Finite Group Databasebeta

← D3×#450 #80144

Subgroups — D3×#450 #80144

Order 384.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 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×D3×D4C211yesmaximal, characteristic
2C2×D3×SD16C212yesmaximal, complemented
3D3×C22⋊Q8C211yesmaximal, characteristic, complemented
4C3×#450C211yesmaximal, characteristic, complemented
5#5218C211yesmaximal, characteristic, complemented
6#5169C212yesmaximal, complemented
7#5046C211yesmaximal, characteristic, complemented
8#4776C212yesmaximal, complemented
9D3×#102C212yesmaximal, complemented
10#4731C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic, complemented
12C2×#450—31nomaximal, Sylow
13C24×D3—21no—
14C22×C3×D4C2211yescharacteristic
15C22×#67—21no—
16C2×D3×Q8C2211yescharacteristic, complemented
17C2×D3×D4—28no—
18C2×D3×D4—21no—
19C2×D3×D4—21no—
20C2×D3×D4C2212yes—
21C2×D3×D4C2212yes—
22C22×D12C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C2×C3×SD16C2212yescomplemented
25C3×C22⋊Q8C2211yescharacteristic, complemented
26#946C2211yescharacteristic, complemented
27C2×#215C2212yescomplemented
28C2×#214C2212yescomplemented
29#924C2211yescharacteristic, complemented
30D3×SD16—28no—
31C2×#204C2212yescomplemented
32C2×D3×C8C2212yescomplemented
33#897C2211yescharacteristic, complemented
34#896—21no—
35D3×C4⋊C4—21no—
36D3×C4⋊C4C2211yescharacteristic, complemented
37D3×C22⋊C4—21no—
38#878—21no—
39C3×#102C2212yescomplemented
40C3×C22⋊C8C2211yescharacteristic, complemented
41#832C2212yescomplemented
42#830C2211yescharacteristic, complemented
43#821C2212yescomplemented
44(C2×D3)⋊C8C2212yescomplemented
45#809C2212yescomplemented
46C23×D4—31no—
47C22×SD16—32no—
48C2×C22⋊Q8—31no—
49#450—31no—
50#450—32no—
51#450—31no—
52#450—32no—
53#450—31no—
54#450D311yescharacteristic, direct product factor
55C2×#102—32no—
56C2×C22⋊C8—31no—
57C24×C3—21no—
58C23×D3—28no—
59C23×D3—21no—
60C23×D3—22no—
61C23×D3—22no—
62C23×D3—44no—
63C23×D3—44no—
64C23×D3D411yescharacteristic
65C2×C3×Q8C2311yescharacteristic, complemented
66C2×C3×D4—24no—
67C2×C3×D4—21no—
68C2×C3×D4C2312yes—
69C22×C3×C4C2311yescharacteristic
70C2×#67—28no—
71C2×#67—22no—
72C2×#67—22no—
73C22×Dic3D411yescharacteristic
74D3×Q8—24no—
75D3×D4—28no—
76D3×D4—28no—
77D3×D4—24no—
78D3×D4—24no—
79D3×D4—24no—
80D3×D4D418yes—
81C2×D12—21no—
82C2×D12—24no—
83C2×D12C2312yes—
84C2×C4×D3—24no—
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3D411yescharacteristic
88C2×C4×D3D411yescharacteristic
89C2×C4×D3C2311yescharacteristic
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C2×Dic6C2311yescharacteristic, complemented
93C3×SD16—24no—
94C2×C3×C8C2312yescomplemented
95C3×C4⋊C4—21no—
96C3×C4⋊C4C2311yescharacteristic, complemented
97C3×C22⋊C4—21no—
98#217—21no—
99#215—24no—
100#214—24no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3C2311yescharacteristic, complemented
105Dic3⋊C4—21no—
106Dic3⋊C4—21no—
107Dic3⋊C4—21no—
108C2×C3⋊C8C2312yescomplemented
109#204—24no—
110D3×C8—24no—
111C25—61no—
112C22×Q8—31no—
113C22×D4—32no—
114C22×D4—32no—
115C22×D4—31no—
116C22×D4—61no—
117C22×D4—61no—
118C22×D4—61no—
119C22×D4—68no—
120C22×D4C2×D311yescharacteristic
121C23×C4—31no—
122C2×SD16—32no—
123C2×SD16—32no—
124C2×SD16—32no—
125C2×SD16—68no—
126C2×SD16C2×D312yescomplemented
127C22×C8—32no—
128C22⋊Q8—31no—
129C22⋊Q8—31no—
130C22⋊Q8—31no—
131C22⋊Q8—61no—
132C22⋊Q8—61no—
133C22⋊Q8C2×D311yescharacteristic, complemented
134C2×C4⋊C4—31no—
135C2×C4⋊C4—61no—
136C2×C22⋊C4—61no—
137#102—32no—
138#102—32no—
139#102—32no—
140#102C2×D312yescomplemented
141C22⋊C8—31no—
142C22⋊C8—32no—
143C22⋊C8C2×D311yescharacteristic, complemented
144C23×C3—24no—
145C23×C3—22no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3C2×D411yescharacteristic
149C22×D3—22no—
150C22×D3—22no—
151C22×D3—24no—
152C22×D3—28no—
153C22×D3—22no—
154C22×D3—28no—
155C22×D3—44no—
156C22×D3—432no—
157C22×D3—42no—
158C22×D3—42no—
159C22×D3—44no—
160C22×D3—44no—
161C22×D3—44no—
162C22×D3SD1614yescomplemented
163C22×D3C2×D411yescharacteristic
164C22×D3C2×D411yescharacteristic
165C3×Q8—22no—
166C3×D4—24no—
167C3×D4—22no—
168C3×D4C2×D414yes—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4C2×D411yescharacteristic
173C2×C3×C4C2411yescharacteristic
174#67—28no—
175#67—28no—
176C2×Dic3—22no—
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3C2×D411yescharacteristic
180C2×Dic3C2×D411yescharacteristic
181D12—24no—
182D12—22no—
183D12C2×D414yes—
184C4×D3—22no—
185C4×D3—22no—
186C4×D3—22no—
187C4×D3—42no—
188C4×D3—42no—
189C4×D3C2×D414yes—
190Dic6—22no—
191Dic6—42no—
192C3×C8—22no—
193C3⋊C8—22no—
194C24—21no—
195C24—31no—
196C24—62no—
197C24—62no—
198C24—68no—
199C24—61no—
200C24—124no—
201C24—124no—
202C2×Q8—31no—
203C2×Q8—61no—
204C2×Q8—64no—
205C2×Q8C22×D311yescharacteristic, complemented
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—38no—
209C2×D4—32no—
210C2×D4—64no—
211C2×D4—64no—
212C2×D4—62no—
213C2×D4—62no—
214C2×D4—64no—
215C2×D4—68no—
216C2×D4—68no—
217C2×D4—68no—
218C2×D4—64no—
219C2×D4—61no—
220C2×D4C22×D312yes—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—64no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4C22×D311yescharacteristic
230SD16—24no—
231SD16—64no—
232SD16—64no—
233SD16—64no—
234C2×C8—32no—
235C2×C8—64no—
236C2×C8C22×D312yescomplemented
237C4⋊C4—21no—
238C4⋊C4—31no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4C22×D311yescharacteristic, complemented
244C22⋊C4—21no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—24no—
251C22×C3—48no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3C2×SD1612yescomplemented
255C22×C3C22×D411yescharacteristic
256C2×D3—21no—
257C2×D3—21no—
258C2×D3—28no—
259C2×D3—24no—
260C2×D3—416no—
261C2×D3—48no—
262C2×D3—42no—
263C2×D3—42no—
264C2×D3—416no—
265C2×D3#12012yes—
266C2×D3C2×SD1612yes—
267C2×D3C2×SD1612yescomplemented
268C2×D3AGL(1,ℤ/8)12yes—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4C22×D412yes—
273Dic3—21no—
274Dic3—41no—
275Dic3—41no—
276Dic3#12012yes—
277C23—24no—
278C23—22no—
279C23—31no—
280C23—34no—
281C23—31no—
282C23—42no—
283C23—42no—
284C23—62no—
285C23—62no—
286C23—68no—
287C23—64no—
288C23—68no—
289C23—62no—
290C23—122no—
291C23—124no—
292C23—124no—
293C23—124no—
294C23—1232no—
295C23—124no—
296C23—122no—
297C23D3×D411yescharacteristic
298Q8—22no—
299Q8—62no—
300Q8—122no—
301D4—24no—
302D4—22no—
303D4—34no—
304D4—64no—
305D4—68no—
306D4—68no—
307D4—62no—
308D4D3×D414yes—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—31no—
313C2×C4—34no—
314C2×C4—31no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4—62no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4D3×D411yescharacteristic
324C2×C4C23×D311yescharacteristic
325C8—22no—
326C8—62no—
327C2×C3—22no—
328C2×C3—44no—
329C2×C3C2×#12011yescharacteristic
330C2×C3C22×SD1611yescharacteristic
331C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
332D3—22no—
333D3—44no—
334D3#45014yesdirect product factor
335C22—21no—
336C22—21no—
337C22—24no—
338C22—32no—
339C22—32no—
340C22—32no—
341C22—32no—
342C22—48no—
343C22—42no—
344C22—42no—
345C22—68no—
346C22—61no—
347C22—61no—
348C22—64no—
349C22—128no—
350C22—1216no—
351C22—1216no—
352C22—122no—
353C22—122no—
354C22D3×SD1612yescomplemented
355C22C2×D3×D411yescharacteristic
356C4—21no—
357C4—32no—
358C4—41no—
359C4—41no—
360C4—61no—
361C4—121no—
362C4—121no—
363C4C2×D3×D412yes—
364C3C2×#45011yescharacteristic, complemented, Sylow
365C2—22no—
366C2—34no—
367C2—44no—
368C2—62no—
369C2—124no—
370C2D3×#12011yescharacteristic
371C2C2×D3×SD1611yescharacteristic
372C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radicalC22×D3×D4C2
hypercenter#450D3
hypocenterC3C2×#450
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×D3×SD16C2semidirect
3. D3×C22⋊Q8C2semidirect
4. C3×#450C2semidirect
5. #5218C2semidirect
6. #5169C2semidirect
7. #5046C2semidirect
8. #4776C2semidirect
9. D3×#102C2semidirect
10. #4731C2semidirect
11. D3×C22⋊C8C2semidirect
16. C2×D3×Q8C22semidirect
24. C2×C3×SD16C22semidirect
25. C3×C22⋊Q8C22semidirect
26. #946C22semidirect
27. C2×#215C22semidirect
28. C2×#214C22semidirect
29. #924C22semidirect
31. C2×#204C22semidirect
32. C2×D3×C8C22semidirect
33. #897C22semidirect
36. D3×C4⋊C4C22semidirect
39. C3×#102C22semidirect
40. C3×C22⋊C8C22semidirect
41. #832C22semidirect
42. #830C22semidirect
43. #821C22semidirect
44. (C2×D3)⋊C8C22semidirect
45. #809C22semidirect
54. #450D3direct (N × H)
65. C2×C3×Q8C23semidirect
92. C2×Dic6C23semidirect
94. C2×C3×C8C23semidirect
96. C3×C4⋊C4C23semidirect
104. C4⋊Dic3C23semidirect
108. C2×C3⋊C8C23semidirect
126. C2×SD16C2×D3semidirect
133. C22⋊Q8C2×D3semidirect
140. #102C2×D3semidirect
143. C22⋊C8C2×D3semidirect
162. C22×D3SD16semidirect
205. C2×Q8C22×D3semidirect
236. C2×C8C22×D3semidirect
243. C4⋊C4C22×D3semidirect
254. C22×C3C2×SD16semidirect
267. C2×D3C2×SD16semidirect
334. D3#450direct (N × H)
354. C22D3×SD16semidirect
364. C3C2×#450semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×D3×SD162
D3×C22⋊Q82
C3×#4502
#52182
#51692
#50462
#47762
D3×#1022
#47312
D3×C22⋊C82
C2×#4503