Finite Group Databasebeta

← #19145512

Subgroups — #19145512

Order 5504.

532 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#14583446C211yesmaximal, characteristic, complemented
2#127×D43C211yesmaximal, characteristic, complemented
3#14583274C212yesmaximal, complemented
4#14583244C212yesmaximal, complemented
5#14583095C211yesmaximal, characteristic, complemented
6#14583094C211yesmaximal, characteristic, complemented
7#14583091C211yesmaximal, characteristic, complemented
8#14583034C212yesmaximal, complemented
9#14583031C212yesmaximal, complemented
10#14582951C211yesmaximal, characteristic, complemented
11C43×#459C211yesmaximal, characteristic, complemented
12C2×#10621898—21no—
13C2×D4×D43—21no—
14C2×D4×D43—21no—
15C2×D4×D43C2212yes—
16C2×D344C2212yescomplemented
17C2×#10621882C2212yescomplemented
18C2×#10621882C2212yescomplemented
19#13063324C2211yescharacteristic, complemented
20#13063322C2211yescharacteristic
21#13063321—21no—
22#13063321—21no—
23#13063321C2212yes—
24#13063321C2211yescharacteristic, complemented
25#13063311—21no—
26#13063300C2211yescharacteristic
27#13063298—21no—
28#13063296C2211yescharacteristic
29C4×D172C2211yescharacteristic, complemented
30C4×C22⋊D43—21no—
31C42×D43C2211yescharacteristic
32C4⋊(C43⋊C8)C2211yescharacteristic, complemented
33#13063247C2212yescomplemented
34#13063246C2212yescomplemented
35#13063245C2212yescomplemented
36(C2×D43)⋊C8C2212yes—
37C8×Dic43C2212yescomplemented
38C2×D8×C43C2212yescomplemented
39#127×C43C2211yescharacteristic
40C4×D4×C43C2211yescharacteristic, complemented
41C4⋊C8×C43C2211yescharacteristic, complemented
42#102×C43C2212yescomplemented
43C23×D43—22no—
44C2×D4×C43—21no—
45C2×D4×C43—21no—
46C2×D4×C43C2312yes—
47C2×D4×C43C2311yescharacteristic
48C22×C4×C43—21no—
49C2×C22⋊D43—21no—
50C2×C22⋊D43—21no—
51C2×C22⋊D43—22no—
52C2×C22⋊D43—22no—
53C2×C22⋊D43—22no—
54C2×C22⋊D43—21no—
55D4×D43—42no—
56D4×D43—42no—
57D4×D43—44no—
58D4×D43—44no—
59#10621898—44no—
60C2×D172—21no—
61C2×D172—21no—
62C2×D172—21no—
63C2×D172C2312yes—
64C2×D172C2311yescharacteristic
65C2×C4×D43—21no—
66C2×C4×D43D411yescharacteristic
67C2×C4×D43D411yescharacteristic
68C2×C4×D43C2311yescharacteristic
69C2×Dic86—21no—
70D8×C43—42no—
71C2×C8×C43C2312yes—
72C4⋊C4×C43C2311yescharacteristic
73C22⋊C4×C43—21no—
74C42×C43C2311yescharacteristic
75C22⋊Dic43—21no—
76#10621882—42no—
77#10621882—42no—
78(C2×D43)⋊C4—21no—
79(C2×D43)⋊C4—21no—
80C4⋊Dic43C2311yescharacteristic
81Dic43⋊C4—21no—
82C4×Dic43D411yescharacteristic
83C4×Dic43D411yescharacteristic
84C4×Dic43C2311yescharacteristic
85C2×C43⋊C8C2312yes—
86D344—42no—
87C23×C43—21no—
88C23×C43—22no—
89C22×D43—21no—
90C22×D43—22no—
91C22×D43—44no—
92C22×D43—44no—
93C22×D43—44no—
94C22×D43C2×D411yescharacteristic
95D4×C43—21no—
96D4×C43—22no—
97D4×C43—21no—
98D4×C43—22no—
99D4×C43—42no—
100D4×C43—42no—
101C2×C4×C43—21no—
102C2×C4×C43—42no—
103C2×C4×C43C2×D411yescharacteristic
104C2×C4×C43C2×D411yescharacteristic
105C2×C4×C43C2411yescharacteristic
106C22⋊D43—44no—
107C22⋊D43—42no—
108C22⋊D43—42no—
109C22⋊D43—44no—
110C22⋊D43—44no—
111C22⋊D43—42no—
112C2×Dic43—21no—
113C2×Dic43C2×D411yescharacteristic
114C2×Dic43C2×D411yescharacteristic
115C2×Dic43C2×D411yescharacteristic
116D172—21no—
117D172—22no—
118D172—21no—
119D172—22no—
120D172—42no—
121D172—42no—
122D172—42no—
123C4×D43—21no—
124C4×D43—21no—
125C4×D43—22no—
126C4×D43—42no—
127C4×D43D814yescomplemented
128Dic86—42no—
129C8×C43—22no—
130C43⋊C8—22no—
131C22×C43—42no—
132C22×C43—41no—
133C22×C43—42no—
134C22×C43—41no—
135C22×C43—42no—
136C22×C43—41no—
137C22×C43C22×D411yescharacteristic
138C2×D43—21no—
139C2×D43—21no—
140C2×D43—42no—
141C2×D43—41no—
142C2×D43—42no—
143C2×D43—41no—
144C2×D43—42no—
145C2×D43—41no—
146C2×D43—88no—
147C2×D43C2×D812yes—
148C4×C43—21no—
149C4×C43—41no—
150C4×C43C2×D812yes—
151C4×C43H(5,𝔽2)11yescharacteristic
152C4×C43H(5,𝔽2)11yescharacteristic
153Dic43—21no—
154Dic43—41no—
155Dic43C2×D812yes—
156Dic43AGL(1,ℤ/8)12yes—
157#3205—431nomaximal, Sylow
158C2×C43—82no—
159C2×C43—81no—
160C2×C43#53511yescharacteristic
161C2×C43C22×D811yescharacteristic
162C2×C43C2×AGL(1,ℤ/8)11yescharacteristic
163D43—22no—
164D43—81no—
165D43—82no—
166#532—431no—
167C2×#127—431no—
168#493—432no—
169#486—432no—
170#459—431no—
171#459—431no—
172#459—431no—
173#459D4311yescharacteristic, complemented
174#447—432no—
175#447—432no—
176#431—431no—
177C43#320511yescharacteristic, complemented, Sylow
178C2×#41—861no—
179C22×D4—432no—
180C22×D4—861no—
181C22×D4—861no—
182C2×D8—432no—
183C2×D8—432no—
184C2×D8—432no—
185C2×D8C2×D4312yescomplemented
186C4⋊Q8—431no—
187#127—432no—
188#127—431no—
189#127—431no—
190#127—861no—
191#127—861no—
192#127C2×D4311yescharacteristic
193#124—861no—
194#121—431no—
195#121—431no—
196#121—861no—
197C4×D4—431no—
198C4×D4—861no—
199C4×D4C2×D4311yescharacteristic, complemented
200C2×C42—431no—
201C4⋊C8—431no—
202C4⋊C8C2×D4311yescharacteristic, complemented
203#102—432no—
204#102—432no—
205#102—432no—
206#102C2×D4312yescomplemented
207C22⋊C8—432no—
208C4×C8—432no—
209C24—862no—
210#41—1724no—
211C2×Q8—861no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—432no—
215C2×D4—431no—
216C2×D4—861no—
217C2×D4—861no—
218C2×D4—861no—
219C2×D4—862no—
220C2×D4—862no—
221C2×D4—862no—
222C2×D4—861no—
223C2×D4—861no—
224C2×D4—861no—
225C2×D4—1724no—
226C2×D4—1722no—
227C2×D4—1722no—
228C2×D4—1724no—
229C2×D4C22×D4312yes—
230C2×D4C22×D4311yescharacteristic
231C22×C4—21no—
232C22×C4—431no—
233C22×C4—431no—
234C22×C4—431no—
235C22×C4—861no—
236D8—42no—
237D8—1722no—
238D8—1722no—
239D8—1722no—
240C2×C8—432no—
241C2×C8C22×D4312yes—
242C4⋊C4—431no—
243C4⋊C4—861no—
244C4⋊C4C22×D4311yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—861no—
247C22⋊C4—861no—
248C22⋊C4—861no—
249C42—431no—
250C42—431no—
251C42—431no—
252C42C22×D4311yescharacteristic
253C23—21no—
254C23—22no—
255C23—431no—
256C23—861no—
257C23—862no—
258C23—1724no—
259C23—1724no—
260C23—1724no—
261Q8—1722no—
262D4—21no—
263D4—22no—
264D4—21no—
265D4—22no—
266D4—42no—
267D4—42no—
268D4—861no—
269D4—862no—
270D4—861no—
271D4—862no—
272D4—1722no—
273D4—1722no—
274D4—1724no—
275D4—1724no—
276D4—1722no—
277D4—1722no—
278D4—1724no—
279D4—1722no—
280D4—1722no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—431no—
284C2×C4—431no—
285C2×C4—434no—
286C2×C4—431no—
287C2×C4—862no—
288C2×C4—861no—
289C2×C4—861no—
290C2×C4—861no—
291C2×C4—1722no—
292C2×C4D4×D4311yescharacteristic
293C2×C4D4×D4311yescharacteristic
294C2×C4C23×D4311yescharacteristic
295C8—22no—
296C8—862no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—41no—
303C22—432no—
304C22—861no—
305C22—861no—
306C22—3448no—
307C22—1721no—
308C22—1722no—
309C22—1722no—
310C22—1721no—
311C22—1722no—
312C22—1721no—
313C22C2×D4×D4311yescharacteristic
314C4—21no—
315C4—41no—
316C4—432no—
317C4—432no—
318C4—861no—
319C4—1721no—
320C4D8×D4312yes—
321C4#1306337011yescharacteristic
322C4#1306337111yescharacteristic
323C2—82no—
324C2—81no—
325C2—862no—
326C2—3442no—
327C2—3441no—
328C2#1458346511yescharacteristic
329C2C2×D8×D4311yescharacteristic
330C2AGL(1,ℤ/8)×D4311yescharacteristic
subgroupG/H
centerC22C2×D4×D43
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#459C2
socleC22×C43C22×D4
hypercenter#459D43
hypocenterC43#3205
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #14583446C2semidirect
2. #127×D43C2semidirect
3. #14583274C2semidirect
4. #14583244C2semidirect
5. #14583095C2semidirect
6. #14583094C2semidirect
7. #14583091C2semidirect
8. #14583034C2semidirect
9. #14583031C2semidirect
10. #14582951C2semidirect
11. C43×#459C2semidirect
16. C2×D344C22semidirect
17. C2×#10621882C22semidirect
18. C2×#10621882C22semidirect
19. #13063324C22semidirect
24. #13063321C22semidirect
29. C4×D172C22semidirect
32. C4⋊(C43⋊C8)C22semidirect
33. #13063247C22semidirect
34. #13063246C22semidirect
35. #13063245C22semidirect
37. C8×Dic43C22semidirect
38. C2×D8×C43C22semidirect
40. C4×D4×C43C22semidirect
41. C4⋊C8×C43C22semidirect
42. #102×C43C22semidirect
127. C4×D43D8semidirect
173. #459D43semidirect
177. C43#3205semidirect
185. C2×D8C2×D43semidirect
199. C4×D4C2×D43semidirect
202. C4⋊C8C2×D43semidirect
206. #102C2×D43semidirect