Finite Group Databasebeta

← #30648189

Subgroups — #30648189

Order 16256.

548 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16861233C212yesmaximal, complemented
2#22550424C211yesmaximal, characteristic, complemented
3C4×D4×D127C211yesmaximal, characteristic, complemented
4#22550090C212yesmaximal, complemented
5#22550089C212yesmaximal, complemented
6#22550082C211yesmaximal, characteristic, complemented
7#22550081C211yesmaximal, characteristic, complemented
8#22550068C211yesmaximal, characteristic, complemented
9#22549898C211yesmaximal, characteristic, complemented
10#22549874C212yesmaximal, complemented
11#459×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yes—
13C2×D4×D127C2211yescharacteristic
14C22×C4×D127—21no—
15#16861233—28no—
16C2×#14523237C2212yes—
17C2×#14523236C2212yes—
18C2×#14523233C2212yes—
19C2×#14523231C2212yes—
20#16861209C2211yescharacteristic, complemented
21#16861196C2212yescomplemented
22#16861180—21no—
23#16861180—21no—
24C4×D508C2211yescharacteristic
25D4×Dic127—21no—
26D4×Dic127—21no—
27D4×Dic127C2211yescharacteristic
28#16861171C2211yescharacteristic
29C4×C22⋊D127—21no—
30(C22⋊D127)⋊C4—21no—
31C4⋊C4×D127C2211yescharacteristic
32C22⋊C4×D127—21no—
33C42×D127C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic, complemented
35Dic127⋊C8C2212yescomplemented
36#16861134C2212yescomplemented
37#16861132C2212yescomplemented
38#16861130C2212yescomplemented
39C2×D8×C127C2212yes—
40#127×C127C2211yescharacteristic, complemented
41C4×D4×C127C2211yescharacteristic
42C4⋊C8×C127C2211yescharacteristic, complemented
43#102×C127C2212yescomplemented
44C23×D127—21no—
45#14523247—24no—
46#14523247—24no—
47C2×Dic254C2312yes—
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—22no—
50C2×C22⋊D127—21no—
51D4×D127—22no—
52D4×D127D414yescomplemented
53C22×Dic127—21no—
54C22×Dic127—22no—
55C2×C4×D127—21no—
56C2×C4×D127—21no—
57C2×C4×D127—21no—
58C2×C4×D127—44no—
59C2×C4×D127D411yescharacteristic
60C2×C4×D127D411yescharacteristic
61C2×C4×D127C2311yescharacteristic
62#14523237—24no—
63#14523236—24no—
64#14523233—24no—
65#14523231—24no—
66C2×C127⋊C8C2312yes—
67C4⋊Dic127—21no—
68C4⋊Dic127—21no—
69C4⋊Dic127C2311yescharacteristic
70Dic127⋊C4—21no—
71C22⋊Dic127—22no—
72C22⋊Dic127—22no—
73C22⋊Dic127—21no—
74(C2×D127)⋊C4—21no—
75(C2×D127)⋊C4—21no—
76C4×Dic127D411yescharacteristic
77C4×Dic127D411yescharacteristic
78C4×Dic127C2311yescharacteristic
79C2×D4×C127—21no—
80C2×D4×C127—21no—
81C2×D4×C127C2312yes—
82C2×D4×C127C2311yescharacteristic
83C22×C4×C127—21no—
84D8×C127—24no—
85C2×C8×C127C2312yes—
86C4⋊C4×C127C2311yescharacteristic
87C22⋊C4×C127—21no—
88C42×C127C2311yescharacteristic
89C22×D127—21no—
90C22×D127—24no—
91C22×D127—42no—
92C22×D127—42no—
93C22×D127C2×D411yescharacteristic
94Dic254—22no—
95Dic254—22no—
96D508—21no—
97D508C2×D412yes—
98C22⋊D127—24no—
99C22⋊D127—44no—
100C2×Dic127—21no—
101C2×Dic127—22no—
102C2×Dic127—42no—
103C2×Dic127—44no—
104C2×Dic127C2×D411yescharacteristic
105C2×Dic127C2×D411yescharacteristic
106C2×Dic127C2×D411yescharacteristic
107C4×D127—22no—
108C4×D127—41no—
109C4×D127—42no—
110C4×D127—42no—
111C4×D127—42no—
112C4×D127C2×D412yes—
113C4×D127#4112yes—
114C127⋊C8—22no—
115C23×C127—21no—
116C23×C127—22no—
117D4×C127—22no—
118D4×C127—21no—
119D4×C127—22no—
120D4×C127—42no—
121D4×C127—42no—
122D4×C127C2×D412yes—
123C2×C4×C127—21no—
124C2×C4×C127—42no—
125C2×C4×C127C2×D411yescharacteristic
126C2×C4×C127C2×D411yescharacteristic
127C2×C4×C127C2411yescharacteristic
128C8×C127—22no—
129C22×C127—22no—
130C22×C127—41no—
131C22×C127—41no—
132C22×C127—42no—
133C22×C127—42no—
134C22×C127—42no—
135C22×C127C22×D411yescharacteristic
136C2×D127—21no—
137C2×D127—21no—
138C2×D127—22no—
139C2×D127—48no—
140C2×D127—41no—
141C2×D127—41no—
142C2×D127#12112yes—
143C4×C127—21no—
144C4×C127—41no—
145C4×C127AGL(1,ℤ/8)12yescomplemented
146C4×C127C22×D411yescharacteristic
147C4×C127C2×#4111yescharacteristic
148Dic127—21no—
149Dic127—21no—
150Dic127—42no—
151Dic127—41no—
152Dic127#12112yes—
153C2×C127—42no—
154C2×C127—82no—
155C2×C127C2×#12111yescharacteristic
156C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
157C2×C127#57511yescharacteristic
158D127—22no—
159D127—42no—
160#3036—1271nomaximal, Sylow
161C127#303611yescharacteristic, complemented, Sylow
162C2×AGL(1,ℤ/8)—1272no—
163#532—1271no—
164C2×C4×D4—1271no—
165#463—1272no—
166#463—1272no—
167#461—1271no—
168#461—1271no—
169#459—1271no—
170#459D12711yescharacteristic, complemented
171#423—1271no—
172#417—1272no—
173C2×#41—1272no—
174C22×D4—1271no—
175C23×C4—2541no—
176AGL(1,ℤ/8)—2548no—
177C2×SD16—1272no—
178C2×SD16—1272no—
179C2×D8—1272no—
180C2×D8C2×D12712yes—
181C2×M16—1272no—
182C4⋊Q8—1271no—
183#127C2×D12711yescharacteristic, complemented
184#124—1272no—
185#121—2541no—
186#121—2541no—
187C4×D4—1271no—
188C4×D4—1271no—
189C4×D4—1271no—
190C4×D4—2541no—
191C4×D4—2541no—
192C4×D4—2541no—
193C4×D4—2541no—
194C4×D4C2×D12711yescharacteristic
195C2×C4⋊C4—1271no—
196C2×C22⋊C4—2541no—
197C2×C42—1271no—
198C4⋊C8—1272no—
199C4⋊C8—1271no—
200C4⋊C8C2×D12711yescharacteristic, complemented
201#103—1272no—
202#103—1272no—
203#102—1272no—
204#102C2×D12712yescomplemented
205C24—2541no—
206#41—2544no—
207#41—2544no—
208C2×Q8—1272no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—1274no—
212C2×D4—1271no—
213C2×D4—2542no—
214C2×D4—2541no—
215C2×D4—2542no—
216C2×D4C22×D12712yes—
217C2×D4C22×D12711yescharacteristic
218C22×C4—21no—
219C22×C4—1271no—
220C22×C4—1271no—
221C22×C4—1271no—
222C22×C4—5084no—
223C22×C4—2541no—
224C22×C4—2541no—
225C22×C4—2541no—
226C22×C4—2542no—
227C22×C4—2541no—
228SD16—2544no—
229SD16—2544no—
230D8—24no—
231D8—2544no—
232M16—2544no—
233C2×C8—1272no—
234C2×C8C22×D12712yes—
235C4⋊C4—1271no—
236C4⋊C4—2541no—
237C4⋊C4—2541no—
238C4⋊C4—2541no—
239C4⋊C4C22×D12711yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—2541no—
242C22⋊C4—2541no—
243C22⋊C4—2542no—
244C22⋊C4—2542no—
245C22⋊C4—2541no—
246C42—1271no—
247C42—1271no—
248C42—1271no—
249C42C22×D12711yescharacteristic
250C23—21no—
251C23—22no—
252C23—1271no—
253C23—5082no—
254C23—5082no—
255C23—2544no—
256C23—2541no—
257Q8—2542no—
258Q8—2542no—
259D4—22no—
260D4—21no—
261D4—22no—
262D4—42no—
263D4—42no—
264D4—1272no—
265D4—5084no—
266D4—2544no—
267D4—2541no—
268D4D4×D12712yes—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—1271no—
272C2×C4—1272no—
273C2×C4—1272no—
274C2×C4—1271no—
275C2×C4—1271no—
276C2×C4—5081no—
277C2×C4—5082no—
278C2×C4—5082no—
279C2×C4—5084no—
280C2×C4—5082no—
281C2×C4—5082no—
282C2×C4—2541no—
283C2×C4—2542no—
284C2×C4—2542no—
285C2×C4D4×D12711yescharacteristic
286C2×C4D4×D12711yescharacteristic
287C2×C4C23×D12711yescharacteristic
288C8—22no—
289C8—2542no—
290C22—22no—
291C22—41no—
292C22—41no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—1272no—
297C22—5088no—
298C22—5081no—
299C22—5081no—
300C22—2542no—
301C22—2541no—
302C22—2541no—
303C22C2×D4×D12711yescharacteristic
304C4—21no—
305C4—41no—
306C4—1272no—
307C4—5082no—
308C4—5081no—
309C4—2541no—
310C4—2541no—
311C4#1686123312yescomplemented
312C4C2×D4×D12711yescharacteristic
313C4#41×D12711yescharacteristic
314C2—42no—
315C2—82no—
316C2—5082no—
317C2—2542no—
318C2#121×D12711yescharacteristic
319C2C2×#1686123311yescharacteristic
320C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3036
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. #22550424C2semidirect
3. C4×D4×D127C2semidirect
4. #22550090C2semidirect
5. #22550089C2semidirect
6. #22550082C2semidirect
7. #22550081C2semidirect
8. #22550068C2semidirect
9. #22549898C2semidirect
10. #22549874C2semidirect
11. #459×C127C2semidirect
20. #16861209C22semidirect
21. #16861196C22semidirect
34. C4⋊(C127⋊C8)C22semidirect
35. Dic127⋊C8C22semidirect
36. #16861134C22semidirect
37. #16861132C22semidirect
38. #16861130C22semidirect
40. #127×C127C22semidirect
42. C4⋊C8×C127C22semidirect
43. #102×C127C22semidirect
52. D4×D127D4semidirect
145. C4×C127AGL(1,ℤ/8)semidirect
161. C127#3036semidirect
170. #459D127semidirect
183. #127C2×D127semidirect
200. C4⋊C8C2×D127semidirect
204. #102C2×D127semidirect
311. C4#16861233semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
#225504242
C4×D4×D1272
#225500902
#225500892
#225500822
#225500812
#225500682
#225498982
#225498742
#459×C1272
#3036127