Finite Group Databasebeta

← #19145351

Subgroups — #19145351

Order 5504.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D43C211yesmaximal, characteristic, complemented
2C2×#13063300C211yesmaximal, characteristic
3#14583229C211yesmaximal, characteristic, complemented
4#14583219C212yesmaximal, complemented
5#14583131C212yesmaximal, complemented
6#14583129C212yesmaximal, complemented
7#14583052C211yesmaximal, characteristic, complemented
8#14583049C211yesmaximal, characteristic, complemented
9#14583039C212yesmaximal, complemented
10C22⋊C8×D43C211yesmaximal, characteristic, complemented
11C43×#483C211yesmaximal, characteristic, complemented
12C2×Q8×D43C2211yescharacteristic, complemented
13C22×D172—21no—
14C22×D172C2211yescharacteristic
15C22×C4×D43C2211yescharacteristic
16C2×#10621873C2212yescomplemented
17C2×#10621884C2212yescomplemented
18C2×C8×D43C2212yescomplemented
19#13063305C2211yescharacteristic, complemented
20#13063304C2211yescharacteristic
21#13063303—21no—
22#13063302—21no—
23#13063301C2211yescharacteristic, complemented
24#13063300—21no—
25#13063300—21no—
26#13063300C2212yes—
27#13063300C2212yes—
28C4⋊C4×D43—21no—
29C4⋊C4×D43C2211yescharacteristic, complemented
30C2×(C2×D43)⋊C4—21no—
31C22⋊C4×D43—21no—
32#13063256C2212yescomplemented
33#13063248C2212yescomplemented
34#13063247C2212yescomplemented
35#13063245C2212yescomplemented
36C22⋊(C43⋊C8)C2211yescharacteristic
37(C2×D43)⋊C8C2212yescomplemented
38C22⋊Q8×C43C2211yescharacteristic
39C2×C4⋊C4×C43C2211yescharacteristic
40#106×C43C2212yescomplemented
41#103×C43C2212yescomplemented
42C22⋊C8×C43C2211yescharacteristic
43C23×D43—21no—
44C23×D43D411yescharacteristic
45C2×Q8×C43C2311yescharacteristic
46C22×C4×C43—21no—
47C22×C4×C43C2311yescharacteristic
48C22×Dic43D411yescharacteristic
49Q8×D43—42no—
50C2×D172—21no—
51C2×D172—22no—
52C2×D172—22no—
53C2×D172—42no—
54C2×D172—44no—
55C2×D172C2312yes—
56C2×C4×D43—21no—
57C2×C4×D43—21no—
58C2×C4×D43—42no—
59C2×C4×D43D411yescharacteristic
60C2×C4×D43D411yescharacteristic
61C2×C4×D43C2311yescharacteristic
62C2×C4×D43C2311yescharacteristic
63C2×Dic86—21no—
64C2×Dic86C2311yescharacteristic
65C2×C8×C43C2312yes—
66C4⋊C4×C43—21no—
67C4⋊C4×C43—21no—
68C4⋊C4×C43C2312yes—
69C4⋊C4×C43C2311yescharacteristic
70C22⋊C4×C43—21no—
71C22⋊Dic43—21no—
72#10621884—42no—
73(C2×D43)⋊C4—22no—
74(C2×D43)⋊C4—22no—
75(C2×D43)⋊C4—21no—
76(C2×D43)⋊C4—21no—
77C4⋊Dic43—21no—
78C4⋊Dic43C2311yescharacteristic
79Dic43⋊C4—21no—
80Dic43⋊C4—21no—
81Dic43⋊C4—21no—
82C2×C43⋊C8C2312yes—
83#10621873—42no—
84C8×D43—42no—
85C23×C43C2×D411yescharacteristic
86C22×D43—22no—
87C22×D43—41no—
88C22×D43—41no—
89C22×D43—42no—
90C22×D43—42no—
91C22×D43—42no—
92C22×D43SD1614yescomplemented
93C22×D43C2×D411yescharacteristic
94C22×D43C2×D411yescharacteristic
95Q8×C43—22no—
96C2×C4×C43—22no—
97C2×C4×C43—21no—
98C2×C4×C43—21no—
99C2×C4×C43—22no—
100C2×C4×C43—42no—
101C2×C4×C43C2×D411yescharacteristic
102C2×C4×C43C2411yescharacteristic
103C2×Dic43—21no—
104C2×Dic43—21no—
105C2×Dic43—41no—
106C2×Dic43C2×D411yescharacteristic
107C2×Dic43C2×D411yescharacteristic
108D172—22no—
109D172—22no—
110D172—42no—
111D172—44no—
112D172—44no—
113C4×D43—22no—
114C4×D43—22no—
115C4×D43—41no—
116C4×D43—41no—
117C4×D43—42no—
118C4×D43—42no—
119Dic86—22no—
120Dic86—42no—
121C8×C43—22no—
122C43⋊C8—22no—
123C22×C43—21no—
124C22×C43—21no—
125C22×C43C2×SD1612yes—
126C22×C43C22×D411yescharacteristic
127C2×D43—21no—
128C2×D43—21no—
129C2×D43—21no—
130C2×D43—21no—
131C2×D43—44no—
132C2×D43—42no—
133C2×D43—42no—
134C2×D43—42no—
135C2×D43—84no—
136C2×D43C2×SD1612yes—
137C2×D43C2×SD1612yes—
138C4×C43—21no—
139C4×C43—41no—
140C4×C43—42no—
141C4×C43—41no—
142C4×C43H(5,𝔽2)12yes—
143Dic43—21no—
144Dic43—21no—
145Dic43—41no—
146Dic43—41no—
147#3187—431nomaximal, Sylow
148C2×C43—22no—
149C2×C43#53411yescharacteristic
150C2×C43C22×SD1611yescharacteristic
151C2×C43#57511yescharacteristic
152D43—22no—
153D43—22no—
154D43—82no—
155C2×C22⋊Q8—431no—
156C2×#121—431no—
157#483—431no—
158#483D4311yescharacteristic, complemented
159#481—432no—
160#465—432no—
161#465—432no—
162#450—431no—
163#450—431no—
164#448—432no—
165C2×C22⋊C8—431no—
166C43#318711yescharacteristic, complemented, Sylow
167C22×Q8—431no—
168C22×D4—431no—
169C22×D4—861no—
170C23×C4—431no—
171C2×SD16—432no—
172C2×SD16—432no—
173C22×C8—432no—
174C22⋊Q8—431no—
175C22⋊Q8—431no—
176C22⋊Q8—431no—
177C22⋊Q8—861no—
178C22⋊Q8—861no—
179C22⋊Q8C2×D4311yescharacteristic
180#121—432no—
181#121—432no—
182#121—861no—
183#121—861no—
184C2×C4⋊C4—431no—
185C2×C4⋊C4—861no—
186C2×C4⋊C4C2×D4311yescharacteristic
187C2×C22⋊C4—861no—
188C2×C22⋊C4—861no—
189#106—432no—
190#106C2×D4312yescomplemented
191#103—432no—
192#103C2×D4312yescomplemented
193#102—432no—
194#102—432no—
195C22⋊C8—431no—
196C22⋊C8—432no—
197C22⋊C8C2×D4311yescharacteristic
198C24—431no—
199C24—861no—
200C2×Q8—431no—
201C2×Q8—861no—
202C2×Q8—1722no—
203C2×Q8C22×D4311yescharacteristic
204C2×D4—432no—
205C2×D4—862no—
206C2×D4—862no—
207C2×D4—861no—
208C2×D4—1722no—
209C2×D4—1724no—
210C22×C4—21no—
211C22×C4—431no—
212C22×C4—431no—
213C22×C4—431no—
214C22×C4—431no—
215C22×C4—431no—
216C22×C4—861no—
217C22×C4—861no—
218C22×C4—1722no—
219C22×C4C22×D4311yescharacteristic
220SD16—1722no—
221SD16—1722no—
222C2×C8—432no—
223C2×C8—1722no—
224C2×C8C22×D4312yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—431no—
228C4⋊C4—861no—
229C4⋊C4—861no—
230C4⋊C4—861no—
231C4⋊C4—861no—
232C4⋊C4C22×D4311yescharacteristic
233C4⋊C4C22×D4312yes—
234C22⋊C4—21no—
235C22⋊C4—862no—
236C22⋊C4—862no—
237C22⋊C4—861no—
238C22⋊C4—861no—
239C22⋊C4—861no—
240C23—431no—
241C23—431no—
242C23—434no—
243C23—862no—
244C23—1721no—
245C23—1721no—
246C23—1722no—
247C23—1722no—
248C23—1722no—
249C23D4×D4311yescharacteristic
250Q8—22no—
251Q8—862no—
252Q8—1722no—
253D4—862no—
254D4—862no—
255D4—1722no—
256D4—1724no—
257D4—1724no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—431no—
264C2×C4—431no—
265C2×C4—862no—
266C2×C4—862no—
267C2×C4—861no—
268C2×C4—861no—
269C2×C4—1721no—
270C2×C4—1721no—
271C2×C4—1722no—
272C2×C4—1722no—
273C2×C4—1721no—
274C2×C4D4×D4311yescharacteristic
275C2×C4C23×D4311yescharacteristic
276C8—22no—
277C8—862no—
278C22—21no—
279C22—21no—
280C22—432no—
281C22—432no—
282C22—861no—
283C22—861no—
284C22—861no—
285C22—861no—
286C22—3444no—
287C22—1724no—
288C22—1722no—
289C22—1722no—
290C22—1722no—
291C22SD16×D4312yes—
292C22C2×D4×D4311yescharacteristic
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—861no—
298C4—861no—
299C4—1721no—
300C4—1721no—
301C4#1306337112yes—
302C2—22no—
303C2—862no—
304C2—862no—
305C2—3442no—
306C2#1458346011yescharacteristic
307C2C2×SD16×D4311yescharacteristic
308C2#1458369811yescharacteristic
subgroupG/H
centerC22C2×D4×D43
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#483C2
socleC22×C43C22×D4
hypercenter#483D43
hypocenterC43#3187
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D43C2semidirect
3. #14583229C2semidirect
4. #14583219C2semidirect
5. #14583131C2semidirect
6. #14583129C2semidirect
7. #14583052C2semidirect
8. #14583049C2semidirect
9. #14583039C2semidirect
10. C22⋊C8×D43C2semidirect
11. C43×#483C2semidirect
12. C2×Q8×D43C22semidirect
16. C2×#10621873C22semidirect
17. C2×#10621884C22semidirect
18. C2×C8×D43C22semidirect
19. #13063305C22semidirect
23. #13063301C22semidirect
29. C4⋊C4×D43C22semidirect
32. #13063256C22semidirect
33. #13063248C22semidirect
34. #13063247C22semidirect
35. #13063245C22semidirect
37. (C2×D43)⋊C8C22semidirect
40. #106×C43C22semidirect
41. #103×C43C22semidirect
92. C22×D43SD16semidirect
158. #483D43semidirect
166. C43#3187semidirect
190. #106C2×D43semidirect
192. #103C2×D43semidirect