Finite Group Databasebeta

← C2×C3×C4×#611 #13153538

Subgroups — C2×C3×C4×#611 #13153538

Order 1728.

1514 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×C9C211yesmaximal, characteristic, complemented
2C22×C3×#611C211yesmaximal, characteristic
3C22×C3×C4×D9C211yesmaximal, characteristic, complemented
4C2×C3×#3676C211yesmaximal, characteristic, complemented
5C3×C4×#611C218yesmaximal, direct product factor
6C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
7C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
8C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
9C2×C3×C4×#67—31nomaximal
10C2×C4×#611C311yesmaximal, characteristic, direct product factor
11C24×C3×C9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9—24no—
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C22×C3×C4×C9C2214yescomplemented
17C2×C3×#611C418yesdirect product factor
18C2×C3×#611C2214yes—
19C22×C3×Dic9C2211yescharacteristic
20C22×C3×Dic9C2211yescharacteristic
21C22×C3×Dic9C2211yescharacteristic
22C2×C3×C4×D9—24no—
23C2×C3×C4×D9C2214yescomplemented
24C3×#3676C2214yescomplemented
25C3×(C2×D9)⋊C4C2214yescomplemented
26C3×Dic9⋊C4C2214yescomplemented
27C3×C4×Dic9C2214yescomplemented
28C23×C32×C4D311yescharacteristic
29C22×C3×#67—31no—
30C22×C3×C4×D3—31no—
31C2×C3×#217—31no—
32C3×C4×#67—38no—
33C2×C3×(C2×D3)⋊C4—31no—
34C2×C3×Dic3⋊C4—31no—
35C2×C3×C4×Dic3—31no—
36C23×C4×C9—21no—
37C23×C4×C9C2×C311yescharacteristic, complemented
38C22×#611C2×C311yescharacteristic
39C22×C4×D9C2×C311yescharacteristic, complemented
40C2×#3676C2×C311yescharacteristic, complemented
41C4×#611C2×C318yesdirect product factor
42C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
43C2×Dic9⋊C4C2×C311yescharacteristic, complemented
44C2×C4×Dic9C2×C311yescharacteristic, complemented
45C23×C3×C9—22no—
46C23×C3×C9—22no—
47C23×C3×C9C2×C414yescomplemented
48C23×C3×C9C2312yes—
49C23×C3×C9C2311yescharacteristic
50C22×C3×D9—22no—
51C22×C3×D9—22no—
52C22×C3×D9C2×C414yescomplemented
53C22×C3×D9C2312yes—
54C2×C3×C4×C9—22no—
55C2×C3×C4×C9—28no—
56C2×C3×C4×C9D414yescomplemented
57C2×C3×C4×C9C2312yes—
58C2×C3×C4×C9C2312yes—
59C3×#611C2×C4116yesdirect product factor
60C2×C3×Dic9—22no—
61C2×C3×Dic9—22no—
62C2×C3×Dic9C2×C414yescomplemented
63C2×C3×Dic9C2312yes—
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66C3×C4×D9—28no—
67C2×C3×C4×D4—91no—
68C2×C4×#67—31no—
69C24×C32C2×D311yescharacteristic
70C23×C3×D3—31no—
71C22×C32×C4—24no—
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C2×D314yes—
74C22×C32×C4C2×D311yescharacteristic
75C2×C3×#67—38no—
76C2×C3×#67—34no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C2×C3×C4×D3—34no—
81C2×C3×C4×D3—64no—
82C24×C9—21no—
83C24×C9C22×C311yescharacteristic
84C23×D9C22×C311yescharacteristic
85C3×#217—34no—
86C3×(C2×D3)⋊C4—34no—
87C3×Dic3⋊C4—34no—
88C3×C4×Dic3—34no—
89C22×C4×C9—28no—
90C22×C4×C9—21no—
91C22×C4×C9—21no—
92C22×C4×C9—24no—
93C22×C4×C9—24no—
94C22×C4×C9C22×C311yescharacteristic
95C22×C4×C9C22×C311yescharacteristic
96C22×C4×C9C22×C314yescomplemented
97C2×#611C3×C418yesdirect product factor
98C2×#611C22×C314yes—
99C22×Dic9C22×C311yescharacteristic
100C22×Dic9C22×C311yescharacteristic
101C22×Dic9C22×C311yescharacteristic
102C2×C4×D9—24no—
103C2×C4×D9C22×C314yescomplemented
104#3676C22×C314yescomplemented
105(C2×D9)⋊C4C22×C314yescomplemented
106Dic9⋊C4C22×C314yescomplemented
107C4×Dic9C22×C314yescomplemented
108C22×C3×C9—28no—
109C22×C3×C9—22no—
110C22×C3×C9—22no—
111C22×C3×C9C22×C412yes—
112C22×C3×C9C22×C414yescomplemented
113C22×C3×C9C2×D412yes—
114C22×C3×C9#4112yes—
115C22×C3×C9C2411yescharacteristic
116C2×C3×D9—28no—
117C2×C3×D9—22no—
118C2×C3×D9—22no—
119C2×C3×D9C22×C414yescomplemented
120C3×C4×C9—22no—
121C3×C4×C9C2×D414yescomplemented
122C3×Dic9—22no—
123C3×Dic9—22no—
124C3×Dic9C22×C414yescomplemented
125C22×C3×D4—91no—
126C23×C3×C4—21no—
127C23×C3×C4—91no—
128C23×C3×C4D911yescharacteristic, complemented
129C23×C3×C4C3×D311yescharacteristic
130C22×#67—31no—
131C22×C4×D3—31no—
132C3×C4×D4—98no—
133C2×C3×C4⋊C4—91no—
134C2×C3×C22⋊C4—91no—
135C2×C3×C22⋊C4—91no—
136C2×C3×C42—91no—
137C2×#217—31no—
138C4×#67—38no—
139C2×(C2×D3)⋊C4—31no—
140C2×Dic3⋊C4—31no—
141C2×C4×Dic3—31no—
142C23×C32—22no—
143C23×C32—22no—
144C23×C32C4×D314yes—
145C23×C32C22×D312yes—
146C23×C32C22×D311yescharacteristic
147C22×C3×D3—34no—
148C22×C3×D3—32no—
149C22×C3×D3—62no—
150C22×C3×D3—62no—
151C2×C32×C4—22no—
152C2×C32×C4—28no—
153C2×C32×C4#6714yes—
154C2×C32×C4C22×D312yes—
155C2×C32×C4C22×D312yes—
156C3×#67—316no—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C3×C4×D3—68no—
164C23×C9—24no—
165C23×C9—24no—
166C23×C9—22no—
167C23×C9—22no—
168C23×C9—24no—
169C23×C9—22no—
170C23×C9—21no—
171C23×C9C2×C3×C414yescomplemented
172C23×C9C23×C312yes—
173C23×C9C23×C311yescharacteristic
174C22×D9—22no—
175C22×D9—22no—
176C22×D9C2×C3×C414yescomplemented
177C22×D9C23×C312yes—
178C2×C4×C9—24no—
179C2×C4×C9—216no—
180C2×C4×C9—22no—
181C2×C4×C9—28no—
182C2×C4×C9—24no—
183C2×C4×C9—22no—
184C2×C4×C9—22no—
185C2×C4×C9C3×D414yescomplemented
186C2×C4×C9C23×C312yes—
187C2×C4×C9C23×C312yes—
188#611C2×C3×C4116yesdirect product factor
189C2×Dic9—22no—
190C2×Dic9—22no—
191C2×Dic9C2×C3×C414yescomplemented
192C2×Dic9C23×C312yes—
193C2×Dic9C23×C312yes—
194C2×Dic9C23×C312yes—
195C4×D9—28no—
196C2×C4×D4—91noSylow
197C2×C3×C9—24no—
198C2×C3×C9C4×D414yescomplemented
199C2×C3×C9C23×C411yescharacteristic
200C2×C3×C9C22×D411yescharacteristic
201C2×C3×C9C2×#4111yescharacteristic
202C3×D9—24no—
203C24×C3—21no—
204C24×C3—91no—
205C24×C3C2×D911yescharacteristic
206C24×C3C2×C3×D311yescharacteristic
207C23×D3—31no—
208C2×C3×D4—98no—
209C2×C3×D4—94no—
210C22×C3×C4—28no—
211C22×C3×C4—24no—
212C22×C3×C4—24no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—24no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C22×C3×C4—94no—
220C22×C3×C4—184no—
221C22×C3×C4C2×D911yescharacteristic
222C22×C3×C4C2×D911yescharacteristic
223C22×C3×C4C2×D914yescomplemented
224C22×C3×C4C2×C3×D311yescharacteristic
225C22×C3×C4C2×C3×D311yescharacteristic
226C22×C3×C4C2×C3×D314yes—
227C2×#67—38no—
228C2×#67—34no—
229C22×Dic3—31no—
230C22×Dic3—31no—
231C22×Dic3—31no—
232C2×C4×D3—34no—
233C2×C4×D3—64no—
234C3×C4⋊C4—94no—
235C3×C22⋊C4—94no—
236C3×C22⋊C4—94no—
237C3×C42—94no—
238#217—34no—
239(C2×D3)⋊C4—34no—
240Dic3⋊C4—34no—
241C4×Dic3—34no—
242C22×C32—28no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32C2×C4×D312yes—
246C22×C32C2×C4×D314yes—
247C22×C32#23512yes—
248C22×C32C2×#6712yes—
249C22×C32C23×D311yescharacteristic
250C2×C3×D3—34no—
251C2×C3×D3—68no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C32×C4—22no—
255C32×C4C2×#6714yes—
256C3×Dic3—34no—
257C3×Dic3—62no—
258C3×Dic3—62no—
259C22×C9—24no—
260C22×C9—216no—
261C22×C9—24no—
262C22×C9—28no—
263C22×C9—22no—
264C22×C9—22no—
265C22×C9—21no—
266C22×C9—22no—
267C22×C9—24no—
268C22×C9—22no—
269C22×C9—22no—
270C22×C9C22×C3×C412yes—
271C22×C9C22×C3×C414yescomplemented
272C22×C9C2×C3×D412yes—
273C22×C9C3×#4112yes—
274C22×C9C24×C311yescharacteristic
275C2×D9—28no—
276C2×D9—22no—
277C2×D9—22no—
278C2×D9C22×C3×C414yescomplemented
279C4×C9—24no—
280C4×C9—22no—
281C4×C9—24no—
282C4×C9C2×C3×D414yescomplemented
283Dic9—22no—
284Dic9—22no—
285Dic9C22×C3×C414yescomplemented
286C22×D4—91no—
287C23×C4—91no—
288C23×C4C3×D911yescharacteristic, complemented
289C4×D4—98no—
290C2×C4⋊C4—91no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4—91no—
293C2×C42—91no—
294C3×C9C2×C4×D411yescharacteristic, complemented, Sylow
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—94no—
305C23×C3—92no—
306C23×C3—182no—
307C23×C3—182no—
308C23×C3C4×D914yescomplemented
309C23×C3C22×D912yes—
310C23×C3C22×D911yescharacteristic
311C23×C3C3×C4×D314yes—
312C23×C3C22×C3×D312yes—
313C23×C3C22×C3×D311yescharacteristic
314C22×D3—32no—
315C22×D3—34no—
316C22×D3—62no—
317C22×D3—62no—
318C3×D4—916no—
319C2×C3×C4—24no—
320C2×C3×C4—216no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—28no—
324C2×C3×C4—28no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—94no—
329C2×C3×C4—92no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4—182no—
333C2×C3×C4—182no—
334C2×C3×C4—188no—
335C2×C3×C4#61114yesdirect product factor
336C2×C3×C4C22×D912yes—
337C2×C3×C4C22×D912yes—
338C2×C3×C4C3×#6714yes—
339C2×C3×C4C22×C3×D312yes—
340C2×C3×C4C22×C3×D312yes—
341#67—316no—
342C2×Dic3—32no—
343C2×Dic3—34no—
344C2×Dic3—32no—
345C2×Dic3—32no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C4×D3—68no—
349C2×C32—24no—
350C2×C32C4×#6714yes—
351C2×C32C22×C4×D311yescharacteristic
352C2×C32C2×#23511yescharacteristic
353C2×C32C22×#6711yescharacteristic
354C3×D3—64no—
355C2×C9—28no—
356C2×C9—24no—
357C2×C9—24no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9C3×C4×D414yescomplemented
362C2×C9C23×C3×C411yescharacteristic
363C2×C9C22×C3×D411yescharacteristic
364C2×C9C2×C3×#4111yescharacteristic
365D9—24no—
366C24—91no—
367C24C2×C3×D911yescharacteristic
368C2×D4—98no—
369C2×D4—94no—
370C22×C4—24no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—94no—
375C22×C4—184no—
376C22×C4C2×C3×D911yescharacteristic
377C22×C4C2×C3×D911yescharacteristic
378C22×C4C2×C3×D914yescomplemented
379C4⋊C4—94no—
380C22⋊C4—94no—
381C22⋊C4—94no—
382C42—94no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—216no—
386C22×C3—28no—
387C22×C3—28no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—21no—
393C22×C3—22no—
394C22×C3—24no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—94no—
398C22×C3—188no—
399C22×C3—182no—
400C22×C3—182no—
401C22×C3C2×C4×D912yes—
402C22×C3C2×C4×D914yescomplemented
403C22×C3#369712yes—
404C22×C3C2×#61112yes—
405C22×C3C23×D911yescharacteristic
406C22×C3C2×C3×C4×D312yes—
407C22×C3C2×C3×C4×D314yes—
408C22×C3C3×#23512yes—
409C22×C3C2×C3×#6712yes—
410C22×C3C23×C3×D311yescharacteristic
411C2×D3—34no—
412C2×D3—68no—
413C2×D3—62no—
414C2×D3—62no—
415C3×C4—24no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—24no—
419C3×C4—94no—
420C3×C4—182no—
421C3×C4—182no—
422C3×C4C2×#61114yesdirect product factor
423C3×C4C2×C3×#6714yes—
424Dic3—34no—
425Dic3—62no—
426Dic3—62no—
427C32C2×C4×#6711yescharacteristic
428C9—21no—
429C9C2×C3×C4×D411yescharacteristic, complemented
430C23—22no—
431C23—22no—
432C23—92no—
433C23—94no—
434C23—182no—
435C23—182no—
436C23C3×C4×D914yescomplemented
437C23C22×C3×D912yes—
438C23C22×C3×D911yescharacteristic
439D4—916no—
440C2×C4—22no—
441C2×C4—28no—
442C2×C4—92no—
443C2×C4—94no—
444C2×C4—92no—
445C2×C4—92no—
446C2×C4—182no—
447C2×C4—182no—
448C2×C4—188no—
449C2×C4C3×#61114yesdirect product factor
450C2×C4C22×C3×D912yes—
451C2×C4C22×C3×D912yes—
452C2×C3—28no—
453C2×C3—24no—
454C2×C3—24no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—24no—
458C2×C3—21no—
459C2×C3—184no—
460C2×C3C4×#61114yesdirect product factor
461C2×C3C22×C4×D911yescharacteristic
462C2×C3C2×#369711yescharacteristic
463C2×C3C22×#61111yescharacteristic
464C2×C3C3×C4×#6714yes—
465C2×C3C22×C3×C4×D311yescharacteristic
466C2×C3C2×C3×#23511yescharacteristic
467C2×C3C22×C3×#6711yescharacteristic
468D3—64no—
469C22—22no—
470C22—28no—
471C22—22no—
472C22—94no—
473C22—182no—
474C22—188no—
475C22—182no—
476C22C2×C3×C4×D912yes—
477C22C2×C3×C4×D914yescomplemented
478C22C3×#369712yes—
479C22C2×C3×#61112yes—
480C22C23×C3×D911yescharacteristic
481C4—22no—
482C4—94no—
483C4—182no—
484C4—182no—
485C4C2×C3×#61114yesdirect product factor
486C3—21no—
487C3C2×C4×#61111yescharacteristic, direct product factor
488C3C2×C3×C4×#6711yescharacteristic
489C2—24no—
490C2—184no—
491C2C3×C4×#61114yesdirect product factor
492C2C22×C3×C4×D911yescharacteristic
493C2C2×C3×#369711yescharacteristic
494C2C22×C3×#61111yescharacteristic
subgroupG/H
centerC22×C3×C4C2×D9
commutatorC2×C9C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC23×C3×C4×C9C2
socleC23×C32C22×D3
prime radicalC22×C3×#611C2
cosocleC22×C9C24×C3
hypercenterC23×C3×C4D9
hypocenterC9C2×C3×C4×D4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3×C4 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
1. C23×C3×C4×C9C2semidirect
3. C22×C3×C4×D9C2semidirect
4. C2×C3×#3676C2semidirect
5. C3×C4×#611C2direct (N × H)
6. C2×C3×(C2×D9)⋊C4C2semidirect
7. C2×C3×Dic9⋊C4C2semidirect
8. C2×C3×C4×Dic9C2semidirect
10. C2×C4×#611C3direct (N × H)
16. C22×C3×C4×C9C22semidirect
17. C2×C3×#611C4direct (N × H)
23. C2×C3×C4×D9C22semidirect
24. C3×#3676C22semidirect
25. C3×(C2×D9)⋊C4C22semidirect
26. C3×Dic9⋊C4C22semidirect
27. C3×C4×Dic9C22semidirect
37. C23×C4×C9C2×C3semidirect
39. C22×C4×D9C2×C3semidirect
40. C2×#3676C2×C3semidirect
41. C4×#611C2×C3direct (N × H)
42. C2×(C2×D9)⋊C4C2×C3semidirect
43. C2×Dic9⋊C4C2×C3semidirect
44. C2×C4×Dic9C2×C3semidirect
47. C23×C3×C9C2×C4semidirect
52. C22×C3×D9C2×C4semidirect
56. C2×C3×C4×C9D4semidirect
59. C3×#611C2×C4direct (N × H)
62. C2×C3×Dic9C2×C4semidirect
96. C22×C4×C9C22×C3semidirect
97. C2×#611C3×C4direct (N × H)
103. C2×C4×D9C22×C3semidirect
104. #3676C22×C3semidirect
105. (C2×D9)⋊C4C22×C3semidirect
106. Dic9⋊C4C22×C3semidirect
107. C4×Dic9C22×C3semidirect
112. C22×C3×C9C22×C4semidirect
119. C2×C3×D9C22×C4semidirect
121. C3×C4×C9C2×D4semidirect
124. C3×Dic9C22×C4semidirect
128. C23×C3×C4D9semidirect
171. C23×C9C2×C3×C4semidirect
176. C22×D9C2×C3×C4semidirect
185. C2×C4×C9C3×D4semidirect
188. #611C2×C3×C4direct (N × H)
191. C2×Dic9C2×C3×C4semidirect
198. C2×C3×C9C4×D4semidirect
223. C22×C3×C4C2×D9semidirect
271. C22×C9C22×C3×C4semidirect
278. C2×D9C22×C3×C4semidirect
282. C4×C9C2×C3×D4semidirect
285. Dic9C22×C3×C4semidirect
288. C23×C4C3×D9semidirect
294. C3×C9C2×C4×D4semidirect
308. C23×C3C4×D9semidirect
335. C2×C3×C4#611direct (N × H)
361. C2×C9C3×C4×D4semidirect
378. C22×C4C2×C3×D9semidirect
402. C22×C3C2×C4×D9semidirect
422. C3×C4C2×#611direct (N × H)
429. C9C2×C3×C4×D4semidirect
436. C23C3×C4×D9semidirect
449. C2×C4C3×#611direct (N × H)
460. C2×C3C4×#611direct (N × H)
477. C22C2×C3×C4×D9semidirect
485. C4C2×C3×#611direct (N × H)
487. C3C2×C4×#611direct (N × H)
491. C2C3×C4×#611direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×C92
C22×C3×#6112
C22×C3×C4×D92
C2×C3×#36762
C3×C4×#6112
C2×C3×(C2×D9)⋊C42
C2×C3×Dic9⋊C42
C2×C3×C4×Dic92
C2×C3×C4×#673
C2×C4×#6113