Finite Group Databasebeta

← #30648209

Subgroups — #30648209

Order 16256.

584 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16861237C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3C2×#16861205C211yesmaximal, characteristic, complemented
4#22550344C211yesmaximal, characteristic, complemented
5#22550075C214yesmaximal, complemented
6#22550070C214yesmaximal, complemented
7#22549899C211yesmaximal, characteristic, complemented
8#22549880C211yesmaximal, characteristic, complemented
9#418×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×D508C2211yescharacteristic
12C22×C22⋊D127—22no—
13#16861237—24no—
14#16861235—24no—
15C2×#14523237C2212yescomplemented
16C2×#14523236C2212yescomplemented
17C2×D1016C2212yescomplemented
18C2×#14523233C2212yescomplemented
19C2×#14523232C2211yescharacteristic
20#16861205—21no—
21#16861205—21no—
22#16861205C2214yes—
23#16861174C2212yescomplemented
24#16861171C2212yescomplemented
25(C22⋊D127)⋊C4—22no—
26#16861166—21no—
27C2×C4×Dic127C2211yescharacteristic
28C42×D127—21no—
29Dic127⋊C8C2214yescomplemented
30#16861129C2214yescomplemented
31C22×D4×C127C2211yescharacteristic
32C2×M16×C127C2211yescharacteristic
33#117×C127C2211yescharacteristic
34#102×C127C2214yescomplemented
35C23×D127—21no—
36#14523248—22no—
37#14523248D414yescomplemented
38C2×Dic254C2311yescharacteristic
39C2×D508—22no—
40C2×D508—21no—
41C2×D508—22no—
42C2×D508C2312yes—
43C2×D508C2311yescharacteristic
44C2×C22⋊D127—24no—
45C2×C22⋊D127—24no—
46C2×C22⋊D127—21no—
47C2×C22⋊D127—48no—
48C22×Dic127D412yes—
49C2×C4×D127—22no—
50C2×C4×D127—21no—
51#14523237—24no—
52#14523236—24no—
53D1016—24no—
54#14523233—24no—
55#14523232—22no—
56C2×C127⋊C8C2312yes—
57Dic127⋊C4—22no—
58(C2×D127)⋊C4—22no—
59C4×Dic127—22no—
60C4×Dic127D411yescharacteristic
61C4×Dic127D411yescharacteristic
62C4×Dic127C2312yes—
63C24×C127—21no—
64C2×D4×C127—22no—
65C2×D4×C127—21no—
66C2×D4×C127—22no—
67C2×D4×C127C2312yes—
68C22×C4×C127C2311yescharacteristic
69M16×C127—22no—
70C2×C8×C127C2312yes—
71C4⋊C4×C127C2312yes—
72C22⋊C4×C127—21no—
73C42×C127—21no—
74C22×D127—21no—
75C22×D127—22no—
76C22×D127—44no—
77C22×D127—42no—
78Dic254—21no—
79Dic254C2×D412yescomplemented
80D508—22no—
81D508—21no—
82D508—22no—
83D508—42no—
84D508C2×D412yescomplemented
85C22⋊D127—24no—
86C22⋊D127—48no—
87C22⋊D127—48no—
88C2×Dic127—21no—
89C2×Dic127—24no—
90C2×Dic127C2×D412yes—
91C2×Dic127C2×D412yes—
92C4×D127—24no—
93C4×D127—44no—
94C127⋊C8—22no—
95C23×C127—22no—
96C23×C127—44no—
97C23×C127—42no—
98C23×C127C2×D411yescharacteristic
99D4×C127—22no—
100D4×C127—22no—
101D4×C127—42no—
102C2×C4×C127—22no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D412yes—
105C2×C4×C127#4112yes—
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127—22no—
109C22×C127—42no—
110C22×C127—44no—
111C22×C127—84no—
112C22×C127#12112yes—
113C22×C127C22×D411yescharacteristic
114C2×D127—22no—
115C2×D127—42no—
116C2×D127—42no—
117C2×D127—44no—
118C2×D127—84no—
119C4×C127—42no—
120C4×C127#12112yes—
121C4×C127C22×D411yescharacteristic
122C4×C127C2×#4111yescharacteristic
123Dic127—22no—
124Dic127—22no—
125Dic127AGL(1,ℤ/8)14yescomplemented
126C2×C127—22no—
127C2×C127—82no—
128C2×C127C2×#12111yescharacteristic
129C2×C127C2×AGL(1,ℤ/8)12yes—
130D127—42no—
131D127—82no—
132#3038—1271nomaximal, Sylow
133C127#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1271no—
135C2×AGL(1,ℤ/8)—1271no—
136C2×#127—1271no—
137C4×#41—1271no—
138#460—1274no—
139#459—1274no—
140#423—1271no—
141#418—1271no—
142#418D12711yescharacteristic, complemented
143C2×#41—1271no—
144C22×D4—1271no—
145C22×D4—2542no—
146C22×D4C2×D12711yescharacteristic
147AGL(1,ℤ/8)—2544no—
148AGL(1,ℤ/8)—2544no—
149C2×SD16—1272no—
150C2×SD16—1272no—
151C2×D8—1272no—
152C2×D8—1272no—
153C2×M16—1271no—
154C2×M16C2×D12711yescharacteristic
155#127—1274no—
156#127—2541no—
157#127—2541no—
158C4×Q8—1272no—
159C4×D4—1272no—
160C4×D4—2542no—
161#117—2541no—
162#117C2×D12711yescharacteristic
163C2×C42—1271no—
164C2×C42—2541no—
165C4⋊C8—1274no—
166#102—1274no—
167#102C2×D12714yescomplemented
168C24—21no—
169C24—2541no—
170#41—1274no—
171#41—2542no—
172C2×Q8—1271no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—1271no—
177C2×D4—1272no—
178C2×D4—5088no—
179C2×D4—2544no—
180C2×D4—2544no—
181C2×D4—2541no—
182C2×D4—2541no—
183C2×D4—2542no—
184C2×D4—2542no—
185C2×D4C22×D12712yes—
186C22×C4—1272no—
187C22×C4—2542no—
188C22×C4—2541no—
189C22×C4C22×D12711yescharacteristic
190SD16—2544no—
191SD16—2544no—
192D8—2544no—
193D8—2544no—
194M16—22no—
195M16—2542no—
196C2×C8—1272no—
197C2×C8C22×D12712yes—
198C4⋊C4—2542no—
199C4⋊C4C22×D12712yes—
200C22⋊C4—21no—
201C22⋊C4—2542no—
202C42—21no—
203C42—1271no—
204C42—1271no—
205C42—1272no—
206C42—2542no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—5084no—
211C23—5082no—
212C23—2542no—
213C23—2541no—
214C23D4×D12711yescharacteristic
215Q8—1272no—
216Q8—2541no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—1272no—
221D4—5088no—
222D4—5088no—
223D4—5082no—
224D4—2544no—
225D4—2542no—
226D4—2542no—
227D4—2541no—
228C2×C4—22no—
229C2×C4—1272no—
230C2×C4—1272no—
231C2×C4—5084no—
232C2×C4—2544no—
233C2×C4—2541no—
234C2×C4—2544no—
235C2×C4D4×D12712yes—
236C2×C4D4×D12711yescharacteristic
237C2×C4#1452324812yes—
238C2×C4C23×D12711yescharacteristic
239C8—22no—
240C8—2542no—
241C22—22no—
242C22—42no—
243C22—44no—
244C22—84no—
245C22—10164no—
246C22—5082no—
247C22—5084no—
248C22—5082no—
249C22—2542no—
250C22#1686118212yes—
251C22C2×D4×D12711yescharacteristic
252C4—42no—
253C4—1274no—
254C4—2542no—
255C4—2542no—
256C4#1686118212yes—
257C4C2×D4×D12711yescharacteristic
258C4C2×#1452324811yescharacteristic
259C2—22no—
260C2—82no—
261C2—10162no—
262C2—5082no—
263C2C2×#1686118211yescharacteristic
264C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#418×C127C2
socleC22×C127C22×D4
hypercenter#418D127
hypocenterC127#3038
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×#16861237C2semidirect
2. C2×#16861235C2semidirect
3. C2×#16861205C2semidirect
4. #22550344C2semidirect
5. #22550075C2semidirect
6. #22550070C2semidirect
7. #22549899C2semidirect
8. #22549880C2semidirect
9. #418×C127C2semidirect
15. C2×#14523237C22semidirect
16. C2×#14523236C22semidirect
17. C2×D1016C22semidirect
18. C2×#14523233C22semidirect
23. #16861174C22semidirect
24. #16861171C22semidirect
29. Dic127⋊C8C22semidirect
30. #16861129C22semidirect
34. #102×C127C22semidirect
37. #14523248D4semidirect
79. Dic254C2×D4semidirect
84. D508C2×D4semidirect
125. Dic127AGL(1,ℤ/8)semidirect
133. C127#3038semidirect
142. #418D127semidirect
167. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612352
C2×#168612052
#225503442
#225500752
#225500702
#225498992
#225498802
#418×C1272
#3038127