Finite Group Databasebeta

← C3×#10591294 #13158215

Subgroups — C3×#10591294 #13158215

Order 1728.

1314 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4×D3C211yesmaximal, characteristic
2C2×C3×(C2×D3)⋊Dic3C212yesmaximal
3C3×#63658C211yesmaximal, characteristic, complemented
4C32×C3⋊#95C211yesmaximal, characteristic, complemented
5C3×#63579C212yesmaximal, complemented
6C3×#5211—31nomaximal
7C3×#4664—31nomaximal
8#10591294C311yesmaximal, 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×(C2×D3)⋊Dic3—24no—
17C23×C32×C4—31no—
18C22×C3×C4×D3—21no—
19C22×C3×C4×D3D311yescharacteristic
20C22×C3×C4×D3C2×C311yescharacteristic
21C2×C3×#217—32no—
22C2×C3×(C2×D3)⋊C4—32no—
23C2×(C2×D3)⋊Dic3C2×C312yes—
24C32×#95—31no—
25#63658C2×C311yescharacteristic, complemented
26C3×C3⋊#95—21no—
27C3×C3⋊#95—31no—
28C3×C3⋊#95—31no—
29C3×C3⋊#95—32no—
30C3×C3⋊#95—32no—
31C3×C3⋊#95—61no—
32C3×C3⋊#95D311yescharacteristic, complemented
33C3×C3⋊#95C2×C311yescharacteristic, complemented
34#63579C2×C312yescomplemented
35C23×C33C2311yescharacteristic
36C22×C32×D3—22no—
37C22×C32×D3—21no—
38C22×C32×D3—21no—
39C22×C32×D3C2×C412yes—
40C22×C32×D3D414yes—
41C2×C33×C4—22no—
42C2×C33×C4C2×C412yes—
43C2×C3×C3⋊Dic3—22no—
44C2×C3×C3⋊Dic3—22no—
45C2×C3×C3⋊Dic3—22no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3—22no—
50C2×C32×Dic3C2×C412yes—
51C32×C4×D3—28no—
52C3×#381—91no—
53#5211—31no—
54#4664—31no—
55C24×C32—31no—
56C23×C3×D3—21no—
57C23×C3×D3C2×D311yescharacteristic
58C23×C3×D3C22×C311yescharacteristic
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4—21no—
62C22×C32×C4—31no—
63C22×C32×C4—34no—
64C22×C32×C4—32no—
65C22×C32×C4—41no—
66C22×C32×C4—64no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C22×C3⋊Dic3C22×C312yes—
71C22×C3×Dic3—21no—
72C22×C3×Dic3—22no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—62no—
76C22×C3×Dic3C2×D312yes—
77C22×C3×Dic3C2×D311yescharacteristic
78C22×C3×Dic3C22×C312yes—
79C22×C3×Dic3C22×C311yescharacteristic
80C2×C3×C4×D3—28no—
81C2×C3×C4×D3—24no—
82C2×C3×C4×D3—24no—
83C2×C3×C4×D3—24no—
84C2×C3×C4×D3Dic314yescomplemented
85C2×C3×C4×D3C3×C414yescomplemented
86C3×#217—64no—
87C3×(C2×D3)⋊C4—64no—
88(C2×D3)⋊Dic3—24no—
89C22×C33C22×C411yescharacteristic
90C22×C33C2×D412yes—
91C22×C33#4112yes—
92C22×C33#4111yescharacteristic
93C22×C33#4111yescharacteristic
94C2×C32×D3—22no—
95C2×C32×D3—22no—
96C2×C32×D3—24no—
97C2×C32×D3—24no—
98C2×C32×D3C22⋊C414yes—
99C33×C4—22no—
100C3×C3⋊Dic3—42no—
101C32×Dic3—22no—
102C32×Dic3—42no—
103C23×C3×C4—31no—
104C23×C3×C4—31no—
105C23×C3×C4—61no—
106C22×C4×D3C3×D311yescharacteristic
107C2×C3×C22⋊C4—92no—
108C2×#217—32no—
109C2×(C2×D3)⋊C4—32no—
110C3×#95—31no—
111C3×#95—31no—
112C3×#95—61no—
113C3×#95—92no—
114C3×#95—91no—
115C3⋊#95—31no—
116C3⋊#95—31no—
117C3⋊#95—32no—
118C3⋊#95—32no—
119C3⋊#95—61no—
120C3⋊#95C3×D311yescharacteristic, complemented
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—32no—
125C23×C32—34no—
126C23×C32—41no—
127C23×C32—62no—
128C23×C32—61no—
129C23×C32—61no—
130C23×C32C22×D311yescharacteristic
131C23×C32C22×D311yescharacteristic
132C23×C32C23×C311yescharacteristic
133C22×C3×D3—22no—
134C22×C3×D3—24no—
135C22×C3×D3—22no—
136C22×C3×D3—21no—
137C22×C3×D3—22no—
138C22×C3×D3—21no—
139C22×C3×D3—22no—
140C22×C3×D3—21no—
141C22×C3×D3—21no—
142C22×C3×D3—22no—
143C22×C3×D3—24no—
144C22×C3×D3C2×Dic312yes—
145C22×C3×D3#6714yes—
146C22×C3×D3C2×C3×C412yes—
147C22×C3×D3C3×D414yes—
148C2×C32×C4—22no—
149C2×C32×C4—24no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—32no—
156C2×C32×C4—42no—
157C2×C32×C4—42no—
158C2×C32×C4—42no—
159C2×C32×C4—44no—
160C2×C32×C4—68no—
161C2×C32×C4—62no—
162C2×C32×C4—62no—
163C2×C32×C4—62no—
164C2×C32×C4—62no—
165C2×C32×C4C4×D312yes—
166C2×C32×C4C2×Dic312yes—
167C2×C32×C4C2×C3×C412yes—
168C2×C3⋊Dic3—22no—
169C2×C3⋊Dic3—22no—
170C2×C3⋊Dic3—22no—
171C2×C3×Dic3—24no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—24no—
175C2×C3×Dic3—22no—
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—42no—
183C2×C3×Dic3—42no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—122no—
192C2×C3×Dic3—122no—
193C2×C3×Dic3C2×Dic312yes—
194C2×C3×Dic3C2×C3×C412yes—
195C3×C4×D3—216no—
196C3×C4×D3—28no—
197C3×C4×D3—28no—
198#381—91noSylow
199C2×C33C2×C22⋊C411yescharacteristic
200C2×C33#11711yescharacteristic
201C2×C33#11711yescharacteristic
202C2×C33#12312yes—
203C2×C33#12312yes—
204C32×D3—24no—
205C24×C3—31no—
206C24×C3—31no—
207C24×C3—61no—
208C23×D3C2×C3×D311yescharacteristic
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—34no—
215C22×C3×C4—34no—
216C22×C3×C4—32no—
217C22×C3×C4—32no—
218C22×C3×C4—41no—
219C22×C3×C4—64no—
220C22×C3×C4—61no—
221C22×C3×C4—62no—
222C22×C3×C4—64no—
223C22×C3×C4—64no—
224C22×C3×C4—68no—
225C22×C3×C4—92no—
226C22×C3×C4D3211yescharacteristic
227C22×C3×C4C2×C3×D311yescharacteristic
228C22×C3×C4C2×C3×D311yescharacteristic
229C22×Dic3—32no—
230C22×Dic3—32no—
231C22×Dic3—62no—
232C22×Dic3C2×C3×D312yes—
233C22×Dic3C2×C3×D311yescharacteristic
234C2×C4×D3—24no—
235C2×C4×D3C3×Dic314yescomplemented
236C3×C22⋊C4—184no—
237#217—64no—
238(C2×D3)⋊C4—64no—
239C22×C32—21no—
240C22×C32—22no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—21no—
249C22×C32—22no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—22no—
254C22×C32—34no—
255C22×C32—41no—
256C22×C32—42no—
257C22×C32—41no—
258C22×C32—42no—
259C22×C32—41no—
260C22×C32—62no—
261C22×C32—64no—
262C22×C32—62no—
263C22×C32—64no—
264C22×C32C2×C4×D311yescharacteristic
265C22×C32#23511yescharacteristic
266C22×C32#23512yes—
267C22×C32#23512yes—
268C22×C32#23511yescharacteristic
269C22×C32D3×D412yes—
270C22×C32#23711yescharacteristic
271C22×C32#23911yescharacteristic
272C22×C32C22×Dic311yescharacteristic
273C22×C32C2×#6712yes—
274C22×C32C22×C3×C411yescharacteristic
275C22×C32C2×C3×D412yes—
276C22×C32C3×#4112yes—
277C22×C32C3×#4111yescharacteristic
278C22×C32C3×#4111yescharacteristic
279C2×C3×D3—24no—
280C2×C3×D3—24no—
281C2×C3×D3—28no—
282C2×C3×D3—28no—
283C2×C3×D3—22no—
284C2×C3×D3—22no—
285C2×C3×D3—24no—
286C2×C3×D3—24no—
287C2×C3×D3—22no—
288C2×C3×D3—22no—
289C2×C3×D3—24no—
290C2×C3×D3—24no—
291C2×C3×D3—24no—
292C2×C3×D3#21714yes—
293C2×C3×D3C3×C22⋊C414yes—
294C32×C4—22no—
295C32×C4—24no—
296C32×C4—22no—
297C32×C4—22no—
298C32×C4—42no—
299C32×C4—42no—
300C32×C4—44no—
301C32×C4—62no—
302C32×C4—122no—
303C3⋊Dic3—42no—
304C3×Dic3—24no—
305C3×Dic3—22no—
306C3×Dic3—22no—
307C3×Dic3—44no—
308C3×Dic3—42no—
309C3×Dic3—42no—
310C3×Dic3—124no—
311C3×Dic3—122no—
312C3×Dic3—122no—
313C23×C4—31no—
314C2×C22⋊C4—92no—
315#95—31no—
316#95—92no—
317#95—91no—
318C33#38111yescharacteristic, complemented, Sylow
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—32no—
323C23×C3—34no—
324C23×C3—32no—
325C23×C3—34no—
326C23×C3—41no—
327C23×C3—62no—
328C23×C3—64no—
329C23×C3—62no—
330C23×C3—64no—
331C23×C3—62no—
332C23×C3—61no—
333C23×C3—62no—
334C23×C3—61no—
335C23×C3—62no—
336C23×C3—61no—
337C23×C3—61no—
338C23×C3C2×D3211yescharacteristic
339C23×C3C22×C3×D311yescharacteristic
340C23×C3C22×C3×D311yescharacteristic
341C22×D3—21no—
342C22×D3—22no—
343C22×D3—21no—
344C22×D3C2×C3×Dic312yes—
345C22×D3C3×#6714yes—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—32no—
354C2×C3×C4—32no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—44no—
358C2×C3×C4—42no—
359C2×C3×C4—62no—
360C2×C3×C4—68no—
361C2×C3×C4—68no—
362C2×C3×C4—616no—
363C2×C3×C4—64no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—64no—
367C2×C3×C4—62no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—122no—
374C2×C3×C4—122no—
375C2×C3×C4—182no—
376C2×C3×C4—182no—
377C2×C3×C4—182no—
378C2×C3×C4D3×Dic312yes—
379C2×C3×C4C3×C4×D312yes—
380C2×C3×C4C2×C3×Dic312yes—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—64no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—62no—
389C2×Dic3—62no—
390C2×Dic3—62no—
391C2×Dic3—62no—
392C2×Dic3—122no—
393C2×Dic3—122no—
394C2×Dic3C2×C3×Dic312yes—
395C4×D3—28no—
396C2×C32—21no—
397C2×C32—22no—
398C2×C32—21no—
399C2×C32—22no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—22no—
403C2×C32—22no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—22no—
407C2×C32—21no—
408C2×C32—22no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—41no—
412C2×C32—41no—
413C2×C32—41no—
414C2×C32—42no—
415C2×C32—42no—
416C2×C32—64no—
417C2×C32#87211yescharacteristic
418C2×C32D3×C22⋊C411yescharacteristic
419C2×C32#88312yes—
420C2×C32#89211yescharacteristic
421C2×C32#89412yes—
422C2×C32#92611yescharacteristic
423C2×C32#92611yescharacteristic
424C2×C32#92912yes—
425C2×C32#92912yes—
426C2×C32C2×#21711yescharacteristic
427C2×C32C2×C3×C22⋊C411yescharacteristic
428C2×C32C3×#11711yescharacteristic
429C2×C32C3×#11711yescharacteristic
430C2×C32C3×#12312yes—
431C2×C32C3×#12312yes—
432C3×D3—28no—
433C3×D3—24no—
434C3×D3—24no—
435C24—31no—
436C22×C4—31no—
437C22×C4—34no—
438C22×C4—32no—
439C22×C4—64no—
440C22×C4—92no—
441C22×C4C3×D3211yescharacteristic
442C22⋊C4—184no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—34no—
459C22×C3—34no—
460C22×C3—41no—
461C22×C3—42no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—64no—
466C22×C3—64no—
467C22×C3—68no—
468C22×C3—64no—
469C22×C3—68no—
470C22×C3—64no—
471C22×C3—62no—
472C22×C3—64no—
473C22×C3—62no—
474C22×C3—64no—
475C22×C3—62no—
476C22×C3—64no—
477C22×C3—62no—
478C22×C3#379511yescharacteristic
479C22×C3#379812yes—
480C22×C3#379911yescharacteristic
481C22×C3C2×D3×Dic311yescharacteristic
482C22×C3D3×#6712yes—
483C22×C3C2×C3×C4×D311yescharacteristic
484C22×C3C3×#23511yescharacteristic
485C22×C3C3×#23512yes—
486C22×C3C3×#23512yes—
487C22×C3C3×#23511yescharacteristic
488C22×C3C3×D3×D412yes—
489C22×C3C3×#23711yescharacteristic
490C22×C3C3×#23911yescharacteristic
491C22×C3C22×C3×Dic311yescharacteristic
492C22×C3C2×C3×#6712yes—
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—24no—
496C2×D3—24no—
497C2×D3C3×#21714yes—
498C3×C4—24no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—42no—
503C3×C4—42no—
504C3×C4—44no—
505C3×C4—64no—
506C3×C4—62no—
507C3×C4—62no—
508C3×C4—124no—
509C3×C4—122no—
510C3×C4—122no—
511C3×C4—362no—
512Dic3—22no—
513Dic3—42no—
514Dic3—124no—
515Dic3—122no—
516Dic3—122no—
517C32—21no—
518C32—21no—
519C32—21no—
520C32—41no—
521C32#466411yescharacteristic, complemented
522C32#521111yescharacteristic, complemented
523C32C3×#38111yescharacteristic, complemented
524C23—32no—
525C23—34no—
526C23—61no—
527C23—62no—
528C23—61no—
529C23C2×C3×D3211yescharacteristic
530C2×C4—22no—
531C2×C4—32no—
532C2×C4—68no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—182no—
538C2×C4—182no—
539C2×C4—182no—
540C2×C4C3×D3×Dic312yes—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—22no—
548C2×C3—22no—
549C2×C3—22no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—22no—
554C2×C3—22no—
555C2×C3—22no—
556C2×C3—41no—
557C2×C3—41no—
558C2×C3—42no—
559C2×C3—41no—
560C2×C3—42no—
561C2×C3—68no—
562C2×C3—64no—
563C2×C3—64no—
564C2×C3#6384111yescharacteristic
565C2×C3#6385012yes—
566C2×C3#6385511yescharacteristic
567C2×C3#6390512yes—
568C2×C3D3×#21711yescharacteristic
569C2×C3C3×#87211yescharacteristic
570C2×C3C3×D3×C22⋊C411yescharacteristic
571C2×C3C3×#88312yes—
572C2×C3C3×#89211yescharacteristic
573C2×C3C3×#89412yes—
574C2×C3C3×#92611yescharacteristic
575C2×C3C3×#92611yescharacteristic
576C2×C3C3×#92912yes—
577C2×C3C3×#92912yes—
578C2×C3C2×C3×#21711yescharacteristic
579D3—24no—
580C22—34no—
581C22—62no—
582C22—64no—
583C22—62no—
584C22—64no—
585C22C3×#379511yescharacteristic
586C22C3×#379812yes—
587C22C3×#379911yescharacteristic
588C22C2×C3×D3×Dic311yescharacteristic
589C22C3×D3×#6712yes—
590C4—22no—
591C4—62no—
592C4—122no—
593C4—362no—
594C3—21no—
595C3—21no—
596C3—21no—
597C3—41no—
598C3#1059129411yescharacteristic, direct product factor
599C3C3×#466411yescharacteristic, complemented
600C3C3×#521111yescharacteristic, complemented
601C2—64no—
602C2C3×#6384111yescharacteristic
603C2C3×#6385012yes—
604C2C3×#6385511yescharacteristic
605C2C3×#6390512yes—
606C2C3×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×#381
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63658C2semidirect
4. C32×C3⋊#95C2semidirect
5. C3×#63579C2semidirect
8. #10591294C3direct (N × H)
15. C2×C32×C4×D3C4semidirect
25. #63658C2×C3semidirect
32. C3×C3⋊#95D3semidirect
33. C3×C3⋊#95C2×C3semidirect
34. #63579C2×C3semidirect
84. C2×C3×C4×D3Dic3semidirect
85. C2×C3×C4×D3C3×C4semidirect
120. C3⋊#95C3×D3semidirect
235. C2×C4×D3C3×Dic3semidirect
318. C33#381semidirect
521. C32#4664semidirect
522. C32#5211semidirect
523. C32C3×#381semidirect
598. C3#10591294direct (N × H)
599. C3C3×#4664semidirect
600. C3C3×#5211semidirect

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×(C2×D3)⋊Dic32
C3×#636582
C32×C3⋊#952
C3×#635792
C3×#52113
C3×#46643
#105912943