Finite Group Databasebeta

← C3×#10590560 #13157547

Subgroups — C3×#10590560 #13157547

Order 1728.

570 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#888C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3C3×#63637C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic
5C3×#63567C211yesmaximal, characteristic
6C3×#63565C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic
8C3×#5093—31nomaximal
9C3×#4792—31nomaximal
10#10590560C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6—21no—
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C32×C4⋊Q8—31no—
23C3×#944—31no—
24C3×#888—21no—
25C3×#888D311yescharacteristic
26C3×#888C2×C311yescharacteristic
27C3×#869—31no—
28#63866C2×C311yescharacteristic
29C32×#103—31no—
30#63637C2×C311yescharacteristic
31C3×#835—31no—
32C3×#816—31no—
33C3×C8×Dic3—31no—
34C3×#810—21no—
35C3×#810—31no—
36C3×#810—31no—
37C3×#810—31no—
38C3×#810—31no—
39C3×#810—61no—
40C3×#810D311yescharacteristic
41C3×#810C2×C311yescharacteristic
42C3×C4×C3⋊C8—31no—
43#63567C2×C311yescharacteristic
44#63565C2×C311yescharacteristic
45Dic3×C3⋊C8C2×C311yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C32×Dic6—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60C3×#487—91no—
61#5093—31no—
62#4792—31no—
63C2×C32×Q8—31no—
64C2×C32×Q8—61no—
65C2×#634C22×C311yescharacteristic
66C2×C3×Dic6—21no—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73C2×C3×Dic6C2×D311yescharacteristic
74C2×C3×Dic6C22×C311yescharacteristic
75C2×C32×C8—31no—
76C32×C4⋊C4—21no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—41no—
81C32×C4⋊C4—61no—
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85C32×C42—31no—
86C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C22×C311yescharacteristic
92C3×Dic3⋊C4—22no—
93C3×Dic3⋊C4—21no—
94C3×Dic3⋊C4—21no—
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C22×C311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106Dic3⋊Dic3—21no—
107Dic3⋊Dic3—21no—
108C22×C33C2×D411yescharacteristic
109C33×C4—41no—
110C33×C4#4111yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—41no—
113C32×Dic3—41no—
114C32×Dic3—41no—
115C32×Dic3SD1612yes—
116C32×Dic3Q1612yes—
117C3×C4⋊Q8—31no—
118C3×C4⋊Q8—31no—
119C3×C4⋊Q8—61no—
120C3×C4⋊Q8—91no—
121#944—31no—
122#888C3×D311yescharacteristic
123#869—31no—
124C3×#103—31no—
125C3×#103—31no—
126C3×#103—61no—
127C3×#103—91no—
128C3×#103—91no—
129C3×#103—91no—
130C3×C4×C8—91no—
131#835—31no—
132#816—31no—
133C8×Dic3—31no—
134#810—31no—
135#810—31no—
136#810—31no—
137#810—31no—
138#810—61no—
139#810C3×D311yescharacteristic
140C4×C3⋊C8—31no—
141C32×Q8—61no—
142C32×Q8—61no—
143C32×Q8—122no—
144C2×C32×C4—21no—
145C2×C32×C4—22no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—31no—
152C2×C32×C4—31no—
153C2×C32×C4—41no—
154C2×C32×C4—41no—
155C2×C32×C4—41no—
156C2×C32×C4—42no—
157C2×C32×C4—61no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C23×C311yescharacteristic
162C2×C3⋊Dic3—21no—
163#634—21no—
164#634—21no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—41no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—121no—
176C2×C3×Dic3#6711yescharacteristic
177C2×C3×Dic3#6711yescharacteristic
178C2×C3×Dic3C3×D411yescharacteristic
179C2×C3×Dic3C3×D411yescharacteristic
180C3×Dic6—22no—
181C3×Dic6—21no—
182C3×Dic6—21no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185C3×Dic6—41no—
186C3×Dic6—44no—
187C3×Dic6—42no—
188C3×Dic6—42no—
189C3×Dic6—62no—
190C3×Dic6—61no—
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—61no—
194C3×Dic6—121no—
195C32×C8—61no—
196C3⋊(C3⋊C8)—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—21no—
199C3×C3⋊C8—41no—
200C3×C3⋊C8—61no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—121no—
203#487—91noSylow
204C2×C33#12411yescharacteristic
205C2×C33C2×SD1611yescharacteristic
206C2×C33C2×Q1611yescharacteristic
207C2×C3×Q8—31no—
208C2×C3×Q8—31no—
209C2×C3×Q8—61no—
210C2×C3×Q8—61no—
211C2×C3×Q8—61no—
212C2×C3×Q8—62no—
213C2×C3×Q8—91no—
214C2×Dic6—21no—
215C2×Dic6—31no—
216C2×Dic6—31no—
217C2×Dic6—61no—
218C2×Dic6C2×C3×D311yescharacteristic
219C2×C3×C8—31no—
220C2×C3×C8—31no—
221C2×C3×C8—61no—
222C2×C3×C8—91no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—41no—
229C3×C4⋊C4—61no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—62no—
233C3×C4⋊C4—181no—
234C3×C4⋊C4—181no—
235C3×C4⋊C4D3211yescharacteristic
236C3×C4⋊C4C2×C3×D311yescharacteristic
237C3×C4⋊C4C2×C3×D311yescharacteristic
238C3×C42—31no—
239C3×C42—31no—
240C3×C42—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3—61no—
243C4⋊Dic3C2×C3×D311yescharacteristic
244Dic3⋊C4—21no—
245Dic3⋊C4—61no—
246Dic3⋊C4—61no—
247C4×Dic3C2×C3×D311yescharacteristic
248C2×C3⋊C8—31no—
249C2×C3⋊C8—31no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8C2×C3×D311yescharacteristic
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32D3×D411yescharacteristic
257C22×C32C2×#6711yescharacteristic
258C22×C32C2×C3×D411yescharacteristic
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—32no—
266C32×C4—32no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—41no—
270C32×C4—42no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—81no—
274C32×C4—81no—
275C32×C4—82no—
276C32×C4—121no—
277C32×C4—121no—
278C32×C4#23511yescharacteristic
279C32×C4#23711yescharacteristic
280C32×C4#23911yescharacteristic
281C32×C4#23911yescharacteristic
282C32×C4C3×#4111yescharacteristic
283C32×C4C3×#4111yescharacteristic
284C3⋊Dic3—41no—
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—42no—
288C3×Dic3—42no—
289C3×Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—41no—
293C3×Dic3—122no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3#21412yes—
297C3×Dic3#21612yes—
298C3×Dic3C3×SD1612yes—
299C3×Dic3C3×Q1612yes—
300C4⋊Q8—31no—
301C4⋊Q8—91no—
302#103—31no—
303#103—91no—
304#103—91no—
305#103—91no—
306C4×C8—91no—
307C33#48711yescharacteristic, complemented, Sylow
308C3×Q8—62no—
309C3×Q8—61no—
310C3×Q8—61no—
311C3×Q8—61no—
312C3×Q8—61no—
313C3×Q8—121no—
314C3×Q8—122no—
315C3×Q8—122no—
316C3×Q8—124no—
317C3×Q8—181no—
318C3×Q8—181no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—31no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—62no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—121no—
342C2×C3×C4—181no—
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C22×C3×D311yescharacteristic
345C2×C3×C4C22×C3×D311yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—121no—
351C2×Dic3C3×#6711yescharacteristic
352C2×Dic3C3×#6711yescharacteristic
353Dic6—21no—
354Dic6—21no—
355Dic6—42no—
356Dic6—62no—
357Dic6—61no—
358Dic6—61no—
359Dic6—61no—
360Dic6—61no—
361Dic6—121no—
362C3×C8—61no—
363C3×C8—61no—
364C3×C8—121no—
365C3×C8—181no—
366C3⋊C8—21no—
367C3⋊C8—61no—
368C3⋊C8—61no—
369C3⋊C8—121no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—41no—
382C2×C32#88511yescharacteristic
383C2×C32D3×SD1611yescharacteristic
384C2×C32D3×Q1611yescharacteristic
385C2×C32C2×#21411yescharacteristic
386C2×C32C2×#21611yescharacteristic
387C2×C32#94711yescharacteristic
388C2×C32C3×#12411yescharacteristic
389C2×C32C2×C3×SD1611yescharacteristic
390C2×C32C2×C3×Q1611yescharacteristic
391C2×Q8—31no—
392C2×Q8—61no—
393C2×Q8—91no—
394C2×C8—31no—
395C2×C8—91no—
396C4⋊C4—31no—
397C4⋊C4—61no—
398C4⋊C4—181no—
399C4⋊C4—181no—
400C4⋊C4C3×D3211yescharacteristic
401C42—31no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3D3×#6711yescharacteristic
407C22×C3C3×D3×D411yescharacteristic
408C22×C3C2×C3×#6711yescharacteristic
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—32no—
416C3×C4—32no—
417C3×C4—32no—
418C3×C4—32no—
419C3×C4—42no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4—62no—
426C3×C4—62no—
427C3×C4—81no—
428C3×C4—81no—
429C3×C4—82no—
430C3×C4—122no—
431C3×C4—122no—
432C3×C4—121no—
433C3×C4—121no—
434C3×C4—121no—
435C3×C4—121no—
436C3×C4—361no—
437C3×C4#379611yescharacteristic
438C3×C4#379911yescharacteristic
439C3×C4C3×#23511yescharacteristic
440C3×C4C3×#23711yescharacteristic
441C3×C4C3×#23911yescharacteristic
442C3×C4C3×#23911yescharacteristic
443Dic3—41no—
444Dic3—41no—
445Dic3—122no—
446Dic3—121no—
447Dic3—121no—
448Dic3C3×#21412yes—
449Dic3C3×#21612yes—
450C32—21no—
451C32—21no—
452C32—21no—
453C32—41no—
454C32#479211yescharacteristic, complemented
455C32#509311yescharacteristic, complemented
456C32C3×#48711yescharacteristic, complemented
457Q8—61no—
458Q8—61no—
459Q8—122no—
460Q8—181no—
461Q8—181no—
462C2×C4—21no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—181no—
468C2×C4C2×C3×D3211yescharacteristic
469C8—61no—
470C8—181no—
471C2×C3—21no—
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—41no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3#6390711yescharacteristic
484C2×C3D3×#21411yescharacteristic
485C2×C3D3×#21611yescharacteristic
486C2×C3C3×#88511yescharacteristic
487C2×C3C3×D3×SD1611yescharacteristic
488C2×C3C3×D3×Q1611yescharacteristic
489C2×C3C2×C3×#21411yescharacteristic
490C2×C3C2×C3×#21611yescharacteristic
491C2×C3C3×#94711yescharacteristic
492C22C3×D3×#6711yescharacteristic
493C4—32no—
494C4—32no—
495C4—41no—
496C4—121no—
497C4—121no—
498C4—361no—
499C4C3×#379611yescharacteristic
500C4C3×#379911yescharacteristic
501C3—21no—
502C3—21no—
503C3—21no—
504C3—41no—
505C3#1059056011yescharacteristic, direct product factor
506C3C3×#479211yescharacteristic, complemented
507C3C3×#509311yescharacteristic, complemented
508C2C3×#6390711yescharacteristic
509C2C3×D3×#21411yescharacteristic
510C2C3×D3×#21611yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#487
LCSG ▷ C2×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
10. #10590560C3direct (N × H)
307. C33#487semidirect
454. C32#4792semidirect
455. C32#5093semidirect
456. C32C3×#487semidirect
505. C3#10590560direct (N × H)
506. C3C3×#4792semidirect
507. C3C3×#5093semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
C3×#638662
C3×#636372
C32×#8102
C3×#635672
C3×#635652
C3×Dic3×C3⋊C82
C3×#50933
C3×#47923
#105905603