Finite Group Databasebeta

← D3×#577 #87205

Subgroups — D3×#577 #87205

Order 384.

1460 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×2−1+4C211yesmaximal, characteristic, complemented
2D3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
3C3×#577C211yesmaximal, characteristic, complemented
4#5837C211yesmaximal, characteristic, complemented
5#5836C211yesmaximal, characteristic, complemented
6#5778C213yesmaximal, complemented
7D3×#137C213yesmaximal, complemented
8#5775C213yesmaximal, complemented
9D3×AGL(1,ℤ/8)C213yesmaximal, complemented
10#5770C213yesmaximal, complemented
11D3×#135C213yesmaximal, complemented
12#5762C213yesmaximal, complemented
13C2×D3×SD16C213yesmaximal, complemented
14#5751C211yesmaximal, characteristic, complemented
15D3×#131C211yesmaximal, characteristic, complemented
16C2×#577—31nomaximal, Sylow
17C3×2−1+4C2211yescharacteristic, complemented
18C3×H(5,𝔽2)C2211yescharacteristic, complemented
19#1010—23no—
20#1010C2213yescomplemented
21#1010C2211yescharacteristic, complemented
22#1009—21no—
23#1009C2213yescomplemented
24#1009C2211yescharacteristic, complemented
25D3×#41—21no—
26D3×#41—23no—
27D3×#41C2213yes—
28D3×#41C2211yescharacteristic
29D3×#41C2213yescomplemented
30#1007—21no—
31#1007C2213yescomplemented
32C2×D3×Q8—21no—
33C2×D3×Q8C2213yescomplemented
34#1004—23no—
35#1004C2213yescomplemented
36C2×D3×D4—23no—
37C2×D3×D4C2213yes—
38C3×#137C2213yescomplemented
39C3×AGL(1,ℤ/8)C2213yescomplemented
40C3×#135C2213yescomplemented
41C2×C3×SD16C2213yescomplemented
42C3×#131C2211yescharacteristic, complemented
43#951C2213yescomplemented
44#950C2213yescomplemented
45#950C2213yescomplemented
46#949C2213yescomplemented
47#948C2211yescharacteristic, complemented
48#942C2213yescomplemented
49C2×#215C2213yescomplemented
50C2×#214C2213yescomplemented
51#932C2213yescomplemented
52#919C2213yescomplemented
53#918C2216yescomplemented
54D3×Q16C2213yescomplemented
55#916C2213yescomplemented
56#916C2213yescomplemented
57#915C2216yescomplemented
58#914C2216yescomplemented
59D3×SD16C2213yescomplemented
60D3×SD16C2216yescomplemented
61D3×SD16C2211yescharacteristic, complemented
62D3×SD16C2216yescomplemented
63D3×SD16C2216yescomplemented
64#912C2213yescomplemented
65#911C2216yescomplemented
66D3×D8C2213yescomplemented
67#909C2213yescomplemented
68#908C2213yescomplemented
69#907C2213yescomplemented
70D3×M16C2213yescomplemented
71#904C2213yescomplemented
72C2×#204C2213yescomplemented
73#901C2213yescomplemented
74C2×D3×C8C2213yescomplemented
75C2×2−1+4—31no—
76C2×H(5,𝔽2)—31no—
77#577—31no—
78#577—31no—
79#577—33no—
80#577—33no—
81#577—33no—
82#577—33no—
83#577—31no—
84#577D311yescharacteristic, direct product factor
85C2×#137—33no—
86C2×AGL(1,ℤ/8)—33no—
87C2×#135—33no—
88C22×SD16—33no—
89C2×#131—31no—
90C23×D3—23no—
91C3×#41—21no—
92C3×#41—23no—
93C3×#41C2313yes—
94C3×#41C2311yescharacteristic
95C3×#41C2313yescomplemented
96C2×C3×Q8—21no—
97C2×C3×Q8C2313yescomplemented
98C2×C3×D4—23no—
99C2×C3×D4C2313yes—
100C2×#67—26no—
101C2×#67—23no—
102#239—21no—
103#239—23no—
104#239D411yescharacteristic
105#239C2313yes—
106#239C2313yescomplemented
107D3×Q8—21no—
108D3×Q8—23no—
109D3×Q8—23no—
110D3×Q8—23no—
111D3×Q8D411yescharacteristic
112D3×Q8C2313yes—
113D3×Q8C2311yescharacteristic
114D3×Q8C2316yescomplemented
115#237—23no—
116#237—26no—
117#237—23no—
118#237—23no—
119#237—23no—
120#237D413yes—
121#237C2313yes—
122#237C2313yescomplemented
123D3×D4—23no—
124D3×D4—23no—
125D3×D4—23no—
126D3×D4—26no—
127D3×D4—23no—
128D3×D4—23no—
129D3×D4—23no—
130D3×D4D413yes—
131D3×D4C2311yescharacteristic
132D3×D4C2313yes—
133D3×D4C2316yes—
134#235—26no—
135#235—23no—
136#235—23no—
137#235—23no—
138#235—23no—
139#235C2313yes—
140#235C2313yes—
141#235C2313yescomplemented
142C2×D12—23no—
143C2×D12C2313yes—
144C2×C4×D3—23no—
145C2×C4×D3—23no—
146C2×C4×D3—23no—
147C2×C4×D3C2313yes—
148C2×Dic6—23no—
149C2×Dic6—23no—
150C2×Dic6C2313yescomplemented
151C3×Q16C2313yescomplemented
152C3×SD16C2313yescomplemented
153C3×SD16C2316yescomplemented
154C3×SD16C2311yescharacteristic, complemented
155C3×D8C2313yescomplemented
156C3×M16C2313yescomplemented
157C2×C3×C8C2313yescomplemented
158#216C2313yescomplemented
159#216C2313yescomplemented
160#215C2313yescomplemented
161#215C2316yescomplemented
162#215C2311yescharacteristic, complemented
163#214C2316yescomplemented
164#214C2311yescharacteristic, complemented
165#214C2313yescomplemented
166#213C2313yescomplemented
167#213C2313yescomplemented
168#208C2313yescomplemented
169C2×C3⋊C8C2313yescomplemented
170Dic12C2313yescomplemented
171D24C2313yescomplemented
172#204C2313yescomplemented
173#204C2316yescomplemented
174#204C2311yescharacteristic, complemented
175ΓL(1,𝔽25)C2316yescomplemented
176D3×C8C2311yescharacteristic
177D3×C8C2313yes—
178D3×C8C2313yescomplemented
179D3×C8C2313yescomplemented
1802−1+4—31no—
1812−1+4—33no—
1822−1+4—33no—
1832−1+4—61no—
1842−1+4—63no—
1852−1+4C2×D311yescharacteristic, complemented
186H(5,𝔽2)—33no—
187H(5,𝔽2)—31no—
188H(5,𝔽2)—33no—
189H(5,𝔽2)—63no—
190H(5,𝔽2)—61no—
191H(5,𝔽2)C2×D311yescharacteristic, complemented
192C2×#41—31no—
193C2×#41—33no—
194C2×#41—33no—
195C2×#41—63no—
196C2×#41—61no—
197C22×Q8—33no—
198C22×Q8—61no—
199C22×D4—33no—
200C22×D4—63no—
201#137—33no—
202#137—33no—
203#137—36no—
204#137—36no—
205#137—33no—
206#137C2×D313yescomplemented
207AGL(1,ℤ/8)—33no—
208AGL(1,ℤ/8)—33no—
209AGL(1,ℤ/8)—36no—
210AGL(1,ℤ/8)—36no—
211AGL(1,ℤ/8)—33no—
212AGL(1,ℤ/8)C2×D313yescomplemented
213#135—33no—
214#135—33no—
215#135—33no—
216#135—33no—
217#135—33no—
218#135—33no—
219#135—33no—
220#135C2×D313yescomplemented
221C2×Q16—33no—
222C2×SD16—33no—
223C2×SD16—33no—
224C2×SD16—33no—
225C2×SD16—36no—
226C2×SD16—36no—
227C2×SD16—36no—
228C2×SD16—31no—
229C2×SD16—33no—
230C2×SD16C2×D313yescomplemented
231C2×D8—33no—
232#131—31no—
233#131—33no—
234#131—33no—
235#131C2×D311yescharacteristic, complemented
236C2×M16—33no—
237C22×C8—33no—
238C23×C3—23no—
239C22×D3—23no—
240C22×D3—21no—
241C22×D3—23no—
242C22×D3—26no—
243C22×D3—86no—
244C22×D3C2×D413yes—
245C3×Q8—23no—
246C3×Q8C2×D411yescharacteristic
247C3×Q8C2413yes—
248C3×Q8C2411yescharacteristic
249C3×D4—23no—
250C3×D4—23no—
251C3×D4—23no—
252C3×D4C2×D413yes—
253C3×D4C2411yescharacteristic
254C3×D4C2413yes—
255C2×C3×C4—23no—
256C2×C3×C4—23no—
257C2×C3×C4—23no—
258C2×C3×C4C2413yes—
259#67—23no—
260#67—23no—
261#67—23no—
262#67—23no—
263#67—21no—
264#67—23no—
265#67—26no—
266#67C2×D416yes—
267C2×Dic3—23no—
268C2×Dic3—23no—
269C2×Dic3—23no—
270C2×Dic3C2×D413yes—
271D12—23no—
272D12—23no—
273D12—23no—
274D12C2×D413yes—
275D12C2413yes—
276D12C2411yescharacteristic
277C4×D3—21no—
278C4×D3—23no—
279C4×D3—26no—
280C4×D3—23no—
281C4×D3—23no—
282C4×D3—23no—
283C4×D3—23no—
284C4×D3C2×D413yes—
285C4×D3C2×D413yes—
286C4×D3C2411yescharacteristic
287C4×D3C2413yes—
288Dic6—23no—
289Dic6—23no—
290Dic6—26no—
291Dic6—21no—
292Dic6C2×D413yes—
293Dic6C2413yes—
294Dic6C2411yescharacteristic
295C3×C8C2411yescharacteristic
296C3×C8C2413yes—
297C3⋊C8C2411yescharacteristic
298C3⋊C8C2413yes—
299C24—63no—
300#41—21no—
301#41—23no—
302#41—31no—
303#41—33no—
304#41—33no—
305#41—33no—
306#41—33no—
307#41—33no—
308#41—33no—
309#41—33no—
310#41—33no—
311#41—63no—
312#41—63no—
313#41—66no—
314#41—63no—
315#41—63no—
316#41—63no—
317#41—63no—
318#41—66no—
319#41—63no—
320#41—63no—
321#41—61no—
322#41—63no—
323#41C22×D313yes—
324#41C22×D311yescharacteristic
325#41C22×D313yescomplemented
326C2×Q8—21no—
327C2×Q8—31no—
328C2×Q8—33no—
329C2×Q8—36no—
330C2×Q8—31no—
331C2×Q8—33no—
332C2×Q8—63no—
333C2×Q8—63no—
334C2×Q8—63no—
335C2×Q8—63no—
336C2×Q8—63no—
337C2×Q8—61no—
338C2×Q8C22×D313yescomplemented
339C2×D4—23no—
340C2×D4—33no—
341C2×D4—31no—
342C2×D4—33no—
343C2×D4—36no—
344C2×D4—33no—
345C2×D4—63no—
346C2×D4—63no—
347C2×D4—63no—
348C2×D4—63no—
349C2×D4—66no—
350C2×D4—63no—
351C2×D4—63no—
352C2×D4—63no—
353C2×D4—66no—
354C2×D4—63no—
355C2×D4C22×D313yes—
356C22×C4—33no—
357C22×C4—63no—
358C22×C4—63no—
359C22×C4—63no—
360Q16—33no—
361Q16—33no—
362Q16—33no—
363Q16C22×D313yescomplemented
364SD16—33no—
365SD16—36no—
366SD16—31no—
367SD16—33no—
368SD16—36no—
369SD16—31no—
370SD16—33no—
371SD16—36no—
372SD16—31no—
373SD16C22×D313yescomplemented
374SD16C22×D316yescomplemented
375SD16C22×D311yescharacteristic, complemented
376D8—33no—
377D8—33no—
378D8—33no—
379D8C22×D313yescomplemented
380M16—33no—
381M16—36no—
382M16C22×D313yescomplemented
383C2×C8—33no—
384C2×C8—31no—
385C2×C8—33no—
386C2×C8—33no—
387C2×C8—33no—
388C2×C8C22×D313yescomplemented
389C22×C3—21no—
390C22×C3—23no—
391C22×C3—83no—
392C22×C3C22×D413yes—
393C2×D3—26no—
394C2×D3—21no—
395C2×D3—23no—
396C2×D3—42no—
397C2×D3—46no—
398C2×D3—86no—
399C2×D3—83no—
400C2×D3C22×D413yes—
401C2×D3C22×D411yescharacteristic
402C3×C4—23no—
403C3×C4—21no—
404C3×C4C22×D413yes—
405C3×C4C2511yescharacteristic
406Dic3—23no—
407Dic3—21no—
408Dic3C22×D411yescharacteristic
409Dic3C22×D413yes—
410C23—23no—
411C23—33no—
412C23—61no—
413C23—63no—
414C23—66no—
415C23—63no—
416C23—246no—
417Q8—23no—
418Q8—33no—
419Q8—33no—
420Q8—31no—
421Q8—63no—
422Q8—66no—
423Q8—61no—
424Q8—63no—
425Q8D3×D411yescharacteristic
426Q8C23×D313yes—
427Q8C23×D311yescharacteristic
428D4—23no—
429D4—23no—
430D4—23no—
431D4—36no—
432D4—33no—
433D4—31no—
434D4—33no—
435D4—63no—
436D4—63no—
437D4—63no—
438D4—66no—
439D4—61no—
440D4—63no—
441D4—63no—
442D4—63no—
443D4—63no—
444D4—63no—
445D4D3×D413yes—
446D4C23×D311yescharacteristic
447D4C23×D313yes—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—23no—
451C2×C4—33no—
452C2×C4—33no—
453C2×C4—33no—
454C2×C4—31no—
455C2×C4—33no—
456C2×C4—63no—
457C2×C4—63no—
458C2×C4—61no—
459C2×C4—63no—
460C2×C4—66no—
461C2×C4—63no—
462C2×C4—63no—
463C2×C4—63no—
464C2×C4—63no—
465C2×C4—63no—
466C2×C4C23×D313yes—
467C8—31no—
468C8—33no—
469C8C23×D311yescharacteristic
470C8C23×D313yes—
471C2×C3—23no—
472C2×C3—43no—
473C2×C3—41no—
474C2×C3C23×D411yescharacteristic
475D3—23no—
476D3—41no—
477D3—43no—
478D3#57712yesdirect product factor
479C22—21no—
480C22—23no—
481C22—31no—
482C22—33no—
483C22—66no—
484C22—61no—
485C22—63no—
486C22—83no—
487C22—126no—
488C22—122no—
489C22—243no—
490C22—246no—
491C22C2×D3×D413yes—
492C4—21no—
493C4—23no—
494C4—31no—
495C4—33no—
496C4—61no—
497C4—63no—
498C4C2×D3×D413yes—
499C4C24×D311yescharacteristic
500C3C2×#57711yescharacteristic, complemented, Sylow
501C2—23no—
502C2—32no—
503C2—43no—
504C2—41no—
505C2—63no—
506C2—123no—
507C2—121no—
508C2C22×D3×D411yescharacteristic
subgroupG/H
centerC2C22×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC3×#577C2
socleC2×C3C23×D4
hypercenter#577D3
hypocenterC3C2×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. D3×2−1+4C2semidirect
2. D3×H(5,𝔽2)C2semidirect
3. C3×#577C2semidirect
4. #5837C2semidirect
5. #5836C2semidirect
6. #5778C2semidirect
7. D3×#137C2semidirect
8. #5775C2semidirect
9. D3×AGL(1,ℤ/8)C2semidirect
10. #5770C2semidirect
11. D3×#135C2semidirect
12. #5762C2semidirect
13. C2×D3×SD16C2semidirect
14. #5751C2semidirect
15. D3×#131C2semidirect
17. C3×2−1+4C22semidirect
18. C3×H(5,𝔽2)C22semidirect
20. #1010C22semidirect
21. #1010C22semidirect
23. #1009C22semidirect
24. #1009C22semidirect
29. D3×#41C22semidirect
31. #1007C22semidirect
33. C2×D3×Q8C22semidirect
35. #1004C22semidirect
38. C3×#137C22semidirect
39. C3×AGL(1,ℤ/8)C22semidirect
40. C3×#135C22semidirect
41. C2×C3×SD16C22semidirect
42. C3×#131C22semidirect
43. #951C22semidirect
44. #950C22semidirect
45. #950C22semidirect
46. #949C22semidirect
47. #948C22semidirect
48. #942C22semidirect
49. C2×#215C22semidirect
50. C2×#214C22semidirect
51. #932C22semidirect
52. #919C22semidirect
53. #918C22semidirect
54. D3×Q16C22semidirect
55. #916C22semidirect
56. #916C22semidirect
57. #915C22semidirect
58. #914C22semidirect
59. D3×SD16C22semidirect
60. D3×SD16C22semidirect
61. D3×SD16C22semidirect
62. D3×SD16C22semidirect
63. D3×SD16C22semidirect
64. #912C22semidirect
65. #911C22semidirect
66. D3×D8C22semidirect
67. #909C22semidirect
68. #908C22semidirect
69. #907C22semidirect
70. D3×M16C22semidirect
71. #904C22semidirect
72. C2×#204C22semidirect
73. #901C22semidirect
74. C2×D3×C8C22semidirect
84. #577D3direct (N × H)
95. C3×#41C23semidirect
97. C2×C3×Q8C23semidirect
106. #239C23semidirect
114. D3×Q8C23semidirect
122. #237C23semidirect
141. #235C23semidirect
150. C2×Dic6C23semidirect
151. C3×Q16C23semidirect
152. C3×SD16C23semidirect
153. C3×SD16C23semidirect
154. C3×SD16C23semidirect
155. C3×D8C23semidirect
156. C3×M16C23semidirect
157. C2×C3×C8C23semidirect
158. #216C23semidirect
159. #216C23semidirect
160. #215C23semidirect
161. #215C23semidirect
162. #215C23semidirect
163. #214C23semidirect
164. #214C23semidirect
165. #214C23semidirect
166. #213C23semidirect
167. #213C23semidirect
168. #208C23semidirect
169. C2×C3⋊C8C23semidirect
170. Dic12C23semidirect
171. D24C23semidirect
172. #204C23semidirect
173. #204C23semidirect
174. #204C23semidirect
175. ΓL(1,𝔽25)C23semidirect
178. D3×C8C23semidirect
179. D3×C8C23semidirect
185. 2−1+4C2×D3semidirect
191. H(5,𝔽2)C2×D3semidirect
206. #137C2×D3semidirect
212. AGL(1,ℤ/8)C2×D3semidirect
220. #135C2×D3semidirect
230. C2×SD16C2×D3semidirect
235. #131C2×D3semidirect
325. #41C22×D3semidirect
338. C2×Q8C22×D3semidirect
363. Q16C22×D3semidirect
373. SD16C22×D3semidirect
374. SD16C22×D3semidirect
375. SD16C22×D3semidirect
379. D8C22×D3semidirect
382. M16C22×D3semidirect
388. C2×C8C22×D3semidirect
478. D3#577direct (N × H)
500. C3C2×#577semidirect

Maximal subgroups

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

subgroupindex
D3×2−1+42
D3×H(5,𝔽2)2
C3×#5772
#58372
#58362
#57782
D3×#1372
#57752
D3×AGL(1,ℤ/8)2
#57702
D3×#1352
#57622
C2×D3×SD162
#57512
D3×#1312
C2×#5773