⋊C4—21no—
37C4×Dic127—22no—
38C4×Dic127—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
48Dic254—21no—
49Dic254—42no—
50Dic254—82no—
51Dic254SD1612yes—
52C2×Dic127—21no—
53C2×Dic127—22no—
54C2×Dic127—21no—
55C127⋊C8—41no—
56C127⋊C8—41no—
57Q8×C127—21no—
58Q8×C127—41no—
59Q8×C127—41no—
60Q8×C127Q1612yescomplemented
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
69C4×C127—22no—
70C4×C127—21no—
71C4×C127—41no—
72C4×C127—81no—
73C4×C127C2×SD1611yescharacteristic
74C4×C127C2×Q1611yescharacteristic
75C4×C127#13711yescharacteristic
76C4×C127#13711yescharacteristic
77Dic127—22no—
78Dic127—42no—
79Dic127—41no—
80C2×C127#44811yescharacteristic
81C2×C127#45111yescharacteristic
82C2×C127#45511yescharacteristic
83#1645—1271nomaximal, Sylow
84C127#164511yescharacteristic, complemented, Sylow
85Q82—1271no—
86#487—1271no—
87#475D12711yescharacteristic
88#462—1271no—
89#333—1271no—
90Q8⋊C8—1271no—
91Q8⋊C8—1271no—
92C2×Q16—2541no—
93C4⋊Q8—1271no—
94C4⋊Q8—2541no—
95C4⋊Q8—2542no—
96C4⋊Q8—2541no—
97C4⋊Q8C2×D12711yescharacteristic
98C4×Q8—1271no—
99C4×Q8—2541no—
100C4×Q8—2541no—
101C4×Q8C2×D12711yescharacteristic
102#106—21no—
103C4⋊C8—1271no—
104C4⋊C8C2×D12711yescharacteristic
105#103—21no—
106#103—2542no—
107#103—2541no—
108C4×C8—1271no—
109C2×Q8—21no—
110C2×Q8—1271no—
111C2×Q8—1272no—
112C2×Q8—2542no—
113C2×Q8C22⋊D12711yescharacteristic
114Q16—5082no—
115C2×C8—21no—
116C2×C8—2541no—
117C2×C8—2541no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—1271no—
122C4⋊C4—2542no—
123C4⋊C4—2542no—
124C4⋊C4—2541no—
125C4⋊C4—2541no—
126C4⋊C4—2541no—
127C4⋊C4C22⋊D12711yescharacteristic
128C42—21no—
129C42—2542no—
130C42—2541no—
131C42C22×D12711yescharacteristic
132Q8—21no—
133Q8—41no—
134Q8—41no—
135Q8—1272no—
136Q8—10162no—
137Q8—5082no—
138Q8—2541no—
139Q8#1452323812yescomplemented
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—2541no—
144C2×C4—2542no—
145C2×C4—2541no—
146C2×C4D4×D12711yescharacteristic
147C2×C4D4×D12711yescharacteristic
148C2×C4C2×C22⋊D12711yescharacteristic
149C8—41no—
150C8—5081no—
151C8—5081no—
152C22#1686117711yescharacteristic
153C4—22no—
154C4—21no—
155C4—41no—
156C4—81no—
157C4—5081no—
158C4—5082no—
159C4—2542no—
160C4SD16×D12711yescharacteristic
161C4C2×#1452323811yescharacteristic
162C4#1686123811yescharacteristic
163C4#1686123911yescharacteristic
164C2#2255001311yescharacteristic
165C2#2255003111yescharacteristic
166C2#2255005211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#475×C127C2
socle, prime radicalC22×C127#120
hypercenter#475D127
hypocenterC127#1645
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
60. Q8×C127Q16semidirect
84. C127#1645semidirect
139. Q8#14523238semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2542
#225502282
#225500832
#225496242
Q8⋊(C127⋊C8)2
#225496062
#475×C1272
#1645127