Finite Group Databasebeta

← C3×#10591477 #13158488

Subgroups — C3×#10591477 #13158488

Order 1728.

702 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#940C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic, complemented
4C32×#832C211yesmaximal, characteristic, complemented
5C3×#63563C211yesmaximal, characteristic, complemented
6C3×#63562C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8C3×#5071—31nomaximal
9C3×#4764—31nomaximal
10#10591477C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×#67—21no—
14C2×C32×D12C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C32×#127—31no—
23C3×#940—21no—
24C3×#940D311yescharacteristic
25C3×#940C2×C311yescharacteristic
26C3×#890—31no—
27C3×#870—31no—
28#63869C2×C311yescharacteristic, complemented
29C32×#102—31no—
30#63659C2×C311yescharacteristic, complemented
31C3×#832—21no—
32C3×#832—31no—
33C3×#832—31no—
34C3×#832—31no—
35C3×#832—31no—
36C3×#832—61no—
37C3×#832D311yescharacteristic, complemented
38C3×#832C2×C311yescharacteristic, complemented
39C3×#821—31no—
40C3×#815—31no—
41C3×#809—31no—
42C3×#803—31no—
43#63563C2×C311yescharacteristic, complemented
44#63562C2×C311yescharacteristic, complemented
45#63554C2×C311yescharacteristic, complemented
46C23×C33—21no—
47C22×C32×D3—21no—
48C33×D4—21no—
49C33×D4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C32×#67—42no—
53C32×#67—42no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×D12—21no—
58C32×D12—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61C3×#490—91no—
62#5071—31no—
63#4764—31no—
64C2×C32×D4—21no—
65C2×C32×D4—21no—
66C2×C32×D4—21no—
67C2×C32×D4—31no—
68C2×C32×D4—41no—
69C2×C32×D4—61no—
70C2×C32×D4—61no—
71C2×C32×D4C2×D311yescharacteristic
72C2×C32×D4C2×D311yescharacteristic
73C2×C32×D4C22×C311yescharacteristic
74C2×C3×#67—22no—
75C2×C3×#67—22no—
76C2×C3×#67—21no—
77C2×C3×#67—21no—
78C2×C3×#67—21no—
79C2×C3×#67—21no—
80C2×C3×D12—21no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C22×C311yescharacteristic
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85C32×C42—31no—
86C4⋊(C3⋊Dic3)C22×C311yescharacteristic
87C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C22×C311yescharacteristic
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×Dic3⋊C4—61no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3C2×D311yescharacteristic
100C3×C4×Dic3C22×C311yescharacteristic
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107#3724—21no—
108Dic3⋊Dic3—21no—
109C22×C33—41no—
110C22×C33—41no—
111C22×C33—41no—
112C22×C33C2×D411yescharacteristic
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C33×C4#4111yescharacteristic
117C33×C4#4111yescharacteristic
118C3×C3⋊Dic3—41no—
119C32×Dic3—21no—
120C32×Dic3—21no—
121C32×Dic3—41no—
122C3×#127—31no—
123C3×#127—31no—
124C3×#127—61no—
125C3×#125—91no—
126#940C3×D311yescharacteristic
127#890—31no—
128#870—31no—
129C3×#102—31no—
130C3×#102—31no—
131C3×#102—61no—
132C3×#102—91no—
133C3×#102—91no—
134C3×#102—91no—
135C3×#97—91no—
136#832—31no—
137#832—31no—
138#832—31no—
139#832—31no—
140#832—61no—
141#832C3×D311yescharacteristic, complemented
142#821—31no—
143#815—31no—
144#809—31no—
145#803—31no—
146C23×C32—21no—
147C23×C32—22no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—41no—
151C23×C32—41no—
152C23×C32—42no—
153C23×C32—61no—
154C22×C3×D3—22no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C32×D4—21no—
158C32×D4—21no—
159C32×D4—22no—
160C32×D4—22no—
161C32×D4—21no—
162C32×D4—21no—
163C32×D4—21no—
164C32×D4—21no—
165C32×D4—41no—
166C32×D4—41no—
167C32×D4—41no—
168C32×D4—41no—
169C32×D4—42no—
170C32×D4—42no—
171C32×D4—61no—
172C32×D4—61no—
173C32×D4—122no—
174C32×D4—122no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—31no—
179C2×C32×C4—31no—
180C2×C32×C4—41no—
181C2×C32×C4—61no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C23×C311yescharacteristic
185C2×C3⋊Dic3—21no—
186C3×#67—44no—
187C3×#67—44no—
188C3×#67—42no—
189C3×#67—42no—
190C3×#67—42no—
191C3×#67—42no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—41no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—121no—
200C2×C3×Dic3#6711yescharacteristic
201C2×C3×Dic3#6711yescharacteristic
202C2×C3×Dic3C3×D411yescharacteristic
203C2×C3×Dic3C3×D411yescharacteristic
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—41no—
209C3×D12—41no—
210C32×C8—61no—
211C3⋊(C3⋊C8)—21no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—41no—
215C3×C3⋊C8—61no—
216C3×C3⋊C8—61no—
217C3×C3⋊C8—121no—
218#490—91noSylow
219C2×C33—81no—
220C2×C33#12411yescharacteristic
221C2×C33AGL(1,ℤ/8)11yescharacteristic
222C2×C33AGL(1,ℤ/8)11yescharacteristic
223C32×D3—81no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—31no—
228C2×C3×D4—31no—
229C2×C3×D4—41no—
230C2×C3×D4—61no—
231C2×C3×D4—61no—
232C2×C3×D4—61no—
233C2×C3×D4—61no—
234C2×C3×D4—61no—
235C2×C3×D4—62no—
236C2×C3×D4—62no—
237C2×C3×D4D3211yescharacteristic
238C2×C3×D4C2×C3×D311yescharacteristic
239C2×C3×D4C2×C3×D311yescharacteristic
240C2×#67—21no—
241C2×#67—21no—
242C2×D12C2×C3×D311yescharacteristic
243C2×C3×C8—31no—
244C2×C3×C8—31no—
245C2×C3×C8—61no—
246C2×C3×C8—91no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—181no—
252C3×C4⋊C4—181no—
253C3×C42—31no—
254C3×C42—31no—
255C3×C42—61no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258C4⋊Dic3—61no—
259C4⋊Dic3—61no—
260C4⋊Dic3C2×C3×D311yescharacteristic
261Dic3⋊C4—61no—
262Dic3⋊C4—61no—
263Dic3⋊C4—61no—
264C4×Dic3C2×C3×D311yescharacteristic
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8C2×C3×D311yescharacteristic
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32—42no—
274C22×C32—42no—
275C22×C32—42no—
276C22×C32—44no—
277C22×C32—41no—
278C22×C32—41no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—41no—
282C22×C32—42no—
283C22×C32—42no—
284C22×C32—41no—
285C22×C32—41no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—81no—
289C22×C32—81no—
290C22×C32—81no—
291C22×C32—81no—
292C22×C32—82no—
293C22×C32—82no—
294C22×C32—121no—
295C22×C32—121no—
296C22×C32—121no—
297C22×C32D3×D411yescharacteristic
298C22×C32C2×#6711yescharacteristic
299C22×C32C2×C3×D411yescharacteristic
300C2×C3×D3—42no—
301C2×C3×D3—42no—
302C2×C3×D3—42no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—61no—
318C32×C4—61no—
319C32×C4—121no—
320C32×C4#23511yescharacteristic
321C32×C4#23711yescharacteristic
322C32×C4#23711yescharacteristic
323C32×C4#23711yescharacteristic
324C32×C4C3×#4111yescharacteristic
325C32×C4C3×#4111yescharacteristic
326C3⋊Dic3—41no—
327C3×Dic3—22no—
328C3×Dic3—22no—
329C3×Dic3—21no—
330C3×Dic3—21no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—81no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C3×Dic3—241no—
339#127—31no—
340#125—91no—
341#102—31no—
342#102—91no—
343#102—91no—
344#102—91no—
345#97—91no—
346C33#49011yescharacteristic, complemented, Sylow
347C23×C3—22no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—42no—
354C23×C3—62no—
355C23×C3—61no—
356C23×C3—61no—
357C22×D3—21no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—21no—
361C3×D4—21no—
362C3×D4—21no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—21no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—42no—
371C3×D4—42no—
372C3×D4—61no—
373C3×D4—61no—
374C3×D4—61no—
375C3×D4—61no—
376C3×D4—121no—
377C3×D4—121no—
378C3×D4—124no—
379C3×D4—124no—
380C3×D4—122no—
381C3×D4—122no—
382C3×D4—122no—
383C3×D4—122no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—31no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—41no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—121no—
397C2×C3×C4—181no—
398C2×C3×C4C2×D3211yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×C3×C4C22×C3×D311yescharacteristic
401#67—42no—
402#67—42no—
403C2×Dic3—21no—
404C2×Dic3—61no—
405C2×Dic3—61no—
406C2×Dic3—121no—
407C2×Dic3C3×#6711yescharacteristic
408C2×Dic3C3×#6711yescharacteristic
409D12—21no—
410D12—21no—
411C3×C8—61no—
412C3×C8—61no—
413C3×C8—121no—
414C3×C8—181no—
415C3⋊C8—21no—
416C3⋊C8—61no—
417C3⋊C8—61no—
418C3⋊C8—121no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32—82no—
432C2×C32—82no—
433C2×C32—82no—
434C2×C32—84no—
435C2×C32—81no—
436C2×C32—81no—
437C2×C32—81no—
438C2×C32—241no—
439C2×C32#88511yescharacteristic
440C2×C32#91111yescharacteristic
441C2×C32#91411yescharacteristic
442C2×C32#93211yescharacteristic
443C2×C32#93211yescharacteristic
444C2×C32#93611yescharacteristic
445C2×C32C3×#12411yescharacteristic
446C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
447C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
448C3×D3—82no—
449C3×D3—81no—
450C3×D3—81no—
451C2×D4—31no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4C3×D3211yescharacteristic
455C2×C8—31no—
456C2×C8—91no—
457C4⋊C4—31no—
458C4⋊C4—91no—
459C4⋊C4—181no—
460C4⋊C4—181no—
461C42—31no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—44no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—122no—
488C22×C3—122no—
489C22×C3—122no—
490C22×C3—121no—
491C22×C3—121no—
492C22×C3—121no—
493C22×C3—121no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3D3×#6711yescharacteristic
497C22×C3C3×D3×D411yescharacteristic
498C22×C3C2×C3×#6711yescharacteristic
499C2×D3—41no—
500C2×D3—41no—
501C2×D3—41no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—62no—
511C3×C4—62no—
512C3×C4—61no—
513C3×C4—61no—
514C3×C4—61no—
515C3×C4—61no—
516C3×C4—121no—
517C3×C4—121no—
518C3×C4—241no—
519C3×C4—361no—
520C3×C4#380411yescharacteristic
521C3×C4#380511yescharacteristic
522C3×C4C3×#23511yescharacteristic
523C3×C4C3×#23711yescharacteristic
524C3×C4C3×#23711yescharacteristic
525C3×C4C3×#23711yescharacteristic
526Dic3—21no—
527Dic3—21no—
528Dic3—41no—
529Dic3—121no—
530Dic3—121no—
531Dic3—241no—
532C32—21no—
533C32—21no—
534C32—21no—
535C32—41no—
536C32#476411yescharacteristic, complemented
537C32#507111yescharacteristic, complemented
538C32C3×#49011yescharacteristic, complemented
539C23—21no—
540C23—61no—
541D4—21no—
542D4—21no—
543D4—61no—
544D4—61no—
545D4—122no—
546D4—122no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—61no—
550C2×C4—181no—
551C2×C4C2×C3×D3211yescharacteristic
552C8—61no—
553C8—181no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—41no—
566C2×C3—82no—
567C2×C3—82no—
568C2×C3—82no—
569C2×C3—84no—
570C2×C3—81no—
571C2×C3—81no—
572C2×C3—81no—
573C2×C3—242no—
574C2×C3—241no—
575C2×C3—241no—
576C2×C3#6393211yescharacteristic
577C2×C3#6393411yescharacteristic
578C2×C3#6395911yescharacteristic
579C2×C3C3×#88511yescharacteristic
580C2×C3C3×#91111yescharacteristic
581C2×C3C3×#91411yescharacteristic
582C2×C3C3×#93211yescharacteristic
583C2×C3C3×#93211yescharacteristic
584C2×C3C3×#93611yescharacteristic
585D3—81no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—121no—
590C22—121no—
591C22—121no—
592C22C3×D3×#6711yescharacteristic
593C4—61no—
594C4—61no—
595C4—121no—
596C4—361no—
597C4C3×#380411yescharacteristic
598C4C3×#380511yescharacteristic
599C3—21no—
600C3—21no—
601C3—21no—
602C3—41no—
603C3#1059147711yescharacteristic, direct product factor
604C3C3×#476411yescharacteristic, complemented
605C3C3×#507111yescharacteristic, complemented
606C2—81no—
607C2—241no—
608C2C3×#6393211yescharacteristic
609C2C3×#6393411yescharacteristic
610C2C3×#6395911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#940C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#490
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
3. C3×#63659C2semidirect
4. C32×#832C2semidirect
5. C3×#63563C2semidirect
6. C3×#63562C2semidirect
7. C3×#63554C2semidirect
10. #10591477C3direct (N × H)
28. #63869C2×C3semidirect
30. #63659C2×C3semidirect
37. C3×#832D3semidirect
38. C3×#832C2×C3semidirect
43. #63563C2×C3semidirect
44. #63562C2×C3semidirect
45. #63554C2×C3semidirect
141. #832C3×D3semidirect
346. C33#490semidirect
536. C32#4764semidirect
537. C32#5071semidirect
538. C32C3×#490semidirect
603. C3#10591477direct (N × H)
604. C3C3×#4764semidirect
605. C3C3×#5071semidirect

Maximal subgroups

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

subgroupindex
C32×#9402
C3×#638692
C3×#636592
C32×#8322
C3×#635632
C3×#635622
C3×#635542
C3×#50713
C3×#47643
#105914773