Finite Group Databasebeta

← #30637883

Subgroups — #30637883

Order 16256.

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

normal subgroups only

This group has 234 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#22550227C211yesmaximal, characteristic, complemented
3#22550167C211yesmaximal, characteristic, complemented
4#22550166C211yesmaximal, characteristic, complemented
5#22549621C211yesmaximal, characteristic, complemented
6#22549605C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9#16861209C2211yescharacteristic
10#16861209C2211yescharacteristic
11#16861193—22no—
12#16861189—21no—
13#16861183—21no—
14C4×D508C2211yescharacteristic
15C2×C4⋊Dic127—21no—
16#16861141—21no—
17#16861140—21no—
18C4⋊(C127⋊C8)C2211yescharacteristic
19#16861134—22no—
20#16861131—21no—
21#16861130—21no—
22C4×D4×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic
25#14523247—42no—
26#14523247—42no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×D508D411yescharacteristic
30C2×C22⋊D127—22no—
31C22×Dic127—22no—
32C2×C4×D127—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—22no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127D412yes—
39C4⋊Dic127D411yescharacteristic
40(C2×D127)⋊C4—22no—
41(C2×D127)⋊C4—21no—
42C2×D4×C127D411yescharacteristic
43C22×C4×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127D411yescharacteristic
47C22⋊C4×C127—21no—
48C42×C127C2311yescharacteristic
49C22×D127—21no—
50Dic254—41no—
51Dic254—41no—
52Dic254—42no—
53D508—21no—
54D508—21no—
55C22⋊D127—44no—
56C2×Dic127—21no—
57C2×Dic127—22no—
58C2×Dic127—44no—
59C2×Dic127—41no—
60C4×D127—41no—
61C4×D127—41no—
62C127⋊C8—41no—
63C23×C127—21no—
64D4×C127—21no—
65D4×C127SD1612yes—
66C2×C4×C127—21no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
73C8×C127—41no—
74C22×C127—22no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127#12011yescharacteristic
78C2×D127—41no—
79C2×D127—41no—
80C2×D127—41no—
81C4×C127—21no—
82C4×C127—41no—
83C4×C127C2×SD1611yescharacteristic
84C4×C127#13511yescharacteristic
85C4×C127AGL(1,ℤ/8)11yescharacteristic
86C4×C127#13711yescharacteristic
87Dic127—41no—
88Dic127—42no—
89Dic127—81no—
90C2×C127—42no—
91C2×C127#45011yescharacteristic
92C2×C127#45211yescharacteristic
93C2×C127#45411yescharacteristic
94D127—81no—
95#1635—1271nomaximal, Sylow
96C127#163511yescharacteristic, complemented, Sylow
97#547—1271no—
98#487—1271no—
99#476—1271no—
100#476—1271no—
101#332—1271no—
102D4⋊C8—1271no—
103D4⋊C8D12711yescharacteristic, complemented
104C2×#41—2541no—
105C4⋊Q8—1271no—
106C4⋊Q8—1271no—
107#123—2542no—
108C22⋊Q8—2541no—
109#121—2541no—
110C4×D4—1271no—
111C4×D4C2×D12711yescharacteristic
112C2×C4⋊C4—2541no—
113#106—2541no—
114#106—2541no—
115C4⋊C8—1271no—
116C4⋊C8C2×D12711yescharacteristic
117#103—2542no—
118#102—2541no—
119#102—2541no—
120C4×C8C2×D12711yescharacteristic
121#41—5082no—
122#41—5082no—
123C2×Q8—2541no—
124C2×Q8—2541no—
125C2×D4—1271no—
126C2×D4—2542no—
127C2×D4D50811yescharacteristic
128C22×C4—21no—
129C22×C4—2542no—
130C22×C4—2541no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2541no—
134C4⋊C4—1272no—
135C4⋊C4—1271no—
136C4⋊C4—2542no—
137C4⋊C4—2541no—
138C4⋊C4—2541no—
139C4⋊C4—2541no—
140C4⋊C4D50811yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—2542no—
143C22⋊C4—2541no—
144C42C22×D12711yescharacteristic
145C23—21no—
146C23—2541no—
147Q8—5082no—
148Q8—5081no—
149Q8—5081no—
150D4—21no—
151D4—5084no—
152D4—2541no—
153D4—2541no—
154D4#1452323712yes—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—5081no—
159C2×C4—5084no—
160C2×C4—5081no—
161C2×C4—5081no—
162C2×C4—2541no—
163C2×C4—2542no—
164C2×C4D4×D12711yescharacteristic
165C2×C4D4×D12711yescharacteristic
166C2×C4C2×D50811yescharacteristic
167C8—41no—
168C8—41no—
169C8—5081no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—5081no—
174C22—5081no—
175C22—5081no—
176C22#1686117911yescharacteristic
177C4—21no—
178C4—41no—
179C4—10161no—
180C4—5081no—
181C4—5082no—
182C4C2×#1452323711yescharacteristic
183C4#1686122811yescharacteristic
184C4#1686123711yescharacteristic
185C4#1686123811yescharacteristic
186C2—42no—
187C2—10161no—
188C2#2255002911yescharacteristic
189C2#2255003911yescharacteristic
190C2#2255005011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127#120
prime radical#22550530C2
hypercenterD4⋊C8D127
hypocenterC127#1635
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22550227C2semidirect
3. #22550167C2semidirect
4. #22550166C2semidirect
5. #22549621C2semidirect
6. #22549605C2semidirect
7. D4⋊C8×C127C2semidirect
96. C127#1635semidirect
103. D4⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
#225505302
#225502272
#225501672
#225501662
#225496212
#225496052
D4⋊C8×C1272
#1635127