Finite Group Databasebeta

← C2×C3×#63365 #13142462

Subgroups — C2×C3×#63365 #13142462

Order 1728.

762 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#611C211yesmaximal, characteristic
2C2×C3×C9×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#3676C211yesmaximal, characteristic, complemented
4C3×#63365C214yesmaximal, direct product factor
5C2×C3×#806—31nomaximal
6C2×#63365C311yesmaximal, characteristic, direct product factor
7C24×C3×C9C2211yescharacteristic
8C23×C3×D9C411yescharacteristic, complemented
9C22×C3×C4×C9—21no—
10C2×C3×#611—21no—
11C2×C3×#611—21no—
12C2×C3×#611—21no—
13C2×C3×#611—21no—
14C2×C3×#611C412yescomplemented
15C2×C3×#611C2212yes—
16C22×C3×Dic9—21no—
17C22×C3×Dic9C411yescharacteristic, complemented
18C3×C9×C22⋊C4—21no—
19C3×C9×C22⋊C4C2212yescomplemented
20C3×#3676—21no—
21C3×#3676C2212yescomplemented
22C22×C3×#67—31no—
23C2×C32×C22⋊C4D311yescharacteristic
24C2×C3×#217—31no—
25C22×#611C2×C311yescharacteristic
26C3×#806—34no—
27C2×C9×C22⋊C4—21no—
28C2×C9×C22⋊C4C2×C311yescharacteristic, complemented
29C2×#3676C2×C311yescharacteristic, complemented
30#63365C2×C314yesdirect product factor
31C23×C3×C9—42no—
32C23×C3×C9C2×C411yescharacteristic
33C23×C3×C9C2×C411yescharacteristic
34C23×C3×C9D411yescharacteristic
35C23×C3×C9D411yescharacteristic
36C23×C3×C9D411yescharacteristic
37C23×C3×C9D411yescharacteristic
38C23×C3×C9C2311yescharacteristic
39C22×C3×D9—21no—
40C22×C3×D9—21no—
41C22×C3×D9—42no—
42C22×C3×D9C2×C412yescomplemented
43C2×C3×C4×C9—22no—
44C2×C3×C4×C9—42no—
45C3×#611—22no—
46C3×#611—22no—
47C3×#611—22no—
48C3×#611—22no—
49C2×C3×Dic9—21no—
50C2×C3×Dic9—21no—
51C2×C3×Dic9—22no—
52C2×C3×Dic9—42no—
53C2×C3×Dic9C2×C412yescomplemented
54C2×C3×#99—91no—
55C2×#806—31no—
56C24×C32C2×D311yescharacteristic
57C23×C3×D3—31no—
58C22×C32×C4—21no—
59C2×C3×#67—32no—
60C2×C3×#67—32no—
61C2×C3×#67—61no—
62C2×C3×#67—61no—
63C2×C3×#67—61no—
64C2×C3×#67—61no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3—61no—
67C24×C9—21no—
68C24×C9C22×C311yescharacteristic
69C23×D9C3×C411yescharacteristic, complemented
70C32×C22⋊C4—21no—
71C32×C22⋊C4C2×D312yes—
72C3×#217—32no—
73C3×#217—61no—
74C22×C4×C9—22no—
75C22×C4×C9—21no—
76C2×#611—21no—
77C2×#611—21no—
78C2×#611—21no—
79C2×#611—21no—
80C2×#611C3×C412yescomplemented
81C2×#611C22×C312yes—
82C22×Dic9—21no—
83C22×Dic9C3×C411yescharacteristic, complemented
84C9×C22⋊C4—22no—
85C9×C22⋊C4—22no—
86C9×C22⋊C4—21no—
87C9×C22⋊C4C22×C312yescomplemented
88#3676—21no—
89#3676C22×C312yescomplemented
90C22×C3×C9—22no—
91C22×C3×C9—22no—
92C22×C3×C9—22no—
93C22×C3×C9—41no—
94C22×C3×C9—41no—
95C22×C3×C9—41no—
96C22×C3×C9—41no—
97C22×C3×C9C22⋊C411yescharacteristic
98C22×C3×C9C22⋊C411yescharacteristic
99C22×C3×C9C22⋊C411yescharacteristic
100C22×C3×C9C22⋊C411yescharacteristic
101C22×C3×C9C22×C411yescharacteristic
102C22×C3×C9C2×D411yescharacteristic
103C22×C3×C9C2×D411yescharacteristic
104C2×C3×D9—22no—
105C2×C3×D9—42no—
106C2×C3×D9—42no—
107C2×C3×D9—42no—
108C3×C4×C9—42no—
109C3×Dic9—22no—
110C3×Dic9—42no—
111C22×C3×D4—91no—
112C22×#67—31no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4D911yescharacteristic, complemented
116C2×C3×C22⋊C4C3×D311yescharacteristic
117C2×#217—31no—
118C3×#99—94no—
119#806—34no—
120C23×C32—42no—
121C23×C32C4×D311yescharacteristic
122C23×C32C4×D311yescharacteristic
123C23×C32D1211yescharacteristic
124C23×C32D1211yescharacteristic
125C23×C32#6711yescharacteristic
126C23×C32#6711yescharacteristic
127C23×C32C22×D311yescharacteristic
128C22×C3×D3—32no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—122no—
132C2×C32×C4—22no—
133C2×C32×C4—42no—
134C3×#67—62no—
135C3×#67—62no—
136C3×#67—62no—
137C3×#67—62no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—122no—
143C23×C9—21no—
144C23×C9—21no—
145C23×C9—21no—
146C23×C9—21no—
147C23×C9—21no—
148C23×C9—21no—
149C23×C9—21no—
150C23×C9—42no—
151C23×C9—44no—
152C23×C9C2×C3×C411yescharacteristic
153C23×C9C2×C3×C411yescharacteristic
154C23×C9C3×D411yescharacteristic
155C23×C9C3×D411yescharacteristic
156C23×C9C3×D411yescharacteristic
157C23×C9C3×D411yescharacteristic
158C23×C9C23×C311yescharacteristic
159C22×D9—21no—
160C22×D9—21no—
161C22×D9—42no—
162C22×D9C2×C3×C412yescomplemented
163C2×C4×C9—24no—
164C2×C4×C9—22no—
165C2×C4×C9—44no—
166C2×C4×C9—42no—
167#611—22no—
168#611—22no—
169#611—22no—
170#611—22no—
171C2×Dic9—21no—
172C2×Dic9—21no—
173C2×Dic9—22no—
174C2×Dic9—42no—
175C2×Dic9C2×C3×C412yescomplemented
176C2×#99—91noSylow
177C2×C3×C9—21no—
178C2×C3×C9—21no—
179C2×C3×C9—21no—
180C2×C3×C9—21no—
181C2×C3×C9—21no—
182C2×C3×C9—21no—
183C2×C3×C9#9912yescomplemented
184C2×C3×C9C2×C22⋊C411yescharacteristic
185C3×D9—42no—
186C24×C3—21no—
187C24×C3—91no—
188C24×C3C2×D911yescharacteristic
189C24×C3C2×C3×D311yescharacteristic
190C23×D3—31no—
191C2×C3×D4—92no—
192C2×C3×D4—92no—
193C2×C3×D4—181no—
194C2×C3×D4—181no—
195C2×C3×D4—181no—
196C2×C3×D4—181no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—91no—
201C22×C3×C4—181no—
202C2×#67—32no—
203C2×#67—32no—
204C2×#67—61no—
205C2×#67—61no—
206C2×#67—61no—
207C2×#67—61no—
208C22×Dic3—31no—
209C22×Dic3—61no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—21no—
212C3×C22⋊C4—21no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—92no—
215C3×C22⋊C4—181no—
216C3×C22⋊C4C2×D912yescomplemented
217C3×C22⋊C4C2×C3×D312yes—
218#217—32no—
219#217—61no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32(C2×D3)⋊C411yescharacteristic
228C22×C32(C2×D3)⋊C411yescharacteristic
229C22×C32(C2×D3)⋊C411yescharacteristic
230C22×C32(C2×D3)⋊C411yescharacteristic
231C22×C32C2×C4×D311yescharacteristic
232C22×C32C2×D1211yescharacteristic
233C22×C32C2×#6711yescharacteristic
234C2×C3×D3—62no—
235C2×C3×D3—122no—
236C2×C3×D3—122no—
237C2×C3×D3—122no—
238C32×C4—42no—
239C3×Dic3—62no—
240C3×Dic3—122no—
241C22×C9—24no—
242C22×C9—22no—
243C22×C9—22no—
244C22×C9—22no—
245C22×C9—21no—
246C22×C9—21no—
247C22×C9—21no—
248C22×C9—21no—
249C22×C9—21no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9—42no—
253C22×C9—42no—
254C22×C9—41no—
255C22×C9—41no—
256C22×C9—41no—
257C22×C9—41no—
258C22×C9—42no—
259C22×C9—42no—
260C22×C9—42no—
261C22×C9—42no—
262C22×C9C3×C22⋊C411yescharacteristic
263C22×C9C3×C22⋊C411yescharacteristic
264C22×C9C3×C22⋊C411yescharacteristic
265C22×C9C3×C22⋊C411yescharacteristic
266C22×C9C22×C3×C411yescharacteristic
267C22×C9C2×C3×D411yescharacteristic
268C22×C9C2×C3×D411yescharacteristic
269C2×D9—22no—
270C2×D9—42no—
271C2×D9—42no—
272C2×D9—42no—
273C4×C9—44no—
274C4×C9—42no—
275Dic9—22no—
276Dic9—42no—
277C22×D4—91no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4C3×D911yescharacteristic, complemented
280#99—94no—
281C3×C9C2×#9911yescharacteristic, complemented, Sylow
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—44no—
292C23×C3—92no—
293C23×C3—181no—
294C23×C3—181no—
295C23×C3—362no—
296C23×C3C4×D911yescharacteristic
297C23×C3C4×D911yescharacteristic
298C23×C3D3611yescharacteristic
299C23×C3D3611yescharacteristic
300C23×C3#61111yescharacteristic
301C23×C3#61111yescharacteristic
302C23×C3C22×D911yescharacteristic
303C23×C3C3×C4×D311yescharacteristic
304C23×C3C3×C4×D311yescharacteristic
305C23×C3C3×D1211yescharacteristic
306C23×C3C3×D1211yescharacteristic
307C23×C3C3×#6711yescharacteristic
308C23×C3C3×#6711yescharacteristic
309C23×C3C22×C3×D311yescharacteristic
310C22×D3—32no—
311C22×D3—61no—
312C22×D3—61no—
313C22×D3—122no—
314C3×D4—182no—
315C3×D4—182no—
316C3×D4—182no—
317C3×D4—182no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4—44no—
323C2×C3×C4—42no—
324C2×C3×C4—92no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4—182no—
328C2×C3×C4—362no—
329#67—62no—
330#67—62no—
331#67—62no—
332#67—62no—
333C2×Dic3—32no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—62no—
337C2×Dic3—122no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32#80612yes—
345C2×C32C2×(C2×D3)⋊C411yescharacteristic
346C3×D3—122no—
347C2×C9—22no—
348C2×C9—22no—
349C2×C9—21no—
350C2×C9—21no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9—22no—
357C2×C9—41no—
358C2×C9—41no—
359C2×C9—41no—
360C2×C9—41no—
361C2×C9C3×#9912yescomplemented
362C2×C9C2×C3×C22⋊C411yescharacteristic
363D9—42no—
364C24—91no—
365C24C2×C3×D911yescharacteristic
366C2×D4—92no—
367C2×D4—92no—
368C2×D4—181no—
369C2×D4—181no—
370C2×D4—181no—
371C2×D4—181no—
372C22×C4—21no—
373C22×C4—91no—
374C22×C4—181no—
375C22⋊C4—21no—
376C22⋊C4—92no—
377C22⋊C4—181no—
378C22⋊C4C2×C3×D912yescomplemented
379C22×C3—24no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—182no—
408C22×C3—362no—
409C22×C3—362no—
410C22×C3—362no—
411C22×C3(C2×D9)⋊C411yescharacteristic
412C22×C3(C2×D9)⋊C411yescharacteristic
413C22×C3(C2×D9)⋊C411yescharacteristic
414C22×C3(C2×D9)⋊C411yescharacteristic
415C22×C3C2×C4×D911yescharacteristic
416C22×C3C2×D3611yescharacteristic
417C22×C3C2×#61111yescharacteristic
418C22×C3C3×(C2×D3)⋊C411yescharacteristic
419C22×C3C3×(C2×D3)⋊C411yescharacteristic
420C22×C3C3×(C2×D3)⋊C411yescharacteristic
421C22×C3C3×(C2×D3)⋊C411yescharacteristic
422C22×C3C2×C3×C4×D311yescharacteristic
423C22×C3C2×C3×D1211yescharacteristic
424C22×C3C2×C3×#6711yescharacteristic
425C2×D3—62no—
426C2×D3—122no—
427C2×D3—122no—
428C2×D3—122no—
429C3×C4—42no—
430C3×C4—44no—
431C3×C4—42no—
432C3×C4—182no—
433C3×C4—362no—
434Dic3—62no—
435Dic3—122no—
436C32C2×#80611yescharacteristic
437C9—21no—
438C9C2×C3×#9911yescharacteristic, complemented
439C23—42no—
440C23—92no—
441C23—181no—
442C23—181no—
443C23—362no—
444C23C3×C4×D911yescharacteristic
445C23C3×C4×D911yescharacteristic
446C23C3×D3611yescharacteristic
447C23C3×D3611yescharacteristic
448C23C3×#61111yescharacteristic
449C23C3×#61111yescharacteristic
450C23C22×C3×D911yescharacteristic
451D4—182no—
452D4—182no—
453D4—182no—
454D4—182no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—92no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—182no—
461C2×C4—362no—
462C2×C3—22no—
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—22no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—362no—
483C2×C3#6336512yesdirect product factor
484C2×C3C2×(C2×D9)⋊C411yescharacteristic
485C2×C3C3×#80612yes—
486C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
487D3—122no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—182no—
496C22—362no—
497C22—362no—
498C22—362no—
499C22C3×(C2×D9)⋊C411yescharacteristic
500C22C3×(C2×D9)⋊C411yescharacteristic
501C22C3×(C2×D9)⋊C411yescharacteristic
502C22C3×(C2×D9)⋊C411yescharacteristic
503C22C2×C3×C4×D911yescharacteristic
504C22C2×C3×D3611yescharacteristic
505C22C2×C3×#61111yescharacteristic
506C4—42no—
507C4—182no—
508C4—362no—
509C3—21no—
510C3C2×#6336511yescharacteristic, direct product factor
511C3C2×C3×#80611yescharacteristic
512C2—21no—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—21no—
517C2—21no—
518C2—362no—
519C2C3×#6336512yesdirect product factor
520C2C2×C3×(C2×D9)⋊C411yescharacteristic
subgroupG/H
centerC22×C3(C2×D9)⋊C4
commutatorC22×C9C22×C3×C4
FrattiniC23×C3C22×C3×D3
FittingC2×C3×C9×C22⋊C4C2
socleC22×C32(C2×D3)⋊C4
prime radicalC22×C3×#611C2
cosocleC23×C9C23×C3
hypercenterC2×C3×C22⋊C4D9
hypocenterC9C2×C3×#99
LCSG ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C9×C22⋊C4C2semidirect
3. C2×C3×#3676C2semidirect
4. C3×#63365C2direct (N × H)
6. C2×#63365C3direct (N × H)
8. C23×C3×D9C4semidirect
14. C2×C3×#611C4semidirect
17. C22×C3×Dic9C4semidirect
19. C3×C9×C22⋊C4C22semidirect
21. C3×#3676C22semidirect
28. C2×C9×C22⋊C4C2×C3semidirect
29. C2×#3676C2×C3semidirect
30. #63365C2×C3direct (N × H)
42. C22×C3×D9C2×C4semidirect
53. C2×C3×Dic9C2×C4semidirect
69. C23×D9C3×C4semidirect
80. C2×#611C3×C4semidirect
83. C22×Dic9C3×C4semidirect
87. C9×C22⋊C4C22×C3semidirect
89. #3676C22×C3semidirect
115. C2×C3×C22⋊C4D9semidirect
162. C22×D9C2×C3×C4semidirect
175. C2×Dic9C2×C3×C4semidirect
183. C2×C3×C9#99semidirect
216. C3×C22⋊C4C2×D9semidirect
279. C2×C22⋊C4C3×D9semidirect
281. C3×C9C2×#99semidirect
361. C2×C9C3×#99semidirect
378. C22⋊C4C2×C3×D9semidirect
438. C9C2×C3×#99semidirect
483. C2×C3#63365direct (N × H)
510. C3C2×#63365direct (N × H)
519. C2C3×#63365direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6112
C2×C3×C9×C22⋊C42
C2×C3×#36762
C3×#633652
C2×C3×#8063
C2×#633653