Finite Group Databasebeta

← #30648253

Subgroups — #30648253

Order 16256.

490 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14523235C211yesmaximal, characteristic, complemented
2#22550440C211yesmaximal, characteristic
3#22550354C211yesmaximal, characteristic, complemented
4#22550100C214yesmaximal, complemented
5#22550074C214yesmaximal, complemented
6#22549904C211yesmaximal, characteristic, complemented
7#22549882C211yesmaximal, characteristic, complemented
8#22549872C211yesmaximal, characteristic, complemented
9C2×#137×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C2×#14523235—24no—
12C2×#14523235C2214yes—
13C22×C127⋊C8C2211yescharacteristic
14#16861205C2212yes—
15#16861198C2212yes—
16#16861183—22no—
17#16861177—22no—
18Q8×Dic127—22no—
19Q8×Dic127C2214yescomplemented
20#16861168—22no—
21#16861168C2211yescharacteristic
22#16861168C2211yescharacteristic
23Dic127⋊C8C2214yescomplemented
24#16861133C2212yescomplemented
25#16861131C2214yes—
26#16861130C2212yescomplemented
27C2×#41×C127C2211yescharacteristic
28C22×Q8×C127C2211yescharacteristic
29#137×C127—24no—
30C2×Q16×C127C2212yescomplemented
31C2×SD16×C127C2212yescomplemented
32C2×M16×C127C2211yescharacteristic
33C23×D127—21no—
34C2×D508—22no—
35C2×D508—21no—
36C2×D508—22no—
37C2×D508C2312yes—
38C2×C22⋊D127—24no—
39#14523235—28no—
40C2×C127⋊C8—22no—
41C2×C127⋊C8C2312yes—
42C4⋊Dic127—24no—
43C4⋊Dic127D412yes—
44C4⋊Dic127C2312yes—
45C22⋊Dic127—21no—
46C22⋊Dic127D412yes—
47(C2×D127)⋊C4—24no—
48C4×Dic127—24no—
49C4×Dic127—21no—
50C4×Dic127C2312yes—
51#41×C127—22no—
52#41×C127—22no—
53C2×Q8×C127—21no—
54C2×Q8×C127—22no—
55C2×Q8×C127D414yescomplemented
56C2×Q8×C127C2311yescharacteristic
57C2×Q8×C127C2312yes—
58C2×D4×C127—21no—
59C2×D4×C127C2311yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63Q16×C127—24no—
64SD16×C127—24no—
65M16×C127—22no—
66C2×C8×C127C2312yes—
67C22×D127—22no—
68C22×D127—42no—
69C22×D127—82no—
70D508—22no—
71D508—22no—
72D508—42no—
73C22⋊D127—84no—
74C2×Dic127—22no—
75C2×Dic127C2×D414yes—
76C127⋊C8—22no—
77C23×C127—21no—
78C23×C127C2×D411yescharacteristic
79Q8×C127—24no—
80Q8×C127—22no—
81Q8×C127—21no—
82Q8×C127—21no—
83Q8×C127C2×D414yes—
84D4×C127—21no—
85D4×C127—21no—
86D4×C127—42no—
87C2×C4×C127—24no—
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—24no—
105C4×C127—41no—
106C4×C127#12112yes—
107C4×C127C22×D411yescharacteristic
108C4×C127C2×#4111yescharacteristic
109Dic127—24no—
110Dic127—42no—
111C2×C127—22no—
112C2×C127—81no—
113C2×C127C2×#12111yescharacteristic
114C2×C127#57712yes—
115D127—82no—
116#3043—1271nomaximal, Sylow
117C127#304311yescharacteristic, complemented, Sylow
118C2×#137D12711yescharacteristic, complemented
119C22×SD16—1271no—
120#535—1271no—
121#519—1271no—
122#464—1274no—
123#460—1274no—
124#424—1271no—
125#418—1271no—
126#416—1271no—
127C2×#41C2×D12711yescharacteristic
128C22×Q8C2×D12711yescharacteristic
129C22×D4—1271no—
130#137—24no—
131C2×Q16C2×D12712yescomplemented
132C2×SD16—1274no—
133C2×SD16—2544no—
134C2×SD16C2×D12712yescomplemented
135C2×M16C2×D12711yescharacteristic
136C22×C8—1271no—
137#127—1272no—
138#124—1272no—
139#121—2542no—
140#120—2542no—
141C4×Q8—1274no—
142C4×Q8—2542no—
143#117—1271no—
144#117—1271no—
145#117—2542no—
146C4⋊C8—1274no—
147#103—1272no—
148#102—1272no—
149#102—1274no—
150C24—2541no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8C22⋊D12714yescomplemented
156C2×Q8C22×D12711yescharacteristic
157C2×Q8C22×D12712yes—
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—21no—
167C22×C4C22×D12711yescharacteristic
168Q16—24no—
169SD16—24no—
170SD16—2548no—
171M16—22no—
172C2×C8—1272no—
173C2×C8—2542no—
174C2×C8C22×D12712yes—
175C4⋊C4—1272no—
176C4⋊C4—1272no—
177C4⋊C4—2544no—
178C22⋊C4—1272no—
179C22⋊C4—2544no—
180C22⋊C4—2541no—
181C42—1272no—
182C42—2544no—
183C42—2541no—
184C23—21no—
185C23—10162no—
186C23—5082no—
187C23—2542no—
188C23D4×D12711yescharacteristic
189Q8—22no—
190Q8—24no—
191Q8—21no—
192Q8—21no—
193Q8C2×C22⋊D12714yes—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—10164no—
198D4—5082no—
199D4—2542no—
200D4—2542no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—1274no—
206C2×C4—2542no—
207C2×C4D4×D12711yescharacteristic
208C2×C4C2×C22⋊D12712yes—
209C2×C4#1452324712yes—
210C2×C4C23×D12711yescharacteristic
211C8—22no—
212C8—2542no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—10164no—
217C22—5082no—
218C22—5084no—
219C22#1686118112yes—
220C22C2×D4×D12711yescharacteristic
221C4—24no—
222C4—41no—
223C4—5082no—
224C4—2544no—
225C4#1686118112yes—
226C4C22×C22⋊D12711yescharacteristic
227C4C2×#1452324711yescharacteristic
228C2—22no—
229C2—81no—
230C2—10162no—
231C2C2×#1686118111yescharacteristic
232C2#2255068512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radical#22550440C2
hypercenterC2×#137D127
hypocenterC127#3043
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#14523235C2semidirect
3. #22550354C2semidirect
4. #22550100C2semidirect
5. #22550074C2semidirect
6. #22549904C2semidirect
7. #22549882C2semidirect
8. #22549872C2semidirect
9. C2×#137×C127C2semidirect
19. Q8×Dic127C22semidirect
23. Dic127⋊C8C22semidirect
24. #16861133C22semidirect
26. #16861130C22semidirect
30. C2×Q16×C127C22semidirect
31. C2×SD16×C127C22semidirect
55. C2×Q8×C127D4semidirect
117. C127#3043semidirect
118. C2×#137D127semidirect
131. C2×Q16C2×D127semidirect
134. C2×SD16C2×D127semidirect
155. C2×Q8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232352
#225504402
#225503542
#225501002
#225500742
#225499042
#225498822
#225498722
C2×#137×C1272
#3043127