Finite Group Databasebeta

← C3×#10591471 #13158483

Subgroups — C3×#10591471 #13158483

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
1C2×C3×#3753C211yesmaximal, characteristic
2C32×D4×Dic3C211yesmaximal, characteristic, complemented
3C2×C32×#214C211yesmaximal, characteristic, complemented
4C3×#63864C211yesmaximal, characteristic, complemented
5C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#5165—31nomaximal
9C3×#5016—31nomaximal
10#10591471C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C22×C32×Dic3—21no—
14C2×C32×Dic6C2211yescharacteristic
15C3×#3753—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—21no—
18C32×#214—22no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3722—21no—
23C3×#3722—21no—
24C2×C32×SD16—31no—
25C32×C4×D4—31no—
26C2×#3753C2×C311yescharacteristic
27C3×#947—31no—
28C3×D4×Dic3—21no—
29C3×D4×Dic3D311yescharacteristic, complemented
30C3×D4×Dic3C2×C311yescharacteristic, complemented
31C2×C3×#214—21no—
32C2×C3×#214D311yescharacteristic, complemented
33C2×C3×#214C2×C311yescharacteristic, complemented
34C2×C3×#213—31no—
35C2×C3×#213—31no—
36C2×C3×#213—61no—
37C3×#875—31no—
38#63864C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#821—31no—
41C3×Dic3⋊C8—31no—
42C3×#810—31no—
43C3×#809—31no—
44C3×C4⋊(C3⋊C8)—31no—
45Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
46#63565C2×C311yescharacteristic, complemented
47#63564C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C33×D4—21no—
51C33×D4D412yes—
52C2×C33×C4C2311yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—42no—
58C2×C32×Dic3D411yescharacteristic
59C2×C32×Dic3D411yescharacteristic
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64C3×#463—91no—
65#5165—31no—
66#5016—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—41no—
72C2×C32×D4C2×D311yescharacteristic
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C22×C311yescharacteristic
75C22×C32×C4—61no—
76C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
77C22×C3×Dic3—22no—
78C22×C3×Dic3—21no—
79C22×C3×Dic3—21no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×SD16—62no—
87C2×C32×C8—31no—
88C32×C4⋊C4—31no—
89C32×C22⋊C4—61no—
90C32×C42—31no—
91#3753—22no—
92C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
93C3×#217—22no—
94C3×#217—21no—
95C3×#217—21no—
96C3×#214—24no—
97C3×#214—22no—
98C3×#214—22no—
99C3×#213—64no—
100C3×#213—62no—
101C3×#213—62no—
102C3×(C2×D3)⋊C4—61no—
103C3×(C2×D3)⋊C4—61no—
104C3×(C2×D3)⋊C4—61no—
105C3×(C2×D3)⋊C4—61no—
106C3×C4⋊Dic3—21no—
107C3×C4⋊Dic3C2×D311yescharacteristic
108C3×C4⋊Dic3C22×C311yescharacteristic
109C3×C4×Dic3—21no—
110C3×C4×Dic3C2×D311yescharacteristic
111C3×C4×Dic3C22×C311yescharacteristic
112C2×C3×C3⋊C8—21no—
113C2×C3×C3⋊C8—31no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—61no—
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C22×C311yescharacteristic
118#3722—21no—
119#3722—21no—
120C22×C33—22no—
121C22×C33—41no—
122C22×C33—41no—
123C22×C33C2×D411yescharacteristic
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C2×C3×C3⋊D3—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C32×Dic3—21no—
130C32×Dic3—21no—
131C32×Dic3—41no—
132C32×Dic3—41no—
133C2×C3×SD16—31no—
134C2×C3×SD16—31no—
135C2×C3×SD16—61no—
136C2×C3×D8—91no—
137C3×#124—91no—
138C3×C4×D4—31no—
139C3×C4×D4—31no—
140C3×C4×D4—61no—
141#947—31no—
142D4×Dic3C3×D311yescharacteristic, complemented
143C2×#214C3×D311yescharacteristic, complemented
144C2×#213—31no—
145C2×#213—31no—
146C2×#213—61no—
147#875—31no—
148C3×C4⋊C8—91no—
149C3×#103—91no—
150C3×#102—91no—
151#835—31no—
152#821—31no—
153Dic3⋊C8—31no—
154#810—31no—
155#809—31no—
156C4⋊(C3⋊C8)—31no—
157C23×C32—21no—
158C23×C32—22no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—41no—
162C23×C32—41no—
163C23×C32—42no—
164C22×C3⋊D3—21no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C22×C3×D3—121no—
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×D4D1212yes—
182C32×D4#6712yes—
183C32×D4C3×D412yes—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—31no—
188C2×C32×C4—31no—
189C2×C32×C4—41no—
190C2×C32×C4—61no—
191C2×C32×C4—61no—
192C2×C32×C4—122no—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C23×C311yescharacteristic
196C4⋊(C3⋊D3)—21no—
197C4⋊(C3⋊D3)—21no—
198C2×C3×Dic3—22no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—21no—
205C2×C3×Dic3—41no—
206C2×C3×Dic3—44no—
207C2×C3×Dic3—42no—
208C2×C3×Dic3—42no—
209C2×C3×Dic3#6711yescharacteristic
210C2×C3×Dic3#6711yescharacteristic
211C2×C3×Dic3C3×D411yescharacteristic
212C2×C3×Dic3C3×D411yescharacteristic
213C3×D12—62no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—61no—
218C3×D12—121no—
219C3×Dic6—22no—
220C3×Dic6—21no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—21no—
224C3×Dic6—41no—
225C32×C8—61no—
226C3⋊(C3⋊C8)—21no—
227C3×C3⋊C8—22no—
228C3×C3⋊C8—21no—
229C3×C3⋊C8—21no—
230C3×C3⋊C8—62no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233#463—91noSylow
234C2×C33—42no—
235C2×C33#12111yescharacteristic
236C2×C33#13511yescharacteristic
237C2×C33AGL(1,ℤ/8)11yescharacteristic
238C3×C3⋊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—41no—
246C2×C3×D4—91no—
247C2×C3×D4D3211yescharacteristic
248C2×C3×D4C2×C3×D311yescharacteristic
249C2×C3×D4C2×C3×D311yescharacteristic
250C22×C3×C4—61no—
251C22×C3×C4—61no—
252C22×C3×C4—62no—
253C22×Dic3—21no—
254C2×D12—31no—
255C2×D12—31no—
256C2×D12—61no—
257C2×Dic6C2×C3×D311yescharacteristic
258C3×SD16—64no—
259C3×SD16—62no—
260C3×SD16—62no—
261C3×D8—182no—
262C2×C3×C8—31no—
263C2×C3×C8—31no—
264C2×C3×C8—61no—
265C2×C3×C8—91no—
266C3×C4⋊C4—31no—
267C3×C4⋊C4—31no—
268C3×C4⋊C4—61no—
269C3×C22⋊C4—61no—
270C3×C22⋊C4—61no—
271C3×C22⋊C4—62no—
272C3×C22⋊C4—181no—
273C3×C22⋊C4—181no—
274C3×C42—31no—
275C3×C42—31no—
276C3×C42—61no—
277#217—21no—
278#214—22no—
279#213—64no—
280#213—62no—
281#213—62no—
282(C2×D3)⋊C4—61no—
283(C2×D3)⋊C4—61no—
284(C2×D3)⋊C4—61no—
285(C2×D3)⋊C4—61no—
286C4⋊Dic3C2×C3×D311yescharacteristic
287C4×Dic3C2×C3×D311yescharacteristic
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
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×C32D3×D411yescharacteristic
318C22×C32C2×#6711yescharacteristic
319C22×C32C2×C3×D411yescharacteristic
320C2×C3⋊D3—41no—
321C2×C3⋊D3—41no—
322C2×C3⋊D3—41no—
323C2×C3×D3—122no—
324C2×C3×D3—121no—
325C2×C3×D3—121no—
326C2×C3×D3—121no—
327C2×C3×D3—121no—
328C2×C3×D3—121no—
329C2×C3×D3—121no—
330C2×C3×D3—241no—
331C2×C3×D3—241no—
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—61no—
341C32×C4—61no—
342C32×C4—121no—
343C32×C4—121no—
344C32×C4C2×D1211yescharacteristic
345C32×C4#23511yescharacteristic
346C32×C4#23711yescharacteristic
347C32×C4C2×#6711yescharacteristic
348C32×C4C2×C3×D411yescharacteristic
349C32×C4C3×#4111yescharacteristic
350C3×Dic3—22no—
351C3×Dic3—21no—
352C3×Dic3—21no—
353C3×Dic3—21no—
354C3×Dic3—21no—
355C3×Dic3—41no—
356C3×Dic3—42no—
357C3×Dic3—42no—
358C3×Dic3—41no—
359C3×Dic3—41no—
360C3×Dic3—41no—
361C3×Dic3—41no—
362C2×SD16—31no—
363C2×D8—91no—
364#124—91no—
365C4×D4—31no—
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—181no—
378C22×D3—61no—
379C22×D3—61no—
380C22×D3—121no—
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—181no—
399C3×D4—181no—
400C3×D4#62612yes—
401C3×D4C3×D1212yes—
402C3×D4C3×#6712yes—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—31no—
407C2×C3×C4—31no—
408C2×C3×C4—31no—
409C2×C3×C4—31no—
410C2×C3×C4—41no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4—62no—
414C2×C3×C4—61no—
415C2×C3×C4—61no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
418C2×C3×C4—121no—
419C2×C3×C4—124no—
420C2×C3×C4—122no—
421C2×C3×C4—122no—
422C2×C3×C4C2×D3211yescharacteristic
423C2×C3×C4C22×C3×D311yescharacteristic
424C2×C3×C4C22×C3×D311yescharacteristic
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—42no—
428C2×Dic3C3×#6711yescharacteristic
429C2×Dic3C3×#6711yescharacteristic
430D12—62no—
431D12—61no—
432D12—61no—
433D12—61no—
434D12—61no—
435D12—121no—
436Dic6—21no—
437Dic6—21no—
438C3×C8—62no—
439C3×C8—61no—
440C3×C8—61no—
441C3×C8—181no—
442C3⋊C8—21no—
443C3⋊C8—62no—
444C3⋊C8—61no—
445C3⋊C8—61no—
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#91411yescharacteristic
466C2×C32#91611yescharacteristic
467C2×C32#93011yescharacteristic
468C2×C32#93211yescharacteristic
469C2×C32#93911yescharacteristic
470C2×C32#95011yescharacteristic
471C2×C32C3×#12111yescharacteristic
472C2×C32C3×#13511yescharacteristic
473C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
474C3⋊D3—81no—
475C3×D3—242no—
476C3×D3—241no—
477C3×D3—241no—
478C2×Q8—31no—
479C2×D4—91no—
480C2×D4C3×D3211yescharacteristic
481C22×C4—61no—
482SD16—62no—
483D8—182no—
484C2×C8—31no—
485C2×C8—91no—
486C4⋊C4—31no—
487C22⋊C4—61no—
488C22⋊C4—181no—
489C22⋊C4—181no—
490C42—31no—
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—361no—
517C22×C3—361no—
518C22×C3—361no—
519C22×C3D3×#6711yescharacteristic
520C22×C3C3×D3×D411yescharacteristic
521C22×C3C2×C3×#6711yescharacteristic
522C2×D3—122no—
523C2×D3—121no—
524C2×D3—121no—
525C2×D3—121no—
526C2×D3—121no—
527C2×D3—121no—
528C2×D3—121no—
529C2×D3—241no—
530C2×D3—241no—
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—62no—
540C3×C4—61no—
541C3×C4—61no—
542C3×C4—61no—
543C3×C4—61no—
544C3×C4—121no—
545C3×C4—122no—
546C3×C4—122no—
547C3×C4—121no—
548C3×C4—121no—
549C3×C4—121no—
550C3×C4—121no—
551C3×C4#380411yescharacteristic
552C3×C4C2×#62611yescharacteristic
553C3×C4C2×C3×D1211yescharacteristic
554C3×C4C3×#23511yescharacteristic
555C3×C4C3×#23711yescharacteristic
556C3×C4C2×C3×#6711yescharacteristic
557Dic3—21no—
558Dic3—21no—
559Dic3—41no—
560Dic3—41no—
561C32—21no—
562C32—21no—
563C32—21no—
564C32—41no—
565C32#501611yescharacteristic, complemented
566C32#516511yescharacteristic, complemented
567C32C3×#46311yescharacteristic, complemented
568C23—21no—
569C23—181no—
570Q8—61no—
571Q8—61no—
572D4—21no—
573D4—181no—
574D4—181no—
575D4C3×#62612yes—
576C2×C4—31no—
577C2×C4—31no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—122no—
581C2×C4C2×C3×D3211yescharacteristic
582C8—61no—
583C8—181no—
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—721no—
604C2×C3#6393411yescharacteristic
605C2×C3#6393511yescharacteristic
606C2×C3#6397811yescharacteristic
607C2×C3C3×#91411yescharacteristic
608C2×C3C3×#91611yescharacteristic
609C2×C3C3×#93011yescharacteristic
610C2×C3C3×#93211yescharacteristic
611C2×C3C3×#93911yescharacteristic
612C2×C3C3×#95011yescharacteristic
613D3—242no—
614D3—241no—
615D3—241no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—361no—
620C22—361no—
621C22—361no—
622C22C3×D3×#6711yescharacteristic
623C4—61no—
624C4—61no—
625C4—121no—
626C4—121no—
627C4C3×#380411yescharacteristic
628C4C2×C3×#62611yescharacteristic
629C3—21no—
630C3—21no—
631C3—21no—
632C3—41no—
633C3#1059147111yescharacteristic, direct product factor
634C3C3×#501611yescharacteristic, complemented
635C3C3×#516511yescharacteristic, complemented
636C2—42no—
637C2—721no—
638C2C3×#6393411yescharacteristic
639C2C3×#6393511yescharacteristic
640C2C3×#6397811yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
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
2. C32×D4×Dic3C2semidirect
3. C2×C32×#214C2semidirect
4. C3×#63864C2semidirect
5. C3×Dic3⋊(C3⋊C8)C2semidirect
6. C3×#63565C2semidirect
7. C3×#63564C2semidirect
10. #10591471C3direct (N × H)
29. C3×D4×Dic3D3semidirect
30. C3×D4×Dic3C2×C3semidirect
32. C2×C3×#214D3semidirect
33. C2×C3×#214C2×C3semidirect
38. #63864C2×C3semidirect
45. Dic3⋊(C3⋊C8)C2×C3semidirect
46. #63565C2×C3semidirect
47. #63564C2×C3semidirect
142. D4×Dic3C3×D3semidirect
143. C2×#214C3×D3semidirect
369. C33#463semidirect
565. C32#5016semidirect
566. C32#5165semidirect
567. C32C3×#463semidirect
633. C3#10591471direct (N × H)
634. C3C3×#5016semidirect
635. C3C3×#5165semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37532
C32×D4×Dic32
C2×C32×#2142
C3×#638642
C3×Dic3⋊(C3⋊C8)2
C3×#635652
C3×#635642
C3×#51653
C3×#50163
#105914713