Finite Group Databasebeta

← #13178669

Subgroups — #13178669

Order 1728.

1814 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C22×#6295C211yesmaximal, characteristic, complemented
3C3×#64146C212yesmaximal, complemented
4#11723728C212yesmaximal, complemented
5#11723727C212yesmaximal, complemented
6#11723717C212yesmaximal, complemented
7#11723715C211yesmaximal, characteristic, complemented
8#11723695C212yesmaximal, complemented
9#11723670C212yesmaximal, complemented
10#10595239—31nomaximal
11#10595238—32nomaximal
12C23×C3×C3⋊D3C2212yes—
13C2×C33×D4C2211yescharacteristic
14C2×C3×#638C2211yescharacteristic
15C2×C3×#638C2211yescharacteristic, complemented
16C2×C3×#638C2214yescomplemented
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
19C3×#3830—24no—
20C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
21C2×C3×#634C2211yescharacteristic, complemented
22C2×#6295—24no—
23C2×#6295C2212yes—
24C2×#6295C2212yes—
25C2×#6295C2214yescomplemented
26C3×#3757C2212yes—
27C3×#3752C2214yescomplemented
28#88815C2212yescomplemented
29#88814C2214yescomplemented
30#88813C2212yescomplemented
31C2×#3830D311yescharacteristic, complemented
32C2×C3×#237—32no—
33C2×C3×#237—61no—
34C22×#626—32no—
35C22×#625—31no—
36#64146D312yescomplemented
37C3×#937—32no—
38C3×#937—32no—
39C3×#937—62no—
40#63981—32no—
41#63980—32no—
42#63979—34no—
43#63979—32no—
44#63978—34no—
45#63977—32no—
46#63976—32no—
47#63957—31no—
48#63955—32no—
49#63913—32no—
50#63912—32no—
51#63907—34no—
52#63906—34no—
53C23×C33C2311yescharacteristic
54C23×C33C2311yescharacteristic
55C22×C3×C3⋊D3—24no—
56C22×C3×C3⋊D3—44no—
57C22×C3×C3⋊D3C2314yes—
58C22×C3×C3⋊D3C2311yescharacteristic
59C33×D4—22no—
60C2×C33×C4C2311yescharacteristic
61C3×#638—22no—
62C3×#638—22no—
63C3×#638—44no—
64C2×C3×C3⋊Dic3—22no—
65C2×C3×C3⋊Dic3D414yescomplemented
66C2×C3×C3⋊Dic3C2314yes—
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C3×C4×C3⋊D3—22no—
71C3×#634—22no—
72#6295—24no—
73#6295—24no—
74#6295—44no—
75#5830—91no—
76#5592—92no—
77C23×C3⋊D3C2×D312yes—
78C23×C3×D3—32no—
79C23×C3×D3—32no—
80C23×C3×D3—62no—
81C2×C32×D4—21no—
82C2×C32×D4—22no—
83C2×C32×D4—41no—
84C2×C32×D4C2×D311yescharacteristic
85C2×C32×D4C2×D312yes—
86C2×#638C2×D311yescharacteristic
87C2×#638C2×D311yescharacteristic, complemented
88C2×#638C2×D314yescomplemented
89C22×C3⋊Dic3C2×D311yescharacteristic
90C22×C3⋊Dic3C2×D311yescharacteristic, complemented
91#3830—24no—
92C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
93C2×#634C2×D311yescharacteristic, complemented
94C2×C3×#67—32no—
95C2×C3×#67—32no—
96C2×C3×#67—34no—
97C2×C3×#67—34no—
98C2×C3×#67—61no—
99C2×C3×#67—61no—
100C2×C3×#67—64no—
101C22×C3×Dic3—32no—
102C22×C3×Dic3—32no—
103C22×C3×Dic3—61no—
104C22×C3×Dic3—61no—
105C3×#237—68no—
106C3×#237—68no—
107C2×C3×C4×D3—32no—
108C2×C3×C4×D3—61no—
109C2×C3×Dic6—32no—
110C2×C3×Dic6—61no—
111C2×#626—34no—
112C2×#626—34no—
113C2×#626—34no—
114C2×#626—34no—
115C2×#626—68no—
116C2×#625—32no—
117C2×#625—32no—
118C2×#625—34no—
119C2×#625—64no—
120#3757C2×D312yes—
121#3752C2×D314yescomplemented
122C3×#217—32no—
123C3×#217—32no—
124C3×#217—62no—
125C3×(C2×D3)⋊C4—34no—
126C3×(C2×D3)⋊C4—34no—
127C3×(C2×D3)⋊C4—64no—
128#3724—32no—
129Dic3⋊Dic3—34no—
130#3722—34no—
131(C2×D3)⋊Dic3—34no—
132(C2×D3)⋊Dic3—34no—
133Dic32—34no—
134Dic32—32no—
135C22×C33—22no—
136C22×C33—21no—
137C22×C33—22no—
138C22×C33C2×D412yes—
139C22×C33C2411yescharacteristic
140C2×C3×C3⋊D3—21no—
141C2×C3×C3⋊D3—24no—
142C2×C3×C3⋊D3—24no—
143C2×C3×C3⋊D3—22no—
144C2×C3×C3⋊D3—24no—
145C2×C3×C3⋊D3—48no—
146C33×C4—21no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—24no—
149C3×C3⋊Dic3C2×D412yes—
150C3×C3⋊Dic3C2×D412yes—
151C2×C3×#41—91no—
152C22×#67—92no—
153C2×#237—32no—
154C2×#237—61no—
155C22×D12—91no—
156C3×#120—92no—
157#953—92no—
158#947—92no—
159#940—92no—
160#940—94no—
161#939—94no—
162#937—32no—
163#937—32no—
164#937—62no—
165#937—92no—
166#930—92no—
167#930—92no—
168#926—91no—
169#885—94no—
170#884—94no—
171#882—92no—
172#877—92no—
173C23×C32—22no—
174C23×C32—21no—
175C23×C32—21no—
176C23×C32—22no—
177C23×C32—41no—
178C23×C32—41no—
179C23×C32C22×D312yes—
180C23×C32C22×D311yescharacteristic
181C23×C32C22×D312yes—
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3—24no—
184C22×C3⋊D3—44no—
185C22×C3⋊D3C22×D314yes—
186C22×C3⋊D3C22×D311yescharacteristic
187C22×C3×D3—34no—
188C22×C3×D3—34no—
189C22×C3×D3—32no—
190C22×C3×D3—64no—
191C22×C3×D3—61no—
192C22×C3×D3—64no—
193C22×C3×D3—64no—
194C22×C3×D3—124no—
195C22×C3×D3—124no—
196C22×C3×D3—124no—
197C22×C3×D3—128no—
198C32×D4—22no—
199C32×D4—24no—
200C32×D4—24no—
201C32×D4—44no—
202C32×D4—44no—
203C2×C32×C4—21no—
204C2×C32×C4—22no—
205C2×C32×C4—41no—
206C2×C32×C4C22×D311yescharacteristic
207C2×C32×C4C22×D312yes—
208#638—22no—
209#638—22no—
210#638—44no—
211C2×C3⋊Dic3—22no—
212C2×C3⋊Dic3#6714yescomplemented
213C2×C3⋊Dic3C22×D314yes—
214C2×C3⋊Dic3C22×D311yescharacteristic
215C2×C3⋊Dic3C22×D311yescharacteristic
216C2×C3⋊Dic3C22×D311yescharacteristic
217C4×C3⋊D3—22no—
218#634—22no—
219C3×#67—64no—
220C3×#67—64no—
221C3×#67—64no—
222C3×#67—64no—
223C3×#67—124no—
224C3×#67—124no—
225C3×#67—128no—
226C2×C3×Dic3—38no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—34no—
230C2×C3×Dic3—34no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—64no—
233C2×C3×Dic3—61no—
234C2×C3×Dic3—61no—
235C2×C3×Dic3—61no—
236C2×C3×Dic3—64no—
237C2×C3×Dic3—64no—
238C2×C3×Dic3—64no—
239C3×C4×D3—64no—
240C3×C4×D3—64no—
241C3×Dic6—64no—
242C3×Dic6—64no—
243#626—68no—
244#626—68no—
245#626—124no—
246#626—124no—
247#625—64no—
248#625—64no—
249#625—124no—
250#535—271noSylow
251C2×C33—22no—
252C2×C33—41no—
253C2×C33C22×D411yescharacteristic
254C2×C33H(5,𝔽2)12yes—
255C3×C3⋊D3—41no—
256C3×C3⋊D3—44no—
257C24×C3—92no—
258C23×D3—32no—
259C23×D3—32no—
260C23×D3—62no—
261C3×#41—184no—
262C2×C3×Q8—91no—
263C2×C3×D4—21no—
264C2×C3×D4—22no—
265C2×C3×D4—41no—
266C2×C3×D4—91no—
267C2×C3×D4—91no—
268C2×C3×D4—94no—
269C2×C3×D4D3211yescharacteristic
270C2×C3×D4D3212yes—
271C22×C3×C4—91no—
272C22×C3×C4—91no—
273C22×C3×C4—91no—
274C2×#67—32no—
275C2×#67—32no—
276C2×#67—34no—
277C2×#67—34no—
278C2×#67—61no—
279C2×#67—61no—
280C2×#67—64no—
281C2×#67—94no—
282C2×#67—94no—
283C2×#67—94no—
284C2×#67—94no—
285C2×#67—188no—
286C22×Dic3—32no—
287C22×Dic3—32no—
288C22×Dic3—61no—
289C22×Dic3—61no—
290#237—68no—
291#237—68no—
292C2×D12—92no—
293C2×D12—92no—
294C2×D12—94no—
295C2×D12—184no—
296C2×C4×D3—32no—
297C2×C4×D3—61no—
298C2×Dic6—32no—
299C2×Dic6—61no—
300C3×C22⋊C4—92no—
301C3×C22⋊C4—94no—
302#217—32no—
303#217—32no—
304#217—62no—
305#217—94no—
306(C2×D3)⋊C4—34no—
307(C2×D3)⋊C4—34no—
308(C2×D3)⋊C4—64no—
309(C2×D3)⋊C4—94no—
310(C2×D3)⋊C4—94no—
311C4⋊Dic3—92no—
312Dic3⋊C4—94no—
313C4×Dic3—94no—
314C4×Dic3—92no—
315C22×C32—21no—
316C22×C32—24no—
317C22×C32—22no—
318C22×C32—22no—
319C22×C32—22no—
320C22×C32—24no—
321C22×C32—22no—
322C22×C32—21no—
323C22×C32—22no—
324C22×C32—24no—
325C22×C32—22no—
326C22×C32—41no—
327C22×C32—42no—
328C22×C32—42no—
329C22×C32—42no—
330C22×C32—44no—
331C22×C32—42no—
332C22×C32—42no—
333C22×C32—44no—
334C22×C32—82no—
335C22×C32—82no—
336C22×C32D3×D414yes—
337C22×C32C2×#6712yes—
338C22×C32C23×D312yes—
339C22×C32C23×D311yescharacteristic
340C2×C3⋊D3—24no—
341C2×C3⋊D3—21no—
342C2×C3⋊D3—24no—
343C2×C3⋊D3—24no—
344C2×C3⋊D3—22no—
345C2×C3⋊D3—48no—
346C2×C3×D3—62no—
347C2×C3×D3—68no—
348C2×C3×D3—64no—
349C2×C3×D3—62no—
350C2×C3×D3—64no—
351C2×C3×D3—64no—
352C2×C3×D3—64no—
353C2×C3×D3—64no—
354C2×C3×D3—64no—
355C2×C3×D3—64no—
356C2×C3×D3—122no—
357C2×C3×D3—124no—
358C2×C3×D3—124no—
359C2×C3×D3—128no—
360C2×C3×D3—128no—
361C2×C3×D3—1216no—
362C32×C4—21no—
363C32×C4—22no—
364C32×C4—22no—
365C32×C4—42no—
366C32×C4—42no—
367C3⋊Dic3—21no—
368C3⋊Dic3—24no—
369C3⋊Dic3C2×#6712yes—
370C3⋊Dic3C2×#6712yes—
371C3×Dic3—34no—
372C3×Dic3—34no—
373C3×Dic3—62no—
374C3×Dic3—62no—
375C3×Dic3—62no—
376C3×Dic3—62no—
377C3×Dic3—64no—
378C3×Dic3—64no—
379C3×Dic3—124no—
380C2×#41—91no—
381C22×D4—271no—
382#127—272no—
383#124—272no—
384#121—272no—
385#121—272no—
386#120—92no—
387#120—272no—
388#117—271no—
389C33#53511yescharacteristic, complemented, Sylow
390C23×C3—22no—
391C23×C3—21no—
392C23×C3—21no—
393C23×C3—22no—
394C23×C3—41no—
395C23×C3—41no—
396C23×C3—94no—
397C23×C3—91no—
398C23×C3—184no—
399C23×C3—364no—
400C23×C3C2×D3212yes—
401C23×C3C2×D3211yescharacteristic
402C23×C3C2×D3212yes—
403C23×C3C2×D3211yescharacteristic
404C22×D3—34no—
405C22×D3—34no—
406C22×D3—32no—
407C22×D3—64no—
408C22×D3—61no—
409C22×D3—64no—
410C22×D3—64no—
411C22×D3—124no—
412C22×D3—124no—
413C22×D3—124no—
414C22×D3—128no—
415C3×Q8—182no—
416C3×D4—24no—
417C3×D4—24no—
418C3×D4—22no—
419C3×D4—44no—
420C3×D4—44no—
421C3×D4—182no—
422C3×D4—182no—
423C3×D4—364no—
424C2×C3×C4—21no—
425C2×C3×C4—22no—
426C2×C3×C4—41no—
427C2×C3×C4—91no—
428C2×C3×C4—91no—
429C2×C3×C4—94no—
430C2×C3×C4—94no—
431C2×C3×C4—91no—
432C2×C3×C4—182no—
433C2×C3×C4—182no—
434C2×C3×C4C2×D3211yescharacteristic
435C2×C3×C4C2×D3212yes—
436#67—64no—
437#67—64no—
438#67—64no—
439#67—64no—
440#67—124no—
441#67—124no—
442#67—128no—
443#67—188no—
444#67—188no—
445#67—364no—
446#67—364no—
447C2×Dic3—32no—
448C2×Dic3—32no—
449C2×Dic3—38no—
450C2×Dic3—34no—
451C2×Dic3—34no—
452C2×Dic3—32no—
453C2×Dic3—64no—
454C2×Dic3—61no—
455C2×Dic3—61no—
456C2×Dic3—61no—
457C2×Dic3—64no—
458C2×Dic3—64no—
459C2×Dic3—64no—
460D12—184no—
461D12—184no—
462D12—364no—
463C4×D3—64no—
464C4×D3—64no—
465Dic6—64no—
466Dic6—64no—
467C2×C32—22no—
468C2×C32—24no—
469C2×C32—22no—
470C2×C32—24no—
471C2×C32—22no—
472C2×C32—21no—
473C2×C32—41no—
474C2×C32—42no—
475C2×C32—44no—
476C2×C32—42no—
477C2×C32—41no—
478C2×C32—42no—
479C2×C32—42no—
480C2×C32—82no—
481C2×C32—82no—
482C2×C32—82no—
483C2×C32C2×D3×D412yes—
484C2×C32#100414yes—
485C2×C32#100912yes—
486C2×C32C22×#6711yescharacteristic
487C3⋊D3—41no—
488C3⋊D3—44no—
489C3×D3—122no—
490C3×D3—128no—
491C3×D3—124no—
492C3×D3—122no—
493C3×D3—124no—
494C24—92no—
495#41—184no—
496C2×Q8—91no—
497C2×D4—91no—
498C2×D4—91no—
499C2×D4—94no—
500C2×D4—272no—
501C2×D4—272no—
502C2×D4—274no—
503C2×D4—544no—
504C2×D4#111311yescharacteristic
505C22×C4—91no—
506C22×C4—91no—
507C22×C4—91no—
508C4⋊C4—272no—
509C22⋊C4—92no—
510C22⋊C4—94no—
511C22⋊C4—274no—
512C42—272no—
513C22×C3—22no—
514C22×C3—24no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—24no—
518C22×C3—22no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—24no—
524C22×C3—44no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—44no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—82no—
533C22×C3—82no—
534C22×C3—182no—
535C22×C3—184no—
536C22×C3—184no—
537C22×C3—184no—
538C22×C3—181no—
539C22×C3—368no—
540C22×C3D3×#6714yes—
541C22×C3#381112yes—
542C22×C3C22×D3211yescharacteristic
543C22×C3C22×D3212yes—
544C2×D3—62no—
545C2×D3—68no—
546C2×D3—64no—
547C2×D3—64no—
548C2×D3—64no—
549C2×D3—64no—
550C2×D3—64no—
551C2×D3—62no—
552C2×D3—64no—
553C2×D3—64no—
554C2×D3—124no—
555C2×D3—124no—
556C2×D3—122no—
557C2×D3—128no—
558C2×D3—128no—
559C2×D3—1216no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—21no—
563C3×C4—42no—
564C3×C4—42no—
565C3×C4—92no—
566C3×C4—92no—
567C3×C4—184no—
568C3×C4—181no—
569Dic3—34no—
570Dic3—34no—
571Dic3—62no—
572Dic3—62no—
573Dic3—62no—
574Dic3—64no—
575Dic3—64no—
576Dic3—62no—
577Dic3—124no—
578C32—22no—
579C32—21no—
580C32—41no—
581C32#559212yescomplemented
582C32#583011yescharacteristic, complemented
583C23—94no—
584C23—91no—
585C23—184no—
586C23—364no—
587C23C2×#111311yescharacteristic
588C23C2×#111311yescharacteristic
589Q8—182no—
590D4—22no—
591D4—182no—
592D4—182no—
593D4—364no—
594D4—544no—
595D4—544no—
596D4—1084no—
597C2×C4—91no—
598C2×C4—91no—
599C2×C4—94no—
600C2×C4—94no—
601C2×C4—91no—
602C2×C4—182no—
603C2×C4—182no—
604C2×C4C2×#111311yescharacteristic
605C2×C3—24no—
606C2×C3—22no—
607C2×C3—24no—
608C2×C3—22no—
609C2×C3—21no—
610C2×C3—22no—
611C2×C3—44no—
612C2×C3—42no—
613C2×C3—42no—
614C2×C3—42no—
615C2×C3—41no—
616C2×C3—42no—
617C2×C3—41no—
618C2×C3—82no—
619C2×C3—82no—
620C2×C3—82no—
621C2×C3—364no—
622C2×C3—361no—
623C2×C3#6431312yes—
624C2×C3#6431414yes—
625C2×C3C2×D3×#6712yes—
626C2×C3C2×#381111yescharacteristic
627D3—128no—
628D3—122no—
629D3—124no—
630D3—122no—
631D3—124no—
632C22—22no—
633C22—22no—
634C22—21no—
635C22—184no—
636C22—182no—
637C22—184no—
638C22—181no—
639C22—184no—
640C22—368no—
641C22#8904912yes—
642C22C22×#111311yescharacteristic
643C4—21no—
644C4—92no—
645C4—92no—
646C4—184no—
647C4—181no—
648C3—22no—
649C3—21no—
650C3—41no—
651C3#1059523812yescomplemented
652C3#1059523911yescharacteristic, complemented
653C2—22no—
654C2—41no—
655C2—364no—
656C2—361no—
657C2#1172462312yes—
658C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C22×#6295C2semidirect
3. C3×#64146C2semidirect
4. #11723728C2semidirect
5. #11723727C2semidirect
6. #11723717C2semidirect
7. #11723715C2semidirect
8. #11723695C2semidirect
9. #11723670C2semidirect
15. C2×C3×#638C22semidirect
16. C2×C3×#638C22semidirect
18. C22×C3×C3⋊Dic3C22semidirect
20. C2×C3×C4×C3⋊D3C22semidirect
21. C2×C3×#634C22semidirect
25. C2×#6295C22semidirect
27. C3×#3752C22semidirect
28. #88815C22semidirect
29. #88814C22semidirect
30. #88813C22semidirect
31. C2×#3830D3semidirect
36. #64146D3semidirect
65. C2×C3×C3⋊Dic3D4semidirect
87. C2×#638C2×D3semidirect
88. C2×#638C2×D3semidirect
90. C22×C3⋊Dic3C2×D3semidirect
92. C2×C4×C3⋊D3C2×D3semidirect
93. C2×#634C2×D3semidirect
121. #3752C2×D3semidirect
212. C2×C3⋊Dic3#67semidirect
389. C33#535semidirect
581. C32#5592semidirect
582. C32#5830semidirect
651. C3#10595238semidirect
652. C3#10595239semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C22×#62952
C3×#641462
#117237282
#117237272
#117237172
#117237152
#117236952
#117236702
#105952393
#105952383