Finite Group Databasebeta

← #30637170

Subgroups — #30637170

Order 16256.

266 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C4×#14523248C211yesmaximal, characteristic
2#22549900C211yesmaximal, characteristic, complemented
3#22549608C212yesmaximal, complemented
4#22549605C212yesmaximal, complemented
5C4×M16×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C411yescharacteristic
7C2×#14523232—21no—
8C4×Dic254C411yescharacteristic
9C4×Dic254C2211yescharacteristic
10C4×D508C411yescharacteristic
11C4×D508C2211yescharacteristic
12C4×C22⋊D127—21no—
13C4×C22⋊D127—21no—
14#16861168C411yescharacteristic
15#16861166—21no—
16C42×D127—21no—
17C4⋊(C127⋊C8)—21no—
18C4⋊(C127⋊C8)C2212yes—
19C2×M16×C127—21no—
20C2×C42×C127C2211yescharacteristic
21#97×C127—21no—
22C4×C8×C127C2212yes—
23#14523248—22no—
24#14523248—22no—
25C2×Dic254C2×C411yescharacteristic
26C2×D508C2×C411yescharacteristic
27C2×C22⋊D127—21no—
28C2×C4×D127—21no—
29C2×C4×D127—21no—
30C2×C4×D127—21no—
31#14523232—42no—
32C2×C127⋊C8—22no—
33C4⋊Dic127C2×C411yescharacteristic
34C4⋊Dic127C2×C411yescharacteristic
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37C22⋊Dic127—21no—
38(C2×D127)⋊C4—21no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C4×Dic127—21no—
43C22×C4×C127C2×C411yescharacteristic
44C22×C4×C127D411yescharacteristic
45C22×C4×C127D411yescharacteristic
46M16×C127—24no—
47C2×C8×C127—22no—
48C42×C127C2×C411yescharacteristic
49C42×C127D411yescharacteristic
50C42×C127D411yescharacteristic
51C42×C127C2311yescharacteristic
52C22×D127—21no—
53Dic254—21no—
54Dic254M1612yescomplemented
55D508—21no—
56D508M1612yescomplemented
57C22⋊D127—24no—
58C2×Dic127—21no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C4×D127—24no—
62C4×D127—42no—
63C4×D127—41no—
64C4×D127—41no—
65C127⋊C8—42no—
66C23×C127C22⋊C411yescharacteristic
67C2×C4×C127—22no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127C22⋊C411yescharacteristic
73C2×C4×C127C22⋊C411yescharacteristic
74C2×C4×C127C22⋊C411yescharacteristic
75C2×C4×C127C22×C411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C8×C127—24no—
79C22×C127—21no—
80C22×C127—21no—
81C22×C127—21no—
82C22×C127C2×C22⋊C411yescharacteristic
83C2×D127—22no—
84C2×D127—41no—
85C2×D127—41no—
86C4×C127—21no—
87C4×C127—21no—
88C4×C127—21no—
89C4×C127#10412yes—
90C4×C127C2×M1611yescharacteristic
91C4×C127C2×M1611yescharacteristic
92C4×C127AGL(1,ℤ/8)11yescharacteristic
93C4×C127#13711yescharacteristic
94Dic127—22no—
95Dic127—41no—
96Dic127—41no—
97C2×C127—41no—
98C2×C127#40711yescharacteristic
99C2×C127#41711yescharacteristic
100C2×C127C2×#10411yescharacteristic
101D127—42no—
102#1489—1271nomaximal, Sylow
103C127#148911yescharacteristic, complemented, Sylow
104C4×#41—1271no—
105#423—1271no—
106C4×M16D12711yescharacteristic, complemented
107Q8⋊C8—1272no—
108D4⋊C8—1272no—
109C2×#41—1271no—
110C2×M16—21no—
111C2×M16—2541no—
112C4×Q8—1271no—
113C4×Q8—1271no—
114C4×D4—1271no—
115C4×D4—1271no—
116C4×D4—2541no—
117C4×D4—2541no—
118#117—1271no—
119#117—2541no—
120C2×C42—2541no—
121C2×C42C2×D12711yescharacteristic
122C4⋊C8—1272no—
123C4⋊C8—2541no—
124#97—21no—
125C4×C8C2×D12712yes—
126#41—2542no—
127#41—2542no—
128C2×Q8—1271no—
129C2×D4—1271no—
130C2×D4—2541no—
131C22×C4—2541no—
132C22×C4—2541no—
133C22×C4—2541no—
134C22×C4C4×D12711yescharacteristic
135C22×C4C22⋊D12711yescharacteristic
136C22×C4D50811yescharacteristic
137M16—24no—
138M16—5082no—
139C2×C8—22no—
140C2×C8—2542no—
141C4⋊C4—1271no—
142C4⋊C4—1271no—
143C4⋊C4—2541no—
144C4⋊C4—2541no—
145C22⋊C4—2541no—
146C22⋊C4—2541no—
147C22⋊C4—2541no—
148C42—2541no—
149C42—2541no—
150C42—2541no—
151C42C4×D12711yescharacteristic
152C42C22⋊D12711yescharacteristic
153C42D50811yescharacteristic
154C42C22×D12711yescharacteristic
155C23—2541no—
156C23(C2×D127)⋊C411yescharacteristic
157Q8—1272no—
158Q8—2541no—
159D4—1272no—
160D4—2544no—
161D4—2541no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—5081no—
168C2×C4—5081no—
169C2×C4—5082no—
170C2×C4—2544no—
171C2×C4—2541no—
172C2×C4—2541no—
173C2×C4—2541no—
174C2×C4(C2×D127)⋊C411yescharacteristic
175C2×C4(C2×D127)⋊C411yescharacteristic
176C2×C4(C2×D127)⋊C411yescharacteristic
177C2×C4C2×C4×D12711yescharacteristic
178C2×C4C2×C22⋊D12711yescharacteristic
179C2×C4C2×D50811yescharacteristic
180C8—24no—
181C8—5082no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—5081no—
186C22—5081no—
187C22—2542no—
188C22C2×(C2×D127)⋊C411yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—5081no—
193C4—5081no—
194C4—2542no—
195C4#1686113712yes—
196C4M16×D12711yescharacteristic
197C4M16×D12711yescharacteristic
198C4#1686123711yescharacteristic
199C4#1686124211yescharacteristic
200C2—41no—
201C2—5082no—
202C2#2254983911yescharacteristic
203C2#2254987811yescharacteristic
204C2C2×#1686113711yescharacteristic
subgroupG/H
centerC2×C4C2×D508
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingC4×M16×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#14523248C4
cosocleC42×C127C23
hypercenterC4×M16D127
hypocenterC127#1489
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. #22549900C2semidirect
3. #22549608C2semidirect
4. #22549605C2semidirect
5. C4×M16×C127C2semidirect
54. Dic254M16semidirect
56. D508M16semidirect
103. C127#1489semidirect
106. C4×M16D127semidirect

Maximal subgroups

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

subgroupindex
C4×#145232482
#225499002
#225496082
#225496052
C4×M16×C1272
#1489127