Finite Group Databasebeta

← C3×#10590271 #13157285

Subgroups — C3×#10590271 #13157285

Order 1728.

626 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64079C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic
3C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C3×#63575C211yesmaximal, characteristic, complemented
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic
7C3×#63563C211yesmaximal, characteristic
8C3×#5009—31nomaximal
9C3×#4847—31nomaximal
10#10590271C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C33×C42C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×#3750—21no—
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C32×C4×D4—31no—
23#64079C2×C311yescharacteristic, complemented
24C3×C4×D12—21no—
25C3×C4×D12D311yescharacteristic
26C3×C4×D12C2×C311yescharacteristic
27C3×#870—31no—
28C3×#870—31no—
29C3×#870—61no—
30C32×C4⋊C8—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#821—31no—
34C3×#821—31no—
35C3×#818—31no—
36C3×#817—31no—
37C3×#808—31no—
38C3×#807—31no—
39C3×C4⋊(C3⋊C8)—21no—
40C3×C4⋊(C3⋊C8)D311yescharacteristic, complemented
41C3×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
42#63575C2×C311yescharacteristic, complemented
43#63573C2×C311yescharacteristic, complemented
44#63563C2×C311yescharacteristic
45#63563C2×C311yescharacteristic
46C22×C32×D3—21no—
47C2×C33×C4D411yescharacteristic
48C2×C33×C4D411yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12Q812yes—
55C32×C4×D3—42no—
56C32×C3⋊C8—21no—
57C32×C3⋊C8—21no—
58C3×#478—91no—
59#5009—31no—
60#4847—31no—
61C2×C32×D4—31no—
62C22×C32×C4—61no—
63C2×C3×D12—21no—
64C2×C3×D12C2×D311yescharacteristic
65C2×C3×D12C22×C311yescharacteristic
66C2×C3×C4×D3—22no—
67C2×C3×C4×D3—21no—
68C2×C3×C4×D3—21no—
69C2×C32×C8—31no—
70C2×C32×C8—31no—
71C32×C4⋊C4—31no—
72C32×C22⋊C4—61no—
73C32×C42—21no—
74C32×C42—21no—
75C32×C42—21no—
76C32×C42—41no—
77C32×C42C2×D311yescharacteristic
78C32×C42C2×D311yescharacteristic
79C32×C42C22×C311yescharacteristic
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C4⋊(C3⋊Dic3)C22×C311yescharacteristic
82#3750—21no—
83#3750—21no—
84C3×(C2×D3)⋊C4—22no—
85C3×(C2×D3)⋊C4—21no—
86C3×(C2×D3)⋊C4—21no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×Dic3⋊C4—62no—
97C3×Dic3⋊C4—62no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×Dic3⋊C4—61no—
101C3×Dic3⋊C4—61no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107C2×C3×C3⋊C8C22×C311yescharacteristic
108C22×C33C2×D411yescharacteristic
109C2×C32×D3—22no—
110C2×C32×D3—41no—
111C2×C32×D3—41no—
112C33×C4—21no—
113C33×C4—21no—
114C33×C4C2×Q811yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—41no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119C3×#125—91no—
120C3×C4×D4—31no—
121C3×C4×D4—31no—
122C3×C4×D4—61no—
123C4×D12C3×D311yescharacteristic
124#870—31no—
125#870—31no—
126#870—61no—
127C3×#107—91no—
128C3×#106—91no—
129C3×C4⋊C8—31no—
130C3×C4⋊C8—31no—
131C3×C4⋊C8—61no—
132C3×#102—91no—
133C3×#102—91no—
134#832—31no—
135#832—31no—
136#821—31no—
137#821—31no—
138#818—31no—
139#817—31no—
140#808—31no—
141#807—31no—
142C4⋊(C3⋊C8)C3×D311yescharacteristic, complemented
143C23×C32—61no—
144C22×C3×D3—22no—
145C22×C3×D3—21no—
146C22×C3×D3—21no—
147C32×D4—32no—
148C32×D4—61no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—41no—
159C2×C32×C4—41no—
160C2×C32×C4—41no—
161C2×C32×C4—61no—
162C2×C32×C4—122no—
163C2×C32×C4D1211yescharacteristic
164C2×C32×C4D1211yescharacteristic
165C2×C32×C4#6711yescharacteristic
166C2×C32×C4#6711yescharacteristic
167C2×C32×C4C3×D411yescharacteristic
168C2×C32×C4C3×D411yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C23×C311yescharacteristic
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—21no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C3×D12—22no—
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—22no—
187C3×D12Dic612yes—
188C3×D12C3×Q812yes—
189C3×C4×D3—44no—
190C3×C4×D3—42no—
191C3×C4×D3—42no—
192C32×C8—61no—
193C32×C8—61no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—41no—
199C3×C3⋊C8—41no—
200#478—91noSylow
201C2×C33C22⋊Q811yescharacteristic
202C2×C33#13511yescharacteristic
203C2×C33AGL(1,ℤ/8)11yescharacteristic
204C32×D3—42no—
205C2×C3×D4—31no—
206C2×C3×D4—31no—
207C2×C3×D4—61no—
208C22×C3×C4—62no—
209C22×C3×C4—61no—
210C22×C3×C4—61no—
211C2×D12C2×C3×D311yescharacteristic
212C2×C4×D3—21no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—61no—
218C2×C3×C8—61no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4—91no—
224C3×C4⋊C4—181no—
225C3×C4⋊C4—181no—
226C3×C22⋊C4—62no—
227C3×C22⋊C4—61no—
228C3×C22⋊C4—61no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—41no—
233C3×C42D3211yescharacteristic
234C3×C42C2×C3×D311yescharacteristic
235C3×C42C2×C3×D311yescharacteristic
236(C2×D3)⋊C4—21no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—31no—
241C4⋊Dic3—61no—
242C4⋊Dic3—61no—
243C4⋊Dic3C2×C3×D311yescharacteristic
244Dic3⋊C4—62no—
245Dic3⋊C4—62no—
246Dic3⋊C4—61no—
247Dic3⋊C4—61no—
248Dic3⋊C4—61no—
249Dic3⋊C4—61no—
250C2×C3⋊C8C2×C3×D311yescharacteristic
251C2×C3⋊C8C2×C3×D311yescharacteristic
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32—62no—
257C22×C32—121no—
258C22×C32—121no—
259C22×C32C2×D1211yescharacteristic
260C22×C32C2×#6711yescharacteristic
261C22×C32C2×C3×D411yescharacteristic
262C2×C3×D3—24no—
263C2×C3×D3—22no—
264C2×C3×D3—22no—
265C2×C3×D3—42no—
266C2×C3×D3—42no—
267C2×C3×D3—41no—
268C2×C3×D3—41no—
269C2×C3×D3—41no—
270C2×C3×D3—41no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—22no—
274C32×C4—22no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—41no—
291C32×C4—42no—
292C32×C4—42no—
293C32×C4—121no—
294C32×C4C2×Dic611yescharacteristic
295C32×C4#23511yescharacteristic
296C32×C4#23711yescharacteristic
297C32×C4D3×Q811yescharacteristic
298C32×C4C2×C3×Q811yescharacteristic
299C32×C4C3×#4111yescharacteristic
300C3⋊Dic3—41no—
301C3⋊Dic3—41no—
302C3×Dic3—42no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—122no—
306C3×Dic3—122no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311#125—91no—
312C4×D4—31no—
313#107—91no—
314#106—91no—
315C4⋊C8—31no—
316#102—91no—
317#102—91no—
318C33#47811yescharacteristic, complemented, Sylow
319C23×C3—62no—
320C23×C3—61no—
321C23×C3—61no—
322C22×D3—21no—
323C3×D4—32no—
324C3×D4—32no—
325C3×D4—62no—
326C3×D4—62no—
327C3×D4—61no—
328C3×D4—61no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—62no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—124no—
345C2×C3×C4—122no—
346C2×C3×C4—122no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4#62611yescharacteristic
350C2×C3×C4#62611yescharacteristic
351C2×C3×C4C3×D1211yescharacteristic
352C2×C3×C4C3×D1211yescharacteristic
353C2×C3×C4C3×#6711yescharacteristic
354C2×C3×C4C3×#6711yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C22×C3×D311yescharacteristic
357C2×C3×C4C22×C3×D311yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365D12—21no—
366D12C3×Dic612yes—
367C4×D3—42no—
368C3×C8—61no—
369C3×C8—61no—
370C3×C8—61no—
371C3×C8—61no—
372C3×C8—121no—
373C3×C8—121no—
374C3⋊C8—21no—
375C3⋊C8—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—41no—
388C2×C32—122no—
389C2×C32#89711yescharacteristic
390C2×C32#90411yescharacteristic
391C2×C32#90811yescharacteristic
392C2×C32#92411yescharacteristic
393C2×C32#93211yescharacteristic
394C2×C32#95011yescharacteristic
395C2×C32C3×C22⋊Q811yescharacteristic
396C2×C32C3×#13511yescharacteristic
397C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
398C3×D3—44no—
399C3×D3—42no—
400C3×D3—42no—
401C2×D4—31no—
402C22×C4—61no—
403C2×C8—31no—
404C2×C8—31no—
405C4⋊C4—31no—
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—181no—
409C4⋊C4—181no—
410C22⋊C4—61no—
411C42C3×D3211yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—64no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—122no—
420C22×C3—122no—
421C22×C3—121no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3C2×#62611yescharacteristic
426C22×C3C2×C3×D1211yescharacteristic
427C22×C3C2×C3×#6711yescharacteristic
428C2×D3—22no—
429C2×D3—41no—
430C2×D3—41no—
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—122no—
454C3×C4—121no—
455C3×C4—121no—
456C3×C4—361no—
457C3×C4—361no—
458C3×C4D3×Dic611yescharacteristic
459C3×C4#379511yescharacteristic
460C3×C4C2×C3×Dic611yescharacteristic
461C3×C4C3×#23511yescharacteristic
462C3×C4C3×#23711yescharacteristic
463C3×C4C3×D3×Q811yescharacteristic
464Dic3—41no—
465Dic3—122no—
466Dic3—122no—
467Dic3—121no—
468Dic3—121no—
469Dic3—121no—
470Dic3—121no—
471C32—21no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32#484711yescharacteristic, complemented
476C32#500911yescharacteristic, complemented
477C32C3×#47811yescharacteristic, complemented
478C23—61no—
479D4—32no—
480D4—61no—
481C2×C4—61no—
482C2×C4—122no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C4C3×#62611yescharacteristic
486C2×C4C3×#62611yescharacteristic
487C2×C4C2×C3×D3211yescharacteristic
488C8—61no—
489C8—61no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—124no—
503C2×C3—122no—
504C2×C3—122no—
505C2×C3#6382911yescharacteristic
506C2×C3#6383011yescharacteristic
507C2×C3#6385711yescharacteristic
508C2×C3C3×#89711yescharacteristic
509C2×C3C3×#90411yescharacteristic
510C2×C3C3×#90811yescharacteristic
511C2×C3C3×#92411yescharacteristic
512C2×C3C3×#93211yescharacteristic
513C2×C3C3×#95011yescharacteristic
514D3—42no—
515C22—62no—
516C22—121no—
517C22—121no—
518C22C2×C3×#62611yescharacteristic
519C4—21no—
520C4—21no—
521C4—121no—
522C4—361no—
523C4—361no—
524C4C3×D3×Dic611yescharacteristic
525C4C3×#379511yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059027111yescharacteristic, direct product factor
531C3C3×#484711yescharacteristic, complemented
532C3C3×#500911yescharacteristic, complemented
533C2—122no—
534C2C3×#6382911yescharacteristic
535C2C3×#6383011yescharacteristic
536C2C3×#6385711yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC3×C42D32
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×#64079C2semidirect
3. C32×C4⋊(C3⋊C8)C2semidirect
4. C3×#63575C2semidirect
5. C3×#63573C2semidirect
10. #10590271C3direct (N × H)
23. #64079C2×C3semidirect
40. C3×C4⋊(C3⋊C8)D3semidirect
41. C3×C4⋊(C3⋊C8)C2×C3semidirect
42. #63575C2×C3semidirect
43. #63573C2×C3semidirect
142. C4⋊(C3⋊C8)C3×D3semidirect
318. C33#478semidirect
475. C32#4847semidirect
476. C32#5009semidirect
477. C32C3×#478semidirect
530. C3#10590271direct (N × H)
531. C3C3×#4847semidirect
532. C3C3×#5009semidirect

Maximal subgroups

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

subgroupindex
C3×#640792
C32×C4×D122
C32×C4⋊(C3⋊C8)2
C3×#635752
C3×#635732
C3×#635632
C3×#50093
C3×#48473
#105902713