Finite Group Databasebeta

← #30649113

Subgroups — #30649113

Order 16256.

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

normal subgroups only

This group has 400 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#22550432C211yesmaximal, characteristic, complemented
2#124×D127C211yesmaximal, characteristic
3#22550237C212yesmaximal, complemented
4#22550234C212yesmaximal, complemented
5#22550233C211yesmaximal, characteristic, complemented
6#22550231C211yesmaximal, characteristic, complemented
7#22550230C211yesmaximal, characteristic, complemented
8#22549883C212yesmaximal, complemented
9#22549879C212yesmaximal
10C4×#14523231C211yesmaximal, characteristic, complemented
11#488×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2211yescharacteristic
13C2×D4×D127C2211yescharacteristic
14C2×#14523231C2212yes—
15#16861208C2212yescomplemented
16#16861201C2211yescharacteristic, complemented
17#16861198C2211yescharacteristic
18#16861197C2211yescharacteristic
19#16861196C2211yescharacteristic
20#16861195—22no—
21#16861187—22no—
22#16861174—22no—
23#16861167C2212yes—
24C22⋊C4×D127—22no—
25C42×D127C2211yescharacteristic
26#16861134C2212yescomplemented
27#16861133C2212yescomplemented
28#16861132C2212yescomplemented
29#16861131C2212yes—
30#16861130C2212yes—
31#16861129C2212yes—
32#16861119C2212yescomplemented
33C4×C127⋊C8C2211yescharacteristic, complemented
34#125×C127C2211yescharacteristic, complemented
35#124×C127C2211yescharacteristic
36#103×C127C2212yescomplemented
37#102×C127C2212yes—
38C4×C8×C127C2211yescharacteristic, complemented
39C23×D127—21no—
40C2×Dic254—22no—
41C2×Dic254—21no—
42C2×Dic254C2311yescharacteristic
43Q8×D127—24no—
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—21no—
46D4×D127—24no—
47C2×C4×D127—21no—
48C2×C4×D127D412yes—
49C2×C4×D127C2311yescharacteristic
50#14523231—24no—
51C2×C127⋊C8C2312yes—
52C4⋊Dic127C2312yes—
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55C22⋊Dic127—22no—
56(C2×D127)⋊C4—22no—
57(C2×D127)⋊C4—22no—
58(C2×D127)⋊C4—22no—
59C4×Dic127—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×D127—44no—
71C22×D127—42no—
72C22×D127C2×D411yescharacteristic
73Dic254—22no—
74Dic254—44no—
75Dic254—42no—
76D508—22no—
77C22⋊D127—42no—
78C2×Dic127—22no—
79C2×Dic127—21no—
80C2×Dic127C2×D412yes—
81C2×Dic127C2×D411yescharacteristic
82C4×D127—42no—
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—84no—
101C2×D127#12412yes—
102C4×C127—22no—
103C4×C127—41no—
104C4×C127—42no—
105C4×C127C2×#4112yes—
106Dic127—22no—
107Dic127—41no—
108Dic127—42no—
109Dic127#12412yes—
110C2×C127—81no—
111C2×C127C2×#12411yescharacteristic
112C2×C127#57512yes—
113D127—22no—
114D127—81no—
115#3140—1271nomaximal, Sylow
116C127#314011yescharacteristic, complemented, Sylow
117#533—1271no—
118C2×#124—1271no—
119#489—1272no—
120#489—1272no—
121#488—1271no—
122#488—1271no—
123#488—1271no—
124#488D12711yescharacteristic, complemented
125#419—1272no—
126#418—1272no—
127C4×M16—1271no—
128C22×Q8—1271no—
129C22×D4—1271no—
130C2×M16—1272no—
131C4⋊Q8—1272no—
132#125—1271no—
133#125C2×D12711yescharacteristic, complemented
134#124—1271no—
135#124—1271no—
136#124—1271no—
137#124—2542no—
138#124C2×D12711yescharacteristic
139C22⋊Q8—2542no—
140C4×Q8—2542no—
141#117—1272no—
142C2×C22⋊C4—2542no—
143C2×C42—1271no—
144#103—1272no—
145#103—1272no—
146#103—1272no—
147#103C2×D12712yescomplemented
148#102—1272no—
149#102—1272no—
150#102—1272no—
151#102C2×D12712yes—
152#97—1272no—
153C4×C8—1271no—
154C4×C8C2×D12711yescharacteristic, complemented
155C24—2541no—
156C2×Q8—1271no—
157C2×Q8—2542no—
158C2×Q8—2541no—
159C2×Q8—2544no—
160C2×Q8C22×D12711yescharacteristic
161C2×D4—1271no—
162C2×D4—2544no—
163C2×D4—2541no—
164C2×D4C22×D12711yescharacteristic
165C22×C4—1271no—
166C22×C4—1272no—
167C22×C4—2541no—
168M16—2544no—
169C2×C8—1272no—
170C2×C8C22×D12712yes—
171C4⋊C4—22no—
172C4⋊C4—1272no—
173C4⋊C4—2542no—
174C4⋊C4—2542no—
175C4⋊C4C22×D12712yes—
176C22⋊C4—22no—
177C22⋊C4—2542no—
178C22⋊C4—2542no—
179C22⋊C4—2542no—
180C22⋊C4—2542no—
181C42—1272no—
182C42—1271no—
183C42—2542no—
184C42C22×D12711yescharacteristic
185C23—21no—
186C23—1271no—
187C23—5082no—
188C23—5084no—
189C23—2541no—
190Q8—22no—
191Q8—5082no—
192Q8—5084no—
193Q8—2542no—
194D4—22no—
195D4—5082no—
196D4—2542no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—1271no—
200C2×C4—1274no—
201C2×C4—1272no—
202C2×C4—5082no—
203C2×C4—5082no—
204C2×C4—2542no—
205C2×C4—2541no—
206C2×C4D4×D12712yes—
207C2×C4C23×D12711yescharacteristic
208C8—22no—
209C8—2542no—
210C22—41no—
211C22—42no—
212C22—1272no—
213C22—10164no—
214C22—5081no—
215C22—5082no—
216C22—2542no—
217C22C2×D4×D12711yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—1272no—
222C4—5081no—
223C4—5082no—
224C4—2542no—
225C4#41×D12712yes—
226C2—81no—
227C2—10161no—
228C2—2542no—
229C2#124×D12711yescharacteristic
230C2#2255067312yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#488×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×D127C22
hypercenter#488D127
hypocenterC127#3140
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
1. #22550432C2semidirect
3. #22550237C2semidirect
4. #22550234C2semidirect
5. #22550233C2semidirect
6. #22550231C2semidirect
7. #22550230C2semidirect
8. #22549883C2semidirect
10. C4×#14523231C2semidirect
11. #488×C127C2semidirect
15. #16861208C22semidirect
16. #16861201C22semidirect
26. #16861134C22semidirect
27. #16861133C22semidirect
28. #16861132C22semidirect
32. #16861119C22semidirect
33. C4×C127⋊C8C22semidirect
34. #125×C127C22semidirect
36. #103×C127C22semidirect
38. C4×C8×C127C22semidirect
116. C127#3140semidirect
124. #488D127semidirect
133. #125C2×D127semidirect
147. #103C2×D127semidirect
154. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504322
#124×D1272
#225502372
#225502342
#225502332
#225502312
#225502302
#225498832
#225498792
C4×#145232312
#488×C1272
#3140127