Finite Group Databasebeta

← #30638013

Subgroups — #30638013

Order 16256.

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

normal subgroups only

This group has 240 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#22550530C211yesmaximal, characteristic
2#22550272C211yesmaximal, characteristic, complemented
3#22550078C211yesmaximal, characteristic, complemented
4#22549623C211yesmaximal, characteristic, complemented
5D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6#22549603C211yesmaximal, characteristic, complemented
7#474×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C2×#14523235—21no—
10#16861209C2211yescharacteristic
11#16861193—22no—
12#16861189—21no—
13#16861183—21no—
14C4×D508C2211yescharacteristic
15C2×C4⋊Dic127—21no—
16#16861140—22no—
17C4⋊(C127⋊C8)C2211yescharacteristic
18#16861133—21no—
19C4×C127⋊C8C2211yescharacteristic
20C4⋊Q8×C127C2211yescharacteristic
21C4×D4×C127C2211yescharacteristic
22#107×C127—21no—
23C4⋊C8×C127C2211yescharacteristic
24#102×C127—21no—
25#14523247—42no—
26#14523247—42no—
27C2×Dic254—21no—
28C2×D508D411yescharacteristic
29C2×C22⋊D127—22no—
30C22×Dic127—22no—
31C2×C4×D127—21no—
32#14523235—42no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—22no—
36C4⋊Dic127—21no—
37C4⋊Dic127D412yes—
38C4⋊Dic127D411yescharacteristic
39(C2×D127)⋊C4—22no—
40(C2×D127)⋊C4—21no—
41C2×Q8×C127—21no—
42C2×D4×C127D411yescharacteristic
43C22×C4×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127D411yescharacteristic
48C22⋊C4×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51Dic254—41no—
52Dic254—41no—
53D508—21no—
54D508—21no—
55C22⋊D127—44no—
56C2×Dic127—21no—
57C2×Dic127—22no—
58C2×Dic127—44no—
59C4×D127—41no—
60C4×D127—41no—
61C127⋊C8—22no—
62C127⋊C8—41no—
63C23×C127—21no—
64Q8×C127—41no—
65Q8×C127—41no—
66D4×C127—21no—
67D4×C127SD1612yescomplemented
68C2×C4×C127—21no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C8×C127—41no—
76C22×C127—22no—
77C22×C127—41no—
78C22×C127—41no—
79C22×C127#12011yescharacteristic
80C2×D127—41no—
81C2×D127—41no—
82C2×D127—41no—
83C4×C127—21no—
84C4×C127—41no—
85C4×C127—81no—
86C4×C127C2×SD1611yescharacteristic
87C4×C127#13511yescharacteristic
88C4×C127AGL(1,ℤ/8)11yescharacteristic
89C4×C127AGL(1,ℤ/8)11yescharacteristic
90Dic127—41no—
91Dic127—42no—
92C2×C127—42no—
93C2×C127#44911yescharacteristic
94C2×C127#45011yescharacteristic
95C2×C127#45611yescharacteristic
96D127—81no—
97#1654—1271nomaximal, Sylow
98C127#165411yescharacteristic, complemented, Sylow
99#547—1271no—
100#498—1271no—
101#474D12711yescharacteristic, complemented
102#461—1271no—
103#332—1271no—
104D4⋊C8—1271no—
105D4⋊C8—1271no—
106C2×#41—2541no—
107C2×SD16—2541no—
108C4⋊Q8—1271no—
109C4⋊Q8C2×D12711yescharacteristic
110#123—2542no—
111C22⋊Q8—2541no—
112#121—2541no—
113C4×D4—1271no—
114C4×D4C2×D12711yescharacteristic
115C2×C4⋊C4—2541no—
116#107—21no—
117#106—2542no—
118C4⋊C8—1271no—
119C4⋊C8C2×D12711yescharacteristic
120#103—2541no—
121#102—21no—
122C4×C8—1271no—
123#41—5082no—
124#41—5082no—
125C2×Q8—21no—
126C2×Q8—2541no—
127C2×D4—1271no—
128C2×D4—2542no—
129C2×D4C22⋊D12711yescharacteristic
130C22×C4—21no—
131C22×C4—2542no—
132C22×C4—2541no—
133SD16—5082no—
134C2×C8—21no—
135C2×C8—2541no—
136C2×C8—2541no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—1271no—
140C4⋊C4—1272no—
141C4⋊C4—2542no—
142C4⋊C4—2541no—
143C4⋊C4C22⋊D12711yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—2542no—
146C22⋊C4—2541no—
147C42C22×D12711yescharacteristic
148C23—21no—
149C23—2541no—
150Q8—41no—
151Q8—41no—
152Q8—5081no—
153Q8—5081no—
154D4—21no—
155D4—5084no—
156D4—2541no—
157D4—2541no—
158D4#1452323512yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—5084no—
164C2×C4—5081no—
165C2×C4—5081no—
166C2×C4—2541no—
167C2×C4—2542no—
168C2×C4D4×D12711yescharacteristic
169C2×C4D4×D12711yescharacteristic
170C2×C4C2×C22⋊D12711yescharacteristic
171C8—41no—
172C8—5081no—
173C8—2542no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—5081no—
178C22—5081no—
179C22—5081no—
180C22#1686117711yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—5081no—
185C4—5082no—
186C4C2×#1452323511yescharacteristic
187C4#1686122811yescharacteristic
188C4#1686123311yescharacteristic
189C4#1686123411yescharacteristic
190C2—42no—
191C2—10161no—
192C2#2255001811yescharacteristic
193C2#2255002611yescharacteristic
194C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#474×C127C2
socleC22×C127#120
prime radical#22550530C2
hypercenter#474D127
hypocenterC127#1654
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #22550272C2semidirect
3. #22550078C2semidirect
4. #22549623C2semidirect
5. D4⋊(C127⋊C8)C2semidirect
6. #22549603C2semidirect
7. #474×C127C2semidirect
67. D4×C127SD16semidirect
98. C127#1654semidirect
101. #474D127semidirect
158. D4#14523235semidirect

Maximal subgroups

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

subgroupindex
#225505302
#225502722
#225500782
#225496232
D4⋊(C127⋊C8)2
#225496032
#474×C1272
#1654127