Finite Group Databasebeta

← #30648194

Subgroups — #30648194

Order 16256.

548 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16861235C212yesmaximal, complemented
2#22550426C211yesmaximal, characteristic, complemented
3#22550091C214yesmaximal, complemented
4#22550081C212yesmaximal, complemented
5#22550071C212yesmaximal, complemented
6#22549900C211yesmaximal, characteristic, complemented
7#22549875C212yesmaximal, complemented
8C2×C4×D4×C127C211yesmaximal, characteristic, complemented
9C2×#14523248C2212yes—
10#16861235—28no—
11C2×#14523236C2214yes—
12C2×#14523233C2214yes—
13C2×#14523232C2212yes—
14#16861209C2211yescharacteristic, complemented
15#16861206C2211yescharacteristic, complemented
16#16861197C2212yescomplemented
17#16861183—21no—
18#16861183—21no—
19C4×D508—21no—
20C4×D508—21no—
21C4⋊(C127⋊C8)C2212yescomplemented
22C4⋊(C127⋊C8)C2212yescomplemented
23#16861132C2214yescomplemented
24#16861129C2214yescomplemented
25C22×D4×C127C2211yescharacteristic
26C23×C4×C127—21no—
27C4×D4×C127—21no—
28C4×D4×C127—21no—
29C4×D4×C127C2212yes—
30C4×D4×C127C2212yes—
31C2×C4⋊C4×C127C2211yescharacteristic
32C2×C22⋊C4×C127—21no—
33C2×C42×C127C2211yescharacteristic
34#14523248—24no—
35#14523248—24no—
36C2×Dic254C2312yes—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508C2312yes—
40C2×C22⋊D127—22no—
41C2×C4×D127—22no—
42#14523236—28no—
43#14523233—28no—
44#14523232—24no—
45C2×C127⋊C8C2314yes—
46C4⋊Dic127—21no—
47C4⋊Dic127—21no—
48(C2×D127)⋊C4—22no—
49(C2×D127)⋊C4—22no—
50C24×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—22no—
53C2×D4×C127D414yescomplemented
54C2×D4×C127C2312yes—
55C22×C4×C127—21no—
56C22×C4×C127—22no—
57C22×C4×C127—22no—
58C22×C4×C127—44no—
59C22×C4×C127D411yescharacteristic
60C22×C4×C127D411yescharacteristic
61C22×C4×C127C2311yescharacteristic
62C4⋊C4×C127—21no—
63C4⋊C4×C127C2312yes—
64C22⋊C4×C127—22no—
65C22⋊C4×C127—22no—
66C42×C127D411yescharacteristic
67C42×C127D411yescharacteristic
68C42×C127C2311yescharacteristic
69C42×C127C2311yescharacteristic
70C22×D127—22no—
71Dic254—22no—
72Dic254—22no—
73D508—22no—
74D508—22no—
75D508—42no—
76D508—42no—
77C22⋊D127—44no—
78C2×Dic127—22no—
79C4×D127—44no—
80C127⋊C8—24no—
81C23×C127—24no—
82C23×C127—22no—
83C23×C127—42no—
84C23×C127—42no—
85C23×C127C2×D411yescharacteristic
86D4×C127—24no—
87D4×C127—22no—
88D4×C127C2×D414yes—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—41no—
92C2×C4×C127—44no—
93C2×C4×C127—42no—
94C2×C4×C127—44no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D412yes—
101C2×C4×C127#4112yes—
102C2×C4×C127C2411yescharacteristic
103C22×C127—21no—
104C22×C127—21no—
105C22×C127—24no—
106C22×C127—48no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127#12112yes—
110C22×C127C22×D411yescharacteristic
111C2×D127—42no—
112C2×D127—42no—
113C2×D127—42no—
114C4×C127—21no—
115C4×C127—21no—
116C4×C127—21no—
117C4×C127—42no—
118C4×C127#12112yes—
119C4×C127AGL(1,ℤ/8)12yescomplemented
120C4×C127C22×D411yescharacteristic
121C4×C127C2×#4111yescharacteristic
122Dic127—42no—
123C2×C127—22no—
124C2×C127—44no—
125C2×C127C2×#12111yescharacteristic
126C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C127#57511yescharacteristic
128D127—82no—
129#3036—1271nomaximal, Sylow
130C127#303611yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—1272no—
132#532—1271no—
133C2×C4×D4D12711yescharacteristic, complemented
134#463—1274no—
135#461—1272no—
136#459—1272no—
137#423—1271no—
138#417—1272no—
139C2×#41—1272no—
140C22×D4C2×D12711yescharacteristic
141C23×C4—21no—
142AGL(1,ℤ/8)—2548no—
143C2×SD16—1274no—
144C2×D8—1274no—
145C2×M16—1272no—
146C4⋊Q8—1271no—
147#127—1271no—
148#124—1272no—
149#121—2541no—
150#121—2541no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—2541no—
154C4×D4—2541no—
155C4×D4C2×D12712yes—
156C4×D4C2×D12712yes—
157C2×C4⋊C4C2×D12711yescharacteristic
158C2×C22⋊C4—21no—
159C2×C42C2×D12711yescharacteristic
160C4⋊C8—1272no—
161C4⋊C8—1272no—
162#103—1274no—
163#102—1274no—
164C24—21no—
165#41—2544no—
166#41—2544no—
167C2×Q8—1272no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—1272no—
171C2×D4—2542no—
172C2×D4—2541no—
173C2×D4—2541no—
174C2×D4D50814yescomplemented
175C2×D4C22×D12712yes—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—2542no—
181C22×C4C22⋊D12711yescharacteristic
182C22×C4C22⋊D12711yescharacteristic
183C22×C4C22×D12711yescharacteristic
184SD16—2548no—
185D8—2548no—
186M16—2544no—
187C2×C8—1274no—
188C4⋊C4—21no—
189C4⋊C4—2541no—
190C4⋊C4—2541no—
191C4⋊C4C22×D12712yes—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—2542no—
195C22⋊C4—2542no—
196C42C22⋊D12711yescharacteristic
197C42C22⋊D12711yescharacteristic
198C42C22×D12711yescharacteristic
199C42C22×D12711yescharacteristic
200C23—22no—
201C23—24no—
202C23—42no—
203C23—42no—
204C23—2542no—
205C23C2×C22⋊D12711yescharacteristic
206Q8—2542no—
207Q8—2542no—
208D4—22no—
209D4—24no—
210D4—5082no—
211D4—5084no—
212D4—5082no—
213D4—2542no—
214D4—2542no—
215D4C2×D50814yes—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—5084no—
223C2×C4—2542no—
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4C2×C22⋊D12711yescharacteristic
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4C2×C22⋊D12711yescharacteristic
228C2×C4C2×C22⋊D12711yescharacteristic
229C2×C4C2×D50812yes—
230C2×C4#1452324812yes—
231C2×C4C23×D12711yescharacteristic
232C8—2544no—
233C22—21no—
234C22—21no—
235C22—24no—
236C22—48no—
237C22—42no—
238C22—42no—
239C22—5082no—
240C22—5082no—
241C22—5082no—
242C22#1686118312yes—
243C22C22×C22⋊D12711yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—5082no—
249C4#1686118312yes—
250C4#1686123512yescomplemented
251C4C22×D50811yescharacteristic
252C4C2×#1452324811yescharacteristic
253C2—22no—
254C2—44no—
255C2—10162no—
256C2C2×#1686118311yescharacteristic
257C2C2×#1686123511yescharacteristic
258C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×D4×C127C2
socleC22×C127C22×D4
hypercenterC2×C4×D4D127
hypocenterC127#3036
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. #22550426C2semidirect
3. #22550091C2semidirect
4. #22550081C2semidirect
5. #22550071C2semidirect
6. #22549900C2semidirect
7. #22549875C2semidirect
8. C2×C4×D4×C127C2semidirect
14. #16861209C22semidirect
15. #16861206C22semidirect
16. #16861197C22semidirect
21. C4⋊(C127⋊C8)C22semidirect
22. C4⋊(C127⋊C8)C22semidirect
23. #16861132C22semidirect
24. #16861129C22semidirect
53. C2×D4×C127D4semidirect
119. C4×C127AGL(1,ℤ/8)semidirect
130. C127#3036semidirect
133. C2×C4×D4D127semidirect
174. C2×D4D508semidirect
250. C4#16861235semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225504262
#225500912
#225500812
#225500712
#225499002
#225498752
C2×C4×D4×C1272
#3036127