Finite Group Databasebeta

← C3×#10591408 #13158500

Subgroups — C3×#10591408 #13158500

Order 1728.

680 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D4×Dic3C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic
4C32×#832C211yesmaximal, characteristic
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5008—31nomaximal
9C3×#4761—31nomaximal
10#10591408C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C22×C32×Dic3—21no—
13C2×C3×#627—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#217—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×C4×D4—31no—
23C3×D4×Dic3—21no—
24C3×D4×Dic3D311yescharacteristic
25C3×D4×Dic3C2×C311yescharacteristic
26C3×#888—31no—
27C3×#869—31no—
28#63918C2×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×#817—31no—
38C3×Dic3⋊C8—31no—
39C3×#808—31no—
40C3×#808—31no—
41C3×#808—31no—
42C3×C4⋊(C3⋊C8)—31no—
43#63574C2×C311yescharacteristic, complemented
44#63573C2×C311yescharacteristic, complemented
45Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
46C23×C33—21no—
47C33×D4—21no—
48C33×D4Q812yescomplemented
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×#627—42no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59C3×#476—91no—
60#5008—31no—
61#4761—31no—
62C2×C32×D4—21no—
63C2×C32×D4—21no—
64C2×C32×D4—21no—
65C2×C32×D4—41no—
66C2×C32×D4C2×D311yescharacteristic
67C2×C32×D4C2×D311yescharacteristic
68C2×C32×D4C22×C311yescharacteristic
69C22×C32×C4—61no—
70C22×C3×Dic3—22no—
71C22×C3×Dic3—21no—
72C22×C3×Dic3—21no—
73C2×C3×Dic6—61no—
74C2×C3×Dic6—61no—
75C2×#627—21no—
76C2×C32×C8—31no—
77C32×C4⋊C4—31no—
78C32×C4⋊C4—31no—
79C32×C22⋊C4—61no—
80C32×C42—31no—
81C4⋊(C3⋊Dic3)C22×C311yescharacteristic
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C3×#217—22no—
84C3×#217—21no—
85C3×#217—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C22×C311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
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—
107C22×C33—22no—
108C22×C33—41no—
109C22×C33—41no—
110C22×C33C2×D411yescharacteristic
111C33×C4C2×Q811yescharacteristic
112C33×C4#4111yescharacteristic
113C3×C3⋊Dic3—41no—
114C32×Dic3—21no—
115C32×Dic3—41no—
116C32×Dic3—41no—
117C32×Dic3SD1612yes—
118C3×C4⋊Q8—91no—
119C3×C4×D4—31no—
120C3×C4×D4—31no—
121C3×C4×D4—61no—
122D4×Dic3C3×D311yescharacteristic
123#888—31no—
124#869—31no—
125C3×#106—91no—
126C3×#106—91no—
127C3×C4⋊C8—91no—
128C3×#102—31no—
129C3×#102—31no—
130C3×#102—61no—
131C3×#102—91no—
132#832—31no—
133#832—31no—
134#832—61no—
135#832C3×D311yescharacteristic
136#817—31no—
137Dic3⋊C8—31no—
138#808—31no—
139#808—31no—
140#808—31no—
141C4⋊(C3⋊C8)—31no—
142C23×C32—21no—
143C23×C32—22no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—41no—
147C23×C32—41no—
148C23×C32—42no—
149C32×D4—21no—
150C32×D4—22no—
151C32×D4—21no—
152C32×D4—21no—
153C32×D4—22no—
154C32×D4—22no—
155C32×D4—22no—
156C32×D4—42no—
157C32×D4—41no—
158C32×D4—41no—
159C32×D4—42no—
160C32×D4Dic612yescomplemented
161C32×D4Dic612yescomplemented
162C32×D4C3×Q812yescomplemented
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—31no—
167C2×C32×C4—31no—
168C2×C32×C4—41no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4—122no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C23×C311yescharacteristic
175C2×C3⋊Dic3—21no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—44no—
185C2×C3×Dic3—42no—
186C2×C3×Dic3—42no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—121no—
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3#6711yescharacteristic
192C2×C3×Dic3C3×D411yescharacteristic
193C2×C3×Dic3C3×D411yescharacteristic
194C3×Dic6—122no—
195C3×Dic6—122no—
196#627—42no—
197C32×C8—61no—
198C3⋊(C3⋊C8)—21no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—41no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—121no—
205#476—91noSylow
206C2×C33—42no—
207C2×C33C22⋊Q811yescharacteristic
208C2×C33C2×SD1611yescharacteristic
209C2×C33AGL(1,ℤ/8)11yescharacteristic
210C2×C3×Q8—181no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—41no—
215C2×C3×D4D3211yescharacteristic
216C2×C3×D4C2×C3×D311yescharacteristic
217C2×C3×D4C2×C3×D311yescharacteristic
218C22×C3×C4—61no—
219C22×C3×C4—61no—
220C22×C3×C4—62no—
221C22×Dic3—21no—
222C2×Dic6—61no—
223C2×Dic6—61no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—61no—
227C2×C3×C8—91no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—181no—
236C3×C22⋊C4—61no—
237C3×C22⋊C4—61no—
238C3×C22⋊C4—62no—
239C3×C42—31no—
240C3×C42—31no—
241C3×C42—61no—
242#217—21no—
243C4⋊Dic3—31no—
244C4⋊Dic3—31no—
245C4⋊Dic3—61no—
246C4⋊Dic3—61no—
247C4⋊Dic3C2×C3×D311yescharacteristic
248C4⋊Dic3C2×C3×D311yescharacteristic
249Dic3⋊C4—61no—
250C4×Dic3C2×C3×D311yescharacteristic
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—61no—
254C2×C3⋊C8C2×C3×D311yescharacteristic
255C22×C32—22no—
256C22×C32—24no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32—42no—
264C22×C32—42no—
265C22×C32—44no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—42no—
269C22×C32—42no—
270C22×C32—41no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32—41no—
274C22×C32—81no—
275C22×C32—81no—
276C22×C32—81no—
277C22×C32—81no—
278C22×C32—82no—
279C22×C32—82no—
280C22×C32D3×D411yescharacteristic
281C22×C32C2×#6711yescharacteristic
282C22×C32C2×C3×D411yescharacteristic
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—32no—
290C32×C4—41no—
291C32×C4—41no—
292C32×C4—61no—
293C32×C4—121no—
294C32×C4—121no—
295C32×C4C2×Dic611yescharacteristic
296C32×C4C2×Dic611yescharacteristic
297C32×C4#23511yescharacteristic
298C32×C4#23711yescharacteristic
299C32×C4C2×C3×Q811yescharacteristic
300C32×C4C3×#4111yescharacteristic
301C3⋊Dic3—41no—
302C3×Dic3—21no—
303C3×Dic3—21no—
304C3×Dic3—22no—
305C3×Dic3—41no—
306C3×Dic3—42no—
307C3×Dic3—41no—
308C3×Dic3—41no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C3×Dic3—81no—
312C3×Dic3—121no—
313C3×Dic3—121no—
314C3×Dic3—241no—
315C3×Dic3#21412yes—
316C3×Dic3C3×SD1612yes—
317C4⋊Q8—91no—
318C4×D4—31no—
319#106—91no—
320#106—91no—
321C4⋊C8—91no—
322#102—31no—
323#102—91no—
324C33#47611yescharacteristic, complemented, Sylow
325C23×C3—22no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—42no—
332C3×Q8—362no—
333C3×D4—22no—
334C3×D4—21no—
335C3×D4—21no—
336C3×D4—21no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—22no—
340C3×D4—41no—
341C3×D4—41no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4#62712yescomplemented
345C3×D4C3×Dic612yescomplemented
346C3×D4C3×Dic612yescomplemented
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—41no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—62no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—121no—
363C2×C3×C4—122no—
364C2×C3×C4—122no—
365C2×C3×C4—124no—
366C2×C3×C4—181no—
367C2×C3×C4C2×D3211yescharacteristic
368C2×C3×C4C22×C3×D311yescharacteristic
369C2×C3×C4C22×C3×D311yescharacteristic
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—42no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—121no—
376C2×Dic3C3×#6711yescharacteristic
377C2×Dic3C3×#6711yescharacteristic
378Dic6—122no—
379Dic6—122no—
380C3×C8—61no—
381C3×C8—61no—
382C3×C8—121no—
383C3×C8—181no—
384C3⋊C8—21no—
385C3⋊C8—61no—
386C3⋊C8—61no—
387C3⋊C8—121no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32—42no—
401C2×C32—44no—
402C2×C32—42no—
403C2×C32—42no—
404C2×C32—82no—
405C2×C32—82no—
406C2×C32—84no—
407C2×C32#87811yescharacteristic
408C2×C32#91111yescharacteristic
409C2×C32D3×SD1611yescharacteristic
410C2×C32#92411yescharacteristic
411C2×C32C2×#21411yescharacteristic
412C2×C32#94911yescharacteristic
413C2×C32C3×C22⋊Q811yescharacteristic
414C2×C32C2×C3×SD1611yescharacteristic
415C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
416C2×Q8—181no—
417C2×D4C3×D3211yescharacteristic
418C22×C4—61no—
419C2×C8—31no—
420C2×C8—91no—
421C4⋊C4—31no—
422C4⋊C4—31no—
423C4⋊C4—91no—
424C4⋊C4—181no—
425C22⋊C4—61no—
426C42—31no—
427C22×C3—24no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—44no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3D3×#6711yescharacteristic
453C22×C3C3×D3×D411yescharacteristic
454C22×C3C2×C3×#6711yescharacteristic
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—32no—
462C3×C4—32no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—62no—
466C3×C4—61no—
467C3×C4—61no—
468C3×C4—121no—
469C3×C4—122no—
470C3×C4—121no—
471C3×C4—121no—
472C3×C4—121no—
473C3×C4—121no—
474C3×C4—241no—
475C3×C4—361no—
476C3×C4#380411yescharacteristic
477C3×C4C2×#62711yescharacteristic
478C3×C4C2×C3×Dic611yescharacteristic
479C3×C4C2×C3×Dic611yescharacteristic
480C3×C4C3×#23511yescharacteristic
481C3×C4C3×#23711yescharacteristic
482Dic3—21no—
483Dic3—41no—
484Dic3—41no—
485Dic3—121no—
486Dic3—121no—
487Dic3—241no—
488Dic3C3×#21412yes—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#476111yescharacteristic, complemented
494C32#500811yescharacteristic, complemented
495C32C3×#47611yescharacteristic, complemented
496C23—21no—
497Q8—362no—
498D4—21no—
499D4C3×#62712yescomplemented
500C2×C4—31no—
501C2×C4—31no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—122no—
505C2×C4—181no—
506C2×C4C2×C3×D3211yescharacteristic
507C8—61no—
508C8—181no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—44no—
519C2×C3—42no—
520C2×C3—42no—
521C2×C3—42no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—82no—
526C2×C3—82no—
527C2×C3—84no—
528C2×C3#6392511yescharacteristic
529C2×C3D3×#21411yescharacteristic
530C2×C3#6396411yescharacteristic
531C2×C3C3×#87811yescharacteristic
532C2×C3C3×#91111yescharacteristic
533C2×C3C3×D3×SD1611yescharacteristic
534C2×C3C3×#92411yescharacteristic
535C2×C3C2×C3×#21411yescharacteristic
536C2×C3C3×#94911yescharacteristic
537C22—22no—
538C22—41no—
539C22—41no—
540C22C3×D3×#6711yescharacteristic
541C4—32no—
542C4—61no—
543C4—121no—
544C4—121no—
545C4—361no—
546C4C3×#380411yescharacteristic
547C4C2×C3×#62711yescharacteristic
548C3—21no—
549C3—21no—
550C3—21no—
551C3—41no—
552C3#1059140811yescharacteristic, direct product factor
553C3C3×#476111yescharacteristic, complemented
554C3C3×#500811yescharacteristic, complemented
555C2—42no—
556C2C3×#6392511yescharacteristic
557C2C3×D3×#21411yescharacteristic
558C2C3×#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×#476
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×#63918C2semidirect
5. C3×#63574C2semidirect
6. C3×#63573C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10591408C3direct (N × H)
28. #63918C2×C3semidirect
43. #63574C2×C3semidirect
44. #63573C2×C3semidirect
45. Dic3⋊(C3⋊C8)C2×C3semidirect
48. C33×D4Q8semidirect
160. C32×D4Dic6semidirect
161. C32×D4Dic6semidirect
162. C32×D4C3×Q8semidirect
324. C33#476semidirect
344. C3×D4#627semidirect
345. C3×D4C3×Dic6semidirect
346. C3×D4C3×Dic6semidirect
493. C32#4761semidirect
494. C32#5008semidirect
495. C32C3×#476semidirect
499. D4C3×#627semidirect
552. C3#10591408direct (N × H)
553. C3C3×#4761semidirect
554. C3C3×#5008semidirect

Maximal subgroups

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

subgroupindex
C32×D4×Dic32
C3×#639182
C3×#636592
C32×#8322
C3×#635742
C3×#635732
C3×Dic3⋊(C3⋊C8)2
C3×#50083
C3×#47613
#105914083