Finite Group Databasebeta

← C3×#10591302 #13158219

Subgroups — C3×#10591302 #13158219

Order 1728.

1410 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×D12C211yesmaximal, characteristic
2C2×C3×#3750C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
4C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
5C32×C3⋊#95C211yesmaximal, characteristic, complemented
6C3×#5222—31nomaximal
7C3×#4672—31nomaximal
8#10591302C311yesmaximal, characteristic, direct product factor
9C23×C32×D3C2212yes—
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C2×C32×D12—24no—
15C2×C32×D12—22no—
16C3×#3750—22no—
17C3×(C2×D3)⋊Dic3—24no—
18C3×(C2×D3)⋊Dic3—24no—
19C22×C32×D4—31no—
20C22×C3×D12—21no—
21C22×C3×D12D311yescharacteristic
22C22×C3×D12C2×C311yescharacteristic
23C2×#3750C2×C311yescharacteristic, complemented
24C2×C3×#217—32no—
25C2×C3×#217—32no—
26C2×C3×(C2×D3)⋊C4—32no—
27C2×C3×(C2×D3)⋊C4—32no—
28C2×C3×Dic3⋊C4—31no—
29C2×C3×Dic3⋊C4—31no—
30C2×C3×Dic3⋊C4—61no—
31C2×(C2×D3)⋊Dic3C2×C312yescomplemented
32C2×(C2×D3)⋊Dic3C2×C312yescomplemented
33C32×#95—31no—
34C3×C3⋊#95—21no—
35C3×C3⋊#95D311yescharacteristic, complemented
36C3×C3⋊#95C2×C311yescharacteristic, complemented
37C23×C33C2311yescharacteristic
38C22×C32×D3—22no—
39C22×C32×D3—22no—
40C22×C32×D3—22no—
41C22×C32×D3—22no—
42C22×C32×D3—22no—
43C22×C32×D3—22no—
44C22×C32×D3D414yes—
45C2×C33×C4—22no—
46C2×C33×C4D412yes—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—22no—
51C2×C3×C3⋊Dic3D412yescomplemented
52C2×C32×Dic3—22no—
53C2×C32×Dic3—22no—
54C2×C32×Dic3—22no—
55C32×D12—28no—
56C3×#394—91no—
57#5222—31no—
58#4672—31no—
59C24×C32—32no—
60C23×C3×D3—22no—
61C23×C3×D3C2×D312yes—
62C23×C3×D3C22×C312yes—
63C2×C32×D4—64no—
64C2×C32×D4—62no—
65C22×C32×C4—21no—
66C22×C32×C4—21no—
67C22×C32×C4—21no—
68C22×C32×C4—32no—
69C22×C32×C4—41no—
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C311yescharacteristic
74C22×C3⋊Dic3C22×C311yescharacteristic
75C22×C3×Dic3—22no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—61no—
81C22×C3×Dic3—61no—
82C22×C3×Dic3C2×D312yes—
83C22×C3×Dic3C22×C312yes—
84C2×C3×D12—28no—
85C2×C3×D12—24no—
86C2×C3×D12—24no—
87C2×C3×D12—24no—
88C2×C3×D12—22no—
89C2×C3×D12—22no—
90#3750—22no—
91C3×#217—64no—
92C3×#217—64no—
93C3×(C2×D3)⋊C4—64no—
94C3×(C2×D3)⋊C4—64no—
95C3×Dic3⋊C4—64no—
96C3×Dic3⋊C4—62no—
97C3×Dic3⋊C4—62no—
98(C2×D3)⋊Dic3—24no—
99(C2×D3)⋊Dic3—24no—
100C22×C33C2×D412yes—
101C22×C33C2×D411yescharacteristic
102C22×C33C2×D411yescharacteristic
103C22×C33#4111yescharacteristic
104C22×C33#4112yes—
105C2×C32×D3—24no—
106C2×C32×D3—24no—
107C2×C32×D3—24no—
108C2×C32×D3—42no—
109C2×C32×D3—42no—
110C2×C32×D3—42no—
111C2×C32×D3—42no—
112C33×C4—22no—
113C3×C3⋊Dic3—22no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—42no—
116C22×C3×D4—31no—
117C22×C3×D4—31no—
118C22×C3×D4—61no—
119C22×D12C3×D311yescharacteristic
120C2×C3×C4⋊C4—91no—
121C2×C3×C22⋊C4—92no—
122C2×C3×C22⋊C4—92no—
123C2×#217—32no—
124C2×#217—32no—
125C2×(C2×D3)⋊C4—32no—
126C2×(C2×D3)⋊C4—32no—
127C2×Dic3⋊C4—31no—
128C2×Dic3⋊C4—31no—
129C2×Dic3⋊C4—61no—
130C3×#95—31no—
131C3×#95—31no—
132C3×#95—61no—
133C3⋊#95C3×D311yescharacteristic, complemented
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—34no—
138C23×C32—41no—
139C23×C32—62no—
140C23×C32—62no—
141C23×C32—62no—
142C23×C32—62no—
143C23×C32—62no—
144C23×C32—62no—
145C23×C32C22×D311yescharacteristic
146C23×C32C22×D311yescharacteristic
147C23×C32C23×C311yescharacteristic
148C22×C3×D3—24no—
149C22×C3×D3—24no—
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—22no—
155C22×C3×D3—22no—
156C22×C3×D3—22no—
157C22×C3×D3—22no—
158C22×C3×D3—22no—
159C22×C3×D3—22no—
160C22×C3×D3—22no—
161C22×C3×D3—22no—
162C22×C3×D3—24no—
163C22×C3×D3—42no—
164C22×C3×D3—42no—
165C22×C3×D3—42no—
166C22×C3×D3—42no—
167C22×C3×D3#6714yes—
168C22×C3×D3C3×D414yes—
169C32×D4—68no—
170C2×C32×C4—22no—
171C2×C32×C4—24no—
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4—22no—
175C2×C32×C4—22no—
176C2×C32×C4—22no—
177C2×C32×C4—42no—
178C2×C32×C4—42no—
179C2×C32×C4—42no—
180C2×C32×C4—44no—
181C2×C32×C4—62no—
182C2×C32×C4—62no—
183C2×C32×C4—62no—
184C2×C32×C4D1212yes—
185C2×C32×C4#6712yes—
186C2×C32×C4C3×D412yes—
187C2×C3⋊Dic3—21no—
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3—22no—
191C2×C3⋊Dic3C3×D412yescomplemented
192C2×C3×Dic3—24no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—22no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3—22no—
198C2×C3×Dic3—22no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—42no—
202C2×C3×Dic3—42no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—61no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—61no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—122no—
216C3×D12—216no—
217C3×D12—28no—
218C3×D12—28no—
219#394—91noSylow
220C2×C33#12011yescharacteristic
221C2×C33#12112yes—
222C2×C33#12111yescharacteristic
223C2×C33#12312yes—
224C2×C33#12411yescharacteristic
225C32×D3—44no—
226C24×C3—32no—
227C24×C3—32no—
228C24×C3—62no—
229C23×D3C2×C3×D312yes—
230C2×C3×D4—64no—
231C2×C3×D4—64no—
232C2×C3×D4—68no—
233C2×C3×D4—64no—
234C2×C3×D4—62no—
235C2×C3×D4—62no—
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—32no—
240C22×C3×C4—32no—
241C22×C3×C4—41no—
242C22×C3×C4—62no—
243C22×C3×C4—91no—
244C22×C3×C4—91no—
245C22×C3×C4D3211yescharacteristic
246C22×C3×C4C2×C3×D311yescharacteristic
247C22×C3×C4C2×C3×D311yescharacteristic
248C22×Dic3—31no—
249C22×Dic3—31no—
250C22×Dic3—31no—
251C22×Dic3—31no—
252C22×Dic3—61no—
253C22×Dic3—61no—
254C22×Dic3C2×C3×D312yes—
255C2×D12—24no—
256C2×D12—22no—
257C3×C4⋊C4—182no—
258C3×C22⋊C4—184no—
259C3×C22⋊C4—184no—
260#217—64no—
261#217—64no—
262(C2×D3)⋊C4—64no—
263(C2×D3)⋊C4—64no—
264Dic3⋊C4—64no—
265Dic3⋊C4—62no—
266Dic3⋊C4—62no—
267C22×C32—22no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—22no—
272C22×C32—21no—
273C22×C32—22no—
274C22×C32—21no—
275C22×C32—22no—
276C22×C32—21no—
277C22×C32—22no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—22no—
282C22×C32—41no—
283C22×C32—42no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32—41no—
287C22×C32—64no—
288C22×C32—64no—
289C22×C32—64no—
290C22×C32—122no—
291C22×C32—122no—
292C22×C32—122no—
293C22×C32—122no—
294C22×C32C2×D1211yescharacteristic
295C22×C32#23512yes—
296C22×C32D3×D411yescharacteristic
297C22×C32D3×D411yescharacteristic
298C22×C32D3×D412yes—
299C22×C32#23712yes—
300C22×C32#23711yescharacteristic
301C22×C32#23911yescharacteristic
302C22×C32C2×#6711yescharacteristic
303C22×C32C2×#6712yes—
304C22×C32C2×C3×D411yescharacteristic
305C22×C32C2×C3×D411yescharacteristic
306C22×C32C2×C3×D412yes—
307C22×C32C3×#4112yes—
308C22×C32C3×#4111yescharacteristic
309C2×C3×D3—28no—
310C2×C3×D3—24no—
311C2×C3×D3—24no—
312C2×C3×D3—24no—
313C2×C3×D3—24no—
314C2×C3×D3—24no—
315C2×C3×D3—24no—
316C2×C3×D3—44no—
317C2×C3×D3—44no—
318C2×C3×D3—44no—
319C2×C3×D3—44no—
320C2×C3×D3—44no—
321C2×C3×D3—44no—
322C2×C3×D3—42no—
323C2×C3×D3—42no—
324C2×C3×D3—42no—
325C2×C3×D3—42no—
326C2×C3×D3—42no—
327C2×C3×D3—42no—
328C2×C3×D3—42no—
329C2×C3×D3—42no—
330C32×C4—22no—
331C32×C4—24no—
332C32×C4—22no—
333C32×C4—22no—
334C32×C4—42no—
335C32×C4—42no—
336C32×C4—44no—
337C32×C4—122no—
338C3⋊Dic3—22no—
339C3⋊Dic3—41no—
340C3×Dic3—44no—
341C3×Dic3—42no—
342C3×Dic3—42no—
343C3×Dic3—64no—
344C3×Dic3—62no—
345C3×Dic3—62no—
346C3×Dic3—122no—
347C3×Dic3—121no—
348C3×Dic3—121no—
349C22×D4—31no—
350C2×C4⋊C4—91no—
351C2×C22⋊C4—92no—
352C2×C22⋊C4—92no—
353#95—31no—
354C33#39411yescharacteristic, complemented, Sylow
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—34no—
359C23×C3—34no—
360C23×C3—41no—
361C23×C3—64no—
362C23×C3—64no—
363C23×C3—64no—
364C23×C3—62no—
365C23×C3—62no—
366C23×C3—62no—
367C23×C3—62no—
368C23×C3—62no—
369C23×C3—62no—
370C23×C3—62no—
371C23×C3—62no—
372C23×C3—62no—
373C23×C3—62no—
374C23×C3—62no—
375C23×C3—62no—
376C23×C3—122no—
377C23×C3—122no—
378C23×C3—122no—
379C23×C3—122no—
380C23×C3C2×D3211yescharacteristic
381C23×C3C22×C3×D311yescharacteristic
382C23×C3C22×C3×D311yescharacteristic
383C22×D3—22no—
384C22×D3—22no—
385C22×D3—22no—
386C22×D3—22no—
387C22×D3—22no—
388C22×D3—22no—
389C22×D3C3×#6714yes—
390C3×D4—616no—
391C3×D4—68no—
392C3×D4—68no—
393C2×C3×C4—24no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—42no—
401C2×C3×C4—42no—
402C2×C3×C4—44no—
403C2×C3×C4—42no—
404C2×C3×C4—64no—
405C2×C3×C4—62no—
406C2×C3×C4—62no—
407C2×C3×C4—62no—
408C2×C3×C4—62no—
409C2×C3×C4—62no—
410C2×C3×C4—62no—
411C2×C3×C4—92no—
412C2×C3×C4—122no—
413C2×C3×C4—122no—
414C2×C3×C4—181no—
415C2×C3×C4—181no—
416C2×C3×C4—181no—
417C2×C3×C4—182no—
418C2×C3×C4#62612yes—
419C2×C3×C4C3×D1212yes—
420C2×C3×C4C3×#6712yes—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—32no—
425C2×Dic3—32no—
426C2×Dic3—62no—
427C2×Dic3—62no—
428C2×Dic3—62no—
429C2×Dic3—61no—
430C2×Dic3—61no—
431C2×Dic3—61no—
432C2×Dic3—61no—
433C2×Dic3—62no—
434C2×Dic3—61no—
435C2×Dic3—62no—
436C2×Dic3—61no—
437C2×Dic3—62no—
438C2×Dic3—122no—
439D12—28no—
440C2×C32—21no—
441C2×C32—22no—
442C2×C32—22no—
443C2×C32—21no—
444C2×C32—22no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—22no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—22no—
451C2×C32—22no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—41no—
456C2×C32—42no—
457C2×C32—41no—
458C2×C32—42no—
459C2×C32—41no—
460C2×C32—124no—
461C2×C32#87511yescharacteristic
462C2×C32#88211yescharacteristic
463C2×C32#88412yes—
464C2×C32#89412yes—
465C2×C32#89511yescharacteristic
466C2×C32#93512yes—
467C2×C32#93611yescharacteristic
468C2×C32#93811yescharacteristic
469C2×C32#93912yes—
470C2×C32#95311yescharacteristic
471C2×C32C3×#12011yescharacteristic
472C2×C32C3×#12112yes—
473C2×C32C3×#12111yescharacteristic
474C2×C32C3×#12312yes—
475C2×C32C3×#12411yescharacteristic
476C3×D3—48no—
477C3×D3—44no—
478C3×D3—44no—
479C24—32no—
480C2×D4—64no—
481C2×D4—62no—
482C22×C4—32no—
483C22×C4—91no—
484C22×C4—91no—
485C22×C4C3×D3211yescharacteristic
486C4⋊C4—182no—
487C22⋊C4—184no—
488C22⋊C4—184no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—42no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—42no—
509C22×C3—68no—
510C22×C3—64no—
511C22×C3—64no—
512C22×C3—64no—
513C22×C3—64no—
514C22×C3—64no—
515C22×C3—64no—
516C22×C3—124no—
517C22×C3—124no—
518C22×C3—124no—
519C22×C3—124no—
520C22×C3—124no—
521C22×C3—124no—
522C22×C3—122no—
523C22×C3—122no—
524C22×C3—122no—
525C22×C3—122no—
526C22×C3—122no—
527C22×C3—122no—
528C22×C3—122no—
529C22×C3—122no—
530C22×C3#379512yes—
531C22×C3#379611yescharacteristic
532C22×C3#380211yescharacteristic
533C22×C3C2×#62611yescharacteristic
534C22×C3D3×#6712yes—
535C22×C3C2×C3×D1211yescharacteristic
536C22×C3C3×#23512yes—
537C22×C3C3×D3×D411yescharacteristic
538C22×C3C3×D3×D411yescharacteristic
539C22×C3C3×D3×D412yes—
540C22×C3C3×#23712yes—
541C22×C3C3×#23711yescharacteristic
542C22×C3C3×#23911yescharacteristic
543C22×C3C2×C3×#6711yescharacteristic
544C22×C3C2×C3×#6712yes—
545C2×D3—24no—
546C2×D3—24no—
547C2×D3—24no—
548C2×D3—42no—
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—42no—
552C3×C4—24no—
553C3×C4—22no—
554C3×C4—22no—
555C3×C4—22no—
556C3×C4—42no—
557C3×C4—42no—
558C3×C4—44no—
559C3×C4—124no—
560C3×C4—122no—
561C3×C4—122no—
562C3×C4—182no—
563C3×C4—361no—
564Dic3—42no—
565Dic3—64no—
566Dic3—62no—
567Dic3—62no—
568Dic3—122no—
569Dic3—121no—
570Dic3—121no—
571C32—21no—
572C32—21no—
573C32—21no—
574C32—41no—
575C32#467211yescharacteristic, complemented
576C32#522211yescharacteristic, complemented
577C32C3×#39411yescharacteristic, complemented
578C23—34no—
579C23—62no—
580C23—62no—
581C23—62no—
582C23—62no—
583C23—62no—
584C23—62no—
585C23C2×C3×D3211yescharacteristic
586D4—68no—
587C2×C4—22no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—92no—
592C2×C4—181no—
593C2×C4—181no—
594C2×C4—181no—
595C2×C4—182no—
596C2×C4C3×#62612yes—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—22no—
604C2×C3—22no—
605C2×C3—22no—
606C2×C3—22no—
607C2×C3—22no—
608C2×C3—22no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—21no—
612C2×C3—41no—
613C2×C3—41no—
614C2×C3—42no—
615C2×C3—42no—
616C2×C3—41no—
617C2×C3—128no—
618C2×C3—124no—
619C2×C3—124no—
620C2×C3#6386011yescharacteristic
621C2×C3#6388412yes—
622C2×C3#6388512yes—
623C2×C3#6391611yescharacteristic
624C2×C3#6397711yescharacteristic
625C2×C3C3×#87511yescharacteristic
626C2×C3C3×#88211yescharacteristic
627C2×C3C3×#88412yes—
628C2×C3C3×#89412yes—
629C2×C3C3×#89511yescharacteristic
630C2×C3C3×#93512yes—
631C2×C3C3×#93611yescharacteristic
632C2×C3C3×#93811yescharacteristic
633C2×C3C3×#93912yes—
634C2×C3C3×#95311yescharacteristic
635D3—44no—
636C22—64no—
637C22—64no—
638C22—64no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22C3×#379512yes—
644C22C3×#379611yescharacteristic
645C22C3×#380211yescharacteristic
646C22C2×C3×#62611yescharacteristic
647C22C3×D3×#6712yes—
648C4—22no—
649C4—122no—
650C4—182no—
651C4—361no—
652C3—21no—
653C3—21no—
654C3—21no—
655C3—41no—
656C3#1059130211yescharacteristic, direct product factor
657C3C3×#467211yescharacteristic, complemented
658C3C3×#522211yescharacteristic, complemented
659C2—124no—
660C2C3×#6386011yescharacteristic
661C2C3×#6388412yes—
662C2C3×#6388512yes—
663C2C3×#6391611yescharacteristic
664C2C3×#6397711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC22×C32×D12C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#394
LCSG ▷ C23×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×#3750C2semidirect
3. C2×C3×(C2×D3)⋊Dic3C2semidirect
4. C2×C3×(C2×D3)⋊Dic3C2semidirect
5. C32×C3⋊#95C2semidirect
8. #10591302C3direct (N × H)
23. C2×#3750C2×C3semidirect
31. C2×(C2×D3)⋊Dic3C2×C3semidirect
32. C2×(C2×D3)⋊Dic3C2×C3semidirect
35. C3×C3⋊#95D3semidirect
36. C3×C3⋊#95C2×C3semidirect
51. C2×C3×C3⋊Dic3D4semidirect
133. C3⋊#95C3×D3semidirect
191. C2×C3⋊Dic3C3×D4semidirect
354. C33#394semidirect
575. C32#4672semidirect
576. C32#5222semidirect
577. C32C3×#394semidirect
656. C3#10591302direct (N × H)
657. C3C3×#4672semidirect
658. C3C3×#5222semidirect

Maximal subgroups

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

subgroupindex
C22×C32×D122
C2×C3×#37502
C2×C3×(C2×D3)⋊Dic32
C32×C3⋊#952
C3×#52223
C3×#46723
#105913023