Finite Group Databasebeta

← C2×C3×#64123 #13179510

Subgroups — C2×C3×#64123 #13179510

Order 1728.

2060 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic, complemented
2C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C33×SD16C211yesmaximal, characteristic, complemented
4C2×C3×#3755C211yesmaximal, characteristic, complemented
5C2×C3×#3753C211yesmaximal, characteristic, complemented
6C3×#64123C218yesmaximal, direct product factor
7C2×C3×#3745C211yesmaximal, characteristic, complemented
8C2×C3×#3743C211yesmaximal, characteristic, complemented
9C2×#64123C311yesmaximal, characteristic, direct product factor
10C2×C3×#914—34nomaximal
11C23×C3×C3⋊D3—21no—
12C2×C33×Q8C2211yescharacteristic, complemented
13C2×C33×D4C2211yescharacteristic
14C2×C3×#638—21no—
15C3×#3832—22no—
16C3×#3832C2214yescomplemented
17C3×D4×C3⋊D3—22no—
18C3×D4×C3⋊D3C2214yescomplemented
19C2×C3×C4⋊(C3⋊D3)—21no—
20C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
22C2×C3×C4×C3⋊D3—21no—
23C2×C3×C4×C3⋊D3C2211yescharacteristic
24C33×SD16C2214yescomplemented
25C2×C33×C8C2211yescharacteristic, complemented
26C3×#3755C2214yescomplemented
27C3×#3753C2214yescomplemented
28C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
29C3×#3745C2214yescomplemented
30C3×#3743C2214yescomplemented
31C2×#3832C2×C311yescharacteristic, complemented
32C2×D4×C3⋊D3C2×C311yescharacteristic, complemented
33C2×C3×#239—34no—
34C2×C3×D3×D4—34no—
35C2×C32×SD16—24no—
36C2×C32×SD16D314yescomplemented
37C2×C32×SD16C2×C311yescharacteristic, complemented
38C2×#3755C2×C311yescharacteristic, complemented
39C2×#3753C2×C311yescharacteristic, complemented
40#64123C2×C318yesdirect product factor
41C2×#3745C2×C311yescharacteristic, complemented
42C2×#3743C2×C311yescharacteristic, complemented
43C2×C3×#215—34no—
44C2×C3×#213—34no—
45C3×#914—332no—
46C2×C3×D24—34no—
47C2×C3×ΓL(1,𝔽25)—34no—
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3—24no—
52C22×C3×C3⋊D3—44no—
53C22×C3×C3⋊D3D411yescharacteristic
54C33×Q8—21no—
55C33×Q8C2312yescomplemented
56C33×D4—21no—
57C33×D4C2312yes—
58C2×C33×C4—21no—
59C2×C33×C4C2311yescharacteristic
60C3×#638—24no—
61C2×C3×C3⋊Dic3D411yescharacteristic
62C3×C4⋊(C3⋊D3)—24no—
63C3×C4⋊(C3⋊D3)—21no—
64C3×C4⋊(C3⋊D3)—21no—
65C3×C4⋊(C3⋊D3)C2312yes—
66C3×C4⋊(C3⋊D3)C2312yescomplemented
67C3×C4×C3⋊D3—24no—
68C3×C4×C3⋊D3D412yes—
69C3×C4×C3⋊D3C2312yes—
70C33×C8C2312yescomplemented
71C3×C3⋊(C3⋊C8)C2312yescomplemented
72C2×C3×AGL(1,ℤ/8)—91no—
73C2×#914—34no—
74C23×C3⋊D3—21no—
75C23×C3×D3—64no—
76C2×C32×Q8—24no—
77C2×C32×Q8C2×D314yescomplemented
78C2×C32×Q8C22×C311yescharacteristic, complemented
79C2×C32×D4—24no—
80C2×C32×D4C2×D314yes—
81C2×C32×D4C22×C311yescharacteristic
82C2×#638—21no—
83#3832—22no—
84#3832C22×C314yescomplemented
85D4×C3⋊D3—22no—
86D4×C3⋊D3C22×C314yescomplemented
87C2×C4⋊(C3⋊D3)—21no—
88C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
89C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
90C2×C4×C3⋊D3—21no—
91C2×C4×C3⋊D3C22×C311yescharacteristic
92C2×C3×#67—64no—
93C3×#239—316no—
94C3×#239—68no—
95C3×D3×D4—316no—
96C3×D3×D4—68no—
97C2×C3×D12—34no—
98C2×C3×D12—34no—
99C2×C3×D12—64no—
100C2×C3×C4×D3—34no—
101C2×C3×C4×D3—64no—
102C32×SD16—216no—
103C32×SD16C2×D3116yescomplemented
104C32×SD16C22×C314yescomplemented
105C2×C32×C8—24no—
106C2×C32×C8C2×D314yescomplemented
107C2×C32×C8C22×C311yescharacteristic, complemented
108#3755C22×C314yescomplemented
109#3753C22×C314yescomplemented
110C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
111#3745C22×C314yescomplemented
112#3743C22×C314yescomplemented
113C3×#215—316no—
114C3×#213—316no—
115C2×C3×C3⋊C8—34no—
116C3×D24—316no—
117C3×ΓL(1,𝔽25)—316no—
118C22×C33—22no—
119C22×C33—42no—
120C22×C33C2×D411yescharacteristic
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—48no—
125C2×C3×C3⋊D3—42no—
126C2×C3×C3⋊D3—42no—
127C2×C3×C3⋊D3C2×D412yes—
128C33×C4—22no—
129C33×C4C2×D411yescharacteristic
130C33×C4C2411yescharacteristic
131C3×C3⋊Dic3C2×D412yes—
132C2×C3×#41—91no—
133C22×C3×D4—91no—
134C2×#239—34no—
135C2×D3×D4—34no—
136C3×AGL(1,ℤ/8)—98no—
137C2×C3×SD16—24no—
138C2×C3×SD16—91no—
139C2×C3×SD16C3×D314yescomplemented
140C2×C3×SD16C3⋊D311yescharacteristic, complemented
141C2×C3×D8—91no—
142C2×C3×D8—91no—
143C2×C3×M16—91no—
144C2×#215—34no—
145C2×#213—34no—
146#914—332no—
147C2×D24—34no—
148C2×ΓL(1,𝔽25)—34no—
149C23×C32—21no—
150C23×C32—24no—
151C23×C32—44no—
152C22×C3⋊D3—21no—
153C22×C3⋊D3—21no—
154C22×C3⋊D3—24no—
155C22×C3⋊D3—44no—
156C22×C3⋊D3C3×D411yescharacteristic
157C22×C3×D3—34no—
158C22×C3×D3—616no—
159C22×C3×D3—64no—
160C22×C3×D3—64no—
161C22×C3×D3—1216no—
162C32×Q8—21no—
163C32×Q8—28no—
164C32×Q8—24no—
165C32×Q8—44no—
166C32×Q8C22×D318yescomplemented
167C32×Q8C23×C312yescomplemented
168C32×D4—21no—
169C32×D4—28no—
170C32×D4—24no—
171C32×D4—44no—
172C32×D4C22×D318yes—
173C32×D4C23×C312yes—
174C2×C32×C4—21no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—44no—
178C2×C32×C4C22×D314yes—
179C2×C32×C4C23×C311yescharacteristic
180#638—24no—
181C2×C3⋊Dic3C3×D411yescharacteristic
182C4⋊(C3⋊D3)—24no—
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)C23×C312yes—
186C4⋊(C3⋊D3)C23×C312yescomplemented
187C4×C3⋊D3—24no—
188C4×C3⋊D3C3×D412yes—
189C4×C3⋊D3C23×C312yes—
190C3×#67—616no—
191C2×C3×Dic3—34no—
192C3×D12—38no—
193C3×D12—38no—
194C3×D12—616no—
195C3×D12—64no—
196C3×D12—64no—
197C3×C4×D3—38no—
198C3×C4×D3—38no—
199C3×C4×D3—616no—
200C32×C8—28no—
201C32×C8C22×D318yescomplemented
202C32×C8C23×C312yescomplemented
203C3⋊(C3⋊C8)C23×C312yescomplemented
204C3×C3⋊C8—38no—
205C2×AGL(1,ℤ/8)—91noSylow
206C2×C33—42no—
207C2×C33AGL(1,ℤ/8)12yescomplemented
208C2×C33C22×D411yescharacteristic
209C3×C3⋊D3—22no—
210C3×C3⋊D3—42no—
211C3×C3⋊D3—42no—
212C24×C3—181no—
213C23×D3—64no—
214C3×#41—94no—
215C3×#41—182no—
216C2×C3×Q8—24no—
217C2×C3×Q8C2×C3×D314yescomplemented
218C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
219C2×C3×D4—24no—
220C2×C3×D4—94no—
221C2×C3×D4—91no—
222C2×C3×D4—91no—
223C2×C3×D4—181no—
224C2×C3×D4—182no—
225C2×C3×D4—181no—
226C2×C3×D4C2×C3×D314yes—
227C2×C3×D4C2×C3⋊D311yescharacteristic
228C22×C3×C4—91no—
229C22×C3×C4—181no—
230C2×#67—64no—
231#239—316no—
232#239—68no—
233D3×D4—316no—
234D3×D4—68no—
235C2×D12—34no—
236C2×D12—34no—
237C2×D12—64no—
238C2×C4×D3—34no—
239C2×C4×D3—64no—
240C3×SD16—216no—
241C3×SD16—94no—
242C3×SD16C2×C3×D3116yescomplemented
243C3×SD16C2×C3⋊D314yescomplemented
244C3×D8—94no—
245C3×D8—94no—
246C3×M16—94no—
247C2×C3×C8—24no—
248C2×C3×C8—91no—
249C2×C3×C8C2×C3×D314yescomplemented
250C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
251#215—316no—
252#213—316no—
253C2×C3⋊C8—34no—
254D24—316no—
255ΓL(1,𝔽25)—316no—
256C22×C32—22no—
257C22×C32—24no—
258C22×C32—28no—
259C22×C32—48no—
260C22×C32—42no—
261C22×C32—48no—
262C22×C32—88no—
263C22×C32D3×D414yes—
264C22×C32C2×C3×D411yescharacteristic
265C2×C3⋊D3—22no—
266C2×C3⋊D3—22no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—48no—
269C2×C3⋊D3—42no—
270C2×C3⋊D3—42no—
271C2×C3⋊D3C2×C3×D412yes—
272C2×C3×D3—38no—
273C2×C3×D3—68no—
274C2×C3×D3—68no—
275C2×C3×D3—68no—
276C2×C3×D3—1232no—
277C2×C3×D3—128no—
278C2×C3×D3—128no—
279C32×C4—22no—
280C32×C4—24no—
281C32×C4—24no—
282C32×C4—28no—
283C32×C4—48no—
284C32×C4D3×D414yes—
285C32×C4C2×C3×D411yescharacteristic
286C32×C4C23×D314yes—
287C32×C4C24×C311yescharacteristic
288C3⋊Dic3C2×C3×D412yes—
289C3×Dic3—38no—
290C2×#41—91no—
291C22×D4—91no—
292AGL(1,ℤ/8)—98no—
293C2×SD16—91no—
294C2×SD16C3×C3⋊D311yescharacteristic, complemented
295C2×D8—91no—
296C2×D8—91no—
297C2×M16—91no—
298C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
299C23×C3—24no—
300C23×C3—21no—
301C23×C3—44no—
302C23×C3—91no—
303C23×C3—184no—
304C23×C3—181no—
305C23×C3—181no—
306C23×C3—364no—
307C22×D3—34no—
308C22×D3—616no—
309C22×D3—64no—
310C22×D3—64no—
311C22×D3—1216no—
312C3×Q8—24no—
313C3×Q8—21no—
314C3×Q8—28no—
315C3×Q8—44no—
316C3×Q8C22×C3×D318yescomplemented
317C3×Q8C22×C3⋊D312yescomplemented
318C3×D4—24no—
319C3×D4—21no—
320C3×D4—28no—
321C3×D4—44no—
322C3×D4—92no—
323C3×D4—92no—
324C3×D4—184no—
325C3×D4—184no—
326C3×D4—181no—
327C3×D4—181no—
328C3×D4C22×C3×D318yes—
329C3×D4C22×C3⋊D312yes—
330C2×C3×C4—24no—
331C2×C3×C4—21no—
332C2×C3×C4—24no—
333C2×C3×C4—44no—
334C2×C3×C4—91no—
335C2×C3×C4—92no—
336C2×C3×C4—92no—
337C2×C3×C4—184no—
338C2×C3×C4C22×C3×D314yes—
339C2×C3×C4C22×C3⋊D311yescharacteristic
340#67—616no—
341C2×Dic3—34no—
342D12—38no—
343D12—38no—
344D12—616no—
345D12—64no—
346D12—64no—
347C4×D3—38no—
348C4×D3—38no—
349C4×D3—616no—
350C3×C8—28no—
351C3×C8—92no—
352C3×C8C22×C3×D318yescomplemented
353C3×C8C22×C3⋊D312yescomplemented
354C3⋊C8—38no—
355C2×C32—28no—
356C2×C32—24no—
357C2×C32—42no—
358C2×C32—48no—
359C2×C32—88no—
360C2×C32#91418yescomplemented
361C2×C32C3×AGL(1,ℤ/8)12yescomplemented
362C2×C32C2×D3×D414yes—
363C2×C32C22×C3×D411yescharacteristic
364C3⋊D3—22no—
365C3⋊D3—42no—
366C3⋊D3—42no—
367C3×D3—68no—
368C3×D3—128no—
369C3×D3—128no—
370C24—181no—
371#41—94no—
372#41—182no—
373C2×Q8C2×C3×C3⋊D311yescharacteristic, complemented
374C2×D4—94no—
375C2×D4—91no—
376C2×D4—91no—
377C2×D4—181no—
378C2×D4—182no—
379C2×D4—181no—
380C2×D4C2×C3×C3⋊D311yescharacteristic
381C22×C4—91no—
382C22×C4—181no—
383SD16—94no—
384SD16C2×C3×C3⋊D314yescomplemented
385D8—94no—
386D8—94no—
387M16—94no—
388C2×C8—91no—
389C2×C8C2×C3×C3⋊D311yescharacteristic, complemented
390C22×C3—28no—
391C22×C3—22no—
392C22×C3—24no—
393C22×C3—48no—
394C22×C3—48no—
395C22×C3—42no—
396C22×C3—88no—
397C22×C3—92no—
398C22×C3—182no—
399C22×C3—182no—
400C22×C3—182no—
401C22×C3—368no—
402C22×C3—362no—
403C22×C3—362no—
404C22×C3C3×D3×D414yes—
405C22×C3D4×C3⋊D311yescharacteristic
406C2×D3—38no—
407C2×D3—68no—
408C2×D3—68no—
409C2×D3—68no—
410C2×D3—1232no—
411C2×D3—128no—
412C2×D3—128no—
413C3×C4—28no—
414C3×C4—22no—
415C3×C4—24no—
416C3×C4—24no—
417C3×C4—48no—
418C3×C4—92no—
419C3×C4C3×D3×D414yes—
420C3×C4D4×C3⋊D311yescharacteristic
421C3×C4C23×C3×D314yes—
422C3×C4C23×C3⋊D311yescharacteristic
423Dic3—38no—
424C32—24no—
425C32C2×#91414yescomplemented
426C32C2×C3×AGL(1,ℤ/8)11yescharacteristic, complemented
427C23—21no—
428C23—91no—
429C23—184no—
430C23—181no—
431C23—181no—
432C23—364no—
433Q8—21no—
434Q8C22×C3×C3⋊D312yescomplemented
435D4—21no—
436D4—92no—
437D4—92no—
438D4—184no—
439D4—184no—
440D4—181no—
441D4—181no—
442D4C22×C3×C3⋊D312yes—
443C2×C4—21no—
444C2×C4—91no—
445C2×C4—92no—
446C2×C4—92no—
447C2×C4—184no—
448C2×C4C22×C3×C3⋊D311yescharacteristic
449C8—92no—
450C8C22×C3×C3⋊D312yescomplemented
451C2×C3—28no—
452C2×C3—24no—
453C2×C3—48no—
454C2×C3—42no—
455C2×C3—88no—
456C2×C3—182no—
457C2×C3—362no—
458C2×C3—362no—
459C2×C3C3×#91418yescomplemented
460C2×C3#6412312yesdirect product factor
461C2×C3C2×C3×D3×D414yes—
462C2×C3C2×D4×C3⋊D311yescharacteristic
463D3—68no—
464D3—128no—
465D3—128no—
466C22—22no—
467C22—42no—
468C22—92no—
469C22—182no—
470C22—182no—
471C22—182no—
472C22—368no—
473C22—362no—
474C22—362no—
475C22C3×D4×C3⋊D311yescharacteristic
476C4—22no—
477C4—92no—
478C4C3×D4×C3⋊D311yescharacteristic
479C4C23×C3×C3⋊D311yescharacteristic
480C3—24no—
481C3C2×C3×#91414yescomplemented
482C3C2×#6412311yescharacteristic, direct product factor
483C2—42no—
484C2—182no—
485C2—362no—
486C2—362no—
487C2C3×#6412312yesdirect product factor
488C2C2×C3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C2×C3×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×C3×D4×C3⋊D3C2semidirect
3. C2×C33×SD16C2semidirect
4. C2×C3×#3755C2semidirect
5. C2×C3×#3753C2semidirect
6. C3×#64123C2direct (N × H)
7. C2×C3×#3745C2semidirect
8. C2×C3×#3743C2semidirect
9. C2×#64123C3direct (N × H)
12. C2×C33×Q8C22semidirect
16. C3×#3832C22semidirect
18. C3×D4×C3⋊D3C22semidirect
21. C2×C3×C4⋊(C3⋊D3)C22semidirect
24. C33×SD16C22semidirect
25. C2×C33×C8C22semidirect
26. C3×#3755C22semidirect
27. C3×#3753C22semidirect
28. C2×C3×C3⋊(C3⋊C8)C22semidirect
29. C3×#3745C22semidirect
30. C3×#3743C22semidirect
31. C2×#3832C2×C3semidirect
32. C2×D4×C3⋊D3C2×C3semidirect
36. C2×C32×SD16D3semidirect
37. C2×C32×SD16C2×C3semidirect
38. C2×#3755C2×C3semidirect
39. C2×#3753C2×C3semidirect
40. #64123C2×C3direct (N × H)
41. C2×#3745C2×C3semidirect
42. C2×#3743C2×C3semidirect
55. C33×Q8C23semidirect
66. C3×C4⋊(C3⋊D3)C23semidirect
70. C33×C8C23semidirect
71. C3×C3⋊(C3⋊C8)C23semidirect
77. C2×C32×Q8C2×D3semidirect
78. C2×C32×Q8C22×C3semidirect
84. #3832C22×C3semidirect
86. D4×C3⋊D3C22×C3semidirect
89. C2×C4⋊(C3⋊D3)C22×C3semidirect
103. C32×SD16C2×D3semidirect
104. C32×SD16C22×C3semidirect
106. C2×C32×C8C2×D3semidirect
107. C2×C32×C8C22×C3semidirect
108. #3755C22×C3semidirect
109. #3753C22×C3semidirect
110. C2×C3⋊(C3⋊C8)C22×C3semidirect
111. #3745C22×C3semidirect
112. #3743C22×C3semidirect
139. C2×C3×SD16C3×D3semidirect
140. C2×C3×SD16C3⋊D3semidirect
166. C32×Q8C22×D3semidirect
167. C32×Q8C23×C3semidirect
186. C4⋊(C3⋊D3)C23×C3semidirect
201. C32×C8C22×D3semidirect
202. C32×C8C23×C3semidirect
203. C3⋊(C3⋊C8)C23×C3semidirect
207. C2×C33AGL(1,ℤ/8)semidirect
217. C2×C3×Q8C2×C3×D3semidirect
218. C2×C3×Q8C2×C3⋊D3semidirect
242. C3×SD16C2×C3×D3semidirect
243. C3×SD16C2×C3⋊D3semidirect
249. C2×C3×C8C2×C3×D3semidirect
250. C2×C3×C8C2×C3⋊D3semidirect
294. C2×SD16C3×C3⋊D3semidirect
298. C33C2×AGL(1,ℤ/8)semidirect
316. C3×Q8C22×C3×D3semidirect
317. C3×Q8C22×C3⋊D3semidirect
352. C3×C8C22×C3×D3semidirect
353. C3×C8C22×C3⋊D3semidirect
360. C2×C32#914semidirect
361. C2×C32C3×AGL(1,ℤ/8)semidirect
373. C2×Q8C2×C3×C3⋊D3semidirect
384. SD16C2×C3×C3⋊D3semidirect
389. C2×C8C2×C3×C3⋊D3semidirect
425. C32C2×#914semidirect
426. C32C2×C3×AGL(1,ℤ/8)semidirect
434. Q8C22×C3×C3⋊D3semidirect
450. C8C22×C3×C3⋊D3semidirect
459. C2×C3C3×#914semidirect
460. C2×C3#64123direct (N × H)
481. C3C2×C3×#914semidirect
482. C3C2×#64123direct (N × H)
487. C2C3×#64123direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×D4×C3⋊D32
C2×C33×SD162
C2×C3×#37552
C2×C3×#37532
C3×#641232
C2×C3×#37452
C2×C3×#37432
C2×#641233
C2×C3×#9143