Finite Group Databasebeta

← #30637166

Subgroups — #30637166

Order 16256.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14523247C211yesmaximal, characteristic
2#22549898C211yesmaximal, characteristic, complemented
3C4×#14523231C211yesmaximal, characteristic, complemented
4#22549608C211yesmaximal, characteristic, complemented
5#22549606C211yesmaximal, characteristic, complemented
6D4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic, complemented
8C2×#14523247C411yescharacteristic
9C2×#14523231—21no—
10C2×#14523231—21no—
11C4×Dic254C2211yescharacteristic
12#16861174C411yescharacteristic
13D4×Dic127C411yescharacteristic
14C4×C22⋊D127—21no—
15(C22⋊D127)⋊C4—21no—
16#16861167C411yescharacteristic
17#16861165—21no—
18C2×C4×Dic127—21no—
19C42×D127C2211yescharacteristic
20C4⋊(C127⋊C8)C2211yescharacteristic
21Dic127⋊C8—21no—
22#16861119—21no—
23C4×C127⋊C8C2211yescharacteristic
24C4×D4×C127C2211yescharacteristic
25C4⋊C8×C127C2211yescharacteristic
26C4×C8×C127C2211yescharacteristic
27#14523247—22no—
28#14523247—22no—
29C2×Dic254C2×C411yescharacteristic
30C2×C22⋊D127—21no—
31C22×Dic127—21no—
32C22×Dic127—21no—
33C2×C4×D127C2×C411yescharacteristic
34C2×C4×D127D411yescharacteristic
35C2×C4×D127D411yescharacteristic
36#14523231—24no—
37#14523231—42no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C4⋊Dic127C2×C411yescharacteristic
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43C22⋊Dic127—21no—
44(C2×D127)⋊C4—21no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C4×Dic127—21no—
48C4×Dic127C2×C411yescharacteristic
49C4×Dic127D411yescharacteristic
50C4×Dic127D411yescharacteristic
51C2×D4×C127C2×C411yescharacteristic
52C22×C4×C127—21no—
53C2×C8×C127—21no—
54C2×C8×C127—21no—
55C4⋊C4×C127C2×C411yescharacteristic
56C22⋊C4×C127—21no—
57C42×C127C2311yescharacteristic
58C22×D127C22⋊C411yescharacteristic
59Dic254—21no—
60Dic254M1612yescomplemented
61C22⋊D127—24no—
62C2×Dic127—24no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—42no—
66C2×Dic127C22⋊C411yescharacteristic
67C2×Dic127C22⋊C411yescharacteristic
68C2×Dic127C22⋊C411yescharacteristic
69C4×D127—21no—
70C4×D127—21no—
71C4×D127—21no—
72C4×D127—21no—
73C4×D127—22no—
74C127⋊C8—22no—
75C127⋊C8—41no—
76C23×C127—21no—
77D4×C127—21no—
78D4×C127M1612yescomplemented
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C8×C127—22no—
86C8×C127—41no—
87C22×C127—22no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127C2×C22⋊C411yescharacteristic
91C2×D127—21no—
92C2×D127—21no—
93C2×D127—21no—
94C4×C127—21no—
95C4×C127—41no—
96C4×C127C2×M1611yescharacteristic
97C4×C127C2×M1611yescharacteristic
98C4×C127AGL(1,ℤ/8)11yescharacteristic
99C4×C127#13711yescharacteristic
100Dic127—21no—
101Dic127—22no—
102Dic127—21no—
103Dic127—41no—
104Dic127#10412yes—
105C2×C127—42no—
106C2×C127#40711yescharacteristic
107C2×C127#41711yescharacteristic
108C2×C127C2×#10411yescharacteristic
109D127—41no—
110#1489—1271nomaximal, Sylow
111C127#148911yescharacteristic, complemented, Sylow
112C4×#41—1271no—
113#423—1271no—
114C4×M16—1271no—
115Q8⋊C8—1271no—
116Q8⋊C8—1271no—
117D4⋊C8—1271no—
118D4⋊C8D12711yescharacteristic, complemented
119C2×#41—1271no—
120C2×M16—2541no—
121C2×M16—2541no—
122C4×Q8—1271no—
123C4×Q8—1271no—
124C4×D4—1271no—
125C4×D4—2541no—
126C4×D4—2541no—
127C4×D4C2×D12711yescharacteristic
128#117—1271no—
129#117—2541no—
130C2×C42—1271no—
131C2×C42—2541no—
132C4⋊C8—1271no—
133C4⋊C8—2541no—
134C4⋊C8C2×D12711yescharacteristic
135#97—2541no—
136C4×C8—1271no—
137C4×C8C2×D12711yescharacteristic
138#41—2542no—
139#41—2542no—
140C2×Q8—1271no—
141C2×D4—2541no—
142C2×D4C4×D12711yescharacteristic
143C22×C4—21no—
144C22×C4—1271no—
145C22×C4—1271no—
146C22×C4—1271no—
147C22×C4—2541no—
148C22×C4—2541no—
149M16—5082no—
150M16—2544no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—2541no—
154C2×C8—2541no—
155C4⋊C4—1271no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4C4×D12711yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—2541no—
161C22⋊C4—2541no—
162C42—1271no—
163C42—1271no—
164C42—1271no—
165C42—2541no—
166C42—2541no—
167C42—2541no—
168C42C22×D12711yescharacteristic
169C23—21no—
170C23—1271no—
171Q8—1272no—
172Q8—2541no—
173D4—21no—
174D4—2544no—
175D4#1452323112yescomplemented
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—1271no—
180C2×C4—1271no—
181C2×C4—1271no—
182C2×C4—5082no—
183C2×C4—2542no—
184C2×C4—2541no—
185C2×C4—2544no—
186C2×C4—2541no—
187C2×C4—2541no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4—2541no—
191C2×C4C2×C4×D12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4D4×D12711yescharacteristic
194C8—22no—
195C8—41no—
196C8—5081no—
197C8—2542no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—2541no—
202C22—2541no—
203C22—2541no—
204C22C22⋊C4×D12711yescharacteristic
205C4—21no—
206C4—41no—
207C4—1272no—
208C4—5081no—
209C4—2541no—
210C4—2542no—
211C4—2541no—
212C4M16×D12711yescharacteristic
213C4C2×#1452323111yescharacteristic
214C4#1686123311yescharacteristic
215C4#1686123811yescharacteristic
216C2—42no—
217C2—5081no—
218C2#2254983811yescharacteristic
219C2#2254987411yescharacteristic
220C2#104×D12711yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingD4⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#14523247C4
cosocleC42×C127C23
hypercenterD4⋊C8D127
hypocenterC127#1489
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22549898C2semidirect
3. C4×#14523231C2semidirect
4. #22549608C2semidirect
5. #22549606C2semidirect
6. D4⋊(C127⋊C8)C2semidirect
7. D4⋊C8×C127C2semidirect
60. Dic254M16semidirect
78. D4×C127M16semidirect
111. C127#1489semidirect
118. D4⋊C8D127semidirect
175. D4#14523231semidirect

Maximal subgroups

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

subgroupindex
C4×#145232472
#225498982
C4×#145232312
#225496082
#225496062
D4⋊(C127⋊C8)2
D4⋊C8×C1272
#1489127