Finite Group Databasebeta

← #13162051

Subgroups — #13162051

Order 1728.

1666 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic
2C2×C3×#3757C212yesmaximal, complemented
3C2×#88815C211yesmaximal, characteristic, complemented
4C2×#88814C212yesmaximal, complemented
5C2×#88813C211yesmaximal, characteristic, complemented
6#10591707—32nomaximal
7#10591704—31nomaximal
8C24×C33C2211yescharacteristic
9C23×C3×C3⋊D3C2211yescharacteristic
10C2×C3×#638—22no—
11C2×C3×#638—22no—
12C2×C3×#638C414yescomplemented
13C22×C3×C3⋊Dic3C2212yes—
14C22×C3×C3⋊Dic3C2212yes—
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C3×#3757—24no—
17#88815—22no—
18#88814—24no—
19#88813—22no—
20C22×#638D311yescharacteristic
21C22×C3×#67—32no—
22C22×C3×#67—61no—
23C2×#3757D312yescomplemented
24C2×C3×#217—32no—
25C2×C3×#217—32no—
26C2×C3×#217—62no—
27C2×#3724—31no—
28C2×Dic3⋊Dic3—32no—
29C2×#3722—32no—
30C2×(C2×D3)⋊Dic3—32no—
31C2×(C2×D3)⋊Dic3—32no—
32C2×Dic32—31no—
33C2×Dic32—32no—
34C23×C33—22no—
35C23×C33—21no—
36C23×C33—21no—
37C23×C33—22no—
38C23×C33C2×C412yes—
39C23×C33C2311yescharacteristic
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3—22no—
44C22×C3×C3⋊D3C2×C412yes—
45C3×#638—28no—
46C2×C3×C3⋊Dic3—22no—
47C2×C3×C3⋊Dic3—22no—
48C2×C3×C3⋊Dic3—22no—
49C2×C3×C3⋊Dic3—22no—
50C2×C3×C3⋊Dic3—22no—
51C2×C3×C3⋊Dic3C2×C412yes—
52C2×C3×C3⋊Dic3D412yes—
53C2×C3×C3⋊Dic3D412yes—
54C2×C3×C3⋊Dic3D414yescomplemented
55#5221—91no—
56#4957—92no—
57C24×C32—22no—
58C24×C32—21no—
59C24×C32—41no—
60C24×C32C2×D312yes—
61C24×C32C2×D311yescharacteristic
62C23×C3⋊D3C2×D311yescharacteristic
63C23×C3×D3—32no—
64C23×C3×D3—61no—
65C2×#638—22no—
66C2×#638—22no—
67C2×#638Dic314yescomplemented
68C22×C3⋊Dic3C2×D312yes—
69C22×C3⋊Dic3C2×D312yes—
70C22×C3⋊Dic3C2×D311yescharacteristic
71C2×C3×#67—38no—
72C2×C3×#67—64no—
73C2×C3×#67—64no—
74C2×C3×#67—64no—
75C2×C3×#67—64no—
76C2×C3×#67—64no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—32no—
79C22×C3×Dic3—32no—
80C22×C3×Dic3—32no—
81C22×C3×Dic3—32no—
82C22×C3×Dic3—61no—
83C22×C3×Dic3—62no—
84C22×C3×Dic3—62no—
85#3757—24no—
86C3×#217—64no—
87C3×#217—64no—
88C3×#217—68no—
89#3724—62no—
90Dic3⋊Dic3—64no—
91#3722—64no—
92(C2×D3)⋊Dic3—64no—
93(C2×D3)⋊Dic3—64no—
94Dic32—64no—
95Dic32—62no—
96C22×C33—22no—
97C22×C33—24no—
98C22×C33—42no—
99C22×C33—41no—
100C22×C33—41no—
101C22×C33C22×C411yescharacteristic
102C22×C33C2×D411yescharacteristic
103C22×C33C2×D411yescharacteristic
104C22×C33C2×D412yes—
105C22×C33#4112yes—
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—24no—
108C2×C3×C3⋊D3—42no—
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C3×C3⋊Dic3—24no—
112C3×C3⋊Dic3—42no—
113C3×C3⋊Dic3C22⋊C414yescomplemented
114C22×C3×D4—91no—
115C22×#67—32no—
116C22×#67—61no—
117C2×C3×C22⋊C4—92no—
118C2×#217—32no—
119C2×#217—32no—
120C2×#217—62no—
121C2×#217—92no—
122C2×(C2×D3)⋊C4—92no—
123C2×(C2×D3)⋊C4—92no—
124C2×C4⋊Dic3—91no—
125C2×Dic3⋊C4—92no—
126C2×C4×Dic3—92no—
127C2×C4×Dic3—91no—
128C23×C32—22no—
129C23×C32—22no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—24no—
133C23×C32—24no—
134C23×C32—22no—
135C23×C32—22no—
136C23×C32—22no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—22no—
142C23×C32—24no—
143C23×C32—22no—
144C23×C32—21no—
145C23×C32—22no—
146C23×C32—42no—
147C23×C32—41no—
148C23×C32—42no—
149C23×C32—42no—
150C23×C32—42no—
151C23×C32—42no—
152C23×C32—42no—
153C23×C32—42no—
154C23×C32—44no—
155C23×C32—42no—
156C23×C32—42no—
157C23×C32—44no—
158C23×C32C4×D314yes—
159C23×C32C2×Dic312yes—
160C23×C32C22×D312yes—
161C23×C32C22×D311yescharacteristic
162C22×C3⋊D3—21no—
163C22×C3⋊D3—21no—
164C22×C3⋊D3—22no—
165C22×C3⋊D3—22no—
166C22×C3⋊D3C2×Dic312yes—
167C22×C3×D3—34no—
168C22×C3×D3—62no—
169C22×C3×D3—62no—
170C22×C3×D3—62no—
171C22×C3×D3—62no—
172C22×C3×D3—62no—
173C22×C3×D3—62no—
174C22×C3×D3—62no—
175C22×C3×D3—62no—
176C22×C3×D3—62no—
177C22×C3×D3—122no—
178C22×C3×D3—122no—
179#638—28no—
180C2×C3⋊Dic3—22no—
181C2×C3⋊Dic3—22no—
182C2×C3⋊Dic3—22no—
183C2×C3⋊Dic3—22no—
184C2×C3⋊Dic3—22no—
185C2×C3⋊Dic3C2×Dic312yes—
186C2×C3⋊Dic3#6712yes—
187C2×C3⋊Dic3#6712yes—
188C2×C3⋊Dic3#6714yescomplemented
189C3×#67—616no—
190C3×#67—616no—
191C2×C3×Dic3—34no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—64no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—64no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3—122no—
212C2×C3×Dic3—122no—
213C2×C3×Dic3—122no—
214C2×C3×Dic3—122no—
215#390—271noSylow
216C2×C33—42no—
217C2×C33C2×C22⋊C411yescharacteristic
218C2×C33C4×D412yes—
219C2×C33#12112yes—
220C2×C33#12411yescharacteristic
221C2×C33#12711yescharacteristic
222C3×C3⋊D3—42no—
223C24×C3—22no—
224C24×C3—21no—
225C24×C3—41no—
226C24×C3—91no—
227C24×C3D3212yes—
228C24×C3D3211yescharacteristic
229C23×D3—32no—
230C23×D3—61no—
231C2×C3×D4—94no—
232C2×C3×D4—182no—
233C2×C3×D4—182no—
234C22×C3×C4—91no—
235C22×C3×C4—92no—
236C22×C3×C4—92no—
237C2×#67—38no—
238C2×#67—64no—
239C2×#67—64no—
240C2×#67—64no—
241C2×#67—64no—
242C2×#67—64no—
243C22×Dic3—32no—
244C22×Dic3—32no—
245C22×Dic3—32no—
246C22×Dic3—32no—
247C22×Dic3—32no—
248C22×Dic3—61no—
249C22×Dic3—62no—
250C22×Dic3—62no—
251C3×C22⋊C4—184no—
252#217—64no—
253#217—64no—
254#217—68no—
255#217—184no—
256(C2×D3)⋊C4—184no—
257(C2×D3)⋊C4—184no—
258C4⋊Dic3—182no—
259Dic3⋊C4—184no—
260C4×Dic3—184no—
261C4×Dic3—182no—
262C22×C32—24no—
263C22×C32—22no—
264C22×C32—24no—
265C22×C32—28no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—24no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—21no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—41no—
282C22×C32—42no—
283C22×C32—41no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32—44no—
287C22×C32—44no—
288C22×C32—44no—
289C22×C32—44no—
290C22×C32—48no—
291C22×C32—42no—
292C22×C32—44no—
293C22×C32—44no—
294C22×C32—44no—
295C22×C32—44no—
296C22×C32—44no—
297C22×C32—42no—
298C22×C32—48no—
299C22×C32—44no—
300C22×C32—44no—
301C22×C32—42no—
302C22×C32—42no—
303C22×C32—41no—
304C22×C32—42no—
305C22×C32—42no—
306C22×C32—42no—
307C22×C32—41no—
308C22×C32C2×C4×D312yes—
309C22×C32#23512yes—
310C22×C32D3×D412yes—
311C22×C32D3×D412yes—
312C22×C32D3×D412yes—
313C22×C32D3×D412yes—
314C22×C32#23712yes—
315C22×C32#23712yes—
316C22×C32C22×Dic311yescharacteristic
317C22×C32C2×#6712yes—
318C22×C32C2×#6711yescharacteristic
319C22×C32C2×#6711yescharacteristic
320C2×C3⋊D3—22no—
321C2×C3⋊D3—24no—
322C2×C3⋊D3—41no—
323C2×C3⋊D3—42no—
324C2×C3⋊D3—41no—
325C2×C3×D3—64no—
326C2×C3×D3—64no—
327C2×C3×D3—64no—
328C2×C3×D3—64no—
329C2×C3×D3—124no—
330C2×C3×D3—122no—
331C2×C3×D3—124no—
332C2×C3×D3—122no—
333C2×C3×D3—122no—
334C2×C3×D3—122no—
335C2×C3×D3—122no—
336C2×C3×D3—122no—
337C3⋊Dic3—24no—
338C3⋊Dic3—42no—
339C3⋊Dic3#21714yescomplemented
340C3×Dic3—38no—
341C3×Dic3—64no—
342C3×Dic3—64no—
343C3×Dic3—64no—
344C3×Dic3—124no—
345C3×Dic3—124no—
346C3×Dic3—122no—
347C3×Dic3—122no—
348C22×D4—91no—
349C2×C4⋊C4—271no—
350C2×C22⋊C4—92no—
351C2×C22⋊C4—272no—
352C2×C42—271no—
353C33#39011yescharacteristic, complemented, Sylow
354C23×C3—24no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—24no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3—22no—
363C23×C3—22no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—24no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—21no—
372C23×C3—42no—
373C23×C3—42no—
374C23×C3—42no—
375C23×C3—42no—
376C23×C3—42no—
377C23×C3—42no—
378C23×C3—44no—
379C23×C3—44no—
380C23×C3—42no—
381C23×C3—42no—
382C23×C3—42no—
383C23×C3—41no—
384C23×C3—92no—
385C23×C3—182no—
386C23×C3—182no—
387C23×C3—181no—
388C23×C3—181no—
389C23×C3D3×Dic314yes—
390C23×C3#62412yes—
391C23×C3C2×D3212yes—
392C23×C3C2×D3211yescharacteristic
393C22×D3—34no—
394C22×D3—62no—
395C22×D3—62no—
396C22×D3—62no—
397C22×D3—62no—
398C22×D3—62no—
399C22×D3—62no—
400C22×D3—62no—
401C22×D3—62no—
402C22×D3—62no—
403C22×D3—122no—
404C22×D3—122no—
405C3×D4—188no—
406C2×C3×C4—92no—
407C2×C3×C4—92no—
408C2×C3×C4—92no—
409C2×C3×C4—94no—
410C2×C3×C4—182no—
411C2×C3×C4—182no—
412C2×C3×C4—182no—
413C2×C3×C4—182no—
414C2×C3×C4—182no—
415#67—616no—
416#67—616no—
417C2×Dic3—34no—
418C2×Dic3—34no—
419C2×Dic3—34no—
420C2×Dic3—34no—
421C2×Dic3—34no—
422C2×Dic3—62no—
423C2×Dic3—62no—
424C2×Dic3—62no—
425C2×Dic3—64no—
426C2×Dic3—62no—
427C2×Dic3—62no—
428C2×Dic3—62no—
429C2×Dic3—62no—
430C2×Dic3—64no—
431C2×Dic3—62no—
432C2×Dic3—62no—
433C2×Dic3—62no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3—62no—
437C2×Dic3—122no—
438C2×Dic3—122no—
439C2×Dic3—122no—
440C2×Dic3—122no—
441C2×C32—22no—
442C2×C32—21no—
443C2×C32—22no—
444C2×C32—22no—
445C2×C32—22no—
446C2×C32—22no—
447C2×C32—22no—
448C2×C32—22no—
449C2×C32—22no—
450C2×C32—21no—
451C2×C32—22no—
452C2×C32—21no—
453C2×C32—41no—
454C2×C32—42no—
455C2×C32—42no—
456C2×C32—41no—
457C2×C32—41no—
458C2×C32—44no—
459C2×C32—48no—
460C2×C32—48no—
461C2×C32—44no—
462C2×C32—42no—
463C2×C32D3×C22⋊C412yes—
464C2×C32#67⋊C412yes—
465C2×C32#88412yes—
466C2×C32#88512yes—
467C2×C32C4×#6712yes—
468C2×C32D4×Dic312yes—
469C2×C32#93611yescharacteristic
470C2×C32#93812yes—
471C2×C32#93912yes—
472C2×C32#94012yes—
473C2×C32#94011yescharacteristic
474C2×C32C2×#21711yescharacteristic
475C3⋊D3—42no—
476C3×D3—124no—
477C3×D3—124no—
478C24—91no—
479C24#111311yescharacteristic
480C2×D4—94no—
481C2×D4—182no—
482C2×D4—182no—
483C22×C4—91no—
484C22×C4—92no—
485C22×C4—92no—
486C4⋊C4—542no—
487C22⋊C4—184no—
488C22⋊C4—544no—
489C42—542no—
490C22×C3—24no—
491C22×C3—28no—
492C22×C3—24no—
493C22×C3—24no—
494C22×C3—24no—
495C22×C3—24no—
496C22×C3—22no—
497C22×C3—21no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—44no—
510C22×C3—44no—
511C22×C3—44no—
512C22×C3—44no—
513C22×C3—48no—
514C22×C3—42no—
515C22×C3—44no—
516C22×C3—44no—
517C22×C3—44no—
518C22×C3—44no—
519C22×C3—44no—
520C22×C3—42no—
521C22×C3—44no—
522C22×C3—48no—
523C22×C3—44no—
524C22×C3—41no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—41no—
530C22×C3—42no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—41no—
534C22×C3—41no—
535C22×C3—42no—
536C22×C3—184no—
537C22×C3—182no—
538C22×C3—361no—
539C22×C3—362no—
540C22×C3—361no—
541C22×C3C2×D3×Dic312yes—
542C22×C3#380412yes—
543C22×C3#380412yes—
544C22×C3#380512yes—
545C22×C3C2×#62411yescharacteristic
546C22×C3D3×#6712yes—
547C22×C3D3×#6712yes—
548C22×C3D3×#6712yes—
549C22×C3D3×#6712yes—
550C22×C3#381111yescharacteristic
551C22×C3#381112yes—
552C22×C3#381111yescharacteristic
553C2×D3—64no—
554C2×D3—64no—
555C2×D3—64no—
556C2×D3—64no—
557C2×D3—124no—
558C2×D3—122no—
559C2×D3—124no—
560C2×D3—122no—
561C2×D3—122no—
562C2×D3—122no—
563C2×D3—122no—
564C2×D3—122no—
565C3×C4—94no—
566C3×C4—184no—
567C3×C4—362no—
568Dic3—38no—
569Dic3—64no—
570Dic3—64no—
571Dic3—64no—
572Dic3—124no—
573Dic3—124no—
574Dic3—122no—
575Dic3—122no—
576C32—21no—
577C32—22no—
578C32—41no—
579C32#495712yescomplemented
580C32#522111yescharacteristic, complemented
581C23—22no—
582C23—22no—
583C23—21no—
584C23—21no—
585C23—92no—
586C23—182no—
587C23—182no—
588C23—181no—
589C23—181no—
590C23#629412yes—
591C23C2×#111311yescharacteristic
592D4—188no—
593C2×C4—92no—
594C2×C4—92no—
595C2×C4—92no—
596C2×C4—94no—
597C2×C4—182no—
598C2×C4—182no—
599C2×C4—182no—
600C2×C4—182no—
601C2×C4—182no—
602C2×C3—21no—
603C2×C3—22no—
604C2×C3—21no—
605C2×C3—22no—
606C2×C3—22no—
607C2×C3—22no—
608C2×C3—22no—
609C2×C3—22no—
610C2×C3—22no—
611C2×C3—22no—
612C2×C3—22no—
613C2×C3—21no—
614C2×C3—44no—
615C2×C3—48no—
616C2×C3—48no—
617C2×C3—42no—
618C2×C3—44no—
619C2×C3—41no—
620C2×C3—41no—
621C2×C3—42no—
622C2×C3—42no—
623C2×C3—41no—
624C2×C3—362no—
625C2×C3#6395212yes—
626C2×C3#6395311yescharacteristic
627C2×C3#6395812yes—
628C2×C3#6395912yes—
629C2×C3D3×#21712yes—
630C2×C3#6397012yes—
631C2×C3#6397112yes—
632C2×C3Dic3×#6712yes—
633C2×C3#67⋊Dic312yes—
634C2×C3#6397411yescharacteristic
635C2×C3#6397911yescharacteristic
636C2×C3#6397912yes—
637D3—124no—
638D3—124no—
639C22—22no—
640C22—24no—
641C22—41no—
642C22—42no—
643C22—41no—
644C22—182no—
645C22—184no—
646C22—361no—
647C22—362no—
648C22—361no—
649C22C2×#629411yescharacteristic
650C22#8904612yes—
651C22#8904911yescharacteristic
652C22#8904911yescharacteristic
653C22#8904912yes—
654C4—94no—
655C4—184no—
656C4—362no—
657C3—21no—
658C3—22no—
659C3—41no—
660C3#1059170411yescharacteristic, complemented
661C3#1059170712yescomplemented
662C2—42no—
663C2—362no—
664C2#1172371811yescharacteristic
665C2#1172372311yescharacteristic
666C2#1172372512yes—
667C2#1172372612yes—
668C2#1172372811yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutatorC22×C33C22×C4
FittingC24×C33C22
socle, cosocleC23×C33C23
prime radicalC22×C3×#638C2
hypercenterC24#1113
hypocenterC33#390
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3757C2semidirect
3. C2×#88815C2semidirect
4. C2×#88814C2semidirect
5. C2×#88813C2semidirect
12. C2×C3×#638C4semidirect
23. C2×#3757D3semidirect
54. C2×C3×C3⋊Dic3D4semidirect
67. C2×#638Dic3semidirect
113. C3×C3⋊Dic3C22⋊C4semidirect
188. C2×C3⋊Dic3#67semidirect
339. C3⋊Dic3#217semidirect
353. C33#390semidirect
579. C32#4957semidirect
580. C32#5221semidirect
660. C3#10591704semidirect
661. C3#10591707semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C2×C3×#37572
C2×#888152
C2×#888142
C2×#888132
#105917073
#105917043