Finite Group Databasebeta

← C3×#10591411 #13158489

Subgroups — C3×#10591411 #13158489

Order 1728.

656 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D4×Dic3C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic
4C32×#832C211yesmaximal, characteristic
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63574C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5009—31nomaximal
9C3×#4766—31nomaximal
10#10591411C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C22×C32×Dic3—21no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×#217—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C32×C4×D4—31no—
23C3×D4×Dic3—21no—
24C3×D4×Dic3D311yescharacteristic
25C3×D4×Dic3C2×C311yescharacteristic
26C3×#890—31no—
27C3×#870—31no—
28#63869C2×C311yescharacteristic, complemented
29C32×#102—31no—
30#63659C2×C311yescharacteristic
31C3×#832—21no—
32C3×#832—31no—
33C3×#832—31no—
34C3×#832—61no—
35C3×#832D311yescharacteristic
36C3×#832C2×C311yescharacteristic
37C3×#818—31no—
38C3×Dic3⋊C8—31no—
39C3×#808—31no—
40C3×#808—31no—
41C3×#807—31no—
42C3×C4⋊(C3⋊C8)—31no—
43#63575C2×C311yescharacteristic, complemented
44#63574C2×C311yescharacteristic, complemented
45Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
46C23×C33—21no—
47C33×D4—21no—
48C33×D4Q812yes—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—42no—
54C2×C32×Dic3D411yescharacteristic
55C2×C32×Dic3D411yescharacteristic
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58C3×#478—91no—
59#5009—31no—
60#4766—31no—
61C2×C32×D4—21no—
62C2×C32×D4—21no—
63C2×C32×D4—21no—
64C2×C32×D4—41no—
65C2×C32×D4C2×D311yescharacteristic
66C2×C32×D4C2×D311yescharacteristic
67C2×C32×D4C22×C311yescharacteristic
68C22×C32×C4—61no—
69C22×C3×Dic3—22no—
70C22×C3×Dic3—21no—
71C22×C3×Dic3—21no—
72C2×C32×C8—31no—
73C32×C4⋊C4—31no—
74C32×C4⋊C4—31no—
75C32×C22⋊C4—61no—
76C32×C42—31no—
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#217—22no—
80C3×#217—21no—
81C3×#217—21no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C22×C311yescharacteristic
91C3×C4⋊Dic3C22×C311yescharacteristic
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C22×C311yescharacteristic
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C22×C311yescharacteristic
104#3724—21no—
105Dic3⋊Dic3—21no—
106C22×C33—22no—
107C22×C33—41no—
108C22×C33—41no—
109C22×C33C2×D411yescharacteristic
110C33×C4C2×Q811yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—41no—
113C32×Dic3—21no—
114C32×Dic3—21no—
115C32×Dic3—41no—
116C32×Dic3—41no—
117C3×#125—91no—
118C3×C4×D4—31no—
119C3×C4×D4—31no—
120C3×C4×D4—61no—
121D4×Dic3C3×D311yescharacteristic
122#890—31no—
123#870—31no—
124C3×#107—91no—
125C3×#106—91no—
126C3×C4⋊C8—91no—
127C3×#102—31no—
128C3×#102—31no—
129C3×#102—61no—
130C3×#102—91no—
131#832—31no—
132#832—31no—
133#832—61no—
134#832C3×D311yescharacteristic
135#818—31no—
136Dic3⋊C8—31no—
137#808—31no—
138#808—31no—
139#807—31no—
140C4⋊(C3⋊C8)—31no—
141C23×C32—21no—
142C23×C32—22no—
143C23×C32—21no—
144C23×C32—21no—
145C23×C32—41no—
146C23×C32—41no—
147C23×C32—42no—
148C32×D4—21no—
149C32×D4—22no—
150C32×D4—21no—
151C32×D4—21no—
152C32×D4—22no—
153C32×D4—22no—
154C32×D4—22no—
155C32×D4—42no—
156C32×D4—41no—
157C32×D4—41no—
158C32×D4—42no—
159C32×D4Dic612yes—
160C32×D4Dic612yes—
161C32×D4C3×Q812yes—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—31no—
166C2×C32×C4—31no—
167C2×C32×C4—41no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C32×C4—122no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C23×C311yescharacteristic
174C2×C3⋊Dic3—21no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—44no—
184C2×C3×Dic3—42no—
185C2×C3×Dic3—42no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—121no—
189C2×C3×Dic3#6711yescharacteristic
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3C3×D411yescharacteristic
192C2×C3×Dic3C3×D411yescharacteristic
193C32×C8—61no—
194C3⋊(C3⋊C8)—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—41no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—61no—
200C3×C3⋊C8—121no—
201#478—91noSylow
202C2×C33—42no—
203C2×C33C22⋊Q811yescharacteristic
204C2×C33#13511yescharacteristic
205C2×C33AGL(1,ℤ/8)11yescharacteristic
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—41no—
210C2×C3×D4D3211yescharacteristic
211C2×C3×D4C2×C3×D311yescharacteristic
212C2×C3×D4C2×C3×D311yescharacteristic
213C22×C3×C4—61no—
214C22×C3×C4—61no—
215C22×C3×C4—62no—
216C22×Dic3—21no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8—61no—
220C2×C3×C8—91no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—181no—
229C3×C4⋊C4—181no—
230C3×C22⋊C4—61no—
231C3×C22⋊C4—61no—
232C3×C22⋊C4—62no—
233C3×C42—31no—
234C3×C42—31no—
235C3×C42—61no—
236#217—21no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3C2×C3×D311yescharacteristic
242C4⋊Dic3C2×C3×D311yescharacteristic
243Dic3⋊C4—61no—
244Dic3⋊C4—61no—
245Dic3⋊C4—61no—
246C4×Dic3C2×C3×D311yescharacteristic
247C2×C3⋊C8—31no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8C2×C3×D311yescharacteristic
251C22×C32—22no—
252C22×C32—24no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32—42no—
260C22×C32—42no—
261C22×C32—44no—
262C22×C32—41no—
263C22×C32—41no—
264C22×C32—42no—
265C22×C32—42no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32—41no—
270C22×C32—81no—
271C22×C32—81no—
272C22×C32—81no—
273C22×C32—81no—
274C22×C32—82no—
275C22×C32—82no—
276C22×C32D3×D411yescharacteristic
277C22×C32C2×#6711yescharacteristic
278C22×C32C2×C3×D411yescharacteristic
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—61no—
288C32×C4—61no—
289C32×C4—121no—
290C32×C4—121no—
291C32×C4C2×Dic611yescharacteristic
292C32×C4C2×Dic611yescharacteristic
293C32×C4#23511yescharacteristic
294C32×C4#23711yescharacteristic
295C32×C4C2×C3×Q811yescharacteristic
296C32×C4C3×#4111yescharacteristic
297C3⋊Dic3—41no—
298C3×Dic3—22no—
299C3×Dic3—21no—
300C3×Dic3—21no—
301C3×Dic3—21no—
302C3×Dic3—21no—
303C3×Dic3—41no—
304C3×Dic3—42no—
305C3×Dic3—41no—
306C3×Dic3—41no—
307C3×Dic3—41no—
308C3×Dic3—41no—
309C3×Dic3—81no—
310C3×Dic3—121no—
311C3×Dic3—121no—
312C3×Dic3—241no—
313#125—91no—
314C4×D4—31no—
315#107—91no—
316#106—91no—
317C4⋊C8—91no—
318#102—31no—
319#102—91no—
320C33#47811yescharacteristic, complemented, Sylow
321C23×C3—22no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—42no—
328C3×D4—22no—
329C3×D4—21no—
330C3×D4—21no—
331C3×D4—21no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4#62712yes—
340C3×D4C3×Dic612yes—
341C3×D4C3×Dic612yes—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—41no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—121no—
358C2×C3×C4—124no—
359C2×C3×C4—122no—
360C2×C3×C4—122no—
361C2×C3×C4—181no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C22×C3×D311yescharacteristic
364C2×C3×C4C22×C3×D311yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—42no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—121no—
371C2×Dic3C3×#6711yescharacteristic
372C2×Dic3C3×#6711yescharacteristic
373C3×C8—61no—
374C3×C8—61no—
375C3×C8—121no—
376C3×C8—181no—
377C3⋊C8—21no—
378C3⋊C8—61no—
379C3⋊C8—61no—
380C3⋊C8—121no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—42no—
394C2×C32—44no—
395C2×C32—42no—
396C2×C32—42no—
397C2×C32—82no—
398C2×C32—82no—
399C2×C32—84no—
400C2×C32#87811yescharacteristic
401C2×C32#91211yescharacteristic
402C2×C32#91411yescharacteristic
403C2×C32#92411yescharacteristic
404C2×C32#93211yescharacteristic
405C2×C32#95011yescharacteristic
406C2×C32C3×C22⋊Q811yescharacteristic
407C2×C32C3×#13511yescharacteristic
408C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
409C2×D4C3×D3211yescharacteristic
410C22×C4—61no—
411C2×C8—31no—
412C2×C8—91no—
413C4⋊C4—31no—
414C4⋊C4—31no—
415C4⋊C4—91no—
416C4⋊C4—181no—
417C4⋊C4—181no—
418C22⋊C4—61no—
419C42—31no—
420C22×C3—24no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—44no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—41no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3D3×#6711yescharacteristic
446C22×C3C3×D3×D411yescharacteristic
447C22×C3C2×C3×#6711yescharacteristic
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—62no—
457C3×C4—61no—
458C3×C4—61no—
459C3×C4—61no—
460C3×C4—61no—
461C3×C4—121no—
462C3×C4—122no—
463C3×C4—121no—
464C3×C4—121no—
465C3×C4—121no—
466C3×C4—121no—
467C3×C4—241no—
468C3×C4—361no—
469C3×C4#380411yescharacteristic
470C3×C4C2×#62711yescharacteristic
471C3×C4C2×C3×Dic611yescharacteristic
472C3×C4C2×C3×Dic611yescharacteristic
473C3×C4C3×#23511yescharacteristic
474C3×C4C3×#23711yescharacteristic
475Dic3—21no—
476Dic3—21no—
477Dic3—41no—
478Dic3—41no—
479Dic3—121no—
480Dic3—121no—
481Dic3—241no—
482C32—21no—
483C32—21no—
484C32—21no—
485C32—41no—
486C32#476611yescharacteristic, complemented
487C32#500911yescharacteristic, complemented
488C32C3×#47811yescharacteristic, complemented
489C23—21no—
490D4—21no—
491D4C3×#62712yes—
492C2×C4—31no—
493C2×C4—31no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—122no—
497C2×C4—181no—
498C2×C4C2×C3×D3211yescharacteristic
499C8—61no—
500C8—181no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—44no—
511C2×C3—42no—
512C2×C3—42no—
513C2×C3—42no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—41no—
517C2×C3—82no—
518C2×C3—82no—
519C2×C3—84no—
520C2×C3#6393311yescharacteristic
521C2×C3#6393411yescharacteristic
522C2×C3#6396411yescharacteristic
523C2×C3C3×#87811yescharacteristic
524C2×C3C3×#91211yescharacteristic
525C2×C3C3×#91411yescharacteristic
526C2×C3C3×#92411yescharacteristic
527C2×C3C3×#93211yescharacteristic
528C2×C3C3×#95011yescharacteristic
529C22—22no—
530C22—41no—
531C22—41no—
532C22C3×D3×#6711yescharacteristic
533C4—61no—
534C4—61no—
535C4—121no—
536C4—121no—
537C4—361no—
538C4C3×#380411yescharacteristic
539C4C2×C3×#62711yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059141111yescharacteristic, direct product factor
545C3C3×#476611yescharacteristic, complemented
546C3C3×#500911yescharacteristic, complemented
547C2—42no—
548C2C3×#6393311yescharacteristic
549C2C3×#6393411yescharacteristic
550C2C3×#6396411yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63869C2semidirect
5. C3×#63575C2semidirect
6. C3×#63574C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10591411C3direct (N × H)
28. #63869C2×C3semidirect
43. #63575C2×C3semidirect
44. #63574C2×C3semidirect
45. Dic3⋊(C3⋊C8)C2×C3semidirect
320. C33#478semidirect
486. C32#4766semidirect
487. C32#5009semidirect
488. C32C3×#478semidirect
544. C3#10591411direct (N × H)
545. C3C3×#4766semidirect
546. C3C3×#5009semidirect

Maximal subgroups

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

subgroupindex
C32×D4×Dic32
C3×#638692
C3×#636592
C32×#8322
C3×#635752
C3×#635742
C3×Dic3⋊(C3⋊C8)2
C3×#50093
C3×#47663
#105914113