Finite Group Databasebeta

← #13162059

Subgroups — #13162059

Order 1728.

1570 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3757C211yesmaximal, characteristic
3C2×C3×#3757C211yesmaximal, characteristic
4C2×#88815C212yesmaximal, complemented
5#11722082C212yesmaximal, complemented
6#10591719—31nomaximal
7#10591718—31nomaximal
8#10591693—31nomaximal
9C24×C33C2211yescharacteristic
10C22×C3×C3⋊Dic3—22no—
11C22×C3×C3⋊Dic3—24no—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2212yes—
15C22×C3×C3⋊Dic3C2212yes—
16C3×#3757—22no—
17C3×#3757C414yescomplemented
18#88815—24no—
19C23×C3⋊Dic3D311yescharacteristic
20C23×C3×Dic3—31no—
21C23×C3×Dic3—31no—
22C23×C3×Dic3—61no—
23C2×#3757D311yescharacteristic
24C2×#3757D311yescharacteristic
25C2×C3×#217—31no—
26C2×C3×#217—31no—
27C2×C3×#217—31no—
28C2×C3×#217—31no—
29C2×C3×#217—61no—
30C2×C3×#217—61no—
31C2×#3724—32no—
32C2×Dic3⋊Dic3—32no—
33C2×Dic3⋊Dic3—32no—
34#63579—32no—
35#63579—32no—
36#63579—32no—
37C23×C33—22no—
38C23×C33—22no—
39C23×C33C2×C412yes—
40C23×C33D412yes—
41C23×C33Q812yes—
42C23×C33C2311yescharacteristic
43C2×C3×C3⋊Dic3—28no—
44C2×C3×C3⋊Dic3—22no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3—22no—
47C2×C3×C3⋊Dic3—22no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—22no—
51C2×C3×C3⋊Dic3—22no—
52C2×C3×C3⋊Dic3C2×C414yescomplemented
53C2×C3×C3⋊Dic3D414yes—
54#5210—91no—
55#4950—91no—
56#4946—91no—
57C24×C32—21no—
58C24×C32—21no—
59C24×C32—21no—
60C24×C32—41no—
61C24×C32C2×D311yescharacteristic
62C24×C32C2×D311yescharacteristic
63C24×C32C2×D311yescharacteristic
64C22×C3⋊Dic3—22no—
65C22×C3⋊Dic3—24no—
66C22×C3⋊Dic3C2×D311yescharacteristic
67C22×C3⋊Dic3C2×D311yescharacteristic
68C22×C3⋊Dic3C2×D312yes—
69C22×C3⋊Dic3C2×D312yes—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—64no—
79C22×C3×Dic3—64no—
80C22×C3×Dic3—62no—
81C22×C3×Dic3—68no—
82C22×C3×Dic3—64no—
83C22×C3×Dic3—62no—
84C22×C3×Dic3—62no—
85C22×C3×Dic3—62no—
86C22×C3×Dic3—61no—
87C22×C3×Dic3—61no—
88#3757—22no—
89#3757Dic314yescomplemented
90C3×#217—34no—
91C3×#217—34no—
92C3×#217—64no—
93C3×#217—64no—
94C3×#217—62no—
95C3×#217—62no—
96#3724—64no—
97Dic3⋊Dic3—64no—
98Dic3⋊Dic3—64no—
99C22×C33—24no—
100C22×C33—22no—
101C22×C33—24no—
102C22×C33—22no—
103C22×C33C4⋊C414yescomplemented
104C22×C33C22×C411yescharacteristic
105C22×C33C2×D411yescharacteristic
106C22×C33C2×D412yes—
107C22×C33C2×Q811yescharacteristic
108C22×C33#4112yes—
109C3×C3⋊Dic3—24no—
110C3×C3⋊Dic3—24no—
111C3×C3⋊Dic3—41no—
112C3×C3⋊Dic3—41no—
113C23×C3×C4—91no—
114C23×Dic3—31no—
115C23×Dic3—31no—
116C23×Dic3—61no—
117C2×C3×C22⋊C4—91no—
118C2×C3×C22⋊C4—91no—
119C2×#217—31no—
120C2×#217—31no—
121C2×#217—31no—
122C2×#217—31no—
123C2×#217—61no—
124C2×#217—61no—
125C2×C4⋊Dic3—92no—
126C2×Dic3⋊C4—92no—
127C2×Dic3⋊C4—92no—
128C3⋊#95—92no—
129C3⋊#95—92no—
130C3⋊#95—92no—
131C23×C32—22no—
132C23×C32—22no—
133C23×C32—24no—
134C23×C32—24no—
135C23×C32—22no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—22no—
142C23×C32—22no—
143C23×C32—22no—
144C23×C32—22no—
145C23×C32—22no—
146C23×C32—22no—
147C23×C32—22no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32—42no—
152C23×C32—42no—
153C23×C32—42no—
154C23×C32—41no—
155C23×C32—42no—
156C23×C32—42no—
157C23×C32—42no—
158C23×C32—42no—
159C23×C32—44no—
160C23×C32—44no—
161C23×C32Dic612yes—
162C23×C32Dic612yes—
163C23×C32Dic612yes—
164C23×C32C4×D312yes—
165C23×C32C4×D312yes—
166C23×C32D1212yes—
167C23×C32C2×Dic312yes—
168C23×C32#6712yes—
169C23×C32#6712yes—
170C23×C32C22×D311yescharacteristic
171C23×C32C22×D311yescharacteristic
172C23×C32C22×D311yescharacteristic
173C2×C3⋊Dic3—28no—
174C2×C3⋊Dic3—22no—
175C2×C3⋊Dic3—22no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—22no—
179C2×C3⋊Dic3—22no—
180C2×C3⋊Dic3—22no—
181C2×C3⋊Dic3—22no—
182C2×C3⋊Dic3C2×Dic314yescomplemented
183C2×C3⋊Dic3#6714yes—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—34no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—616no—
191C2×C3×Dic3—68no—
192C2×C3×Dic3—68no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—61no—
213C2×C3×Dic3—122no—
214C2×C3×Dic3—122no—
215C2×C3×Dic3—122no—
216C2×C3×Dic3—122no—
217#385—271noSylow
218C2×C33—24no—
219C2×C33C2×C4⋊C411yescharacteristic
220C2×C33C4×D412yes—
221C2×C33#12011yescharacteristic
222C2×C33C22⋊Q812yes—
223C2×C33#12311yescharacteristic
224C24×C3—21no—
225C24×C3—21no—
226C24×C3—21no—
227C24×C3—41no—
228C24×C3D3211yescharacteristic
229C24×C3D3211yescharacteristic
230C24×C3D3211yescharacteristic
231C22×C3×C4—92no—
232C22×C3×C4—92no—
233C22×C3×C4—91no—
234C22×C3×C4—91no—
235C22×C3×C4—184no—
236C22×C3×C4—182no—
237C22×Dic3—32no—
238C22×Dic3—32no—
239C22×Dic3—32no—
240C22×Dic3—32no—
241C22×Dic3—31no—
242C22×Dic3—31no—
243C22×Dic3—31no—
244C22×Dic3—31no—
245C22×Dic3—62no—
246C22×Dic3—62no—
247C22×Dic3—61no—
248C22×Dic3—61no—
249C22×Dic3—64no—
250C22×Dic3—64no—
251C22×Dic3—68no—
252C22×Dic3—64no—
253C22×Dic3—62no—
254C22×Dic3—62no—
255C3×C22⋊C4—94no—
256C3×C22⋊C4—182no—
257#217—34no—
258#217—34no—
259#217—64no—
260#217—64no—
261#217—62no—
262#217—62no—
263C4⋊Dic3—184no—
264Dic3⋊C4—184no—
265Dic3⋊C4—184no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—24no—
271C22×C32—24no—
272C22×C32—28no—
273C22×C32—28no—
274C22×C32—24no—
275C22×C32—24no—
276C22×C32—24no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—24no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32—24no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—22no—
288C22×C32—22no—
289C22×C32—22no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—22no—
298C22×C32—22no—
299C22×C32—22no—
300C22×C32—44no—
301C22×C32—41no—
302C22×C32—41no—
303C22×C32—42no—
304C22×C32—41no—
305C22×C32—42no—
306C22×C32—44no—
307C22×C32—42no—
308C22×C32—44no—
309C22×C32—44no—
310C22×C32—42no—
311C22×C32—44no—
312C22×C32—42no—
313C22×C32—42no—
314C22×C32—44no—
315C22×C32—48no—
316C22×C32—48no—
317C22×C32—44no—
318C22×C32Dic3⋊C414yescomplemented
319C22×C32Dic3⋊C414yescomplemented
320C22×C32C4⋊Dic314yescomplemented
321C22×C32C2×Dic611yescharacteristic
322C22×C32C2×Dic611yescharacteristic
323C22×C32C2×Dic611yescharacteristic
324C22×C32C2×C4×D311yescharacteristic
325C22×C32C2×C4×D311yescharacteristic
326C22×C32C2×D1211yescharacteristic
327C22×C32#23512yes—
328C22×C32D3×D412yes—
329C22×C32D3×D412yes—
330C22×C32#23712yes—
331C22×C32#23712yes—
332C22×C32C22×Dic311yescharacteristic
333C22×C32C2×#6711yescharacteristic
334C22×C32C2×#6711yescharacteristic
335C22×C32C2×#6712yes—
336C3⋊Dic3—24no—
337C3⋊Dic3—24no—
338C3⋊Dic3—41no—
339C3⋊Dic3—41no—
340C3×Dic3—68no—
341C3×Dic3—64no—
342C3×Dic3—64no—
343C3×Dic3—64no—
344C3×Dic3—64no—
345C3×Dic3—124no—
346C3×Dic3—122no—
347C3×Dic3—122no—
348C3×Dic3—121no—
349C3×Dic3—121no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C23×C4—91no—
353C2×C4⋊C4—272no—
354C2×C22⋊C4—91no—
355C2×C22⋊C4—91no—
356#95—272no—
357C33#38511yescharacteristic, complemented, Sylow
358C23×C3—24no—
359C23×C3—24no—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3—22no—
363C23×C3—22no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—22no—
367C23×C3—22no—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—42no—
379C23×C3—42no—
380C23×C3—42no—
381C23×C3—42no—
382C23×C3—44no—
383C23×C3—44no—
384C23×C3—42no—
385C23×C3—42no—
386C23×C3—42no—
387C23×C3—41no—
388C23×C3D3×Dic312yes—
389C23×C3D3×Dic312yes—
390C23×C3#62412yes—
391C23×C3#62512yes—
392C23×C3#62612yes—
393C23×C3#62612yes—
394C23×C3#62712yes—
395C23×C3#62712yes—
396C23×C3#62712yes—
397C23×C3C2×D3211yescharacteristic
398C23×C3C2×D3211yescharacteristic
399C23×C3C2×D3211yescharacteristic
400C2×C3×C4—94no—
401C2×C3×C4—94no—
402C2×C3×C4—188no—
403C2×C3×C4—182no—
404C2×C3×C4—182no—
405C2×C3×C4—182no—
406C2×C3×C4—182no—
407C2×C3×C4—182no—
408C2×C3×C4—181no—
409C2×C3×C4—181no—
410C2×C3×C4—182no—
411C2×Dic3—34no—
412C2×Dic3—34no—
413C2×Dic3—34no—
414C2×Dic3—34no—
415C2×Dic3—64no—
416C2×Dic3—64no—
417C2×Dic3—616no—
418C2×Dic3—68no—
419C2×Dic3—68no—
420C2×Dic3—64no—
421C2×Dic3—64no—
422C2×Dic3—62no—
423C2×Dic3—62no—
424C2×Dic3—62no—
425C2×Dic3—62no—
426C2×Dic3—62no—
427C2×Dic3—62no—
428C2×Dic3—62no—
429C2×Dic3—62no—
430C2×Dic3—62no—
431C2×Dic3—62no—
432C2×Dic3—62no—
433C2×Dic3—62no—
434C2×Dic3—61no—
435C2×Dic3—61no—
436C2×Dic3—61no—
437C2×Dic3—61no—
438C2×Dic3—62no—
439C2×Dic3—62no—
440C2×Dic3—122no—
441C2×Dic3—122no—
442C2×Dic3—122no—
443C2×Dic3—122no—
444C2×C32—24no—
445C2×C32—28no—
446C2×C32—24no—
447C2×C32—24no—
448C2×C32—22no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—22no—
452C2×C32—22no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—22no—
458C2×C32—22no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—21no—
462C2×C32—22no—
463C2×C32—41no—
464C2×C32—42no—
465C2×C32—41no—
466C2×C32—42no—
467C2×C32—41no—
468C2×C32—44no—
469C2×C32—44no—
470C2×C32—48no—
471C2×C32#87812yes—
472C2×C32#87812yes—
473C2×C32#67⋊C412yes—
474C2×C32#88211yescharacteristic
475C2×C32#88611yescharacteristic
476C2×C32C2×Dic3⋊C411yescharacteristic
477C2×C32C2×Dic3⋊C411yescharacteristic
478C2×C32#92412yes—
479C2×C32C2×C4⋊Dic311yescharacteristic
480C2×C32C4×#6712yes—
481C2×C32#92911yescharacteristic
482C2×C32D4×Dic312yes—
483C2×C32#93511yescharacteristic
484C2×C32#93711yescharacteristic
485C2×C32#95311yescharacteristic
486C24#111311yescharacteristic
487C22×C4—92no—
488C22×C4—92no—
489C22×C4—91no—
490C22×C4—91no—
491C22×C4—184no—
492C22×C4—182no—
493C4⋊C4—544no—
494C22⋊C4—94no—
495C22⋊C4—182no—
496C22×C3—24no—
497C22×C3—24no—
498C22×C3—28no—
499C22×C3—28no—
500C22×C3—24no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—24no—
504C22×C3—24no—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—24no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—24no—
528C22×C3—24no—
529C22×C3—24no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—44no—
533C22×C3—44no—
534C22×C3—44no—
535C22×C3—44no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—44no—
539C22×C3—48no—
540C22×C3—48no—
541C22×C3—44no—
542C22×C3—41no—
543C22×C3—42no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3—42no—
547C22×C3—44no—
548C22×C3Dic3⋊Dic314yescomplemented
549C22×C3Dic3⋊Dic314yescomplemented
550C22×C3#372414yescomplemented
551C22×C3C2×D3×Dic311yescharacteristic
552C22×C3C2×D3×Dic311yescharacteristic
553C22×C3#380412yes—
554C22×C3#380412yes—
555C22×C3#380512yes—
556C22×C3C2×#62411yescharacteristic
557C22×C3C2×#62511yescharacteristic
558C22×C3C2×#62611yescharacteristic
559C22×C3C2×#62611yescharacteristic
560C22×C3C2×#62711yescharacteristic
561C22×C3C2×#62711yescharacteristic
562C22×C3C2×#62711yescharacteristic
563C22×C3D3×#6712yes—
564C22×C3D3×#6712yes—
565C22×C3#381112yes—
566C3×C4—184no—
567C3×C4—184no—
568C3×C4—361no—
569C3×C4—361no—
570Dic3—68no—
571Dic3—64no—
572Dic3—64no—
573Dic3—64no—
574Dic3—64no—
575Dic3—124no—
576Dic3—122no—
577Dic3—122no—
578Dic3—121no—
579Dic3—121no—
580Dic3—121no—
581Dic3—121no—
582C32—21no—
583C32—21no—
584C32—21no—
585C32—41no—
586C32#494611yescharacteristic, complemented
587C32#495011yescharacteristic, complemented
588C32#521011yescharacteristic, complemented
589C23—22no—
590C23—22no—
591C23#629412yes—
592C23#629512yes—
593C23#629612yes—
594C23C2×#111311yescharacteristic
595C2×C4—94no—
596C2×C4—94no—
597C2×C4—188no—
598C2×C4—182no—
599C2×C4—182no—
600C2×C4—182no—
601C2×C4—182no—
602C2×C4—182no—
603C2×C4—181no—
604C2×C4—181no—
605C2×C4—182no—
606C2×C3—28no—
607C2×C3—24no—
608C2×C3—24no—
609C2×C3—24no—
610C2×C3—21no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—22no—
614C2×C3—22no—
615C2×C3—22no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—22no—
620C2×C3—22no—
621C2×C3—22no—
622C2×C3—21no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—44no—
626C2×C3—44no—
627C2×C3—48no—
628C2×C3—41no—
629C2×C3—42no—
630C2×C3—41no—
631C2×C3—42no—
632C2×C3—41no—
633C2×C3#6395212yes—
634C2×C3#6396111yescharacteristic
635C2×C3#6396211yescharacteristic
636C2×C3#6396412yes—
637C2×C3#6396412yes—
638C2×C3C2×Dic3⋊Dic311yescharacteristic
639C2×C3C2×Dic3⋊Dic311yescharacteristic
640C2×C3#6396611yescharacteristic
641C2×C3C2×#372411yescharacteristic
642C2×C3Dic3×#6712yes—
643C2×C3#67⋊Dic312yes—
644C2×C3#6397611yescharacteristic
645C2×C3#6397711yescharacteristic
646C2×C3#6398111yescharacteristic
647C2×C3#6398212yes—
648C22—22no—
649C22—24no—
650C22—24no—
651C22—22no—
652C22#8881514yescomplemented
653C22C2×#629411yescharacteristic
654C22#8904612yes—
655C22C2×#629511yescharacteristic
656C22C2×#629611yescharacteristic
657C22#8904912yes—
658C4—184no—
659C4—184no—
660C4—361no—
661C4—361no—
662C3—21no—
663C3—21no—
664C3—21no—
665C3—41no—
666C3#1059169311yescharacteristic, complemented
667C3#1059171811yescharacteristic, complemented
668C3#1059171911yescharacteristic, complemented
669C2—24no—
670C2#1172372011yescharacteristic
671C2#1172372112yes—
672C2C2×#8881511yescharacteristic
673C2#1172372612yes—
674C2#1172372711yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutatorC22×C33C22×C4
Fitting, prime radicalC24×C33C22
socle, cosocleC23×C33C23
hypercenterC24#1113
hypocenterC33#385
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#88815C2semidirect
5. #11722082C2semidirect
17. C3×#3757C4semidirect
52. C2×C3×C3⋊Dic3C2×C4semidirect
89. #3757Dic3semidirect
103. C22×C33C4⋊C4semidirect
182. C2×C3⋊Dic3C2×Dic3semidirect
318. C22×C32Dic3⋊C4semidirect
319. C22×C32Dic3⋊C4semidirect
320. C22×C32C4⋊Dic3semidirect
357. C33#385semidirect
548. C22×C3Dic3⋊Dic3semidirect
549. C22×C3Dic3⋊Dic3semidirect
550. C22×C3#3724semidirect
586. C32#4946semidirect
587. C32#4950semidirect
588. C32#5210semidirect
652. C22#88815semidirect
666. C3#10591693semidirect
667. C3#10591718semidirect
668. C3#10591719semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×C3×#37572
C2×#888152
#117220822
#105917193
#105917183
#105916933