Finite Group Databasebeta

← C3×#10591310 #13158225

Subgroups — C3×#10591310 #13158225

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Dic6C211yesmaximal, characteristic, complemented
2C2×C3×#3752C211yesmaximal, characteristic
3C2×C3×#3722C212yesmaximal
4C32×C3⋊#95C211yesmaximal, characteristic, complemented
5C3×#63579C212yesmaximal, complemented
6C3×#5231—31nomaximal
7C3×#4673—31nomaximal
8#10591310C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2212yes—
14C2×C32×Dic6—22no—
15C2×C32×Dic6—24no—
16C3×#3752—22no—
17C3×#3722—24no—
18C22×C32×Q8—31no—
19C22×C3×Dic6—21no—
20C22×C3×Dic6D311yescharacteristic, complemented
21C22×C3×Dic6C2×C311yescharacteristic, complemented
22C2×#3752C2×C311yescharacteristic
23C2×C3×(C2×D3)⋊C4—31no—
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×(C2×D3)⋊C4—32no—
26C2×C3×(C2×D3)⋊C4—32no—
27C2×C3×(C2×D3)⋊C4—61no—
28C2×#3722C2×C312yes—
29C32×#95—31no—
30C3×C3⋊#95—21no—
31C3×C3⋊#95—32no—
32C3×C3⋊#95—32no—
33C3×C3⋊#95D311yescharacteristic, complemented
34C3×C3⋊#95C2×C311yescharacteristic, complemented
35#63579C2×C312yescomplemented
36C23×C33C2311yescharacteristic
37C22×C3×C3⋊D3—21no—
38C22×C3×C3⋊D3—22no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—22no—
41C22×C3×C3⋊D3Q812yescomplemented
42C2×C33×C4—22no—
43C2×C33×C4D412yes—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—22no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3—22no—
50C2×C32×Dic3—22no—
51C2×C32×Dic3D414yes—
52C32×Dic6—28no—
53C3×#393—91no—
54#5231—31no—
55#4673—31no—
56C23×C3⋊D3C22×C311yescharacteristic
57C23×C3×D3—31no—
58C23×C3×D3—31no—
59C23×C3×D3—61no—
60C2×C32×Q8—64no—
61C2×C32×Q8—62no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—32no—
66C22×C32×C4—32no—
67C22×C32×C4—41no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C22×C3⋊Dic3C22×C311yescharacteristic
72C22×C3×Dic3—22no—
73C22×C3×Dic3—22no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—61no—
77C22×C3×Dic3C2×D312yes—
78C22×C3×Dic3C2×D312yes—
79C22×C3×Dic3C22×C312yes—
80C22×C3×Dic3C22×C312yes—
81C2×C3×Dic6—24no—
82C2×C3×Dic6—28no—
83C2×C3×Dic6—22no—
84C2×C3×Dic6—22no—
85C2×C3×Dic6—24no—
86C2×C3×Dic6—24no—
87#3752—22no—
88C3×(C2×D3)⋊C4—64no—
89C3×(C2×D3)⋊C4—64no—
90C3×(C2×D3)⋊C4—62no—
91C3×(C2×D3)⋊C4—62no—
92C3×(C2×D3)⋊C4—64no—
93#3722—24no—
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D412yes—
96C22×C33C2×Q811yescharacteristic
97C22×C33#4112yes—
98C22×C33#4111yescharacteristic
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—24no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3—41no—
103C2×C3×C3⋊D3—42no—
104C33×C4—22no—
105C3×C3⋊Dic3—41no—
106C32×Dic3—24no—
107C32×Dic3—42no—
108C22×C3×Q8—31no—
109C22×C3×Q8—31no—
110C22×C3×Q8—61no—
111C22×Dic6C3×D311yescharacteristic, complemented
112C2×C3×C22⋊C4—92no—
113C2×C3×C22⋊C4—91no—
114C2×(C2×D3)⋊C4—31no—
115C2×(C2×D3)⋊C4—31no—
116C2×(C2×D3)⋊C4—32no—
117C2×(C2×D3)⋊C4—32no—
118C2×(C2×D3)⋊C4—61no—
119C3×#95—31no—
120C3×#95—31no—
121C3×#95—61no—
122C3×#95—92no—
123C3⋊#95—32no—
124C3⋊#95—32no—
125C3⋊#95C3×D311yescharacteristic, complemented
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—41no—
130C23×C32C22×D311yescharacteristic
131C23×C32C22×D311yescharacteristic
132C23×C32C23×C311yescharacteristic
133C22×C3⋊D3—21no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3—22no—
136C22×C3⋊D3—22no—
137C22×C3⋊D3C3×Q812yescomplemented
138C22×C3×D3—32no—
139C22×C3×D3—32no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—61no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—62no—
149C22×C3×D3—61no—
150C22×C3×D3—62no—
151C22×C3×D3—122no—
152C22×C3×D3—122no—
153C32×Q8—68no—
154C2×C32×C4—22no—
155C2×C32×C4—24no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—22no—
160C2×C32×C4—22no—
161C2×C32×C4—34no—
162C2×C32×C4—42no—
163C2×C32×C4—42no—
164C2×C32×C4—42no—
165C2×C32×C4—44no—
166C2×C32×C4—62no—
167C2×C32×C4—62no—
168C2×C32×C4—62no—
169C2×C32×C4—62no—
170C2×C32×C4—62no—
171C2×C32×C4D1212yes—
172C2×C32×C4#6712yes—
173C2×C32×C4C3×D412yes—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—22no—
176C2×C3×Dic3—24no—
177C2×C3×Dic3—24no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—24no—
190C2×C3×Dic3—42no—
191C2×C3×Dic3—42no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—122no—
198C2×C3×Dic3#6714yes—
199C2×C3×Dic3C3×D414yes—
200C3×Dic6—216no—
201C3×Dic6—28no—
202C3×Dic6—28no—
203#393—91noSylow
204C2×C33#12011yescharacteristic
205C2×C33C22⋊Q812yes—
206C2×C33C22⋊Q811yescharacteristic
207C2×C33#12411yescharacteristic
208C2×C33#12412yes—
209C3×C3⋊D3—42no—
210C24×C3—91no—
211C23×D3—31no—
212C23×D3—31no—
213C23×D3—61no—
214C2×C3×Q8—64no—
215C2×C3×Q8—64no—
216C2×C3×Q8—68no—
217C2×C3×Q8—64no—
218C2×C3×Q8—62no—
219C2×C3×Q8—62no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—32no—
224C22×C3×C4—32no—
225C22×C3×C4—32no—
226C22×C3×C4—32no—
227C22×C3×C4—41no—
228C22×C3×C4—62no—
229C22×C3×C4—62no—
230C22×C3×C4—91no—
231C22×C3×C4D3211yescharacteristic
232C22×C3×C4C2×C3×D311yescharacteristic
233C22×C3×C4C2×C3×D311yescharacteristic
234C22×Dic3—31no—
235C22×Dic3—31no—
236C22×Dic3—61no—
237C22×Dic3C2×C3×D312yes—
238C22×Dic3C2×C3×D312yes—
239C2×Dic6—22no—
240C2×Dic6—24no—
241C3×C22⋊C4—184no—
242C3×C22⋊C4—182no—
243(C2×D3)⋊C4—64no—
244(C2×D3)⋊C4—64no—
245(C2×D3)⋊C4—62no—
246(C2×D3)⋊C4—62no—
247(C2×D3)⋊C4—64no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—22no—
251C22×C32—21no—
252C22×C32—22no—
253C22×C32—21no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—22no—
263C22×C32—41no—
264C22×C32—42no—
265C22×C32—41no—
266C22×C32—41no—
267C22×C32—42no—
268C22×C32C2×D1211yescharacteristic
269C22×C32#23512yes—
270C22×C32D3×D412yes—
271C22×C32#23711yescharacteristic
272C22×C32D3×Q811yescharacteristic
273C22×C32D3×Q811yescharacteristic
274C22×C32#23911yescharacteristic
275C22×C32#23912yes—
276C22×C32C2×#6711yescharacteristic
277C22×C32C2×#6712yes—
278C22×C32C2×C3×D411yescharacteristic
279C22×C32C2×C3×D412yes—
280C22×C32C2×C3×Q811yescharacteristic
281C22×C32C3×#4112yes—
282C22×C32C3×#4111yescharacteristic
283C2×C3⋊D3—22no—
284C2×C3⋊D3—24no—
285C2×C3⋊D3—41no—
286C2×C3⋊D3—42no—
287C2×C3⋊D3—41no—
288C2×C3×D3—64no—
289C2×C3×D3—64no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—64no—
293C2×C3×D3—124no—
294C2×C3×D3—122no—
295C2×C3×D3—124no—
296C2×C3×D3—122no—
297C2×C3×D3—121no—
298C2×C3×D3—121no—
299C2×C3×D3—122no—
300C2×C3×D3—121no—
301C2×C3×D3—122no—
302C2×C3×D3—121no—
303C32×C4—22no—
304C32×C4—24no—
305C32×C4—22no—
306C32×C4—22no—
307C32×C4—42no—
308C32×C4—42no—
309C32×C4—44no—
310C32×C4—64no—
311C32×C4—122no—
312C3⋊Dic3—41no—
313C3×Dic3—28no—
314C3×Dic3—24no—
315C3×Dic3—24no—
316C3×Dic3—44no—
317C3×Dic3—42no—
318C3×Dic3—42no—
319C3×Dic3—122no—
320C3×Dic3—121no—
321C3×Dic3—121no—
322C22×Q8—31no—
323C2×C22⋊C4—92no—
324C2×C22⋊C4—91no—
325#95—31no—
326#95—92no—
327C33#39311yescharacteristic, complemented, Sylow
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—41no—
332C23×C3—92no—
333C23×C3—181no—
334C23×C3—182no—
335C23×C3—181no—
336C23×C3—182no—
337C23×C3C2×D3211yescharacteristic
338C23×C3C22×C3×D311yescharacteristic
339C23×C3C22×C3×D311yescharacteristic
340C22×D3—32no—
341C22×D3—32no—
342C22×D3—62no—
343C22×D3—62no—
344C22×D3—62no—
345C22×D3—61no—
346C22×D3—62no—
347C22×D3—62no—
348C22×D3—61no—
349C22×D3—61no—
350C22×D3—62no—
351C22×D3—61no—
352C22×D3—62no—
353C22×D3—122no—
354C22×D3—122no—
355C3×Q8—616no—
356C3×Q8—68no—
357C3×Q8—68no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—34no—
366C2×C3×C4—34no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—44no—
370C2×C3×C4—42no—
371C2×C3×C4—64no—
372C2×C3×C4—64no—
373C2×C3×C4—64no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4—62no—
377C2×C3×C4—62no—
378C2×C3×C4—64no—
379C2×C3×C4—62no—
380C2×C3×C4—62no—
381C2×C3×C4—62no—
382C2×C3×C4—62no—
383C2×C3×C4—62no—
384C2×C3×C4—62no—
385C2×C3×C4—122no—
386C2×C3×C4—122no—
387C2×C3×C4—182no—
388C2×C3×C4—181no—
389C2×C3×C4#62612yes—
390C2×C3×C4C3×D1212yes—
391C2×C3×C4C3×#6712yes—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—62no—
398C2×Dic3—61no—
399C2×Dic3—62no—
400C2×Dic3—62no—
401C2×Dic3—61no—
402C2×Dic3—122no—
403C2×Dic3C3×#6714yes—
404Dic6—28no—
405C2×C32—21no—
406C2×C32—22no—
407C2×C32—22no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—22no—
411C2×C32—21no—
412C2×C32—22no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—22no—
417C2×C32—22no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—41no—
421C2×C32—41no—
422C2×C32—42no—
423C2×C32—41no—
424C2×C32—42no—
425C2×C32#87511yescharacteristic
426C2×C32#88211yescharacteristic
427C2×C32#88512yes—
428C2×C32#89612yes—
429C2×C32#89711yescharacteristic
430C2×C32#94611yescharacteristic
431C2×C32#94612yes—
432C2×C32#94711yescharacteristic
433C2×C32#94712yes—
434C2×C32#95311yescharacteristic
435C2×C32C3×#12011yescharacteristic
436C2×C32C3×C22⋊Q812yes—
437C2×C32C3×C22⋊Q811yescharacteristic
438C2×C32C3×#12412yes—
439C2×C32C3×#12411yescharacteristic
440C3⋊D3—42no—
441C3×D3—124no—
442C3×D3—122no—
443C3×D3—122no—
444C24—91no—
445C2×Q8—64no—
446C2×Q8—62no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—91no—
450C22×C4C3×D3211yescharacteristic
451C22⋊C4—184no—
452C22⋊C4—182no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—42no—
471C22×C3—41no—
472C22×C3—42no—
473C22×C3—182no—
474C22×C3—184no—
475C22×C3—361no—
476C22×C3—362no—
477C22×C3—361no—
478C22×C3#379611yescharacteristic
479C22×C3#379711yescharacteristic
480C22×C3#379912yes—
481C22×C3C2×#62611yescharacteristic
482C22×C3D3×#6712yes—
483C22×C3C2×C3×D1211yescharacteristic
484C22×C3C3×#23512yes—
485C22×C3C3×D3×D412yes—
486C22×C3C3×#23711yescharacteristic
487C22×C3C3×D3×Q811yescharacteristic
488C22×C3C3×D3×Q811yescharacteristic
489C22×C3C3×#23911yescharacteristic
490C22×C3C3×#23912yes—
491C22×C3C2×C3×#6711yescharacteristic
492C22×C3C2×C3×#6712yes—
493C2×D3—64no—
494C2×D3—64no—
495C2×D3—62no—
496C2×D3—62no—
497C2×D3—64no—
498C2×D3—124no—
499C2×D3—122no—
500C2×D3—122no—
501C2×D3—124no—
502C2×D3—121no—
503C2×D3—122no—
504C2×D3—121no—
505C2×D3—122no—
506C2×D3—121no—
507C2×D3—121no—
508C3×C4—24no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—42no—
513C3×C4—42no—
514C3×C4—44no—
515C3×C4—68no—
516C3×C4—64no—
517C3×C4—64no—
518C3×C4—124no—
519C3×C4—122no—
520C3×C4—122no—
521C3×C4—361no—
522Dic3—24no—
523Dic3—42no—
524Dic3—122no—
525Dic3—121no—
526Dic3—121no—
527C32—21no—
528C32—21no—
529C32—21no—
530C32—41no—
531C32#467311yescharacteristic, complemented
532C32#523111yescharacteristic, complemented
533C32C3×#39311yescharacteristic, complemented
534C23—92no—
535C23—181no—
536C23—182no—
537C23—181no—
538C23—182no—
539C23C2×C3×D3211yescharacteristic
540Q8—68no—
541C2×C4—22no—
542C2×C4—34no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—182no—
549C2×C4—181no—
550C2×C4C3×#62612yes—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—22no—
561C2×C3—22no—
562C2×C3—22no—
563C2×C3—22no—
564C2×C3—22no—
565C2×C3—22no—
566C2×C3—41no—
567C2×C3—41no—
568C2×C3—41no—
569C2×C3—42no—
570C2×C3—42no—
571C2×C3—362no—
572C2×C3#6386411yescharacteristic
573C2×C3#6387111yescharacteristic
574C2×C3#6388812yes—
575C2×C3#6390712yes—
576C2×C3#6397711yescharacteristic
577C2×C3C3×#87511yescharacteristic
578C2×C3C3×#88211yescharacteristic
579C2×C3C3×#88512yes—
580C2×C3C3×#89612yes—
581C2×C3C3×#89711yescharacteristic
582C2×C3C3×#94611yescharacteristic
583C2×C3C3×#94612yes—
584C2×C3C3×#94711yescharacteristic
585C2×C3C3×#94712yes—
586C2×C3C3×#95311yescharacteristic
587D3—124no—
588D3—122no—
589D3—122no—
590C22—182no—
591C22—184no—
592C22—361no—
593C22—361no—
594C22—362no—
595C22C3×#379611yescharacteristic
596C22C3×#379711yescharacteristic
597C22C3×#379912yes—
598C22C2×C3×#62611yescharacteristic
599C22C3×D3×#6712yes—
600C4—22no—
601C4—64no—
602C4—122no—
603C4—361no—
604C3—21no—
605C3—21no—
606C3—21no—
607C3—41no—
608C3#1059131011yescharacteristic, direct product factor
609C3C3×#467311yescharacteristic, complemented
610C3C3×#523111yescharacteristic, complemented
611C2—362no—
612C2C3×#6386411yescharacteristic
613C2C3×#6387111yescharacteristic
614C2C3×#6388812yes—
615C2C3×#6390712yes—
616C2C3×#6397711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypercenterC22×C3×C4D32
hypocenterC32C3×#393
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C32×Dic6C2semidirect
4. C32×C3⋊#95C2semidirect
5. C3×#63579C2semidirect
8. #10591310C3direct (N × H)
20. C22×C3×Dic6D3semidirect
21. C22×C3×Dic6C2×C3semidirect
33. C3×C3⋊#95D3semidirect
34. C3×C3⋊#95C2×C3semidirect
35. #63579C2×C3semidirect
41. C22×C3×C3⋊D3Q8semidirect
111. C22×Dic6C3×D3semidirect
125. C3⋊#95C3×D3semidirect
137. C22×C3⋊D3C3×Q8semidirect
327. C33#393semidirect
531. C32#4673semidirect
532. C32#5231semidirect
533. C32C3×#393semidirect
608. C3#10591310direct (N × H)
609. C3C3×#4673semidirect
610. C3C3×#5231semidirect

Maximal subgroups

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

subgroupindex
C22×C32×Dic62
C2×C3×#37522
C2×C3×#37222
C32×C3⋊#952
C3×#635792
C3×#52313
C3×#46733
#105913103