Finite Group Databasebeta

← C3×#10591594 #13158596

Subgroups — C3×#10591594 #13158596

Order 1728.

888 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×C32×D3×Q8C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3C3×#63856C211yesmaximal, characteristic
4C2×C3×#3718C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8C3×#5228—31nomaximal
9C3×#5186—31nomaximal
10#10591594C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C32×D3×Q8—22no—
13C32×D3×Q8—22no—
14C2×C32×C4×D3—21no—
15C2×C32×C4×D3C2211yescharacteristic
16C2×C32×Dic6—21no—
17C2×C32×Dic6C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×#216—22no—
22C2×C32×C3⋊C8C2211yescharacteristic
23C3×#3724—21no—
24C3×(C2×D3)⋊Dic3—21no—
25C3×#3718—22no—
26C22×C32×Q8—31no—
27C2×C3×D3×Q8—21no—
28C2×C3×D3×Q8D311yescharacteristic
29C2×C3×D3×Q8C2×C311yescharacteristic
30C2×C32×Q16—31no—
31C3×#946—31no—
32C2×C3×#216—21no—
33C2×C3×#216—31no—
34C2×C3×#216—31no—
35C2×C3×#216D311yescharacteristic, complemented
36C2×C3×#216C2×C311yescharacteristic, complemented
37C3×#924—31no—
38#63856C2×C311yescharacteristic
39C2×#3718C2×C311yescharacteristic, complemented
40#63662C2×C311yescharacteristic, complemented
41C3×#835—31no—
42C3×#835—31no—
43C3×#835—31no—
44C3×#835—61no—
45C3×#830—31no—
46C3×(C2×D3)⋊C8—31no—
47C3×#816—31no—
48#63566C2×C311yescharacteristic, complemented
49(C2×D3)⋊(C3⋊C8)C2×C311yescharacteristic
50C22×C32×D3D411yescharacteristic
51C33×Q8—21no—
52C33×Q8D412yes—
53C2×C33×C4—21no—
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3D411yescharacteristic
59C32×C4×D3—24no—
60C32×C4×D3—21no—
61C32×C4×D3—21no—
62C32×Dic6—24no—
63C32×Dic6—21no—
64C32×Dic6—21no—
65C32×Dic6—21no—
66C32×Dic6D412yes—
67C3×C3⋊(C3⋊C8)—21no—
68C32×C3⋊C8—21no—
69C3×#451—91no—
70#5228—31no—
71#5186—31no—
72C2×C32×Q8—21no—
73C2×C32×Q8—21no—
74C2×C32×Q8—21no—
75C2×C32×Q8—31no—
76C2×C32×Q8—31no—
77C2×C32×Q8—41no—
78C2×C32×Q8—61no—
79C2×C32×Q8—62no—
80C2×C32×Q8—62no—
81C2×C32×Q8C2×D311yescharacteristic
82C2×C32×Q8C2×D311yescharacteristic
83C2×C32×Q8C22×C311yescharacteristic
84C22×C32×C4—31no—
85C22×C32×C4—61no—
86C3×D3×Q8—24no—
87C3×D3×Q8—24no—
88C3×D3×Q8—22no—
89C3×D3×Q8—22no—
90C3×D3×Q8—22no—
91C3×D3×Q8—22no—
92C2×C3×C4×D3—22no—
93C2×C3×C4×D3—21no—
94C2×C3×C4×D3—21no—
95C2×C3×C4×D3—21no—
96C2×C3×C4×D3C2×D311yescharacteristic
97C2×C3×C4×D3C22×C311yescharacteristic
98C2×C3×Dic6—22no—
99C2×C3×Dic6—21no—
100C2×C3×Dic6—21no—
101C2×C3×Dic6—21no—
102C2×C3×Dic6—21no—
103C2×C3×Dic6C2×D311yescharacteristic
104C2×C3×Dic6C2×D311yescharacteristic
105C2×C3×Dic6C22×C311yescharacteristic
106C2×C3×Dic6C22×C311yescharacteristic
107C32×Q16—62no—
108C2×C32×C8—31no—
109C4⋊(C3⋊Dic3)C22×C311yescharacteristic
110C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
111C3×#217—61no—
112C3×#216—24no—
113C3×#216—22no—
114C3×#216—22no—
115C3×#216—62no—
116C3×#216—62no—
117C3×(C2×D3)⋊C4—61no—
118C3×C4⋊Dic3—31no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—61no—
121C3×Dic3⋊C4—61no—
122C3×Dic3⋊C4—61no—
123C2×C3×C3⋊C8—21no—
124C2×C3×C3⋊C8—31no—
125C2×C3×C3⋊C8—31no—
126C2×C3×C3⋊C8—61no—
127C2×C3×C3⋊C8C2×D311yescharacteristic
128C2×C3×C3⋊C8C22×C311yescharacteristic
129#3724—21no—
130(C2×D3)⋊Dic3—21no—
131#3718—22no—
132C22×C33C2×D411yescharacteristic
133C2×C32×D3—21no—
134C2×C32×D3—21no—
135C2×C32×D3Q1612yescomplemented
136C33×C4—22no—
137C33×C4C2×D411yescharacteristic
138C33×C4C2×D411yescharacteristic
139C3×C3⋊Dic3—41no—
140C32×Dic3—21no—
141C32×Dic3—22no—
142C32×Dic3—41no—
143C22×C3×Q8—31no—
144C22×C3×Q8—31no—
145C22×C3×Q8—61no—
146C2×D3×Q8C3×D311yescharacteristic
147C2×C3×Q16—31no—
148C2×C3×Q16—31no—
149C2×C3×Q16—61no—
150C2×C3×Q16—91no—
151C3×C22⋊Q8—91no—
152#946—31no—
153C2×#216—31no—
154C2×#216—31no—
155C2×#216C3×D311yescharacteristic, complemented
156#924—31no—
157C3×#103—91no—
158C3×#103—91no—
159C3×C22⋊C8—91no—
160#835—31no—
161#835—31no—
162#835—31no—
163#835—61no—
164#830—31no—
165(C2×D3)⋊C8—31no—
166#816—31no—
167C23×C32—31no—
168C22×C3×D3—21no—
169C22×C3×D3#6711yescharacteristic
170C22×C3×D3C3×D411yescharacteristic
171C32×Q8—21no—
172C32×Q8—22no—
173C32×Q8—21no—
174C32×Q8—21no—
175C32×Q8—22no—
176C32×Q8—22no—
177C32×Q8—22no—
178C32×Q8—32no—
179C32×Q8—42no—
180C32×Q8—41no—
181C32×Q8—41no—
182C32×Q8—42no—
183C32×Q8—64no—
184C32×Q8—61no—
185C32×Q8—61no—
186C32×Q8—61no—
187C32×Q8#6712yes—
188C32×Q8#6712yes—
189C32×Q8C3×D412yes—
190C2×C32×C4—21no—
191C2×C32×C4—22no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4—31no—
198C2×C32×C4—41no—
199C2×C32×C4—41no—
200C2×C32×C4—41no—
201C2×C32×C4—42no—
202C2×C32×C4—64no—
203C2×C32×C4—61no—
204C2×C32×C4—61no—
205C2×C32×C4—61no—
206C2×C32×C4—61no—
207C2×C32×C4C22×D311yescharacteristic
208C2×C32×C4C22×D311yescharacteristic
209C2×C32×C4C23×C311yescharacteristic
210C2×C3⋊Dic3—21no—
211C2×C3×Dic3—22no—
212C2×C3×Dic3—21no—
213C2×C3×Dic3—21no—
214C2×C3×Dic3—21no—
215C2×C3×Dic3—21no—
216C2×C3×Dic3—21no—
217C2×C3×Dic3—41no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—121no—
221C2×C3×Dic3#6711yescharacteristic
222C2×C3×Dic3C3×D411yescharacteristic
223C3×C4×D3—28no—
224C3×C4×D3—24no—
225C3×C4×D3—24no—
226C3×C4×D3—22no—
227C3×C4×D3—22no—
228C3×C4×D3—21no—
229C3×C4×D3—21no—
230C3×C4×D3—21no—
231C3×C4×D3—21no—
232C3×Dic6—28no—
233C3×Dic6—24no—
234C3×Dic6—24no—
235C3×Dic6—22no—
236C3×Dic6—21no—
237C3×Dic6—21no—
238C3×Dic6—22no—
239C3×Dic6—21no—
240C3×Dic6—21no—
241C3×Dic6—21no—
242C3×Dic6—21no—
243C3×Dic6—22no—
244C3×Dic6—41no—
245C3×Dic6#6712yes—
246C3×Dic6C3×D412yes—
247C32×C8—61no—
248C3⋊(C3⋊C8)—21no—
249C3×C3⋊C8—22no—
250C3×C3⋊C8—21no—
251C3×C3⋊C8—21no—
252C3×C3⋊C8—61no—
253C3×C3⋊C8—61no—
254C3×C3⋊C8—121no—
255#451—91noSylow
256C2×C33#12011yescharacteristic
257C2×C33C2×Q1611yescharacteristic
258C2×C33#13711yescharacteristic
259C32×D3—22no—
260C2×C3×Q8—21no—
261C2×C3×Q8—21no—
262C2×C3×Q8—21no—
263C2×C3×Q8—31no—
264C2×C3×Q8—31no—
265C2×C3×Q8—31no—
266C2×C3×Q8—31no—
267C2×C3×Q8—41no—
268C2×C3×Q8—61no—
269C2×C3×Q8—61no—
270C2×C3×Q8—61no—
271C2×C3×Q8—61no—
272C2×C3×Q8—62no—
273C2×C3×Q8—62no—
274C2×C3×Q8—62no—
275C2×C3×Q8—64no—
276C2×C3×Q8—64no—
277C2×C3×Q8—62no—
278C2×C3×Q8—62no—
279C2×C3×Q8D3211yescharacteristic
280C2×C3×Q8C2×C3×D311yescharacteristic
281C2×C3×Q8C2×C3×D311yescharacteristic
282C22×C3×C4—31no—
283C22×C3×C4—31no—
284C22×C3×C4—61no—
285C22×C3×C4—61no—
286C22×C3×C4—61no—
287C22×C3×C4—62no—
288D3×Q8—22no—
289D3×Q8—22no—
290C2×C4×D3—21no—
291C2×C4×D3C2×C3×D311yescharacteristic
292C2×Dic6—21no—
293C2×Dic6C2×C3×D311yescharacteristic
294C2×Dic6C2×C3×D311yescharacteristic
295C3×Q16—64no—
296C3×Q16—62no—
297C3×Q16—62no—
298C3×Q16—182no—
299C2×C3×C8—31no—
300C2×C3×C8—31no—
301C2×C3×C8—61no—
302C2×C3×C8—91no—
303C3×C4⋊C4—91no—
304C3×C4⋊C4—181no—
305C3×C22⋊C4—181no—
306#217—61no—
307#216—22no—
308#216—62no—
309#216—62no—
310(C2×D3)⋊C4—61no—
311C4⋊Dic3—31no—
312C4⋊Dic3—31no—
313C4⋊Dic3—61no—
314Dic3⋊C4—61no—
315Dic3⋊C4—61no—
316C2×C3⋊C8—31no—
317C2×C3⋊C8—31no—
318C2×C3⋊C8—61no—
319C2×C3⋊C8C2×C3×D311yescharacteristic
320C22×C32—21no—
321C22×C32—21no—
322C22×C32—21no—
323C22×C32—32no—
324C22×C32—41no—
325C22×C32—61no—
326C22×C32—61no—
327C22×C32D3×D411yescharacteristic
328C22×C32C2×#6711yescharacteristic
329C22×C32C2×C3×D411yescharacteristic
330C2×C3×D3—22no—
331C2×C3×D3—22no—
332C2×C3×D3—21no—
333C2×C3×D3—21no—
334C2×C3×D3—21no—
335C2×C3×D3—21no—
336C2×C3×D3—22no—
337C2×C3×D3#21612yescomplemented
338C2×C3×D3C3×Q1612yescomplemented
339C32×C4—22no—
340C32×C4—24no—
341C32×C4—22no—
342C32×C4—22no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—21no—
349C32×C4—41no—
350C32×C4—41no—
351C32×C4—42no—
352C32×C4—42no—
353C32×C4—44no—
354C32×C4—61no—
355C32×C4—62no—
356C32×C4—121no—
357C32×C4D3×D411yescharacteristic
358C32×C4C2×#6711yescharacteristic
359C32×C4C2×#6711yescharacteristic
360C32×C4C2×#6711yescharacteristic
361C32×C4C2×C3×D411yescharacteristic
362C32×C4C2×C3×D411yescharacteristic
363C3⋊Dic3—41no—
364C3×Dic3—22no—
365C3×Dic3—24no—
366C3×Dic3—21no—
367C3×Dic3—22no—
368C3×Dic3—21no—
369C3×Dic3—22no—
370C3×Dic3—42no—
371C3×Dic3—41no—
372C3×Dic3—41no—
373C3×Dic3—121no—
374C3×Dic3—121no—
375C3×Dic3—241no—
376C22×Q8—31no—
377C2×Q16—31no—
378C2×Q16—91no—
379C22⋊Q8—91no—
380#103—91no—
381#103—91no—
382C22⋊C8—91no—
383C33#45111yescharacteristic, complemented, Sylow
384C23×C3—31no—
385C23×C3—31no—
386C23×C3—61no—
387C22×D3C3×#6711yescharacteristic
388C3×Q8—22no—
389C3×Q8—21no—
390C3×Q8—21no—
391C3×Q8—21no—
392C3×Q8—22no—
393C3×Q8—22no—
394C3×Q8—22no—
395C3×Q8—32no—
396C3×Q8—32no—
397C3×Q8—41no—
398C3×Q8—41no—
399C3×Q8—42no—
400C3×Q8—42no—
401C3×Q8—62no—
402C3×Q8—68no—
403C3×Q8—64no—
404C3×Q8—64no—
405C3×Q8—62no—
406C3×Q8—62no—
407C3×Q8—61no—
408C3×Q8—61no—
409C3×Q8—61no—
410C3×Q8—61no—
411C3×Q8—61no—
412C3×Q8—61no—
413C3×Q8—121no—
414C3×Q8#62512yes—
415C3×Q8C3×#6712yes—
416C3×Q8C3×#6712yes—
417C2×C3×C4—22no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—31no—
425C2×C3×C4—31no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—42no—
429C2×C3×C4—41no—
430C2×C3×C4—61no—
431C2×C3×C4—68no—
432C2×C3×C4—64no—
433C2×C3×C4—64no—
434C2×C3×C4—62no—
435C2×C3×C4—62no—
436C2×C3×C4—62no—
437C2×C3×C4—61no—
438C2×C3×C4—61no—
439C2×C3×C4—61no—
440C2×C3×C4—61no—
441C2×C3×C4—61no—
442C2×C3×C4—61no—
443C2×C3×C4—61no—
444C2×C3×C4—61no—
445C2×C3×C4—121no—
446C2×C3×C4—181no—
447C2×C3×C4C2×D3211yescharacteristic
448C2×C3×C4C22×C3×D311yescharacteristic
449C2×C3×C4C22×C3×D311yescharacteristic
450C2×Dic3—21no—
451C2×Dic3—21no—
452C2×Dic3—61no—
453C2×Dic3—61no—
454C2×Dic3—121no—
455C2×Dic3C3×#6711yescharacteristic
456C4×D3—24no—
457C4×D3—21no—
458C4×D3—21no—
459Dic6—24no—
460Dic6—21no—
461Dic6—21no—
462Dic6—21no—
463Dic6C3×#6712yes—
464C3×C8—62no—
465C3×C8—61no—
466C3×C8—61no—
467C3×C8—181no—
468C3⋊C8—21no—
469C3⋊C8—61no—
470C3⋊C8—61no—
471C3⋊C8—121no—
472C2×C32—21no—
473C2×C32—21no—
474C2×C32—21no—
475C2×C32—21no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—21no—
481C2×C32—41no—
482C2×C32—41no—
483C2×C32—41no—
484C2×C32—62no—
485C2×C32D3×Q1611yescharacteristic
486C2×C32#91811yescharacteristic
487C2×C32#93711yescharacteristic
488C2×C32#94211yescharacteristic
489C2×C32C2×#21611yescharacteristic
490C2×C32#95311yescharacteristic
491C2×C32C3×#12011yescharacteristic
492C2×C32C2×C3×Q1611yescharacteristic
493C2×C32C3×#13711yescharacteristic
494C3×D3—24no—
495C3×D3—22no—
496C3×D3—22no—
497C2×Q8—31no—
498C2×Q8—31no—
499C2×Q8—61no—
500C2×Q8—62no—
501C2×Q8—62no—
502C2×Q8C3×D3211yescharacteristic
503C22×C4—31no—
504C22×C4—61no—
505Q16—62no—
506Q16—182no—
507C2×C8—31no—
508C2×C8—91no—
509C4⋊C4—91no—
510C4⋊C4—181no—
511C22⋊C4—181no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—32no—
516C22×C3—32no—
517C22×C3—41no—
518C22×C3—62no—
519C22×C3—62no—
520C22×C3—62no—
521C22×C3—61no—
522C22×C3—61no—
523C22×C3—61no—
524C22×C3—61no—
525C22×C3D3×#6711yescharacteristic
526C22×C3C3×D3×D411yescharacteristic
527C22×C3C2×C3×#6711yescharacteristic
528C2×D3—21no—
529C2×D3—21no—
530C2×D3C3×#21612yescomplemented
531C3×C4—24no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—22no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—42no—
542C3×C4—42no—
543C3×C4—44no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—62no—
547C3×C4—64no—
548C3×C4—61no—
549C3×C4—62no—
550C3×C4—61no—
551C3×C4—62no—
552C3×C4—122no—
553C3×C4—121no—
554C3×C4—121no—
555C3×C4—361no—
556C3×C4C2×#62511yescharacteristic
557C3×C4D3×#6711yescharacteristic
558C3×C4C3×D3×D411yescharacteristic
559C3×C4C2×C3×#6711yescharacteristic
560C3×C4C2×C3×#6711yescharacteristic
561C3×C4C2×C3×#6711yescharacteristic
562Dic3—21no—
563Dic3—22no—
564Dic3—41no—
565Dic3—121no—
566Dic3—121no—
567Dic3—241no—
568C32—21no—
569C32—21no—
570C32—21no—
571C32—41no—
572C32#518611yescharacteristic, complemented
573C32#522811yescharacteristic, complemented
574C32C3×#45111yescharacteristic, complemented
575C23—31no—
576Q8—21no—
577Q8—32no—
578Q8—64no—
579Q8—61no—
580Q8—61no—
581Q8—61no—
582Q8C3×#62512yes—
583C2×C4—21no—
584C2×C4—31no—
585C2×C4—64no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—181no—
591C2×C4C2×C3×D3211yescharacteristic
592C8—61no—
593C8—181no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—41no—
604C2×C3—41no—
605C2×C3—41no—
606C2×C3—64no—
607C2×C3—62no—
608C2×C3—62no—
609C2×C3D3×#21611yescharacteristic
610C2×C3#6394811yescharacteristic
611C2×C3#6397611yescharacteristic
612C2×C3C3×D3×Q1611yescharacteristic
613C2×C3C3×#91811yescharacteristic
614C2×C3C3×#93711yescharacteristic
615C2×C3C3×#94211yescharacteristic
616C2×C3C2×C3×#21611yescharacteristic
617C2×C3C3×#95311yescharacteristic
618D3—22no—
619C22—32no—
620C22—61no—
621C22—61no—
622C22C3×D3×#6711yescharacteristic
623C4—22no—
624C4—61no—
625C4—62no—
626C4—121no—
627C4—361no—
628C4C2×C3×#62511yescharacteristic
629C4C3×D3×#6711yescharacteristic
630C3—21no—
631C3—21no—
632C3—21no—
633C3—41no—
634C3#1059159411yescharacteristic, direct product factor
635C3C3×#518611yescharacteristic, complemented
636C3C3×#522811yescharacteristic, complemented
637C2—62no—
638C2C3×D3×#21611yescharacteristic
639C2C3×#6394811yescharacteristic
640C2C3×#6397611yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C32×#216C2semidirect
4. C2×C3×#3718C2semidirect
5. C3×#63662C2semidirect
6. C3×#63566C2semidirect
10. #10591594C3direct (N × H)
35. C2×C3×#216D3semidirect
36. C2×C3×#216C2×C3semidirect
39. C2×#3718C2×C3semidirect
40. #63662C2×C3semidirect
48. #63566C2×C3semidirect
135. C2×C32×D3Q16semidirect
155. C2×#216C3×D3semidirect
337. C2×C3×D3#216semidirect
338. C2×C3×D3C3×Q16semidirect
383. C33#451semidirect
530. C2×D3C3×#216semidirect
572. C32#5186semidirect
573. C32#5228semidirect
574. C32C3×#451semidirect
634. C3#10591594direct (N × H)
635. C3C3×#5186semidirect
636. C3C3×#5228semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×Q82
C2×C32×#2162
C3×#638562
C2×C3×#37182
C3×#636622
C3×#635662
C3×(C2×D3)⋊(C3⋊C8)2
C3×#52283
C3×#51863
#105915943