Finite Group Databasebeta

← C3×#10591476 #13158486

Subgroups — C3×#10591476 #13158486

Order 1728.

788 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C32×#214C211yesmaximal, characteristic, complemented
3C2×C32×#213C211yesmaximal, characteristic
4C3×#63863C211yesmaximal, characteristic, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8C3×#5165—31nomaximal
9C3×#5153—31nomaximal
10#10591476C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#214—22no—
19C32×#213—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×(C2×D3)⋊Dic3—21no—
23C3×(C2×D3)⋊Dic3—21no—
24C2×C32×SD16—31no—
25C2×C32×D8—31no—
26D4×C3⋊Dic3C2×C311yescharacteristic, complemented
27C3×#947—31no—
28C3×#936—31no—
29C3×D4×Dic3—31no—
30C3×D4×Dic3—31no—
31C3×D4×Dic3—61no—
32C2×C3×#214—21no—
33C2×C3×#214D311yescharacteristic, complemented
34C2×C3×#214C2×C311yescharacteristic, complemented
35C2×C3×#213—21no—
36C2×C3×#213D311yescharacteristic
37C2×C3×#213C2×C311yescharacteristic
38#63863C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#832—31no—
41C3×#821—31no—
42C3×#816—31no—
43C3×Dic3⋊C8—31no—
44C3×Dic3⋊C8—31no—
45#63572C2×C311yescharacteristic, complemented
46#63566C2×C311yescharacteristic, complemented
47#63563C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C32×D3—21no—
50C33×D4—21no—
51C33×D4D412yes—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—42no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C2×C3×C3⋊Dic3D411yescharacteristic
57C2×C32×Dic3—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64C3×#463—91no—
65#5165—31no—
66#5153—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—31no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C22×C311yescharacteristic
76C22×C3⋊Dic3—21no—
77C22×C3×Dic3—61no—
78C22×C3×Dic3—61no—
79C22×C3×Dic3—62no—
80C2×C3×D12—21no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C22×C311yescharacteristic
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×SD16—62no—
87C32×D8—62no—
88C2×C32×C8—31no—
89C2×C32×C8—31no—
90#3757—21no—
91C4⋊(C3⋊Dic3)C22×C311yescharacteristic
92C4×C3⋊Dic3C22×C311yescharacteristic
93C3×#217—61no—
94C3×#217—61no—
95C3×#217—61no—
96C3×#217—61no—
97C3×#217—62no—
98C3×#214—24no—
99C3×#214—22no—
100C3×#214—22no—
101C3×#213—24no—
102C3×#213—22no—
103C3×#213—22no—
104C3×(C2×D3)⋊C4—61no—
105C3×(C2×D3)⋊C4—61no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—61no—
109C3×C4×Dic3—31no—
110C3×C4×Dic3—31no—
111C3×C4×Dic3—61no—
112C2×C3×C3⋊C8—21no—
113C2×C3×C3⋊C8—21no—
114C2×C3×C3⋊C8C2×D311yescharacteristic
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C22×C311yescharacteristic
117C2×C3×C3⋊C8C22×C311yescharacteristic
118(C2×D3)⋊Dic3—21no—
119(C2×D3)⋊Dic3—21no—
120C22×C33—22no—
121C22×C33—41no—
122C22×C33—41no—
123C22×C33C2×D411yescharacteristic
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3—21no—
131C3×C3⋊Dic3—41no—
132C32×Dic3—41no—
133C2×C3×SD16—31no—
134C2×C3×SD16—31no—
135C2×C3×SD16—61no—
136C2×C3×D8—31no—
137C2×C3×D8—31no—
138C2×C3×D8—61no—
139C3×#124—91no—
140C3×C4×D4—91no—
141#947—31no—
142#936—31no—
143D4×Dic3—31no—
144D4×Dic3—31no—
145D4×Dic3—61no—
146C2×#214C3×D311yescharacteristic, complemented
147C2×#213C3×D311yescharacteristic
148C3×C4⋊C8—91no—
149C3×#103—91no—
150C3×#102—91no—
151#835—31no—
152#832—31no—
153#821—31no—
154#816—31no—
155Dic3⋊C8—31no—
156Dic3⋊C8—31no—
157C23×C32—21no—
158C23×C32—22no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—41no—
162C23×C32—41no—
163C23×C32—42no—
164C23×C32—61no—
165C22×C3×D3—21no—
166C22×C3×D3—21no—
167C22×C3×D3—41no—
168C32×Q8—61no—
169C32×Q8—61no—
170C32×D4—21no—
171C32×D4—22no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—22no—
175C32×D4—22no—
176C32×D4—22no—
177C32×D4—42no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—42no—
181C32×D4—61no—
182C32×D4—61no—
183C32×D4#6712yes—
184C32×D4#6712yes—
185C32×D4C3×D412yes—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—41no—
190C2×C32×C4—61no—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C23×C311yescharacteristic
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—42no—
196C2×C3⋊Dic3C3×D411yescharacteristic
197C2×C3⋊Dic3C3×D411yescharacteristic
198C2×C3×Dic3—21no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—41no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—61no—
210C2×C3×Dic3—124no—
211C2×C3×Dic3—122no—
212C2×C3×Dic3—122no—
213C3×D12—22no—
214C3×D12—21no—
215C3×D12—21no—
216C3×D12—21no—
217C3×D12—21no—
218C3×D12—41no—
219C3×Dic6—22no—
220C3×Dic6—21no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—21no—
224C3×Dic6—41no—
225C32×C8—61no—
226C32×C8—61no—
227C3×C3⋊C8—22no—
228C3×C3⋊C8—22no—
229C3×C3⋊C8—21no—
230C3×C3⋊C8—21no—
231C3×C3⋊C8—21no—
232C3×C3⋊C8—21no—
233#463—91noSylow
234C2×C33—42no—
235C2×C33#12111yescharacteristic
236C2×C33#13511yescharacteristic
237C2×C33AGL(1,ℤ/8)11yescharacteristic
238C32×D3—81no—
239C2×C3×Q8—31no—
240C2×C3×Q8—31no—
241C2×C3×Q8—61no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4—21no—
245C2×C3×D4—31no—
246C2×C3×D4—31no—
247C2×C3×D4—41no—
248C2×C3×D4—61no—
249C2×C3×D4D3211yescharacteristic
250C2×C3×D4C2×C3×D311yescharacteristic
251C2×C3×D4C2×C3×D311yescharacteristic
252C22×C3×C4—181no—
253C22×Dic3—61no—
254C22×Dic3—61no—
255C22×Dic3—62no—
256C2×D12C2×C3×D311yescharacteristic
257C2×Dic6C2×C3×D311yescharacteristic
258C3×SD16—64no—
259C3×SD16—62no—
260C3×SD16—62no—
261C3×D8—64no—
262C3×D8—62no—
263C3×D8—62no—
264C2×C3×C8—31no—
265C2×C3×C8—31no—
266C2×C3×C8—31no—
267C2×C3×C8—31no—
268C2×C3×C8—61no—
269C2×C3×C8—61no—
270C3×C4⋊C4—91no—
271C3×C22⋊C4—181no—
272C3×C22⋊C4—181no—
273C3×C22⋊C4—181no—
274C3×C42—91no—
275#217—61no—
276#217—61no—
277#217—61no—
278#217—61no—
279#217—62no—
280#214—22no—
281#213—22no—
282(C2×D3)⋊C4—61no—
283(C2×D3)⋊C4—61no—
284C4⋊Dic3—31no—
285C4⋊Dic3—31no—
286C4⋊Dic3—61no—
287C4×Dic3—31no—
288C4×Dic3—31no—
289C4×Dic3—61no—
290C2×C3⋊C8C2×C3×D311yescharacteristic
291C2×C3⋊C8C2×C3×D311yescharacteristic
292C22×C32—22no—
293C22×C32—24no—
294C22×C32—22no—
295C22×C32—22no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—41no—
300C22×C32—42no—
301C22×C32—42no—
302C22×C32—44no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32—42no—
306C22×C32—42no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32—41no—
311C22×C32—81no—
312C22×C32—81no—
313C22×C32—81no—
314C22×C32—81no—
315C22×C32—82no—
316C22×C32—82no—
317C22×C32—121no—
318C22×C32—121no—
319C22×C32—121no—
320C22×C32D3×D411yescharacteristic
321C22×C32D3×D411yescharacteristic
322C22×C32C2×C3×D411yescharacteristic
323C2×C3×D3—42no—
324C2×C3×D3—41no—
325C2×C3×D3—41no—
326C2×C3×D3—41no—
327C2×C3×D3—41no—
328C2×C3×D3—41no—
329C2×C3×D3—41no—
330C2×C3×D3—81no—
331C2×C3×D3—81no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—41no—
339C32×C4—41no—
340C32×C4—121no—
341C32×C4#23711yescharacteristic
342C32×C4#23711yescharacteristic
343C32×C4C2×#6711yescharacteristic
344C32×C4C2×#6711yescharacteristic
345C32×C4C2×C3×D411yescharacteristic
346C32×C4C3×#4111yescharacteristic
347C3⋊Dic3—21no—
348C3⋊Dic3—21no—
349C3⋊Dic3—41no—
350C3×Dic3—42no—
351C3×Dic3—41no—
352C3×Dic3—41no—
353C3×Dic3—62no—
354C3×Dic3—61no—
355C3×Dic3—61no—
356C3×Dic3—61no—
357C3×Dic3—61no—
358C3×Dic3—121no—
359C3×Dic3—122no—
360C3×Dic3—121no—
361C3×Dic3—121no—
362C2×SD16—31no—
363C2×D8—31no—
364#124—91no—
365C4×D4—91no—
366C4⋊C8—91no—
367#103—91no—
368#102—91no—
369C33#46311yescharacteristic, complemented, Sylow
370C23×C3—22no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—42no—
377C23×C3—61no—
378C23×C3—61no—
379C23×C3—121no—
380C22×D3—21no—
381C3×Q8—62no—
382C3×Q8—61no—
383C3×Q8—61no—
384C3×Q8—61no—
385C3×Q8—61no—
386C3×Q8—121no—
387C3×D4—22no—
388C3×D4—21no—
389C3×D4—21no—
390C3×D4—21no—
391C3×D4—22no—
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—62no—
399C3×D4—61no—
400C3×D4—61no—
401C3×D4—61no—
402C3×D4—61no—
403C3×D4—121no—
404C3×D4#62512yes—
405C3×D4C3×#6712yes—
406C3×D4C3×#6712yes—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—41no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4—91no—
414C2×C3×C4—91no—
415C2×C3×C4—121no—
416C2×C3×C4—181no—
417C2×C3×C4—362no—
418C2×C3×C4C2×D3211yescharacteristic
419C2×C3×C4C22×C3×D311yescharacteristic
420C2×C3×C4C22×C3×D311yescharacteristic
421C2×Dic3—21no—
422C2×Dic3—31no—
423C2×Dic3—31no—
424C2×Dic3—31no—
425C2×Dic3—31no—
426C2×Dic3—61no—
427C2×Dic3—61no—
428C2×Dic3—62no—
429C2×Dic3—61no—
430C2×Dic3—61no—
431C2×Dic3—124no—
432C2×Dic3—122no—
433C2×Dic3—122no—
434D12—21no—
435D12—21no—
436Dic6—21no—
437Dic6—21no—
438C3×C8—62no—
439C3×C8—62no—
440C3×C8—61no—
441C3×C8—61no—
442C3×C8—61no—
443C3×C8—61no—
444C3⋊C8—21no—
445C3⋊C8—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—41no—
456C2×C32—41no—
457C2×C32—41no—
458C2×C32—42no—
459C2×C32—44no—
460C2×C32—42no—
461C2×C32—42no—
462C2×C32—82no—
463C2×C32—82no—
464C2×C32—84no—
465C2×C32—241no—
466C2×C32#91111yescharacteristic
467C2×C32#91211yescharacteristic
468C2×C32#91411yescharacteristic
469C2×C32#91611yescharacteristic
470C2×C32#93911yescharacteristic
471C2×C32#93911yescharacteristic
472C2×C32C3×#12111yescharacteristic
473C2×C32C3×#13511yescharacteristic
474C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
475C3×D3—82no—
476C3×D3—81no—
477C3×D3—81no—
478C2×Q8—31no—
479C2×D4—31no—
480C2×D4C3×D3211yescharacteristic
481C22×C4—181no—
482SD16—62no—
483D8—62no—
484C2×C8—31no—
485C2×C8—31no—
486C4⋊C4—91no—
487C22⋊C4—181no—
488C22⋊C4—181no—
489C22⋊C4—181no—
490C42—91no—
491C22×C3—24no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—44no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—82no—
515C22×C3—82no—
516C22×C3—122no—
517C22×C3—121no—
518C22×C3—121no—
519C22×C3—121no—
520C22×C3—121no—
521C22×C3—121no—
522C22×C3—121no—
523C22×C3—241no—
524C22×C3—241no—
525C22×C3#381111yescharacteristic
526C22×C3C3×D3×D411yescharacteristic
527C22×C3C3×D3×D411yescharacteristic
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—122no—
540C3×C4—121no—
541C3×C4—121no—
542C3×C4—181no—
543C3×C4—181no—
544C3×C4—361no—
545C3×C4#380511yescharacteristic
546C3×C4C2×#62511yescharacteristic
547C3×C4C3×#23711yescharacteristic
548C3×C4C3×#23711yescharacteristic
549C3×C4C2×C3×#6711yescharacteristic
550C3×C4C2×C3×#6711yescharacteristic
551Dic3—41no—
552Dic3—62no—
553Dic3—61no—
554Dic3—61no—
555Dic3—61no—
556Dic3—61no—
557Dic3—121no—
558Dic3—122no—
559Dic3—121no—
560Dic3—121no—
561C32—21no—
562C32—21no—
563C32—21no—
564C32—41no—
565C32#515311yescharacteristic, complemented
566C32#516511yescharacteristic, complemented
567C32C3×#46311yescharacteristic, complemented
568C23—21no—
569C23—61no—
570Q8—61no—
571Q8—61no—
572D4—21no—
573D4—61no—
574D4—61no—
575D4C3×#62512yes—
576C2×C4—61no—
577C2×C4—91no—
578C2×C4—91no—
579C2×C4—181no—
580C2×C4—362no—
581C2×C4C2×C3×D3211yescharacteristic
582C8—61no—
583C8—61no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—44no—
594C2×C3—42no—
595C2×C3—42no—
596C2×C3—42no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—41no—
600C2×C3—82no—
601C2×C3—82no—
602C2×C3—84no—
603C2×C3—242no—
604C2×C3—241no—
605C2×C3—241no—
606C2×C3#6393611yescharacteristic
607C2×C3#6393711yescharacteristic
608C2×C3#6398011yescharacteristic
609C2×C3C3×#91111yescharacteristic
610C2×C3C3×#91211yescharacteristic
611C2×C3C3×#91411yescharacteristic
612C2×C3C3×#91611yescharacteristic
613C2×C3C3×#93911yescharacteristic
614C2×C3C3×#93911yescharacteristic
615D3—81no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—121no—
620C22—121no—
621C22—121no—
622C22C3×#381111yescharacteristic
623C4—121no—
624C4—181no—
625C4—181no—
626C4—361no—
627C4C3×#380511yescharacteristic
628C4C2×C3×#62511yescharacteristic
629C3—21no—
630C3—21no—
631C3—21no—
632C3—41no—
633C3#1059147611yescharacteristic, direct product factor
634C3C3×#515311yescharacteristic, complemented
635C3C3×#516511yescharacteristic, complemented
636C2—42no—
637C2—241no—
638C2C3×#6393611yescharacteristic
639C2C3×#6393711yescharacteristic
640C2C3×#6398011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C32×#213C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#463
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
1. C3×D4×C3⋊Dic3C2semidirect
2. C2×C32×#214C2semidirect
4. C3×#63863C2semidirect
5. C3×#63572C2semidirect
6. C3×#63566C2semidirect
7. C3×#63563C2semidirect
10. #10591476C3direct (N × H)
26. D4×C3⋊Dic3C2×C3semidirect
33. C2×C3×#214D3semidirect
34. C2×C3×#214C2×C3semidirect
38. #63863C2×C3semidirect
45. #63572C2×C3semidirect
46. #63566C2×C3semidirect
47. #63563C2×C3semidirect
146. C2×#214C3×D3semidirect
369. C33#463semidirect
565. C32#5153semidirect
566. C32#5165semidirect
567. C32C3×#463semidirect
633. C3#10591476direct (N × H)
634. C3C3×#5153semidirect
635. C3C3×#5165semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C2×C32×#2142
C2×C32×#2132
C3×#638632
C3×#635722
C3×#635662
C3×#635632
C3×#51653
C3×#51533
#105914763