Finite Group Databasebeta

← #19145398

Subgroups — #19145398

Order 5504.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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×#13063296C211yesmaximal, characteristic
3#14583235C211yesmaximal, characteristic, complemented
4#14583232C211yesmaximal, characteristic, complemented
5#14583211C212yesmaximal, complemented
6#14583144C212yesmaximal, complemented
7#14583142C212yesmaximal, complemented
8#14583050C211yesmaximal, characteristic, complemented
9#14583036C212yesmaximal, complemented
10#14582861C211yesmaximal, characteristic, complemented
11C43×#450C211yesmaximal, characteristic, complemented
12C2×Q8×D43C2211yescharacteristic, complemented
13C22×C22⋊D43—21no—
14C22×C4×D43C2211yescharacteristic
15C2×#10621883C2212yescomplemented
16C2×#10621872C2212yescomplemented
17#13063305C2211yescharacteristic, complemented
18#13063304C2211yescharacteristic
19#13063303—21no—
20#13063302—21no—
21#13063301C2211yescharacteristic, complemented
22#13063296—21no—
23#13063296—21no—
24#13063296C2212yes—
25#13063296C2212yes—
26C2×C4⋊Dic43C2211yescharacteristic
27C4⋊C4×D43—21no—
28C4⋊C4×D43C2211yescharacteristic, complemented
29C2×C22⋊Dic43—21no—
30C22⋊C4×D43—21no—
31#13063259C2212yescomplemented
32#13063258C2212yescomplemented
33#13063250C2212yescomplemented
34#13063249C2212yescomplemented
35#13063246C2212yescomplemented
36C22⋊(C43⋊C8)C2211yescharacteristic
37(C2×D43)⋊C8C2212yescomplemented
38C22×D4×C43C2211yescharacteristic
39C2×SD16×C43C2212yescomplemented
40C22⋊Q8×C43C2211yescharacteristic
41#102×C43C2212yescomplemented
42C22⋊C8×C43C2211yescharacteristic
43C24×C43—21no—
44C23×D43D411yescharacteristic
45C2×Q8×C43C2311yescharacteristic
46C2×D4×C43—21no—
47C2×D4×C43—42no—
48C2×D4×C43C2312yes—
49C22×C4×C43C2311yescharacteristic
50C2×C22⋊D43—22no—
51C2×C22⋊D43—22no—
52C2×C22⋊D43—44no—
53C22×Dic43—21no—
54C22×Dic43D411yescharacteristic
55Q8×D43—42no—
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
65SD16×C43—42no—
66C2×C8×C43C2312yes—
67C4⋊C4×C43—21no—
68C4⋊C4×C43C2311yescharacteristic
69C22⋊C4×C43—21no—
70C22⋊Dic43—22no—
71C22⋊Dic43—22no—
72C22⋊Dic43—21no—
73#10621883—42no—
74(C2×D43)⋊C4—21no—
75(C2×D43)⋊C4—21no—
76C4⋊Dic43—21no—
77C4⋊Dic43—21no—
78C4⋊Dic43C2311yescharacteristic
79C4⋊Dic43C2312yes—
80Dic43⋊C4—21no—
81Dic43⋊C4—21no—
82Dic43⋊C4—21no—
83C2×C43⋊C8C2312yes—
84#10621872—42no—
85C23×C43—22no—
86C23×C43—42no—
87C23×C43—42no—
88C23×C43—42no—
89C23×C43C2×D411yescharacteristic
90C22×D43—22no—
91C22×D43—22no—
92C22×D43—41no—
93C22×D43C2×D411yescharacteristic
94C22×D43C2×D411yescharacteristic
95Q8×C43—22no—
96D4×C43—22no—
97D4×C43—22no—
98D4×C43—42no—
99C2×C4×C43—22no—
100C2×C4×C43—21no—
101C2×C4×C43—21no—
102C2×C4×C43C2×D411yescharacteristic
103C2×C4×C43C2411yescharacteristic
104C22⋊D43—44no—
105C22⋊D43—44no—
106C2×Dic43—21no—
107C2×Dic43—21no—
108C2×Dic43—22no—
109C2×Dic43—41no—
110C2×Dic43—42no—
111C2×Dic43C2×D411yescharacteristic
112C2×Dic43C2×D411yescharacteristic
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×C43—42no—
126C22×C43—42no—
127C22×C43—42no—
128C22×C43—84no—
129C22×C43#13712yes—
130C22×C43C22×D411yescharacteristic
131C2×D43—21no—
132C2×D43—21no—
133C2×D43—21no—
134C2×D43—21no—
135C2×D43—21no—
136C2×D43—44no—
137C2×D43AGL(1,ℤ/8)12yes—
138C4×C43—21no—
139C4×C43—41no—
140C4×C43—41no—
141C4×C43H(5,𝔽2)12yes—
142Dic43—21no—
143Dic43—21no—
144Dic43—41no—
145Dic43—41no—
146Dic43—42no—
147#3193—431nomaximal, Sylow
148C2×C43—22no—
149C2×C43—82no—
150C2×C43#53411yescharacteristic
151C2×C43C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C43C2×#13711yescharacteristic
153D43—22no—
154D43—41no—
155C2×C22⋊Q8—431no—
156C2×#121—431no—
157#484—431no—
158#484—431no—
159#480—432no—
160#468—432no—
161#468—432no—
162#450—431no—
163#450D4311yescharacteristic, complemented
164#448—432no—
165#407—431no—
166C43#319311yescharacteristic, complemented, Sylow
167C22×Q8—431no—
168C22×D4—861no—
169C22×D4C2×D4311yescharacteristic
170C23×C4—431no—
171C2×SD16—432no—
172C2×SD16C2×D4312yescomplemented
173C2×M16—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—431no—
186C2×C4⋊C4—861no—
187C2×C22⋊C4—861no—
188C2×C22⋊C4—861no—
189#107—432no—
190#107—432no—
191#103—432no—
192#103—432no—
193#102—432no—
194#102C2×D4312yescomplemented
195C22⋊C8—431no—
196C22⋊C8—432no—
197C22⋊C8C2×D4311yescharacteristic
198C24—21no—
199C24—431no—
200C2×Q8—431no—
201C2×Q8—861no—
202C2×Q8—1722no—
203C2×Q8C22×D4311yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—862no—
207C2×D4—862no—
208C2×D4—1724no—
209C2×D4C22×D4312yes—
210C22×C4—431no—
211C22×C4—431no—
212C22×C4—431no—
213C22×C4—431no—
214C22×C4—431no—
215C22×C4—861no—
216C22×C4—861no—
217C22×C4—861no—
218C22×C4—1722no—
219C22×C4C22×D4311yescharacteristic
220SD16—42no—
221SD16—1722no—
222M16—1722no—
223C2×C8—432no—
224C2×C8C22×D4312yes—
225C4⋊C4—21no—
226C4⋊C4—431no—
227C4⋊C4—432no—
228C4⋊C4—861no—
229C4⋊C4—861no—
230C4⋊C4—861no—
231C4⋊C4—861no—
232C4⋊C4—861no—
233C4⋊C4C22×D4311yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—861no—
236C22⋊C4—861no—
237C22⋊C4—862no—
238C22⋊C4—862no—
239C22⋊C4—861no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—431no—
245C23—431no—
246C23—862no—
247C23—862no—
248C23—1721no—
249C23D4×D4311yescharacteristic
250Q8—22no—
251Q8—862no—
252Q8—1722no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—1724no—
257D4—1724no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—431no—
262C2×C4—431no—
263C2×C4—862no—
264C2×C4—862no—
265C2×C4—861no—
266C2×C4—862no—
267C2×C4—861no—
268C2×C4—1721no—
269C2×C4—1721no—
270C2×C4—1722no—
271C2×C4—1722no—
272C2×C4—1721no—
273C2×C4—1722no—
274C2×C4D4×D4311yescharacteristic
275C2×C4C23×D4311yescharacteristic
276C8—22no—
277C8—862no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—432no—
285C22—861no—
286C22—861no—
287C22—861no—
288C22—861no—
289C22—861no—
290C22—1724no—
291C22#1306335412yes—
292C22C2×D4×D4311yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—861no—
297C4—861no—
298C4—1722no—
299C4—1721no—
300C4—1721no—
301C4#1306337012yes—
302C2—22no—
303C2—82no—
304C2—862no—
305C2—1721no—
306C2#1458345911yescharacteristic
307C2AGL(1,ℤ/8)×D4311yescharacteristic
308C2C2×#1306335411yescharacteristic
subgroupG/H
centerC22C2×D4×D43
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#450C2
socleC22×C43C22×D4
hypercenter#450D43
hypocenterC43#3193
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D43C2semidirect
3. #14583235C2semidirect
4. #14583232C2semidirect
5. #14583211C2semidirect
6. #14583144C2semidirect
7. #14583142C2semidirect
8. #14583050C2semidirect
9. #14583036C2semidirect
10. #14582861C2semidirect
11. C43×#450C2semidirect
12. C2×Q8×D43C22semidirect
15. C2×#10621883C22semidirect
16. C2×#10621872C22semidirect
17. #13063305C22semidirect
21. #13063301C22semidirect
28. C4⋊C4×D43C22semidirect
31. #13063259C22semidirect
32. #13063258C22semidirect
33. #13063250C22semidirect
34. #13063249C22semidirect
35. #13063246C22semidirect
37. (C2×D43)⋊C8C22semidirect
39. C2×SD16×C43C22semidirect
41. #102×C43C22semidirect
163. #450D43semidirect
166. C43#3193semidirect
172. C2×SD16C2×D43semidirect
194. #102C2×D43semidirect