Finite Group Databasebeta

← C2×C3×#64120 #13179506

Subgroups — C2×C3×#64120 #13179506

Order 1728.

2092 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C33×D8C211yesmaximal, characteristic, complemented
4C2×C3×#3754C211yesmaximal, characteristic, complemented
5C2×C3×#3753C211yesmaximal, characteristic, complemented
6C3×#64120C218yesmaximal, direct product factor
7C2×C3×#3744C211yesmaximal, characteristic, complemented
8C2×C3×#3743C211yesmaximal, characteristic, complemented
9C2×#64120C311yesmaximal, characteristic, direct product factor
10C2×C3×#911—34nomaximal
11C23×C3×C3⋊D3—21no—
12C2×C33×D4C2211yescharacteristic
13C2×C33×D4C2211yescharacteristic, complemented
14C2×C3×#638—21no—
15C2×C3×#638—21no—
16C22×C3×C3⋊Dic3—21no—
17C3×#3830—22no—
18C3×#3830C2214yescomplemented
19C3×D4×C3⋊D3—22no—
20C3×D4×C3⋊D3C2214yescomplemented
21C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C3×C4×C3⋊D3C2211yescharacteristic
23C2×C3×#634C2211yescharacteristic, complemented
24C33×D8C2214yescomplemented
25C2×C33×C8C2211yescharacteristic, complemented
26C3×#3754C2214yescomplemented
27C3×#3753C2214yescomplemented
28C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
29C3×#3744C2214yescomplemented
30C3×#3743C2214yescomplemented
31C2×#3830C2×C311yescharacteristic, complemented
32C2×D4×C3⋊D3C2×C311yescharacteristic, complemented
33C2×C3×#237—34no—
34C2×C3×D3×D4—34no—
35C2×C32×D8—24no—
36C2×C32×D8D314yescomplemented
37C2×C32×D8C2×C311yescharacteristic, complemented
38C2×#3754C2×C311yescharacteristic, complemented
39C2×#3753C2×C311yescharacteristic, complemented
40#64120C2×C318yesdirect product factor
41C2×#3744C2×C311yescharacteristic, complemented
42C2×#3743C2×C311yescharacteristic, complemented
43C2×C3×#214—34no—
44C2×C3×#213—34no—
45C3×#911—332no—
46C2×C3×#204—34no—
47C2×C3×ΓL(1,𝔽25)—34no—
48C23×C33—21no—
49C23×C33—21no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3—24no—
52C22×C3×C3⋊D3—44no—
53C22×C3×C3⋊D3D411yescharacteristic
54C33×D4—21no—
55C33×D4—21no—
56C33×D4C2312yes—
57C33×D4C2312yescomplemented
58C2×C33×C4C2311yescharacteristic
59C3×#638—24no—
60C3×#638—24no—
61C2×C3×C3⋊Dic3—21no—
62C2×C3×C3⋊Dic3—24no—
63C2×C3×C3⋊Dic3D411yescharacteristic
64C3×C4⋊(C3⋊D3)—21no—
65C3×C4⋊(C3⋊D3)C2312yes—
66C3×C4×C3⋊D3D412yes—
67C3×C4×C3⋊D3C2312yes—
68C3×#634—21no—
69C3×#634C2312yescomplemented
70C33×C8C2312yescomplemented
71C3×C3⋊(C3⋊C8)C2312yescomplemented
72C2×C3×AGL(1,ℤ/8)—91no—
73C2×#911—34no—
74C23×C3⋊D3—21no—
75C23×C3×D3—64no—
76C2×C32×D4—24no—
77C2×C32×D4—24no—
78C2×C32×D4C2×D314yes—
79C2×C32×D4C2×D314yescomplemented
80C2×C32×D4C22×C311yescharacteristic
81C2×C32×D4C22×C311yescharacteristic, complemented
82C2×#638—21no—
83C2×#638—21no—
84C22×C3⋊Dic3—21no—
85#3830—22no—
86#3830C22×C314yescomplemented
87D4×C3⋊D3—22no—
88D4×C3⋊D3C22×C314yescomplemented
89C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
90C2×C4×C3⋊D3C22×C311yescharacteristic
91C2×#634C22×C311yescharacteristic, complemented
92C2×C3×#67—64no—
93C2×C3×#67—64no—
94C22×C3×Dic3—64no—
95C3×#237—316no—
96C3×#237—68no—
97C3×D3×D4—316no—
98C3×D3×D4—68no—
99C2×C3×D12—34no—
100C2×C3×C4×D3—34no—
101C2×C3×Dic6—34no—
102C32×D8—216no—
103C32×D8C2×D3116yescomplemented
104C32×D8C22×C314yescomplemented
105C2×C32×C8—24no—
106C2×C32×C8C2×D314yescomplemented
107C2×C32×C8C22×C311yescharacteristic, complemented
108#3754C22×C314yescomplemented
109#3753C22×C314yescomplemented
110C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
111#3744C22×C314yescomplemented
112#3743C22×C314yescomplemented
113C3×#214—316no—
114C3×#213—316no—
115C2×C3×C3⋊C8—34no—
116C3×#204—316no—
117C3×ΓL(1,𝔽25)—316no—
118C22×C33—22no—
119C22×C33—22no—
120C22×C33—42no—
121C22×C33—42no—
122C22×C33C2×D411yescharacteristic
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3—48no—
126C2×C3×C3⋊D3—42no—
127C2×C3×C3⋊D3C2×D412yes—
128C33×C4C2×D411yescharacteristic
129C33×C4C2411yescharacteristic
130C3×C3⋊Dic3—22no—
131C3×C3⋊Dic3C2×D412yes—
132C2×C3×#41—91no—
133C22×C3×D4—91no—
134C2×#237—34no—
135C2×D3×D4—34no—
136C3×AGL(1,ℤ/8)—98no—
137C2×C3×SD16—91no—
138C2×C3×SD16—91no—
139C2×C3×D8—24no—
140C2×C3×D8—91no—
141C2×C3×D8C3×D314yescomplemented
142C2×C3×D8C3⋊D311yescharacteristic, complemented
143C2×C3×M16—91no—
144C2×#214—34no—
145C2×#213—34no—
146#911—332no—
147C2×#204—34no—
148C2×ΓL(1,𝔽25)—34no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32—24no—
152C23×C32—24no—
153C23×C32—44no—
154C23×C32—44no—
155C22×C3⋊D3—21no—
156C22×C3⋊D3—24no—
157C22×C3⋊D3—44no—
158C22×C3⋊D3C3×D411yescharacteristic
159C22×C3×D3—34no—
160C22×C3×D3—616no—
161C22×C3×D3—64no—
162C22×C3×D3—1216no—
163C32×D4—21no—
164C32×D4—21no—
165C32×D4—28no—
166C32×D4—28no—
167C32×D4—24no—
168C32×D4—24no—
169C32×D4—44no—
170C32×D4—44no—
171C32×D4C22×D318yes—
172C32×D4C22×D318yescomplemented
173C32×D4C23×C312yes—
174C32×D4C23×C312yescomplemented
175C2×C32×C4—24no—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C23×C311yescharacteristic
178#638—24no—
179#638—24no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—24no—
182C2×C3⋊Dic3C3×D411yescharacteristic
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)C23×C312yes—
185C4×C3⋊D3C3×D412yes—
186C4×C3⋊D3C23×C312yes—
187#634—21no—
188#634C23×C312yescomplemented
189C3×#67—616no—
190C3×#67—616no—
191C2×C3×Dic3—34no—
192C2×C3×Dic3—616no—
193C2×C3×Dic3—64no—
194C3×D12—38no—
195C3×D12—64no—
196C3×C4×D3—38no—
197C3×C4×D3—38no—
198C3×Dic6—38no—
199C3×Dic6—64no—
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×C33—42no—
208C2×C33AGL(1,ℤ/8)12yescomplemented
209C2×C33C22×D411yescharacteristic
210C3×C3⋊D3—22no—
211C3×C3⋊D3—42no—
212C24×C3—181no—
213C23×D3—64no—
214C3×#41—94no—
215C3×#41—182no—
216C2×C3×Q8—91no—
217C2×C3×D4—24no—
218C2×C3×D4—24no—
219C2×C3×D4—94no—
220C2×C3×D4—91no—
221C2×C3×D4—182no—
222C2×C3×D4—181no—
223C2×C3×D4—181no—
224C2×C3×D4C2×C3×D314yes—
225C2×C3×D4C2×C3×D314yescomplemented
226C2×C3×D4C2×C3⋊D311yescharacteristic
227C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
228C22×C3×C4—91no—
229C22×C3×C4—181no—
230C2×#67—64no—
231C2×#67—64no—
232C22×Dic3—64no—
233#237—316no—
234#237—68no—
235D3×D4—316no—
236D3×D4—68no—
237C2×D12—34no—
238C2×C4×D3—34no—
239C2×Dic6—34no—
240C3×SD16—94no—
241C3×SD16—94no—
242C3×D8—216no—
243C3×D8—94no—
244C3×D8C2×C3×D3116yescomplemented
245C3×D8C2×C3⋊D314yescomplemented
246C3×M16—94no—
247C2×C3×C8—24no—
248C2×C3×C8—91no—
249C2×C3×C8C2×C3×D314yescomplemented
250C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
251#214—316no—
252#213—316no—
253C2×C3⋊C8—34no—
254#204—316no—
255ΓL(1,𝔽25)—316no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—24no—
259C22×C32—28no—
260C22×C32—28no—
261C22×C32—48no—
262C22×C32—48no—
263C22×C32—42no—
264C22×C32—42no—
265C22×C32—48no—
266C22×C32—48no—
267C22×C32—88no—
268C22×C32—88no—
269C22×C32D3×D414yes—
270C22×C32C2×C3×D411yescharacteristic
271C2×C3⋊D3—22no—
272C2×C3⋊D3—22no—
273C2×C3⋊D3—48no—
274C2×C3⋊D3—42no—
275C2×C3⋊D3C2×C3×D412yes—
276C2×C3×D3—38no—
277C2×C3×D3—68no—
278C2×C3×D3—68no—
279C2×C3×D3—1232no—
280C2×C3×D3—128no—
281C32×C4—24no—
282C32×C4—24no—
283C32×C4D3×D414yes—
284C32×C4C2×C3×D411yescharacteristic
285C32×C4C23×D314yes—
286C32×C4C24×C311yescharacteristic
287C3⋊Dic3—22no—
288C3⋊Dic3C2×C3×D412yes—
289C3×Dic3—38no—
290C3×Dic3—68no—
291C2×#41—91no—
292C22×D4—91no—
293AGL(1,ℤ/8)—98no—
294C2×SD16—91no—
295C2×SD16—91no—
296C2×D8—91no—
297C2×D8C3×C3⋊D311yescharacteristic, complemented
298C2×M16—91no—
299C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—44no—
305C23×C3—44no—
306C23×C3—91no—
307C23×C3—184no—
308C23×C3—181no—
309C23×C3—364no—
310C22×D3—34no—
311C22×D3—616no—
312C22×D3—64no—
313C22×D3—1216no—
314C3×Q8—92no—
315C3×Q8—181no—
316C3×D4—24no—
317C3×D4—24no—
318C3×D4—21no—
319C3×D4—21no—
320C3×D4—28no—
321C3×D4—28no—
322C3×D4—44no—
323C3×D4—44no—
324C3×D4—92no—
325C3×D4—184no—
326C3×D4—184no—
327C3×D4—181no—
328C3×D4C22×C3×D318yes—
329C3×D4C22×C3×D318yescomplemented
330C3×D4C22×C3⋊D312yes—
331C3×D4C22×C3⋊D312yescomplemented
332C2×C3×C4—24no—
333C2×C3×C4—91no—
334C2×C3×C4—92no—
335C2×C3×C4—92no—
336C2×C3×C4—184no—
337C2×C3×C4—181no—
338C2×C3×C4C22×C3×D314yes—
339C2×C3×C4C22×C3⋊D311yescharacteristic
340#67—616no—
341#67—616no—
342C2×Dic3—34no—
343C2×Dic3—616no—
344C2×Dic3—64no—
345D12—38no—
346D12—64no—
347C4×D3—38no—
348C4×D3—38no—
349Dic6—38no—
350Dic6—64no—
351C3×C8—28no—
352C3×C8—92no—
353C3×C8C22×C3×D318yescomplemented
354C3×C8C22×C3⋊D312yescomplemented
355C3⋊C8—38no—
356C2×C32—28no—
357C2×C32—24no—
358C2×C32—42no—
359C2×C32—42no—
360C2×C32—48no—
361C2×C32—48no—
362C2×C32—88no—
363C2×C32—88no—
364C2×C32#91118yescomplemented
365C2×C32C3×AGL(1,ℤ/8)12yescomplemented
366C2×C32C2×D3×D414yes—
367C2×C32C22×C3×D411yescharacteristic
368C3⋊D3—22no—
369C3⋊D3—42no—
370C3×D3—68no—
371C3×D3—128no—
372C24—181no—
373#41—94no—
374#41—182no—
375C2×Q8—91no—
376C2×D4—94no—
377C2×D4—91no—
378C2×D4—182no—
379C2×D4—181no—
380C2×D4—181no—
381C2×D4C2×C3×C3⋊D311yescharacteristic
382C2×D4C2×C3×C3⋊D311yescharacteristic, complemented
383C22×C4—91no—
384C22×C4—181no—
385SD16—94no—
386SD16—94no—
387D8—94no—
388D8C2×C3×C3⋊D314yescomplemented
389M16—94no—
390C2×C8—91no—
391C2×C8C2×C3×C3⋊D311yescharacteristic, complemented
392C22×C3—28no—
393C22×C3—28no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—24no—
397C22×C3—48no—
398C22×C3—48no—
399C22×C3—48no—
400C22×C3—48no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—88no—
404C22×C3—88no—
405C22×C3—92no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—368no—
409C22×C3—362no—
410C22×C3C3×D3×D414yes—
411C22×C3D4×C3⋊D311yescharacteristic
412C2×D3—38no—
413C2×D3—68no—
414C2×D3—68no—
415C2×D3—1232no—
416C2×D3—128no—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—92no—
420C3×C4—182no—
421C3×C4C3×D3×D414yes—
422C3×C4D4×C3⋊D311yescharacteristic
423C3×C4C23×C3×D314yes—
424C3×C4C23×C3⋊D311yescharacteristic
425Dic3—38no—
426Dic3—68no—
427C32—24no—
428C32C2×#91114yescomplemented
429C32C2×C3×AGL(1,ℤ/8)11yescharacteristic, complemented
430C23—21no—
431C23—21no—
432C23—91no—
433C23—184no—
434C23—181no—
435C23—364no—
436Q8—92no—
437Q8—181no—
438D4—21no—
439D4—21no—
440D4—92no—
441D4—184no—
442D4—184no—
443D4—181no—
444D4C22×C3×C3⋊D312yes—
445D4C22×C3×C3⋊D312yescomplemented
446C2×C4—91no—
447C2×C4—92no—
448C2×C4—92no—
449C2×C4—184no—
450C2×C4—181no—
451C2×C4C22×C3×C3⋊D311yescharacteristic
452C8—92no—
453C8C22×C3×C3⋊D312yescomplemented
454C2×C3—28no—
455C2×C3—24no—
456C2×C3—48no—
457C2×C3—48no—
458C2×C3—42no—
459C2×C3—42no—
460C2×C3—88no—
461C2×C3—88no—
462C2×C3—182no—
463C2×C3—362no—
464C2×C3C3×#91118yescomplemented
465C2×C3#6412012yesdirect product factor
466C2×C3C2×C3×D3×D414yes—
467C2×C3C2×D4×C3⋊D311yescharacteristic
468D3—68no—
469D3—128no—
470C22—22no—
471C22—22no—
472C22—42no—
473C22—42no—
474C22—92no—
475C22—182no—
476C22—182no—
477C22—368no—
478C22—362no—
479C22C3×D4×C3⋊D311yescharacteristic
480C4—92no—
481C4—182no—
482C4C3×D4×C3⋊D311yescharacteristic
483C4C23×C3×C3⋊D311yescharacteristic
484C3—24no—
485C3C2×C3×#91114yescomplemented
486C3C2×#6412011yescharacteristic, direct product factor
487C2—42no—
488C2—42no—
489C2—182no—
490C2—362no—
491C2C3×#6412012yesdirect product factor
492C2C2×C3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC2×C33×D8C2
socleC22×C33C2×D4
hypercenterC2×C3×D8C3⋊D3
hypocenterC32C2×C3×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C2×C3×D4×C3⋊D3C2semidirect
3. C2×C33×D8C2semidirect
4. C2×C3×#3754C2semidirect
5. C2×C3×#3753C2semidirect
6. C3×#64120C2direct (N × H)
7. C2×C3×#3744C2semidirect
8. C2×C3×#3743C2semidirect
9. C2×#64120C3direct (N × H)
13. C2×C33×D4C22semidirect
18. C3×#3830C22semidirect
20. C3×D4×C3⋊D3C22semidirect
23. C2×C3×#634C22semidirect
24. C33×D8C22semidirect
25. C2×C33×C8C22semidirect
26. C3×#3754C22semidirect
27. C3×#3753C22semidirect
28. C2×C3×C3⋊(C3⋊C8)C22semidirect
29. C3×#3744C22semidirect
30. C3×#3743C22semidirect
31. C2×#3830C2×C3semidirect
32. C2×D4×C3⋊D3C2×C3semidirect
36. C2×C32×D8D3semidirect
37. C2×C32×D8C2×C3semidirect
38. C2×#3754C2×C3semidirect
39. C2×#3753C2×C3semidirect
40. #64120C2×C3direct (N × H)
41. C2×#3744C2×C3semidirect
42. C2×#3743C2×C3semidirect
57. C33×D4C23semidirect
69. C3×#634C23semidirect
70. C33×C8C23semidirect
71. C3×C3⋊(C3⋊C8)C23semidirect
79. C2×C32×D4C2×D3semidirect
81. C2×C32×D4C22×C3semidirect
86. #3830C22×C3semidirect
88. D4×C3⋊D3C22×C3semidirect
91. C2×#634C22×C3semidirect
103. C32×D8C2×D3semidirect
104. C32×D8C22×C3semidirect
106. C2×C32×C8C2×D3semidirect
107. C2×C32×C8C22×C3semidirect
108. #3754C22×C3semidirect
109. #3753C22×C3semidirect
110. C2×C3⋊(C3⋊C8)C22×C3semidirect
111. #3744C22×C3semidirect
112. #3743C22×C3semidirect
141. C2×C3×D8C3×D3semidirect
142. C2×C3×D8C3⋊D3semidirect
172. C32×D4C22×D3semidirect
174. C32×D4C23×C3semidirect
188. #634C23×C3semidirect
201. C32×C8C22×D3semidirect
202. C32×C8C23×C3semidirect
203. C3⋊(C3⋊C8)C23×C3semidirect
208. C2×C33AGL(1,ℤ/8)semidirect
225. C2×C3×D4C2×C3×D3semidirect
227. C2×C3×D4C2×C3⋊D3semidirect
244. C3×D8C2×C3×D3semidirect
245. C3×D8C2×C3⋊D3semidirect
249. C2×C3×C8C2×C3×D3semidirect
250. C2×C3×C8C2×C3⋊D3semidirect
297. C2×D8C3×C3⋊D3semidirect
299. C33C2×AGL(1,ℤ/8)semidirect
329. C3×D4C22×C3×D3semidirect
331. C3×D4C22×C3⋊D3semidirect
353. C3×C8C22×C3×D3semidirect
354. C3×C8C22×C3⋊D3semidirect
364. C2×C32#911semidirect
365. C2×C32C3×AGL(1,ℤ/8)semidirect
382. C2×D4C2×C3×C3⋊D3semidirect
388. D8C2×C3×C3⋊D3semidirect
391. C2×C8C2×C3×C3⋊D3semidirect
428. C32C2×#911semidirect
429. C32C2×C3×AGL(1,ℤ/8)semidirect
445. D4C22×C3×C3⋊D3semidirect
453. C8C22×C3×C3⋊D3semidirect
464. C2×C3C3×#911semidirect
465. C2×C3#64120direct (N × H)
485. C3C2×C3×#911semidirect
486. C3C2×#64120direct (N × H)
491. C2C3×#64120direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C3×D4×C3⋊D32
C2×C33×D82
C2×C3×#37542
C2×C3×#37532
C3×#641202
C2×C3×#37442
C2×C3×#37432
C2×#641203
C2×C3×#9113