Finite Group Databasebeta

← C3×#10590572 #13157565

Subgroups — C3×#10590572 #13157565

Order 1728.

630 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#892C211yesmaximal, characteristic
2C3×#63841C211yesmaximal, characteristic
3C2×C3×D3×C3⋊C8C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6C3×#63576C211yesmaximal, characteristic, complemented
7C3×#63573C211yesmaximal, characteristic, complemented
8C3×#4998—31nomaximal
9C3×#4860—31nomaximal
10#10590572C311yesmaximal, characteristic, direct product factor
11C2×C32×C4×D3C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×(C2×D3)⋊C4—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C3×Dic32—21no—
22C3×D3×C3⋊C8C414yescomplemented
23C32×#117—31no—
24C3×#926—31no—
25C22×C3×C3⋊C8—31no—
26C2×C3×D3×C8—31no—
27C3×#892—21no—
28C3×#892—31no—
29C3×#892D311yescharacteristic
30C3×#892C2×C311yescharacteristic
31#63841C2×C311yescharacteristic
32C2×D3×C3⋊C8C2×C311yescharacteristic
33C32×#106—31no—
34#63634C2×C311yescharacteristic, complemented
35C3×#817—31no—
36C3×#808—21no—
37C3×#808—31no—
38C3×#808D311yescharacteristic, complemented
39C3×#808C2×C311yescharacteristic, complemented
40C3×#807—31no—
41C3×#807—31no—
42C3×#807—31no—
43C3×#807—31no—
44C3×#807—61no—
45#63576C2×C311yescharacteristic, complemented
46#63573C2×C311yescharacteristic, complemented
47C22×C32×D3D411yescharacteristic
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C32×C4×D3C2×C412yes—
55C32×C4×D3Q812yes—
56C3×C3⋊(C3⋊C8)C2×C412yes—
57C32×C3⋊C8C2×C412yes—
58C3×#427—91no—
59#4998—31no—
60#4860—31no—
61C22×C32×C4—31no—
62C2×C3×C4×D3—21no—
63C2×C3×C4×D3C2×D311yescharacteristic
64C2×C3×C4×D3C22×C311yescharacteristic
65C2×C32×C8—31no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4—21no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4—31no—
70C32×C4⋊C4—31no—
71C32×C4⋊C4—41no—
72C32×C4⋊C4C2×D311yescharacteristic
73C32×C4⋊C4C2×D311yescharacteristic
74C32×C4⋊C4C22×C311yescharacteristic
75C32×C22⋊C4—61no—
76C32×C42—61no—
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#217—61no—
80C3×(C2×D3)⋊C4—22no—
81C3×(C2×D3)⋊C4—21no—
82C3×(C2×D3)⋊C4—21no—
83C3×(C2×D3)⋊C4—61no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C22×C311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×C4×Dic3—22no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3—61no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—34no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105C3×D3×C8—34no—
106(C2×D3)⋊Dic3—21no—
107Dic32—21no—
108D3×C3⋊C8C3×C414yescomplemented
109C22×C33C2×D411yescharacteristic
110C2×C32×D3—21no—
111C2×C32×D3—21no—
112C2×C32×D3C4⋊C412yes—
113C33×C4—41no—
114C33×C4C22×C411yescharacteristic
115C33×C4C2×Q811yescharacteristic
116C3×C3⋊Dic3—41no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119C32×Dic3C4⋊C412yes—
120C22×C3×C8—91no—
121C3×#117—31no—
122C3×#117—31no—
123C3×#117—61no—
124C3×#117—91no—
125#926—31no—
126C22×C3⋊C8—31no—
127C2×D3×C8—31no—
128#892—31no—
129#892C3×D311yescharacteristic
130C3×#107—91no—
131C3×#107—91no—
132C3×#106—31no—
133C3×#106—31no—
134C3×#106—61no—
135C3×#106—91no—
136#817—31no—
137#808—31no—
138#808C3×D311yescharacteristic, complemented
139#807—31no—
140#807—31no—
141#807—31no—
142#807—31no—
143#807—61no—
144C23×C32—31no—
145C22×C3×D3—21no—
146C22×C3×D3#6711yescharacteristic
147C22×C3×D3C3×D411yescharacteristic
148C2×C32×C4—21no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—31no—
156C2×C32×C4—32no—
157C2×C32×C4—32no—
158C2×C32×C4—41no—
159C2×C32×C4—41no—
160C2×C32×C4—41no—
161C2×C32×C4—42no—
162C2×C32×C4—61no—
163C2×C32×C4—61no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C23×C311yescharacteristic
167C2×C3⋊Dic3—21no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—121no—
178C2×C3×Dic3#6711yescharacteristic
179C2×C3×Dic3C3×D411yescharacteristic
180C3×C4×D3—22no—
181C3×C4×D3—22no—
182C3×C4×D3Dic612yes—
183C3×C4×D3C4×D312yes—
184C3×C4×D3C2×C3×C412yes—
185C3×C4×D3C3×Q812yes—
186C32×C8—32no—
187C3⋊(C3⋊C8)C2×C3×C412yes—
188C3×C3⋊C8—22no—
189C3×C3⋊C8—32no—
190C3×C3⋊C8—32no—
191C3×C3⋊C8—62no—
192C3×C3⋊C8C4×D312yes—
193C3×C3⋊C8C2×C3×C412yes—
194#427—91noSylow
195C2×C33C2×C4⋊C411yescharacteristic
196C2×C33#13511yescharacteristic
197C2×C33#13511yescharacteristic
198C32×D3—22no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—61no—
202C2×C4×D3C2×C3×D311yescharacteristic
203C2×C3×C8—31no—
204C2×C3×C8—31no—
205C2×C3×C8—61no—
206C2×C3×C8—94no—
207C2×C3×C8—91no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—31no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—41no—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—61no—
218C3×C4⋊C4—91no—
219C3×C4⋊C4D3211yescharacteristic
220C3×C4⋊C4C2×C3×D311yescharacteristic
221C3×C4⋊C4C2×C3×D311yescharacteristic
222C3×C22⋊C4—61no—
223C3×C22⋊C4—61no—
224C3×C22⋊C4—62no—
225C3×C22⋊C4—181no—
226C3×C42—61no—
227C3×C42—61no—
228C3×C42—62no—
229C3×C42—181no—
230#217—61no—
231(C2×D3)⋊C4—21no—
232(C2×D3)⋊C4—61no—
233C4⋊Dic3—31no—
234C4⋊Dic3—31no—
235C4⋊Dic3—61no—
236C4⋊Dic3C2×C3×D311yescharacteristic
237C4⋊Dic3C2×C3×D311yescharacteristic
238C4×Dic3—21no—
239C4×Dic3—61no—
240C4×Dic3—61no—
241C2×C3⋊C8—31no—
242C2×C3⋊C8—31no—
243C2×C3⋊C8—34no—
244C2×C3⋊C8—61no—
245C2×C3⋊C8C2×C3×D311yescharacteristic
246D3×C8—34no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—32no—
251C22×C32—41no—
252C22×C32—61no—
253C22×C32—61no—
254C22×C32D3×D411yescharacteristic
255C22×C32C2×#6711yescharacteristic
256C22×C32C2×C3×D411yescharacteristic
257C2×C3×D3—21no—
258C2×C3×D3—21no—
259C2×C3×D3—21no—
260C2×C3×D3—21no—
261C2×C3×D3—22no—
262C2×C3×D3—41no—
263C2×C3×D3—41no—
264C2×C3×D3Dic3⋊C412yes—
265C2×C3×D3C3×C4⋊C412yes—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—32no—
273C32×C4—41no—
274C32×C4—41no—
275C32×C4—41no—
276C32×C4—42no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—81no—
280C32×C4—81no—
281C32×C4—82no—
282C32×C4—121no—
283C32×C4—121no—
284C32×C4C2×Dic611yescharacteristic
285C32×C4C2×C4×D311yescharacteristic
286C32×C4C2×C4×D311yescharacteristic
287C32×C4D3×Q811yescharacteristic
288C32×C4C22×C3×C411yescharacteristic
289C32×C4C2×C3×Q811yescharacteristic
290C3⋊Dic3—41no—
291C3×Dic3—22no—
292C3×Dic3—42no—
293C3×Dic3—41no—
294C3×Dic3—41no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—81no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—241no—
301C3×Dic3Dic3⋊C412yes—
302C3×Dic3C3×C4⋊C412yes—
303C22×C8—91no—
304#117—31no—
305#117—91no—
306#107—91no—
307#107—91no—
308#106—31no—
309#106—91no—
310C33#42711yescharacteristic, complemented, Sylow
311C23×C3—31no—
312C23×C3—31no—
313C23×C3—61no—
314C22×D3C3×#6711yescharacteristic
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—31no—
323C2×C3×C4—31no—
324C2×C3×C4—32no—
325C2×C3×C4—32no—
326C2×C3×C4—32no—
327C2×C3×C4—32no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—61no—
335C2×C3×C4—62no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—121no—
341C2×C3×C4—181no—
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C22×C3×D311yescharacteristic
344C2×C3×C4C22×C3×D311yescharacteristic
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—121no—
350C2×Dic3C3×#6711yescharacteristic
351C4×D3C3×Dic612yes—
352C4×D3C3×C4×D312yes—
353C3×C8—32no—
354C3×C8—32no—
355C3×C8—62no—
356C3×C8—92no—
357C3⋊C8—32no—
358C3⋊C8—32no—
359C3⋊C8—62no—
360C3⋊C8C3×C4×D312yes—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—62no—
374C2×C32D3×C4⋊C411yescharacteristic
375C2×C32#91611yescharacteristic
376C2×C32#91611yescharacteristic
377C2×C32C2×Dic3⋊C411yescharacteristic
378C2×C32#95011yescharacteristic
379C2×C32#95011yescharacteristic
380C2×C32C2×C3×C4⋊C411yescharacteristic
381C2×C32C3×#13511yescharacteristic
382C2×C32C3×#13511yescharacteristic
383C3×D3—22no—
384C3×D3—22no—
385C3×D3—42no—
386C22×C4—31no—
387C2×C8—31no—
388C2×C8—94no—
389C2×C8—91no—
390C4⋊C4—31no—
391C4⋊C4—31no—
392C4⋊C4—91no—
393C4⋊C4C3×D3211yescharacteristic
394C22⋊C4—61no—
395C22⋊C4—181no—
396C42—61no—
397C42—181no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—32no—
402C22×C3—32no—
403C22×C3—41no—
404C22×C3—62no—
405C22×C3—61no—
406C22×C3—61no—
407C22×C3—61no—
408C22×C3—61no—
409C22×C3—121no—
410C22×C3—121no—
411C22×C3D3×#6711yescharacteristic
412C22×C3C3×D3×D411yescharacteristic
413C22×C3C2×C3×#6711yescharacteristic
414C2×D3—21no—
415C2×D3—21no—
416C2×D3C3×Dic3⋊C412yes—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—32no—
424C3×C4—32no—
425C3×C4—42no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—62no—
432C3×C4—81no—
433C3×C4—81no—
434C3×C4—82no—
435C3×C4—122no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—121no—
439C3×C4—121no—
440C3×C4—241no—
441C3×C4—361no—
442C3×C4D3×Dic611yescharacteristic
443C3×C4C4×D3211yescharacteristic
444C3×C4C2×C3×Dic611yescharacteristic
445C3×C4C2×C3×C4×D311yescharacteristic
446C3×C4C2×C3×C4×D311yescharacteristic
447C3×C4C3×D3×Q811yescharacteristic
448Dic3—41no—
449Dic3—41no—
450Dic3—121no—
451Dic3—121no—
452Dic3—241no—
453Dic3C3×Dic3⋊C412yes—
454C32—21no—
455C32—21no—
456C32—21no—
457C32—41no—
458C32#486011yescharacteristic, complemented
459C32#499811yescharacteristic, complemented
460C32C3×#42711yescharacteristic, complemented
461C23—31no—
462C2×C4—21no—
463C2×C4—31no—
464C2×C4—32no—
465C2×C4—32no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—181no—
469C2×C4C2×C3×D3211yescharacteristic
470C8—32no—
471C8—92no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—62no—
485C2×C3—62no—
486C2×C3—122no—
487C2×C3D3×Dic3⋊C411yescharacteristic
488C2×C3#6393511yescharacteristic
489C2×C3#6394711yescharacteristic
490C2×C3C3×D3×C4⋊C411yescharacteristic
491C2×C3C3×#91611yescharacteristic
492C2×C3C3×#91611yescharacteristic
493C2×C3C2×C3×Dic3⋊C411yescharacteristic
494C2×C3C3×#95011yescharacteristic
495C2×C3C3×#95011yescharacteristic
496D3—22no—
497C22—32no—
498C22—61no—
499C22—61no—
500C22C3×D3×#6711yescharacteristic
501C4—32no—
502C4—41no—
503C4—121no—
504C4—121no—
505C4—361no—
506C4C3×D3×Dic611yescharacteristic
507C4C3×C4×D3211yescharacteristic
508C3—21no—
509C3—21no—
510C3—21no—
511C3—41no—
512C3#1059057211yescharacteristic, direct product factor
513C3C3×#486011yescharacteristic, complemented
514C3C3×#499811yescharacteristic, complemented
515C2—62no—
516C2C3×D3×Dic3⋊C411yescharacteristic
517C2C3×#6393511yescharacteristic
518C2C3×#6394711yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#427
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#63634C2semidirect
5. C32×#808C2semidirect
6. C3×#63576C2semidirect
7. C3×#63573C2semidirect
10. #10590572C3direct (N × H)
22. C3×D3×C3⋊C8C4semidirect
34. #63634C2×C3semidirect
38. C3×#808D3semidirect
39. C3×#808C2×C3semidirect
45. #63576C2×C3semidirect
46. #63573C2×C3semidirect
108. D3×C3⋊C8C3×C4semidirect
138. #808C3×D3semidirect
310. C33#427semidirect
458. C32#4860semidirect
459. C32#4998semidirect
460. C32C3×#427semidirect
512. C3#10590572direct (N × H)
513. C3C3×#4860semidirect
514. C3C3×#4998semidirect

Maximal subgroups

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

subgroupindex
C32×#8922
C3×#638412
C2×C3×D3×C3⋊C82
C3×#636342
C32×#8082
C3×#635762
C3×#635732
C3×#49983
C3×#48603
#105905723