Finite Group Databasebeta

← #30648223

Subgroups — #30648223

Order 16256.

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

normal subgroups only

This group has 500 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×#16861242C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic
3C2×#16861196C211yesmaximal, characteristic, complemented
4#22550344C211yesmaximal, characteristic, complemented
5#22550097C214yesmaximal, complemented
6#22550090C214yesmaximal, complemented
7#22549899C211yesmaximal, characteristic, complemented
8#22549884C211yesmaximal, characteristic, complemented
9#418×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×Dic254C2211yescharacteristic
12#16861242—24no—
13#16861235—24no—
14C2×Dic508C2212yescomplemented
15C2×#14523237C2212yescomplemented
16C2×#14523236C2212yes—
17C2×#14523233C2212yes—
18C2×#14523232C2211yescharacteristic
19#16861196—21no—
20#16861196—21no—
21#16861196C2214yes—
22#16861174C2212yescomplemented
23#16861171C2212yescomplemented
24(C22⋊D127)⋊C4—22no—
25#16861166—21no—
26C2×C22⋊Dic127—22no—
27C2×C4×Dic127C2211yescharacteristic
28C42×D127—21no—
29Dic127⋊C8C2214yescomplemented
30#16861132C2214yescomplemented
31C22×D4×C127C2211yescharacteristic
32C2×M16×C127C2211yescharacteristic
33#117×C127C2211yescharacteristic
34#102×C127C2214yes—
35#14523248—22no—
36#14523248D414yes—
37C2×Dic254—21no—
38C2×Dic254—22no—
39C2×Dic254—22no—
40C2×Dic254C2312yes—
41C2×Dic254C2311yescharacteristic
42C2×D508C2311yescharacteristic
43C2×C22⋊D127—21no—
44C22×Dic127—21no—
45C22×Dic127D412yes—
46C2×C4×D127—22no—
47C2×C4×D127—21no—
48Dic508—24no—
49#14523237—24no—
50#14523236—24no—
51#14523233—24no—
52#14523232—22no—
53C2×C127⋊C8C2312yes—
54Dic127⋊C4—22no—
55C22⋊Dic127—24no—
56C22⋊Dic127—24no—
57(C2×D127)⋊C4—22no—
58C4×Dic127—22no—
59C4×Dic127D411yescharacteristic
60C4×Dic127D411yescharacteristic
61C4×Dic127C2312yes—
62C24×C127—21no—
63C2×D4×C127—22no—
64C2×D4×C127—21no—
65C2×D4×C127—22no—
66C2×D4×C127C2312yes—
67C22×C4×C127C2311yescharacteristic
68M16×C127—22no—
69C2×C8×C127C2312yes—
70C4⋊C4×C127C2312yes—
71C22⋊C4×C127—21no—
72C42×C127—21no—
73C22×D127—21no—
74Dic254—21no—
75Dic254—22no—
76Dic254—22no—
77Dic254—42no—
78Dic254C2×D412yes—
79D508—21no—
80D508C2×D412yes—
81C22⋊D127—24no—
82C2×Dic127—21no—
83C2×Dic127—22no—
84C2×Dic127—42no—
85C2×Dic127—42no—
86C2×Dic127C2×D412yes—
87C2×Dic127C2×D412yes—
88C4×D127—24no—
89C4×D127—44no—
90C127⋊C8—22no—
91C23×C127—22no—
92C23×C127—44no—
93C23×C127—42no—
94C23×C127C2×D411yescharacteristic
95D4×C127—22no—
96D4×C127—22no—
97D4×C127—42no—
98C2×C4×C127—22no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D412yes—
101C2×C4×C127#4112yes—
102C2×C4×C127C2411yescharacteristic
103C8×C127—22no—
104C22×C127—22no—
105C22×C127—42no—
106C22×C127—44no—
107C22×C127—84no—
108C22×C127#12112yes—
109C22×C127C22×D411yescharacteristic
110C2×D127—22no—
111C2×D127—42no—
112C4×C127—42no—
113C4×C127#12112yes—
114C4×C127C22×D411yescharacteristic
115C4×C127C2×#4111yescharacteristic
116Dic127—22no—
117Dic127—22no—
118Dic127—22no—
119Dic127—42no—
120C2×C127—22no—
121C2×C127—82no—
122C2×C127C2×#12111yescharacteristic
123C2×C127#57512yes—
124D127—42no—
125#3040—1271nomaximal, Sylow
126C127#304011yescharacteristic, complemented, Sylow
127C2×#137—1271no—
128C2×AGL(1,ℤ/8)—1271no—
129C2×#124—1271no—
130C4×#41—1271no—
131#464—1274no—
132#463—1274no—
133#423—1271no—
134#419—1271no—
135#418D12711yescharacteristic, complemented
136C2×#41—1271no—
137C22×Q8—1271no—
138C22×D4C2×D12711yescharacteristic
139#137—2544no—
140AGL(1,ℤ/8)—2544no—
141C2×Q16—1272no—
142C2×SD16—1272no—
143C2×SD16—1272no—
144C2×D8—1272no—
145C2×M16—1271no—
146C2×M16C2×D12711yescharacteristic
147#124—1274no—
148#124—2541no—
149#124—2541no—
150C4×Q8—1272no—
151C4×D4—1272no—
152C4×D4—2542no—
153#117—2541no—
154#117C2×D12711yescharacteristic
155C2×C22⋊C4—2542no—
156C2×C42—1271no—
157C2×C42—2541no—
158C4⋊C8—1274no—
159#103—1274no—
160#102C2×D12714yes—
161C24—21no—
162#41—1274no—
163#41—2542no—
164C2×Q8—1271no—
165C2×Q8—1272no—
166C2×Q8—2542no—
167C2×Q8—2542no—
168C2×Q8—2541no—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—1271no—
173C2×D4—2541no—
174C2×D4C22×D12712yes—
175C22×C4—1272no—
176C22×C4—2542no—
177C22×C4—2541no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180Q16—2544no—
181SD16—2544no—
182SD16—2544no—
183D8—2544no—
184M16—22no—
185M16—2542no—
186C2×C8—1272no—
187C2×C8C22×D12712yes—
188C4⋊C4—2542no—
189C4⋊C4C22×D12712yes—
190C22⋊C4—21no—
191C22⋊C4—2542no—
192C22⋊C4—2544no—
193C22⋊C4—2544no—
194C42—21no—
195C42—1271no—
196C42—1271no—
197C42—1272no—
198C42—2542no—
199C23—22no—
200C23—44no—
201C23—42no—
202C23—2541no—
203C23D4×D12711yescharacteristic
204Q8—1272no—
205Q8—5082no—
206Q8—2542no—
207Q8—2542no—
208Q8—2541no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—1272no—
213D4—2544no—
214D4—2541no—
215C2×C4—22no—
216C2×C4—1272no—
217C2×C4—1272no—
218C2×C4—5084no—
219C2×C4—5082no—
220C2×C4—5082no—
221C2×C4—2542no—
222C2×C4—2541no—
223C2×C4—2544no—
224C2×C4D4×D12712yes—
225C2×C4D4×D12711yescharacteristic
226C2×C4#1452324812yes—
227C2×C4C23×D12711yescharacteristic
228C8—22no—
229C8—2542no—
230C22—22no—
231C22—42no—
232C22—44no—
233C22—84no—
234C22—5082no—
235C22—2542no—
236C22#1686118212yes—
237C22C2×D4×D12711yescharacteristic
238C4—42no—
239C4—5082no—
240C4—2542no—
241C4—2542no—
242C4—2542no—
243C4#1686118212yes—
244C4C2×D4×D12711yescharacteristic
245C4C2×#1452324811yescharacteristic
246C2—22no—
247C2—82no—
248C2—5082no—
249C2C2×#1686118211yescharacteristic
250C2#2255067212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861235C2
hypercenter#418D127
hypocenterC127#3040
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
3. C2×#16861196C2semidirect
4. #22550344C2semidirect
5. #22550097C2semidirect
6. #22550090C2semidirect
7. #22549899C2semidirect
8. #22549884C2semidirect
9. #418×C127C2semidirect
14. C2×Dic508C22semidirect
15. C2×#14523237C22semidirect
22. #16861174C22semidirect
23. #16861171C22semidirect
29. Dic127⋊C8C22semidirect
30. #16861132C22semidirect
126. C127#3040semidirect
135. #418D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×#168612352
C2×#168611962
#225503442
#225500972
#225500902
#225498992
#225498842
#418×C1272
#3040127