Finite Group Databasebeta

← #30649127

Subgroups — #30649127

Order 16256.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550425C211yesmaximal, characteristic
2#125×D127C211yesmaximal, characteristic, complemented
3#22550243C212yesmaximal, complemented
4#22550239C212yesmaximal, complemented
5#22550233C211yesmaximal, characteristic, complemented
6#22550231C211yesmaximal, characteristic, complemented
7#22550230C211yesmaximal, characteristic, complemented
8#22549876C212yesmaximal, complemented
9#22549874C212yesmaximal, complemented
10C4×#14523231C211yesmaximal, characteristic, complemented
11#488×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2211yescharacteristic
13C2×#14523247C2211yescharacteristic
14C2×#14523231C2212yes—
15#16861207C2211yescharacteristic
16#16861205C2211yescharacteristic
17#16861201C2211yescharacteristic
18#16861200C2212yes—
19#16861199—22no—
20#16861197C2211yescharacteristic
21#16861182—22no—
22(C22⋊D127)⋊C4—22no—
23C4⋊C4×D127—22no—
24C4⋊C4×D127C2212yes—
25C42×D127C2211yescharacteristic
26#16861134C2212yes—
27#16861133C2212yes—
28#16861132C2212yes—
29#16861131C2212yes—
30#16861130C2212yes—
31#16861129C2212yes—
32#16861119C2212yes—
33C4×C127⋊C8C2211yescharacteristic
34#125×C127C2211yescharacteristic
35#124×C127C2211yescharacteristic
36#103×C127C2212yes—
37#102×C127C2212yes—
38C4×C8×C127C2211yescharacteristic
39#14523249—24no—
40#14523247—24no—
41C2×Dic254C2311yescharacteristic
42C2×D508—21no—
43C2×D508C2311yescharacteristic
44C2×C22⋊D127—21no—
45C2×C22⋊D127—22no—
46C22×Dic127—21no—
47C2×C4×D127—22no—
48C2×C4×D127—21no—
49C2×C4×D127D412yes—
50C2×C4×D127C2311yescharacteristic
51#14523231—24no—
52C2×C127⋊C8C2312yes—
53C4⋊Dic127—22no—
54C4⋊Dic127C2312yes—
55Dic127⋊C4—22no—
56Dic127⋊C4—22no—
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59(C2×D127)⋊C4—22no—
60C4×Dic127D412yes—
61C4×Dic127C2311yescharacteristic
62C2×Q8×C127C2311yescharacteristic
63C2×D4×C127C2311yescharacteristic
64C2×C8×C127C2312yes—
65C4⋊C4×C127—22no—
66C4⋊C4×C127C2312yes—
67C22⋊C4×C127—22no—
68C42×C127C2311yescharacteristic
69C22×D127—21no—
70C22×D127C2×D411yescharacteristic
71Dic254—22no—
72D508—22no—
73D508—42no—
74C22⋊D127—44no—
75C22⋊D127—42no—
76C2×Dic127—21no—
77C2×Dic127—22no—
78C2×Dic127—42no—
79C2×Dic127C2×D412yes—
80C2×Dic127C2×D411yescharacteristic
81C4×D127—42no—
82C4×D127—44no—
83C4×D127—42no—
84C4×D127#4114yes—
85C127⋊C8—22no—
86C23×C127—21no—
87Q8×C127—22no—
88D4×C127—22no—
89C2×C4×C127—21no—
90C2×C4×C127—22no—
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2411yescharacteristic
93C8×C127—22no—
94C22×C127—41no—
95C22×C127—42no—
96C22×C127C22×D411yescharacteristic
97C2×D127—22no—
98C2×D127—41no—
99C2×D127—42no—
100C2×D127#12412yes—
101C4×C127—22no—
102C4×C127—41no—
103C4×C127—42no—
104C4×C127C2×#4112yes—
105Dic127—22no—
106Dic127—41no—
107Dic127—42no—
108Dic127#12412yes—
109C2×C127—81no—
110C2×C127C2×#12411yescharacteristic
111C2×C127#57512yes—
112D127—22no—
113D127—81no—
114#3142—1271nomaximal, Sylow
115C127#314211yescharacteristic, complemented, Sylow
116#532—1271no—
117C2×#125—1271no—
118#491—1272no—
119#490—1272no—
120#488—1271no—
121#488—1271no—
122#488—1271no—
123#488D12711yescharacteristic, complemented
124#417—1272no—
125#417—1272no—
126C4×M16—1271no—
127C2×#41—1271no—
128C2×#41—1271no—
129C2×M16—1272no—
130C4⋊Q8—1271no—
131#127—1271no—
132#125—1272no—
133#125—1271no—
134#125—2542no—
135#125C2×D12711yescharacteristic
136#124—1271no—
137#124C2×D12711yescharacteristic
138#121—2542no—
139C4×D4—2542no—
140C2×C4⋊C4—1272no—
141C2×C4⋊C4—2542no—
142C2×C42—1271no—
143#103—1272no—
144#103—1272no—
145#103—1272no—
146#103C2×D12712yes—
147#102—1272no—
148#102—1272no—
149#102—1272no—
150#102C2×D12712yes—
151#97—1272no—
152C4×C8—1271no—
153C4×C8C2×D12711yescharacteristic
154#41—2544no—
155#41—2544no—
156C2×Q8—1271no—
157C2×Q8C22×D12711yescharacteristic
158C2×D4—1271no—
159C2×D4—2541no—
160C2×D4—2542no—
161C2×D4—2541no—
162C2×D4C22×D12711yescharacteristic
163C22×C4—1271no—
164C22×C4—1272no—
165C22×C4—2541no—
166C22×C4—2542no—
167C22×C4—2541no—
168M16—2544no—
169C2×C8—1272no—
170C2×C8C22×D12712yes—
171C4⋊C4—22no—
172C4⋊C4—1272no—
173C4⋊C4—2542no—
174C4⋊C4—2542no—
175C4⋊C4—2542no—
176C4⋊C4—2542no—
177C4⋊C4—2542no—
178C4⋊C4C22×D12712yes—
179C22⋊C4—22no—
180C22⋊C4—2542no—
181C42—1272no—
182C42—1271no—
183C42C22×D12711yescharacteristic
184C23—21no—
185C23—1271no—
186C23—2541no—
187Q8—22no—
188Q8—2542no—
189D4—22no—
190D4—5082no—
191D4—5082no—
192D4—5084no—
193D4—2542no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—1271no—
197C2×C4—1274no—
198C2×C4—1272no—
199C2×C4—5082no—
200C2×C4—5082no—
201C2×C4—5084no—
202C2×C4—5082no—
203C2×C4—2541no—
204C2×C4—2542no—
205C2×C4D4×D12712yes—
206C2×C4C23×D12711yescharacteristic
207C8—22no—
208C8—2542no—
209C22—41no—
210C22—42no—
211C22—1272no—
212C22—5081no—
213C22—5082no—
214C22—2542no—
215C22C2×D4×D12711yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—1272no—
220C4—5081no—
221C4—5082no—
222C4—2542no—
223C4#41×D12712yes—
224C2—81no—
225C2—10161no—
226C2—2542no—
227C2#124×D12711yescharacteristic
228C2#2255067312yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#488×C127C2
socleC22×C127C22×D4
prime radical#22550425C2
hypercenter#488D127
hypocenterC127#3142
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D127C2semidirect
3. #22550243C2semidirect
4. #22550239C2semidirect
5. #22550233C2semidirect
6. #22550231C2semidirect
7. #22550230C2semidirect
8. #22549876C2semidirect
9. #22549874C2semidirect
10. C4×#14523231C2semidirect
11. #488×C127C2semidirect
115. C127#3142semidirect
123. #488D127semidirect

Maximal subgroups

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

subgroupindex
#225504252
#125×D1272
#225502432
#225502392
#225502332
#225502312
#225502302
#225498762
#225498742
C4×#145232312
#488×C1272
#3142127