Finite Group Databasebeta

← #30649125

Subgroups — #30649125

Order 16256.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550427C211yesmaximal, characteristic
2C4⋊Q8×D127C211yesmaximal, characteristic, complemented
3#22550237C214yesmaximal, complemented
4#22550226C211yesmaximal, characteristic, complemented
5#22550224C211yesmaximal, characteristic, complemented
6#22550223C211yesmaximal, characteristic, complemented
7#22549876C214yesmaximal, complemented
8C4×#14523231C211yesmaximal, characteristic, complemented
9#487×C127C211yesmaximal, characteristic, complemented
10C2×#14523249C2212yes—
11C2×Q8×D127—21no—
12C2×#14523231C2212yes—
13#16861209C2211yescharacteristic
14#16861208—21no—
15#16861208C2212yes—
16#16861207—21no—
17#16861206C2211yescharacteristic
18#16861198C2212yes—
19#16861184—21no—
20#16861184—21no—
21#16861171—21no—
22#16861171—21no—
23C4⋊C4×D127—21no—
24C4⋊C4×D127C2212yes—
25C42×D127C2211yescharacteristic
26#16861133C2214yes—
27#16861131C2214yes—
28#16861129C2214yes—
29#16861119C2212yes—
30C4×C127⋊C8C2211yescharacteristic
31C4⋊Q8×C127C2211yescharacteristic
32C4⋊Q8×C127C2211yescharacteristic
33#103×C127C2214yes—
34C4×C8×C127C2211yescharacteristic
35#14523249—28no—
36C2×Dic254—22no—
37C2×Dic254—21no—
38Q8×D127—44no—
39C2×D508—22no—
40C2×D508—21no—
41C2×D508—21no—
42C2×D508C2312yes—
43C2×C4×D127—22no—
44C2×C4×D127—22no—
45C2×C4×D127D411yescharacteristic
46C2×C4×D127D411yescharacteristic
47C2×C4×D127C2311yescharacteristic
48#14523231—24no—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127—21no—
51C4⋊Dic127C2312yes—
52Dic127⋊C4—22no—
53Dic127⋊C4—22no—
54(C2×D127)⋊C4—22no—
55(C2×D127)⋊C4—22no—
56C4×Dic127D411yescharacteristic
57C4×Dic127D411yescharacteristic
58C4×Dic127C2311yescharacteristic
59C2×Q8×C127—21no—
60C2×Q8×C127C2312yes—
61C2×C8×C127C2312yes—
62C4⋊C4×C127—21no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127—21no—
65C4⋊C4×C127C2312yes—
66C42×C127C2311yescharacteristic
67C22×D127—22no—
68C22×D127C2×D411yescharacteristic
69Dic254—44no—
70Dic254—42no—
71D508—24no—
72D508—44no—
73D508—42no—
74D508—42no—
75C2×Dic127—22no—
76C2×Dic127C2×D411yescharacteristic
77C2×Dic127C2×D411yescharacteristic
78C2×Dic127C2×D411yescharacteristic
79C4×D127—22no—
80C4×D127—22no—
81C4×D127—44no—
82C4×D127—44no—
83C4×D127#4114yes—
84C127⋊C8—22no—
85Q8×C127—24no—
86Q8×C127—42no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2411yescharacteristic
92C8×C127—22no—
93C22×C127C22×D411yescharacteristic
94C2×D127—21no—
95C2×D127—21no—
96C2×D127—42no—
97C2×D127—42no—
98C2×D127—42no—
99C2×D127#12412yes—
100C4×C127—42no—
101C4×C127—42no—
102C4×C127AGL(1,ℤ/8)12yes—
103C4×C127#13712yes—
104C4×C127C2×#4112yes—
105Dic127—21no—
106Dic127—21no—
107Dic127—42no—
108Dic127#12412yes—
109C2×C127C2×#12411yescharacteristic
110C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C127C2×#13711yescharacteristic
112D127—22no—
113D127—82no—
114#3141—1271nomaximal, Sylow
115C127#314111yescharacteristic, complemented, Sylow
116#532—1271no—
117C2×C4⋊Q8—1271no—
118#489—1274no—
119#487—1271no—
120#487D12711yescharacteristic, complemented
121#486—1271no—
122#486—1271no—
123#417—1274no—
124C4×M16—1271no—
125C2×#41—1272no—
126C22×Q8—2541no—
127C2×M16—1272no—
128C4⋊Q8—1272no—
129C4⋊Q8—1271no—
130C4⋊Q8—2541no—
131C4⋊Q8—2541no—
132C4⋊Q8C2×D12711yescharacteristic
133C4⋊Q8C2×D12711yescharacteristic
134#127—1271no—
135#124—1272no—
136#121—2541no—
137#121—2541no—
138C4×D4—2541no—
139C4×D4—2541no—
140C2×C4⋊C4—1272no—
141C2×C4⋊C4—2541no—
142C2×C42—1271no—
143#103—1274no—
144#103C2×D12714yes—
145#102—1274no—
146#102—1274no—
147#97—1272no—
148C4×C8—1271no—
149C4×C8C2×D12711yescharacteristic
150#41—2548no—
151C2×Q8—21no—
152C2×Q8—5084no—
153C2×Q8—2542no—
154C2×Q8—2541no—
155C2×Q8C22×D12712yes—
156C2×D4—1272no—
157C2×D4—2542no—
158C2×D4—2541no—
159C2×D4—2541no—
160C22×C4—1271no—
161C22×C4—1271no—
162C22×C4—1271no—
163C22×C4—2542no—
164C22×C4—2542no—
165M16—2544no—
166C2×C8—1272no—
167C2×C8C22×D12712yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—1272no—
172C4⋊C4—2542no—
173C4⋊C4—2542no—
174C4⋊C4—2541no—
175C4⋊C4C22×D12712yes—
176C22⋊C4—2542no—
177C22⋊C4—2542no—
178C42—1271no—
179C42—1271no—
180C42—1271no—
181C42C22×D12711yescharacteristic
182C23—1271no—
183C23—2542no—
184Q8—24no—
185Q8—42no—
186Q8—5084no—
187Q8—5082no—
188D4—5084no—
189D4—5082no—
190D4—5082no—
191D4—2544no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—1271no—
195C2×C4—1274no—
196C2×C4—1271no—
197C2×C4—1271no—
198C2×C4—5084no—
199C2×C4—5084no—
200C2×C4—2542no—
201C2×C4—2542no—
202C2×C4—2542no—
203C2×C4D4×D12711yescharacteristic
204C2×C4D4×D12711yescharacteristic
205C2×C4C23×D12711yescharacteristic
206C8—22no—
207C8—2542no—
208C22—1272no—
209C22—5082no—
210C22—5082no—
211C22—5082no—
212C22—2541no—
213C22—2541no—
214C22C2×D4×D12711yescharacteristic
215C4—42no—
216C4—42no—
217C4—1272no—
218C4—5082no—
219C4—2541no—
220C4—2541no—
221C4#1686123612yes—
222C4#1686124112yes—
223C4#41×D12712yes—
224C2—10162no—
225C2—2542no—
226C2#124×D12711yescharacteristic
227C2C2×#1686123611yescharacteristic
228C2C2×#1686124111yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#487×C127C2
socleC22×C127C22×D4
prime radical#22550427C2
hypercenter#487D127
hypocenterC127#3141
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D127C2semidirect
3. #22550237C2semidirect
4. #22550226C2semidirect
5. #22550224C2semidirect
6. #22550223C2semidirect
7. #22549876C2semidirect
8. C4×#14523231C2semidirect
9. #487×C127C2semidirect
115. C127#3141semidirect
120. #487D127semidirect

Maximal subgroups

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

subgroupindex
#225504272
C4⋊Q8×D1272
#225502372
#225502262
#225502242
#225502232
#225498762
C4×#145232312
#487×C1272
#3141127