Finite Group Databasebeta

← C3×#10591295 #13158270

Subgroups — C3×#10591295 #13158270

Order 1728.

1378 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4×D3C211yesmaximal, characteristic
2C2×C3×Dic3⋊Dic3C212yesmaximal, complemented
3C2×C3×(C2×D3)⋊Dic3C212yesmaximal
4C3×#63658C211yesmaximal, characteristic, complemented
5C32×C3⋊#95C211yesmaximal, characteristic, complemented
6C3×#5212—31nomaximal
7C3×#4667—31nomaximal
8#10591295C311yesmaximal, characteristic, direct product factor
9C23×C32×D3C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2211yescharacteristic
14C2×C32×C4×D3—24no—
15C2×C32×C4×D3C414yescomplemented
16C3×Dic3⋊Dic3—24no—
17C3×(C2×D3)⋊Dic3—24no—
18C23×C32×C4—31no—
19C22×C3×C4×D3—21no—
20C22×C3×C4×D3D311yescharacteristic
21C22×C3×C4×D3C2×C311yescharacteristic
22C2×C3×#217—32no—
23C2×C3×(C2×D3)⋊C4—32no—
24C2×C3×C4⋊Dic3—32no—
25C2×C3×Dic3⋊C4—32no—
26C2×Dic3⋊Dic3C2×C312yescomplemented
27C2×(C2×D3)⋊Dic3C2×C312yes—
28C32×#95—31no—
29#63658C2×C311yescharacteristic, complemented
30C3×C3⋊#95—21no—
31C3×C3⋊#95—31no—
32C3×C3⋊#95—31no—
33C3×C3⋊#95—61no—
34C3×C3⋊#95D311yescharacteristic, complemented
35C3×C3⋊#95C2×C311yescharacteristic, complemented
36C23×C33C2311yescharacteristic
37C22×C32×D3—22no—
38C22×C32×D3—22no—
39C22×C32×D3C2×C412yes—
40C22×C32×D3D412yes—
41C22×C32×D3Q812yes—
42C2×C33×C4—22no—
43C2×C33×C4C2×C412yes—
44C2×C3×C3⋊Dic3—22no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3—22no—
50C2×C32×Dic3C2×C412yes—
51C2×C32×Dic3D414yes—
52C32×C4×D3—28no—
53C3×#380—91no—
54#5212—31no—
55#4667—31no—
56C24×C32—31no—
57C23×C3×D3—21no—
58C23×C3×D3C2×D311yescharacteristic
59C23×C3×D3C22×C311yescharacteristic
60C22×C32×C4—21no—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4—31no—
64C22×C32×C4—34no—
65C22×C32×C4—32no—
66C22×C32×C4—41no—
67C22×C32×C4—64no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C22×C3⋊Dic3C22×C312yes—
72C22×C3×Dic3—21no—
73C22×C3×Dic3—22no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—32no—
76C22×C3×Dic3—62no—
77C22×C3×Dic3C2×D312yes—
78C22×C3×Dic3C2×D311yescharacteristic
79C22×C3×Dic3C22×C312yes—
80C22×C3×Dic3C22×C311yescharacteristic
81C2×C3×C4×D3—28no—
82C2×C3×C4×D3—24no—
83C2×C3×C4×D3—24no—
84C2×C3×C4×D3—24no—
85C2×C3×C4×D3Dic314yescomplemented
86C2×C3×C4×D3C3×C414yescomplemented
87C3×#217—64no—
88C3×(C2×D3)⋊C4—64no—
89C3×C4⋊Dic3—64no—
90C3×Dic3⋊C4—64no—
91Dic3⋊Dic3—24no—
92(C2×D3)⋊Dic3—24no—
93C22×C33C22×C411yescharacteristic
94C22×C33C2×D412yes—
95C22×C33C2×D411yescharacteristic
96C22×C33C2×Q811yescharacteristic
97C22×C33#4112yes—
98C2×C32×D3—22no—
99C2×C32×D3—22no—
100C2×C32×D3—24no—
101C2×C32×D3—24no—
102C2×C32×D3C4⋊C414yescomplemented
103C33×C4—22no—
104C3×C3⋊Dic3—42no—
105C32×Dic3—42no—
106C32×Dic3C22⋊C414yescomplemented
107C23×C3×C4—31no—
108C23×C3×C4—31no—
109C23×C3×C4—61no—
110C22×C4×D3C3×D311yescharacteristic
111C2×C3×C4⋊C4—92no—
112C2×C3×C22⋊C4—92no—
113C2×#217—32no—
114C2×(C2×D3)⋊C4—32no—
115C2×C4⋊Dic3—32no—
116C2×Dic3⋊C4—32no—
117C3×#95—31no—
118C3×#95—31no—
119C3×#95—61no—
120C3×#95—91no—
121C3⋊#95—31no—
122C3⋊#95—31no—
123C3⋊#95—61no—
124C3⋊#95C3×D311yescharacteristic, complemented
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—32no—
129C23×C32—32no—
130C23×C32—32no—
131C23×C32—41no—
132C23×C32—62no—
133C23×C32—62no—
134C23×C32C22×D311yescharacteristic
135C23×C32C22×D311yescharacteristic
136C23×C32C23×C311yescharacteristic
137C22×C3×D3—24no—
138C22×C3×D3—24no—
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—22no—
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3—22no—
146C22×C3×D3Dic612yes—
147C22×C3×D3D1212yes—
148C22×C3×D3C2×Dic312yes—
149C22×C3×D3C2×C3×C412yes—
150C22×C3×D3C3×D412yes—
151C22×C3×D3C3×Q812yes—
152C2×C32×C4—22no—
153C2×C32×C4—24no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—32no—
160C2×C32×C4—34no—
161C2×C32×C4—42no—
162C2×C32×C4—42no—
163C2×C32×C4—42no—
164C2×C32×C4—44no—
165C2×C32×C4—68no—
166C2×C32×C4—62no—
167C2×C32×C4—62no—
168C2×C32×C4—62no—
169C2×C32×C4C4×D312yes—
170C2×C32×C4C2×Dic312yes—
171C2×C32×C4C2×C3×C412yes—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3—22no—
175C2×C3×Dic3—24no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—42no—
185C2×C3×Dic3—42no—
186C2×C3×Dic3—64no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—122no—
194C2×C3×Dic3—122no—
195C2×C3×Dic3C2×Dic312yes—
196C2×C3×Dic3#6714yes—
197C2×C3×Dic3C2×C3×C412yes—
198C2×C3×Dic3C3×D414yes—
199C3×C4×D3—216no—
200C3×C4×D3—28no—
201C3×C4×D3—28no—
202#380—91noSylow
203C2×C33C2×C22⋊C411yescharacteristic
204C2×C33C2×C4⋊C411yescharacteristic
205C2×C33#11711yescharacteristic
206C2×C33#12112yes—
207C2×C33C22⋊Q812yes—
208C32×D3—24no—
209C24×C3—31no—
210C24×C3—31no—
211C24×C3—61no—
212C23×D3C2×C3×D311yescharacteristic
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—31no—
217C22×C3×C4—31no—
218C22×C3×C4—34no—
219C22×C3×C4—34no—
220C22×C3×C4—32no—
221C22×C3×C4—32no—
222C22×C3×C4—41no—
223C22×C3×C4—64no—
224C22×C3×C4—61no—
225C22×C3×C4—62no—
226C22×C3×C4—64no—
227C22×C3×C4—64no—
228C22×C3×C4—68no—
229C22×C3×C4—92no—
230C22×C3×C4D3211yescharacteristic
231C22×C3×C4C2×C3×D311yescharacteristic
232C22×C3×C4C2×C3×D311yescharacteristic
233C22×Dic3—32no—
234C22×Dic3—32no—
235C22×Dic3—62no—
236C22×Dic3C2×C3×D312yes—
237C22×Dic3C2×C3×D311yescharacteristic
238C2×C4×D3—24no—
239C2×C4×D3C3×Dic314yescomplemented
240C3×C4⋊C4—184no—
241C3×C22⋊C4—184no—
242#217—64no—
243(C2×D3)⋊C4—64no—
244C4⋊Dic3—64no—
245Dic3⋊C4—64no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—22no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—21no—
257C22×C32—22no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—22no—
261C22×C32—34no—
262C22×C32—41no—
263C22×C32—42no—
264C22×C32—41no—
265C22×C32—41no—
266C22×C32—42no—
267C22×C32—62no—
268C22×C32—62no—
269C22×C32—64no—
270C22×C32—64no—
271C22×C32C2×Dic611yescharacteristic
272C22×C32C2×C4×D311yescharacteristic
273C22×C32C2×D1211yescharacteristic
274C22×C32#23512yes—
275C22×C32#23512yes—
276C22×C32D3×D411yescharacteristic
277C22×C32D3×D412yes—
278C22×C32D3×Q811yescharacteristic
279C22×C32C22×Dic311yescharacteristic
280C22×C32C2×#6712yes—
281C22×C32C22×C3×C411yescharacteristic
282C22×C32C2×C3×D411yescharacteristic
283C22×C32C2×C3×D412yes—
284C22×C32C2×C3×Q811yescharacteristic
285C22×C32C3×#4112yes—
286C2×C3×D3—24no—
287C2×C3×D3—28no—
288C2×C3×D3—24no—
289C2×C3×D3—28no—
290C2×C3×D3—24no—
291C2×C3×D3—22no—
292C2×C3×D3—24no—
293C2×C3×D3—22no—
294C2×C3×D3—22no—
295C2×C3×D3—22no—
296C2×C3×D3—24no—
297C2×C3×D3—24no—
298C2×C3×D3—24no—
299C2×C3×D3C4⋊Dic314yescomplemented
300C2×C3×D3C3×C4⋊C414yescomplemented
301C32×C4—22no—
302C32×C4—24no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—34no—
306C32×C4—42no—
307C32×C4—42no—
308C32×C4—44no—
309C32×C4—122no—
310C3⋊Dic3—42no—
311C3×Dic3—24no—
312C3×Dic3—44no—
313C3×Dic3—42no—
314C3×Dic3—42no—
315C3×Dic3—124no—
316C3×Dic3—122no—
317C3×Dic3—122no—
318C3×Dic3#21714yescomplemented
319C3×Dic3C3×C22⋊C414yescomplemented
320C23×C4—31no—
321C2×C4⋊C4—92no—
322C2×C22⋊C4—92no—
323#95—31no—
324#95—91no—
325C33#38011yescharacteristic, complemented, Sylow
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—32no—
330C23×C3—32no—
331C23×C3—32no—
332C23×C3—32no—
333C23×C3—32no—
334C23×C3—32no—
335C23×C3—41no—
336C23×C3—62no—
337C23×C3—62no—
338C23×C3—62no—
339C23×C3—64no—
340C23×C3—64no—
341C23×C3—62no—
342C23×C3—62no—
343C23×C3—62no—
344C23×C3—62no—
345C23×C3C2×D3211yescharacteristic
346C23×C3C22×C3×D311yescharacteristic
347C23×C3C22×C3×D311yescharacteristic
348C22×D3—22no—
349C22×D3—22no—
350C22×D3C3×Dic612yes—
351C22×D3C3×D1212yes—
352C22×D3C2×C3×Dic312yes—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—32no—
361C2×C3×C4—34no—
362C2×C3×C4—32no—
363C2×C3×C4—34no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—44no—
367C2×C3×C4—42no—
368C2×C3×C4—62no—
369C2×C3×C4—64no—
370C2×C3×C4—68no—
371C2×C3×C4—68no—
372C2×C3×C4—616no—
373C2×C3×C4—64no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4—62no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—122no—
381C2×C3×C4—122no—
382C2×C3×C4—182no—
383C2×C3×C4—182no—
384C2×C3×C4—182no—
385C2×C3×C4D3×Dic312yes—
386C2×C3×C4C3×C4×D312yes—
387C2×C3×C4C2×C3×Dic312yes—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—64no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—62no—
395C2×Dic3—62no—
396C2×Dic3—62no—
397C2×Dic3—62no—
398C2×Dic3—122no—
399C2×Dic3—122no—
400C2×Dic3C2×C3×Dic312yes—
401C2×Dic3C3×#6714yes—
402C4×D3—28no—
403C2×C32—21no—
404C2×C32—22no—
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—22no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—41no—
419C2×C32—41no—
420C2×C32—41no—
421C2×C32—42no—
422C2×C32—42no—
423C2×C32—64no—
424C2×C32#87211yescharacteristic
425C2×C32D3×C22⋊C411yescharacteristic
426C2×C32#88412yes—
427C2×C32D3×C4⋊C411yescharacteristic
428C2×C32#89612yes—
429C2×C32#92412yes—
430C2×C32C2×C4⋊Dic311yescharacteristic
431C2×C32#92611yescharacteristic
432C2×C32#93012yes—
433C2×C32C2×#21711yescharacteristic
434C2×C32C2×C3×C22⋊C411yescharacteristic
435C2×C32C2×C3×C4⋊C411yescharacteristic
436C2×C32C3×#11711yescharacteristic
437C2×C32C3×#12112yes—
438C2×C32C3×C22⋊Q812yes—
439C3×D3—28no—
440C3×D3—24no—
441C3×D3—24no—
442C24—31no—
443C22×C4—31no—
444C22×C4—34no—
445C22×C4—32no—
446C22×C4—64no—
447C22×C4—92no—
448C22×C4C3×D3211yescharacteristic
449C4⋊C4—184no—
450C22⋊C4—184no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—34no—
467C22×C3—34no—
468C22×C3—41no—
469C22×C3—42no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—42no—
473C22×C3—64no—
474C22×C3—64no—
475C22×C3—68no—
476C22×C3—64no—
477C22×C3—68no—
478C22×C3—62no—
479C22×C3—64no—
480C22×C3—64no—
481C22×C3—64no—
482C22×C3—62no—
483C22×C3—64no—
484C22×C3—62no—
485C22×C3—62no—
486C22×C3D3×Dic611yescharacteristic
487C22×C3#379812yes—
488C22×C3D3×D1211yescharacteristic
489C22×C3C2×D3×Dic311yescharacteristic
490C22×C3D3×#6712yes—
491C22×C3C2×C3×Dic611yescharacteristic
492C22×C3C2×C3×C4×D311yescharacteristic
493C22×C3C2×C3×D1211yescharacteristic
494C22×C3C3×#23512yes—
495C22×C3C3×#23512yes—
496C22×C3C3×D3×D411yescharacteristic
497C22×C3C3×D3×D412yes—
498C22×C3C3×D3×Q811yescharacteristic
499C22×C3C22×C3×Dic311yescharacteristic
500C22×C3C2×C3×#6712yes—
501C2×D3—24no—
502C2×D3—22no—
503C2×D3—24no—
504C2×D3—22no—
505C2×D3C3×C4⋊Dic314yescomplemented
506C3×C4—24no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—34no—
511C3×C4—34no—
512C3×C4—42no—
513C3×C4—42no—
514C3×C4—44no—
515C3×C4—64no—
516C3×C4—124no—
517C3×C4—122no—
518C3×C4—122no—
519C3×C4—362no—
520Dic3—42no—
521Dic3—124no—
522Dic3—122no—
523Dic3—122no—
524Dic3C3×#21714yescomplemented
525C32—21no—
526C32—21no—
527C32—21no—
528C32—41no—
529C32#466711yescharacteristic, complemented
530C32#521211yescharacteristic, complemented
531C32C3×#38011yescharacteristic, complemented
532C23—32no—
533C23—32no—
534C23—32no—
535C23—62no—
536C23—62no—
537C23C2×C3×D3211yescharacteristic
538C2×C4—22no—
539C2×C4—32no—
540C2×C4—34no—
541C2×C4—68no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—182no—
546C2×C4—182no—
547C2×C4—182no—
548C2×C4C3×D3×Dic312yes—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—22no—
553C2×C3—22no—
554C2×C3—22no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—22no—
562C2×C3—22no—
563C2×C3—22no—
564C2×C3—41no—
565C2×C3—42no—
566C2×C3—41no—
567C2×C3—41no—
568C2×C3—42no—
569C2×C3—68no—
570C2×C3—64no—
571C2×C3—64no—
572C2×C3#6385112yes—
573C2×C3#6385511yescharacteristic
574C2×C3D3×C4⋊Dic311yescharacteristic
575C2×C3#6390612yes—
576C2×C3D3×#21711yescharacteristic
577C2×C3C3×#87211yescharacteristic
578C2×C3C3×D3×C22⋊C411yescharacteristic
579C2×C3C3×#88412yes—
580C2×C3C3×D3×C4⋊C411yescharacteristic
581C2×C3C3×#89612yes—
582C2×C3C3×#92412yes—
583C2×C3C2×C3×C4⋊Dic311yescharacteristic
584C2×C3C3×#92611yescharacteristic
585C2×C3C3×#93012yes—
586C2×C3C2×C3×#21711yescharacteristic
587D3—24no—
588C22—34no—
589C22—62no—
590C22—64no—
591C22—62no—
592C22—64no—
593C22C3×D3×Dic611yescharacteristic
594C22C3×#379812yes—
595C22C3×D3×D1211yescharacteristic
596C22C2×C3×D3×Dic311yescharacteristic
597C22C3×D3×#6712yes—
598C4—22no—
599C4—34no—
600C4—122no—
601C4—362no—
602C3—21no—
603C3—21no—
604C3—21no—
605C3—41no—
606C3#1059129511yescharacteristic, direct product factor
607C3C3×#466711yescharacteristic, complemented
608C3C3×#521211yescharacteristic, complemented
609C2—64no—
610C2C3×#6385112yes—
611C2C3×#6385511yescharacteristic
612C2C3×D3×C4⋊Dic311yescharacteristic
613C2C3×#6390612yes—
614C2C3×D3×#21711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C32×D3C22
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#380
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic3⋊Dic3C2semidirect
4. C3×#63658C2semidirect
5. C32×C3⋊#95C2semidirect
8. #10591295C3direct (N × H)
15. C2×C32×C4×D3C4semidirect
26. C2×Dic3⋊Dic3C2×C3semidirect
29. #63658C2×C3semidirect
34. C3×C3⋊#95D3semidirect
35. C3×C3⋊#95C2×C3semidirect
85. C2×C3×C4×D3Dic3semidirect
86. C2×C3×C4×D3C3×C4semidirect
102. C2×C32×D3C4⋊C4semidirect
106. C32×Dic3C22⋊C4semidirect
124. C3⋊#95C3×D3semidirect
239. C2×C4×D3C3×Dic3semidirect
299. C2×C3×D3C4⋊Dic3semidirect
300. C2×C3×D3C3×C4⋊C4semidirect
318. C3×Dic3#217semidirect
319. C3×Dic3C3×C22⋊C4semidirect
325. C33#380semidirect
505. C2×D3C3×C4⋊Dic3semidirect
524. Dic3C3×#217semidirect
529. C32#4667semidirect
530. C32#5212semidirect
531. C32C3×#380semidirect
606. C3#10591295direct (N × H)
607. C3C3×#4667semidirect
608. C3C3×#5212semidirect

Maximal subgroups

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

subgroupindex
C22×C32×C4×D32
C2×C3×Dic3⋊Dic32
C2×C3×(C2×D3)⋊Dic32
C3×#636582
C32×C3⋊#952
C3×#52123
C3×#46673
#105912953