Finite Group Databasebeta

← #30648254

Subgroups — #30648254

Order 16256.

490 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16861239C211yesmaximal, characteristic, complemented
2C22×#14523237C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic
4#22550353C211yesmaximal, characteristic, complemented
5#22550101C214yesmaximal, complemented
6#22550075C214yesmaximal, complemented
7#22549902C211yesmaximal, characteristic, complemented
8#22549880C211yesmaximal, characteristic, complemented
9#416×C127C211yesmaximal, characteristic, complemented
10C22×Dic254C2211yescharacteristic
11C22×D508C2211yescharacteristic
12#16861239—24no—
13C2×#14523238C2212yescomplemented
14C2×#14523237—24no—
15C2×#14523237C2214yes—
16C2×#14523236C2212yescomplemented
17C2×#14523232C2211yescharacteristic
18#16861205C2212yes—
19#16861198C2212yes—
20#16861183—22no—
21#16861177—22no—
22C4×Dic254—22no—
23#16861174C2214yescomplemented
24#16861168C2211yescharacteristic
25#16861165—22no—
26Dic127⋊C8C2214yescomplemented
27#16861129C2214yes—
28C2×#41×C127C2211yescharacteristic
29C22×C8×C127C2211yescharacteristic
30#117×C127C2211yescharacteristic
31#103×C127C2212yescomplemented
32#102×C127C2212yescomplemented
33C23×D127—21no—
34C2×Dic254—21no—
35C2×Dic254—22no—
36C2×Dic254D414yescomplemented
37C2×Dic254C2312yes—
38C2×D508—22no—
39C2×D508—21no—
40C2×D508—22no—
41C2×D508C2312yes—
42C2×C22⋊D127—24no—
43C22×Dic127—21no—
44#14523238—24no—
45#14523237—28no—
46#14523236—24no—
47#14523232—22no—
48C2×C127⋊C8C2312yes—
49C4⋊Dic127D412yes—
50Dic127⋊C4—24no—
51C22⋊Dic127D412yes—
52(C2×D127)⋊C4—24no—
53C4×Dic127—24no—
54C4×Dic127C2312yes—
55#41×C127—22no—
56#41×C127—22no—
57C2×Q8×C127C2311yescharacteristic
58C2×D4×C127—21no—
59C2×D4×C127C2311yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62C2×C8×C127—22no—
63C2×C8×C127C2312yes—
64C4⋊C4×C127C2312yes—
65C22⋊C4×C127—21no—
66C42×C127—21no—
67C22×D127—22no—
68C22×D127—42no—
69C22×D127—82no—
70Dic254—24no—
71Dic254—22no—
72Dic254C2×D414yes—
73D508—22no—
74D508—22no—
75D508—42no—
76C22⋊D127—84no—
77C2×Dic127—24no—
78C2×Dic127—22no—
79C2×Dic127C2×D414yes—
80C127⋊C8—22no—
81C23×C127—21no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—21no—
84Q8×C127—21no—
85D4×C127—21no—
86D4×C127—21no—
87D4×C127—42no—
88C2×C4×C127—22no—
89C2×C4×C127—21no—
90C2×C4×C127—42no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D412yes—
93C2×C4×C127#4112yes—
94C2×C4×C127C2411yescharacteristic
95C8×C127—22no—
96C22×C127—22no—
97C22×C127—41no—
98C22×C127—42no—
99C22×C127#12112yes—
100C22×C127C22×D411yescharacteristic
101C2×D127—42no—
102C2×D127—44no—
103C2×D127—84no—
104C4×C127—41no—
105C4×C127—42no—
106C4×C127#12112yes—
107C4×C127C22×D411yescharacteristic
108C4×C127C2×#4111yescharacteristic
109Dic127—24no—
110Dic127—24no—
111C2×C127—22no—
112C2×C127—81no—
113C2×C127C2×#12111yescharacteristic
114C2×C127#57712yes—
115D127—82no—
116#3043—1271nomaximal, Sylow
117C127#304311yescharacteristic, complemented, Sylow
118C2×#137—1271no—
119C22×SD16—1271no—
120#535—1271no—
121#519—1271no—
122#464—1274no—
123#460—1274no—
124#424—1271no—
125#418—1271no—
126#416D12711yescharacteristic, complemented
127C2×#41C2×D12711yescharacteristic
128C22×Q8—1271no—
129C22×D4—1271no—
130#137—2544no—
131C2×Q16—1272no—
132C2×SD16—1272no—
133C2×SD16—1274no—
134C2×SD16—2544no—
135C2×M16—1271no—
136C22×C8C2×D12711yescharacteristic
137#127—1272no—
138#124—1272no—
139#121—2542no—
140#120—2542no—
141C4×Q8—1274no—
142C4×Q8—2542no—
143#117—1271no—
144#117—2542no—
145#117C2×D12711yescharacteristic
146C4⋊C8—1274no—
147#103C2×D12712yescomplemented
148#102—1274no—
149#102C2×D12712yescomplemented
150C24—2541no—
151#41—22no—
152#41—22no—
153C2×Q8—1274no—
154C2×Q8—1272no—
155C2×Q8—2541no—
156C2×Q8—2542no—
157C2×Q8C22×D12711yescharacteristic
158C2×D4—21no—
159C2×D4—1272no—
160C2×D4—2544no—
161C2×D4—2541no—
162C2×D4—2542no—
163C2×D4—2542no—
164C2×D4C22×D12711yescharacteristic
165C22×C4—21no—
166C22×C4—2541no—
167C22×C4C22×D12711yescharacteristic
168Q16—2544no—
169SD16—2548no—
170SD16—2544no—
171M16—2542no—
172C2×C8—22no—
173C2×C8—1272no—
174C2×C8C22×D12712yes—
175C4⋊C4—1272no—
176C4⋊C4—2544no—
177C4⋊C4C22×D12712yes—
178C22⋊C4—21no—
179C22⋊C4—1272no—
180C22⋊C4—2544no—
181C42—21no—
182C42—1272no—
183C42—2544no—
184C23—21no—
185C23—10162no—
186C23—5082no—
187C23—2542no—
188C23D4×D12711yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—1274no—
192Q8—2542no—
193Q8—2544no—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—10164no—
198D4—5082no—
199D4—2542no—
200D4—2542no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—1274no—
205C2×C4—2542no—
206C2×C4—2544no—
207C2×C4D4×D12712yes—
208C2×C4D4×D12711yescharacteristic
209C2×C4#1452324812yes—
210C2×C4C23×D12711yescharacteristic
211C8—22no—
212C8—2542no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—10164no—
217C22—5082no—
218C22—5084no—
219C22#1686118212yes—
220C22C2×D4×D12711yescharacteristic
221C4—41no—
222C4—42no—
223C4—2544no—
224C4—2544no—
225C4#1686118212yes—
226C4C2×D4×D12711yescharacteristic
227C4C2×#1452324811yescharacteristic
228C2—22no—
229C2—81no—
230C2—10162no—
231C2C2×#1686118211yescharacteristic
232C2#2255068312yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
prime radical#22550440C2
hypercenter#416D127
hypocenterC127#3043
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C22×#14523237C2semidirect
4. #22550353C2semidirect
5. #22550101C2semidirect
6. #22550075C2semidirect
7. #22549902C2semidirect
8. #22549880C2semidirect
9. #416×C127C2semidirect
13. C2×#14523238C22semidirect
16. C2×#14523236C22semidirect
23. #16861174C22semidirect
26. Dic127⋊C8C22semidirect
31. #103×C127C22semidirect
32. #102×C127C22semidirect
36. C2×Dic254D4semidirect
117. C127#3043semidirect
126. #416D127semidirect
147. #103C2×D127semidirect
149. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232372
#225504402
#225503532
#225501012
#225500752
#225499022
#225498802
#416×C1272
#3043127