Finite Group Databasebeta

← #30648222

Subgroups — #30648222

Order 16256.

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

normal subgroups only

This group has 500 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
1C2×#16861239C211yesmaximal, characteristic, complemented
2C2×#16861196C211yesmaximal, characteristic, complemented
3#41×Dic127C211yesmaximal, characteristic, complemented
4#22550096C214yesmaximal, complemented
5#22550089C214yesmaximal, complemented
6#22549899C211yesmaximal, characteristic, complemented
7#22549886C211yesmaximal, characteristic, complemented
8#22549881C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861239—24no—
12C2×#14523238C2212yescomplemented
13C2×#14523236C2212yescomplemented
14C2×#14523232C2211yescharacteristic
15#16861196—21no—
16#16861196—21no—
17#16861196C2214yes—
18Q8×Dic127C2212yescomplemented
19D4×Dic127—22no—
20D4×Dic127C2212yescomplemented
21#16861168—21no—
22#16861168C2211yescharacteristic
23C2×C22⋊Dic127—22no—
24C2×C4×Dic127—21no—
25C2×C4×Dic127C2211yescharacteristic
26Dic127⋊C8C2214yescomplemented
27#16861134C2214yescomplemented
28#16861130C2214yes—
29C2×#41×C127C2211yescharacteristic
30C22×D4×C127C2211yescharacteristic
31AGL(1,ℤ/8)×C127—24no—
32C2×SD16×C127C2212yes—
33C2×D8×C127C2212yes—
34C2×M16×C127C2211yescharacteristic
35C2×Dic254—21no—
36C2×Dic254—22no—
37C2×Dic254—22no—
38C2×Dic254C2312yes—
39C22×Dic127—22no—
40C22×Dic127—21no—
41C22×Dic127D412yes—
42#14523238—24no—
43#14523236—24no—
44#14523232—22no—
45C2×C127⋊C8C2312yes—
46C4⋊Dic127—22no—
47C4⋊Dic127C2312yes—
48C22⋊Dic127—24no—
49C22⋊Dic127—24no—
50C22⋊Dic127—22no—
51C22⋊Dic127—21no—
52C4×Dic127—22no—
53C4×Dic127—21no—
54C4×Dic127D411yescharacteristic
55C4×Dic127D411yescharacteristic
56C4×Dic127C2312yes—
57C24×C127—21no—
58#41×C127—22no—
59#41×C127D414yes—
60C2×Q8×C127C2311yescharacteristic
61C2×D4×C127—21no—
62C2×D4×C127—22no—
63C2×D4×C127—21no—
64C2×D4×C127—22no—
65C2×D4×C127C2312yes—
66C2×D4×C127C2311yescharacteristic
67C22×C4×C127—21no—
68C22×C4×C127C2311yescharacteristic
69SD16×C127—24no—
70D8×C127—24no—
71M16×C127—22no—
72C2×C8×C127C2312yes—
73Dic254—22no—
74Dic254—22no—
75Dic254—42no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—42no—
79C2×Dic127—44no—
80C2×Dic127—42no—
81C2×Dic127C2×D412yes—
82C2×Dic127C2×D412yes—
83C127⋊C8—22no—
84C23×C127—21no—
85C23×C127—22no—
86C23×C127—44no—
87C23×C127—42no—
88C23×C127C2×D411yescharacteristic
89Q8×C127—21no—
90Q8×C127C2×D412yes—
91D4×C127—24no—
92D4×C127—22no—
93D4×C127—21no—
94D4×C127—22no—
95D4×C127—42no—
96D4×C127C2×D412yes—
97C2×C4×C127—21no—
98C2×C4×C127—24no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D412yes—
101C2×C4×C127#4112yes—
102C2×C4×C127C2411yescharacteristic
103C8×C127—22no—
104C22×C127—22no—
105C22×C127—22no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127—44no—
109C22×C127—84no—
110C22×C127#12112yes—
111C22×C127C22×D411yescharacteristic
112C4×C127—22no—
113C4×C127#12112yes—
114C4×C127C22×D411yescharacteristic
115C4×C127C2×#4111yescharacteristic
116Dic127—22no—
117Dic127—22no—
118Dic127—42no—
119Dic127—42no—
120C2×C127—22no—
121C2×C127—42no—
122C2×C127—82no—
123C2×C127C2×#12111yescharacteristic
124C2×C127#57512yes—
125#3040—1271nomaximal, Sylow
126C127#304011yescharacteristic, complemented, Sylow
127C2×#137—1271no—
128C2×AGL(1,ℤ/8)D12711yescharacteristic
129C2×#124—1271no—
130C4×#41—1271no—
131#464—1274no—
132#463—1274no—
133#423—1271no—
134#419—1271no—
135#418—1271no—
136C2×#41C2×D12711yescharacteristic
137C22×Q8—1271no—
138C22×D4C2×D12711yescharacteristic
139#137—2544no—
140AGL(1,ℤ/8)—24no—
141C2×Q16—1272no—
142C2×SD16—1272no—
143C2×SD16C2×D12712yes—
144C2×D8C2×D12712yes—
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—1271no—
154#117—2541no—
155C2×C22⋊C4—2542no—
156C2×C42—1271no—
157C2×C42—2541no—
158C4⋊C8—1274no—
159#103—1274no—
160#102—1274no—
161C24—21no—
162#41—22no—
163#41C22⋊D12714yes—
164C2×Q8—1272no—
165C2×Q8—2542no—
166C2×Q8—2542no—
167C2×Q8—2541no—
168C2×Q8C22×D12711yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4C22×D12712yes—
174C2×D4C22×D12711yescharacteristic
175C22×C4—21no—
176C22×C4—1272no—
177C22×C4—2542no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180Q16—2544no—
181SD16—24no—
182SD16—2544no—
183D8—24no—
184M16—22no—
185M16—2542no—
186C2×C8—1272no—
187C2×C8C22×D12712yes—
188C4⋊C4—1272no—
189C4⋊C4—2542no—
190C22⋊C4—2544no—
191C22⋊C4—2544no—
192C22⋊C4—2542no—
193C22⋊C4—2541no—
194C42—1271no—
195C42—1271no—
196C42—1272no—
197C42—2542no—
198C42—2541no—
199C23—21no—
200C23—22no—
201C23—44no—
202C23—42no—
203C23D4×D12711yescharacteristic
204Q8—21no—
205Q8—5082no—
206Q8—2542no—
207Q8—2542no—
208Q8C2×C22⋊D12712yes—
209D4—24no—
210D4—22no—
211D4—21no—
212D4—22no—
213D4—42no—
214D4C2×C22⋊D12712yes—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—1272no—
218C2×C4—1272no—
219C2×C4—5082no—
220C2×C4—5084no—
221C2×C4—5082no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4D4×D12711yescharacteristic
225C2×C4C2×C22⋊D12712yes—
226C2×C4#1452324712yes—
227C2×C4C23×D12711yescharacteristic
228C8—22no—
229C8—2542no—
230C22—22no—
231C22—22no—
232C22—42no—
233C22—42no—
234C22—44no—
235C22—84no—
236C22#1686118112yes—
237C22C2×D4×D12711yescharacteristic
238C4—22no—
239C4—5082no—
240C4—5082no—
241C4—2542no—
242C4—2542no—
243C4#1686118112yes—
244C4C22×C22⋊D12711yescharacteristic
245C4C2×#1452324711yescharacteristic
246C2—22no—
247C2—42no—
248C2—82no—
249C2C2×#1686118111yescharacteristic
250C2#2255067212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3040
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861196C2semidirect
3. #41×Dic127C2semidirect
4. #22550096C2semidirect
5. #22550089C2semidirect
6. #22549899C2semidirect
7. #22549886C2semidirect
8. #22549881C2semidirect
12. C2×#14523238C22semidirect
13. C2×#14523236C22semidirect
18. Q8×Dic127C22semidirect
20. D4×Dic127C22semidirect
26. Dic127⋊C8C22semidirect
27. #16861134C22semidirect
126. C127#3040semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168611962
#41×Dic1272
#225500962
#225500892
#225498992
#225498862
#225498812
C2×AGL(1,ℤ/8)×C1272
#3040127