Finite Group Databasebeta

← C3×#10595079 #13174676

Subgroups — C3×#10595079 #13174676

Order 1728.

1814 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C3×#3811C211yesmaximal, characteristic
3C32×#929C212yesmaximal, complemented
4C3×#63974C211yesmaximal, characteristic, complemented
5C3×#63962C212yesmaximal, complemented
6C3×#63921C212yesmaximal, complemented
7C3×#63883C212yesmaximal, complemented
8C3×#63853C214yesmaximal, complemented
9C3×D3×#123—32nomaximal
10#10595079C311yesmaximal, characteristic, direct product factor
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C3×D32C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic, complemented
14C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
15C2×C3×C4×C3⋊D3—24no—
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C3×C4×C3⋊D3C2212yescomplemented
18C2×C32×#67C2212yes—
19C3×#3811—24no—
20C2×C3×#626C2212yes—
21C2×C3×#624C2211yescharacteristic
22C2×C3×#624C2212yes—
23C32×#217C2212yescomplemented
24C32×(C2×D3)⋊C4C2214yescomplemented
25C32×Dic3⋊C4C2214yescomplemented
26C3×Dic3⋊Dic3C2214yescomplemented
27C3×#3722C2212yescomplemented
28C3×(C2×D3)⋊Dic3C2214yescomplemented
29C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
30C2×C3×D3×D4—32no—
31C22×C3×C4×D3—32no—
32C22×C3×C4×D3—61no—
33C2×#3811C2×C311yescharacteristic
34C32×#123—32no—
35C3×#935—32no—
36C3×#929—22no—
37C3×#929D312yescomplemented
38C3×#929C2×C312yescomplemented
39C3×#894—34no—
40C3×D3×C4⋊C4—34no—
41C3×#886—32no—
42C3×#883—34no—
43C3×D3×C22⋊C4—32no—
44C3×D3×C22⋊C4—34no—
45#63974C2×C311yescharacteristic, complemented
46#63962C2×C312yescomplemented
47#63921C2×C312yescomplemented
48#63883C2×C312yescomplemented
49#63853C2×C314yescomplemented
50C23×C33C2311yescharacteristic
51C22×C3×C3⋊D3—24no—
52C22×C3×C3⋊D3D414yes—
53C22×C3×C3⋊D3C2311yescharacteristic
54C22×C3×C3⋊D3C2311yescharacteristic
55C22×C32×D3C2312yes—
56C2×C3×D32—22no—
57C2×C3×D32—24no—
58C2×C33×C4—22no—
59C2×C33×C4C2312yes—
60C2×C3×C3⋊Dic3—22no—
61C2×C3×C3⋊Dic3C2312yes—
62C3×C4×C3⋊D3—24no—
63C3×C4×C3⋊D3—24no—
64C32×#67—24no—
65C2×C32×Dic3C2314yes—
66C2×C32×Dic3C2312yes—
67C3×#626—24no—
68C3×#624—22no—
69C3×#624—24no—
70C2×C3×#123—91no—
71D3×#123—32no—
72C23×C3⋊D3C22×C311yescharacteristic
73C23×C3×D3—32no—
74C23×C3×D3—32no—
75C23×C3×D3—61no—
76C22×D32C22×C311yescharacteristic
77C2×C32×D4—32no—
78C22×C32×C4—21no—
79C22×C32×C4—22no—
80C22×C32×C4—41no—
81C22×C32×C4C2×D312yescomplemented
82C22×C32×C4C22×C311yescharacteristic, complemented
83C22×C3⋊Dic3C22×C311yescharacteristic, complemented
84C2×C4×C3⋊D3—24no—
85C2×C4×C3⋊D3C22×C312yes—
86C2×C4×C3⋊D3C22×C312yescomplemented
87C2×C3×#67—22no—
88C2×C3×#67—32no—
89C2×C3×#67C2×D312yes—
90C2×C3×#67C22×C312yes—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—61no—
93C3×D3×D4—68no—
94C2×C3×D12—32no—
95C2×C3×C4×D3—34no—
96C2×C3×C4×D3—34no—
97C2×C3×C4×D3—34no—
98C2×C3×C4×D3—32no—
99C2×C3×C4×D3—68no—
100C2×C3×C4×D3—62no—
101C2×C3×C4×D3—62no—
102C2×C3×C4×D3—68no—
103#3811—24no—
104C2×#626C22×C312yes—
105C2×#624C22×C311yescharacteristic
106C2×#624C22×C312yes—
107C32×C4⋊C4—34no—
108C32×C22⋊C4—32no—
109C32×C22⋊C4—34no—
110C3×#217—22no—
111C3×#217—34no—
112C3×#217C2×D312yescomplemented
113C3×#217C22×C312yescomplemented
114C3×(C2×D3)⋊C4—24no—
115C3×(C2×D3)⋊C4—34no—
116C3×(C2×D3)⋊C4—34no—
117C3×(C2×D3)⋊C4C2×D314yescomplemented
118C3×(C2×D3)⋊C4C22×C314yescomplemented
119C3×C4⋊Dic3—34no—
120C3×Dic3⋊C4—24no—
121C3×Dic3⋊C4—34no—
122C3×Dic3⋊C4C2×D314yescomplemented
123C3×Dic3⋊C4C22×C314yescomplemented
124Dic3⋊Dic3C22×C314yescomplemented
125#3722C22×C312yescomplemented
126(C2×D3)⋊Dic3C22×C314yescomplemented
127C22×C33—22no—
128C22×C33C2×D412yes—
129C22×C33C2411yescharacteristic
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—28no—
132C2×C3×C3⋊D3C2×D412yes—
133C2×C3×C3⋊D3C2×D412yes—
134C2×C3×C3⋊D3#4114yes—
135C2×C32×D3—22no—
136C2×C32×D3—24no—
137C3×D32—44no—
138C33×C4—22no—
139C3×C3⋊Dic3—22no—
140C32×Dic3—22no—
141C32×Dic3—24no—
142C22×C3×D4—91no—
143C23×C3×C4—91no—
144C2×D3×D4—32no—
145C22×C4×D3—32no—
146C22×C4×D3—61no—
147C3×#123—32no—
148C3×#123—32no—
149C3×#123—62no—
150C3×#123—94no—
151C3×#123—92no—
152C2×C3×C4⋊C4—92no—
153C2×C3×C22⋊C4—91no—
154C2×C3×C22⋊C4—92no—
155#935—32no—
156#929C3×D312yescomplemented
157#894—34no—
158D3×C4⋊C4—34no—
159#886—32no—
160#883—34no—
161D3×C22⋊C4—32no—
162D3×C22⋊C4—34no—
163C23×C32—21no—
164C23×C32—22no—
165C23×C32—32no—
166C23×C32—41no—
167C23×C32C22×D312yes—
168C23×C32C23×C311yescharacteristic
169C22×C3⋊D3—24no—
170C22×C3⋊D3C3×D414yes—
171C22×C3⋊D3C23×C311yescharacteristic
172C22×C3⋊D3C23×C311yescharacteristic
173C22×C3×D3—22no—
174C22×C3×D3—32no—
175C22×C3×D3—38no—
176C22×C3×D3—32no—
177C22×C3×D3—64no—
178C22×C3×D3—61no—
179C22×C3×D3—61no—
180C22×C3×D3—68no—
181C22×C3×D3—64no—
182C22×C3×D3—68no—
183C22×C3×D3—68no—
184C22×C3×D3C22×D312yes—
185C22×C3×D3C23×C312yes—
186C2×D32—22no—
187C2×D32—24no—
188C32×D4—64no—
189C2×C32×C4—22no—
190C2×C32×C4—24no—
191C2×C32×C4—24no—
192C2×C32×C4—22no—
193C2×C32×C4—24no—
194C2×C32×C4—32no—
195C2×C32×C4—34no—
196C2×C32×C4—42no—
197C2×C32×C4—44no—
198C2×C32×C4—44no—
199C2×C32×C4C22×D314yes—
200C2×C32×C4C23×C312yes—
201C2×C3⋊Dic3—22no—
202C2×C3⋊Dic3C23×C312yes—
203C4×C3⋊D3—24no—
204C4×C3⋊D3—24no—
205C3×#67—24no—
206C3×#67—24no—
207C3×#67—44no—
208C3×#67—64no—
209C2×C3×Dic3—22no—
210C2×C3×Dic3—24no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—64no—
214C2×C3×Dic3—64no—
215C2×C3×Dic3C22×D314yes—
216C2×C3×Dic3C22×D312yes—
217C2×C3×Dic3C23×C314yes—
218C2×C3×Dic3C23×C312yes—
219C3×D12—64no—
220C3×C4×D3—68no—
221C3×C4×D3—64no—
222C3×C4×D3—68no—
223C3×C4×D3—68no—
224C3×C4×D3—68no—
225C3×C4×D3—68no—
226#626—24no—
227#624—22no—
228#624—24no—
229C2×#123—91noSylow
230C2×C33—22no—
231C2×C33C22×D411yescharacteristic
232C2×C33C2×#4112yes—
233C3×C3⋊D3—22no—
234C3×C3⋊D3#12314yescomplemented
235C32×D3—42no—
236C24×C3—91no—
237C24×C3—91no—
238C23×D3—32no—
239C23×D3—32no—
240C23×D3—61no—
241C2×C3×D4—32no—
242C2×C3×D4—32no—
243C2×C3×D4—62no—
244C2×C3×D4—92no—
245C2×C3×D4—184no—
246C22×C3×C4—21no—
247C22×C3×C4—22no—
248C22×C3×C4—41no—
249C22×C3×C4—92no—
250C22×C3×C4—91no—
251C22×C3×C4—91no—
252C22×C3×C4—92no—
253C22×C3×C4—92no—
254C22×C3×C4—184no—
255C22×C3×C4D3211yescharacteristic, complemented
256C22×C3×C4C2×C3×D312yescomplemented
257C2×#67—32no—
258C2×#67C2×C3×D312yes—
259C22×Dic3—32no—
260C22×Dic3—61no—
261D3×D4—68no—
262C2×D12—32no—
263C2×C4×D3—34no—
264C2×C4×D3—34no—
265C2×C4×D3—34no—
266C2×C4×D3—32no—
267C2×C4×D3—62no—
268C2×C4×D3—62no—
269C2×C4×D3—68no—
270C2×C4×D3—68no—
271C3×C4⋊C4—34no—
272C3×C4⋊C4—34no—
273C3×C4⋊C4—64no—
274C3×C4⋊C4—94no—
275C3×C22⋊C4—32no—
276C3×C22⋊C4—32no—
277C3×C22⋊C4—34no—
278C3×C22⋊C4—34no—
279C3×C22⋊C4—64no—
280C3×C22⋊C4—62no—
281C3×C22⋊C4—92no—
282C3×C22⋊C4—94no—
283#217—34no—
284#217C2×C3×D312yescomplemented
285(C2×D3)⋊C4—34no—
286(C2×D3)⋊C4—34no—
287(C2×D3)⋊C4C2×C3×D314yescomplemented
288C4⋊Dic3—34no—
289Dic3⋊C4—34no—
290Dic3⋊C4C2×C3×D314yescomplemented
291C22×C32—22no—
292C22×C32—24no—
293C22×C32—24no—
294C22×C32—21no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—24no—
298C22×C32—42no—
299C22×C32—41no—
300C22×C32—44no—
301C22×C32—44no—
302C22×C32—62no—
303C22×C32—64no—
304C22×C32D3×D414yes—
305C22×C32C2×C3×D412yes—
306C22×C32C23×D312yes—
307C22×C32C24×C311yescharacteristic
308C2×C3⋊D3—22no—
309C2×C3⋊D3—28no—
310C2×C3⋊D3C2×C3×D412yes—
311C2×C3⋊D3C2×C3×D412yes—
312C2×C3⋊D3C3×#4114yes—
313C2×C3×D3—22no—
314C2×C3×D3—24no—
315C2×C3×D3—22no—
316C2×C3×D3—24no—
317C2×C3×D3—34no—
318C2×C3×D3—38no—
319C2×C3×D3—34no—
320C2×C3×D3—42no—
321C2×C3×D3—44no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—64no—
325C2×C3×D3—616no—
326C2×C3×D3—616no—
327C2×C3×D3—64no—
328C2×C3×D3—64no—
329C2×C3×D3—128no—
330D32—44no—
331C32×C4—22no—
332C32×C4—24no—
333C32×C4—24no—
334C32×C4—44no—
335C32×C4—44no—
336C32×C4—62no—
337C32×C4—64no—
338C3⋊Dic3—22no—
339C3×Dic3—22no—
340C3×Dic3—24no—
341C3×Dic3—22no—
342C3×Dic3—24no—
343C3×Dic3—44no—
344C3×Dic3—42no—
345C3×Dic3—64no—
346C3×Dic3—64no—
347C22×D4—91no—
348C23×C4—91no—
349#123—32no—
350#123—94no—
351#123—92no—
352C2×C4⋊C4—92no—
353C2×C22⋊C4—91no—
354C2×C22⋊C4—92no—
355C33C2×#12311yescharacteristic, complemented, Sylow
356C23×C3—21no—
357C23×C3—22no—
358C23×C3—32no—
359C23×C3—32no—
360C23×C3—41no—
361C23×C3—62no—
362C23×C3—91no—
363C23×C3—91no—
364C23×C3—94no—
365C23×C3—184no—
366C23×C3—182no—
367C23×C3—184no—
368C23×C3C2×D3211yescharacteristic
369C23×C3C22×C3×D312yes—
370C22×D3—32no—
371C22×D3—32no—
372C22×D3—38no—
373C22×D3—64no—
374C22×D3—61no—
375C22×D3—61no—
376C22×D3—64no—
377C22×D3—68no—
378C22×D3—68no—
379C22×D3—68no—
380C22×D3C22×C3×D312yes—
381C3×D4—64no—
382C3×D4—64no—
383C3×D4—124no—
384C3×D4—184no—
385C2×C3×C4—24no—
386C2×C3×C4—22no—
387C2×C3×C4—24no—
388C2×C3×C4—22no—
389C2×C3×C4—24no—
390C2×C3×C4—34no—
391C2×C3×C4—32no—
392C2×C3×C4—32no—
393C2×C3×C4—34no—
394C2×C3×C4—44no—
395C2×C3×C4—44no—
396C2×C3×C4—42no—
397C2×C3×C4—64no—
398C2×C3×C4—62no—
399C2×C3×C4—92no—
400C2×C3×C4—184no—
401C2×C3×C4—184no—
402C2×C3×C4—184no—
403C2×C3×C4—182no—
404C2×C3×C4—182no—
405C2×C3×C4C2×D3212yes—
406C2×C3×C4C22×C3×D314yes—
407#67—24no—
408#67—64no—
409C2×Dic3—34no—
410C2×Dic3—62no—
411C2×Dic3—64no—
412C2×Dic3—64no—
413C2×Dic3C22×C3×D314yes—
414C2×Dic3C22×C3×D312yes—
415D12—64no—
416C4×D3—68no—
417C4×D3—64no—
418C4×D3—68no—
419C4×D3—68no—
420C4×D3—68no—
421C4×D3—68no—
422C2×C32—22no—
423C2×C32—24no—
424C2×C32—24no—
425C2×C32—22no—
426C2×C32—21no—
427C2×C32—22no—
428C2×C32—24no—
429C2×C32—41no—
430C2×C32—42no—
431C2×C32—44no—
432C2×C32—44no—
433C2×C32—122no—
434C2×C32C2×D3×D412yes—
435C2×C32D3×#4114yes—
436C2×C32C22×C3×D411yescharacteristic
437C2×C32C2×C3×#4112yes—
438C3⋊D3—22no—
439C3⋊D3C3×#12314yescomplemented
440C3×D3—38no—
441C3×D3—42no—
442C3×D3—42no—
443C3×D3—64no—
444C3×D3—64no—
445C3×D3—64no—
446C3×D3—82no—
447C24—91no—
448C24—91no—
449C2×D4—32no—
450C2×D4—92no—
451C2×D4—184no—
452C22×C4—92no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—92no—
456C22×C4—92no—
457C22×C4—184no—
458C22×C4C3×D3211yescharacteristic, complemented
459C4⋊C4—34no—
460C4⋊C4—94no—
461C22⋊C4—32no—
462C22⋊C4—34no—
463C22⋊C4—92no—
464C22⋊C4—94no—
465C22×C3—24no—
466C22×C3—22no—
467C22×C3—24no—
468C22×C3—21no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—24no—
472C22×C3—44no—
473C22×C3—44no—
474C22×C3—41no—
475C22×C3—42no—
476C22×C3—64no—
477C22×C3—62no—
478C22×C3—62no—
479C22×C3—64no—
480C22×C3—94no—
481C22×C3—92no—
482C22×C3—92no—
483C22×C3—122no—
484C22×C3—124no—
485C22×C3—188no—
486C22×C3—182no—
487C22×C3—364no—
488C22×C3#381112yes—
489C22×C3C3×D3×D414yes—
490C22×C3C22×D3211yescharacteristic
491C22×C3C23×C3×D312yes—
492C2×D3—22no—
493C2×D3—24no—
494C2×D3—34no—
495C2×D3—38no—
496C2×D3—34no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—64no—
500C2×D3—616no—
501C2×D3—616no—
502C2×D3—64no—
503C2×D3—64no—
504C2×D3—128no—
505C3×C4—24no—
506C3×C4—22no—
507C3×C4—24no—
508C3×C4—44no—
509C3×C4—44no—
510C3×C4—64no—
511C3×C4—62no—
512C3×C4—62no—
513C3×C4—64no—
514C3×C4—122no—
515C3×C4—124no—
516C3×C4—182no—
517Dic3—22no—
518Dic3—24no—
519Dic3—64no—
520Dic3—64no—
521C32—21no—
522C32—22no—
523C32—41no—
524C32D3×#12312yescomplemented
525C32C2×C3×#12311yescharacteristic, complemented
526C23—32no—
527C23—91no—
528C23—91no—
529C23—94no—
530C23—184no—
531C23—182no—
532C23—184no—
533C23C2×C3×D3211yescharacteristic
534D4—64no—
535D4—184no—
536C2×C4—22no—
537C2×C4—34no—
538C2×C4—32no—
539C2×C4—92no—
540C2×C4—184no—
541C2×C4—184no—
542C2×C4—182no—
543C2×C4—184no—
544C2×C4—182no—
545C2×C4C2×C3×D3212yes—
546C2×C3—24no—
547C2×C3—22no—
548C2×C3—24no—
549C2×C3—21no—
550C2×C3—22no—
551C2×C3—24no—
552C2×C3—22no—
553C2×C3—44no—
554C2×C3—44no—
555C2×C3—41no—
556C2×C3—42no—
557C2×C3—94no—
558C2×C3—122no—
559C2×C3—122no—
560C2×C3—182no—
561C2×C3—242no—
562C2×C3#6430612yes—
563C2×C3C2×#381111yescharacteristic
564C2×C3C2×C3×D3×D412yes—
565C2×C3C3×D3×#4114yes—
566D3—38no—
567D3—42no—
568D3—64no—
569D3—64no—
570D3—64no—
571C22—22no—
572C22—62no—
573C22—64no—
574C22—92no—
575C22—94no—
576C22—92no—
577C22—188no—
578C22—182no—
579C22—364no—
580C22C3×#381112yes—
581C22C22×C3×D3211yescharacteristic
582C4—22no—
583C4—62no—
584C4—64no—
585C4—182no—
586C3—21no—
587C3—22no—
588C3—41no—
589C3#1059507911yescharacteristic, direct product factor
590C3C3×D3×#12312yescomplemented
591C2—22no—
592C2—94no—
593C2—122no—
594C2—182no—
595C2C3×#6430612yes—
596C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC2×C3×#3811C2
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×C3⋊D3C2semidirect
3. C32×#929C2semidirect
4. C3×#63974C2semidirect
5. C3×#63962C2semidirect
6. C3×#63921C2semidirect
7. C3×#63883C2semidirect
8. C3×#63853C2semidirect
10. #10595079C3direct (N × H)
13. C22×C33×C4C22semidirect
14. C22×C3×C3⋊Dic3C22semidirect
17. C2×C3×C4×C3⋊D3C22semidirect
23. C32×#217C22semidirect
24. C32×(C2×D3)⋊C4C22semidirect
25. C32×Dic3⋊C4C22semidirect
26. C3×Dic3⋊Dic3C22semidirect
27. C3×#3722C22semidirect
28. C3×(C2×D3)⋊Dic3C22semidirect
29. C22×C4×C3⋊D3C2×C3semidirect
37. C3×#929D3semidirect
38. C3×#929C2×C3semidirect
45. #63974C2×C3semidirect
46. #63962C2×C3semidirect
47. #63921C2×C3semidirect
48. #63883C2×C3semidirect
49. #63853C2×C3semidirect
81. C22×C32×C4C2×D3semidirect
82. C22×C32×C4C22×C3semidirect
83. C22×C3⋊Dic3C22×C3semidirect
86. C2×C4×C3⋊D3C22×C3semidirect
112. C3×#217C2×D3semidirect
113. C3×#217C22×C3semidirect
117. C3×(C2×D3)⋊C4C2×D3semidirect
118. C3×(C2×D3)⋊C4C22×C3semidirect
122. C3×Dic3⋊C4C2×D3semidirect
123. C3×Dic3⋊C4C22×C3semidirect
124. Dic3⋊Dic3C22×C3semidirect
125. #3722C22×C3semidirect
126. (C2×D3)⋊Dic3C22×C3semidirect
156. #929C3×D3semidirect
234. C3×C3⋊D3#123semidirect
255. C22×C3×C4D32semidirect
256. C22×C3×C4C2×C3×D3semidirect
284. #217C2×C3×D3semidirect
287. (C2×D3)⋊C4C2×C3×D3semidirect
290. Dic3⋊C4C2×C3×D3semidirect
355. C33C2×#123semidirect
439. C3⋊D3C3×#123semidirect
458. C22×C4C3×D32semidirect
524. C32D3×#123semidirect
525. C32C2×C3×#123semidirect
589. C3#10595079direct (N × H)
590. C3C3×D3×#123semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C3×#38112
C32×#9292
C3×#639742
C3×#639622
C3×#639212
C3×#638832
C3×#638532
C3×D3×#1233
#105950793