Finite Group Databasebeta

← #30637825

Subgroups — #30637825

Order 16256.

210 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550605C211yesmaximal, characteristic, complemented
2#22550222C211yesmaximal, characteristic
3#22550073C211yesmaximal, characteristic
4#22549621C211yesmaximal, characteristic, complemented
5Q8⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549606C211yesmaximal, characteristic, complemented
7#460×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9#16861209—21no—
10#16861209C2211yescharacteristic
11#16861200—22no—
12#16861200—21no—
13Q8×Dic127—21no—
14C4×Dic254—21no—
15C4×Dic254C2211yescharacteristic
16C4⋊(C127⋊C8)C2211yescharacteristic
17#16861130—21no—
18#16861130—22no—
19C4×C127⋊C8C2211yescharacteristic
20C2×SD16×C127—21no—
21#127×C127C2211yescharacteristic
22C4×Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24#102×C127—21no—
25C2×Dic254—21no—
26C2×Dic254D411yescharacteristic
27#14523236—42no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—22no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127D412yes—
34C4⋊Dic127D411yescharacteristic
35Dic127⋊C4—22no—
36Dic127⋊C4—21no—
37C4×Dic127—22no—
38C4×Dic127—21no—
39C2×Q8×C127D411yescharacteristic
40C2×D4×C127—21no—
41C2×D4×C127—21no—
42C2×D4×C127—21no—
43SD16×C127—42no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127D411yescharacteristic
47C42×C127—21no—
48C42×C127C2311yescharacteristic
49Dic254—21no—
50Dic254—21no—
51Dic254—41no—
52Dic254—41no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—22no—
56C127⋊C8—41no—
57C127⋊C8—41no—
58C23×C127—41no—
59Q8×C127—21no—
60Q8×C127SD1612yescomplemented
61D4×C127—42no—
62D4×C127—41no—
63D4×C127—41no—
64D4×C127—41no—
65D4×C127—41no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—41no—
72C22×C127—81no—
73C22×C127—81no—
74C22×C127—81no—
75C22×C127#12011yescharacteristic
76C4×C127—22no—
77C4×C127—21no—
78C4×C127—41no—
79C4×C127C2×SD1611yescharacteristic
80C4×C127#13511yescharacteristic
81C4×C127AGL(1,ℤ/8)11yescharacteristic
82C4×C127#13711yescharacteristic
83Dic127—41no—
84Dic127—42no—
85Dic127—41no—
86C2×C127—161no—
87C2×C127#44811yescharacteristic
88C2×C127#44911yescharacteristic
89C2×C127#45411yescharacteristic
90#1626—1271nomaximal, Sylow
91C127#162611yescharacteristic, complemented, Sylow
92#557—1271no—
93#486—1271no—
94#460—1271no—
95#460D12711yescharacteristic
96#332—1271no—
97Q8⋊C8—1271no—
98Q8⋊C8—1271no—
99C2×SD16—21no—
100C2×SD16—2541no—
101C4⋊Q8—1271no—
102C4⋊Q8—2541no—
103#127C2×D12711yescharacteristic
104#125—2542no—
105#125—2541no—
106C4×Q8—1271no—
107C4×Q8—2541no—
108C4×Q8—2541no—
109C4×Q8C2×D12711yescharacteristic
110C4⋊C8—1271no—
111C4⋊C8C2×D12711yescharacteristic
112#102—21no—
113#102—2541no—
114#102—2542no—
115C4×C8—1271no—
116C2×Q8—1271no—
117C2×Q8—2541no—
118C2×Q8C22⋊D12711yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122SD16—42no—
123SD16—5082no—
124C2×C8—21no—
125C2×C8—2541no—
126C2×C8—2541no—
127C4⋊C4—21no—
128C4⋊C4—1271no—
129C4⋊C4—1272no—
130C4⋊C4—2542no—
131C4⋊C4—2542no—
132C4⋊C4—2542no—
133C4⋊C4—2541no—
134C4⋊C4—2541no—
135C4⋊C4C22⋊D12711yescharacteristic
136C42—21no—
137C42—2542no—
138C42—2541no—
139C42C22×D12711yescharacteristic
140C23—41no—
141Q8—21no—
142Q8—5081no—
143Q8—5081no—
144Q8—2541no—
145Q8—2541no—
146Q8#1452323612yescomplemented
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—2541no—
155C2×C4—2542no—
156C2×C4—2541no—
157C2×C4D4×D12711yescharacteristic
158C2×C4D4×D12711yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C8—41no—
161C8—5081no—
162C8—5081no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#1686117711yescharacteristic
167C4—21no—
168C4—22no—
169C4—41no—
170C4—5082no—
171C4—5081no—
172C4—5081no—
173C4C2×#1452323611yescharacteristic
174C4#1686123011yescharacteristic
175C4#1686123611yescharacteristic
176C4#1686123911yescharacteristic
177C2—161no—
178C2#2255001411yescharacteristic
179C2#2255002011yescharacteristic
180C2#2255004711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#460×C127C2
socleC22×C127#120
prime radical#127×C127C22
hypercenter#460D127
hypocenterC127#1626
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #22550605C2semidirect
4. #22549621C2semidirect
5. Q8⋊(C127⋊C8)C2semidirect
6. #22549606C2semidirect
60. Q8×C127SD16semidirect
91. C127#1626semidirect
146. Q8#14523236semidirect

Maximal subgroups

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

subgroupindex
#225506052
#225502222
#225500732
#225496212
Q8⋊(C127⋊C8)2
#225496062
#460×C1272
#1626127