Finite Group Databasebeta

← #13161456

Subgroups — #13161456

Order 1728.

2244 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#67C211yesmaximal, characteristic
2C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#88864C211yesmaximal, characteristic, complemented
4C2×C3×#3757C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6#11722061C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592919—31nomaximal
9#10591721—34nomaximal
10C24×C33C2211yescharacteristic
11C23×C32×D3C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C2×C32×#67—22no—
17C2×C32×#67—22no—
18C2×C32×#67C414yescomplemented
19C22×C32×Dic3C2211yescharacteristic
20C2×D3×C3⋊Dic3—22no—
21C2×D3×C3⋊Dic3—22no—
22C2×D3×C3⋊Dic3—22no—
23#88864—22no—
24C3×#3757—22no—
25(C2×D3)⋊(C3⋊Dic3)—22no—
26C22×C32×D4—31no—
27C23×C3⋊Dic3—31no—
28C22×C3×#67D314yes—
29C22×D3×Dic3—34no—
30C2×#3757—31no—
31C2×#3757—31no—
32C2×#3757—34no—
33C2×#3757—64no—
34C2×#3757D311yescharacteristic, complemented
35C2×C3×#217—34no—
36C2×(C2×D3)⋊Dic3—34no—
37#63658—31no—
38#63658—31no—
39#63579—34no—
40#63579—34no—
41C23×C33—21no—
42C23×C33—21no—
43C23×C33—21no—
44C23×C33—21no—
45C23×C33—21no—
46C23×C33—21no—
47C23×C33C2×C412yescomplemented
48C23×C33C2311yescharacteristic
49C22×C32×D3—21no—
50C22×C32×D3—21no—
51C22×C32×D3—21no—
52C22×C32×D3—21no—
53C22×C32×D3C2×C412yes—
54C22×C32×D3D412yes—
55C22×C32×D3D412yes—
56C2×C3⋊(C3⋊Dic3)—21no—
57C2×C3⋊(C3⋊Dic3)—21no—
58C2×C3⋊(C3⋊Dic3)—21no—
59C2×C3⋊(C3⋊Dic3)—21no—
60C2×C3⋊(C3⋊Dic3)—21no—
61C2×C3⋊(C3⋊Dic3)D412yescomplemented
62C2×C3×C3⋊Dic3—21no—
63C2×C3×C3⋊Dic3—21no—
64C2×C3×C3⋊Dic3—21no—
65C2×C3×C3⋊Dic3—21no—
66C2×C3×C3⋊Dic3—21no—
67C2×C3×C3⋊Dic3D412yescomplemented
68C32×#67—28no—
69C2×C32×Dic3—21no—
70C2×C32×Dic3—21no—
71C2×C32×Dic3C2×C412yescomplemented
72D3×C3⋊Dic3—28no—
73#5220—94no—
74#4956—91no—
75C24×C32—24no—
76C24×C32—31no—
77C24×C32C2×D311yescharacteristic
78C24×C32C2×D314yes—
79C23×C3×D3C2×D314yes—
80C2×C32×D4—34no—
81C2×C32×D4—62no—
82C2×C32×D4—62no—
83C22×C32×C4—31no—
84C22×C3⋊Dic3—31no—
85C22×C3⋊Dic3—34no—
86C22×C3⋊Dic3—31no—
87C22×C3⋊Dic3—34no—
88C22×C3⋊Dic3—62no—
89C22×C3⋊Dic3—62no—
90C22×C3⋊Dic3—62no—
91C22×C3⋊Dic3—64no—
92C22×C3⋊Dic3—64no—
93C22×C3⋊Dic3C2×D311yescharacteristic
94C22×C3⋊Dic3C2×D311yescharacteristic
95C2×C3×#67—28no—
96C2×C3×#67—28no—
97C2×C3×#67Dic3116yescomplemented
98C22×C3×Dic3—34no—
99C22×C3×Dic3—34no—
100C22×C3×Dic3C2×D314yes—
101C2×D3×Dic3—68no—
102C2×D3×Dic3—68no—
103C2×D3×Dic3—68no—
104#3757—22no—
105#3757—62no—
106#3757—62no—
107#3757—68no—
108#3757—616no—
109C3×#217—68no—
110(C2×D3)⋊Dic3—68no—
111C22×C33—22no—
112C22×C33—22no—
113C22×C33—22no—
114C22×C33—41no—
115C22×C33—41no—
116C22×C33—41no—
117C22×C33—41no—
118C22×C33C22×C411yescharacteristic
119C22×C33C2×D411yescharacteristic
120C22×C33C2×D411yescharacteristic
121C22×C33C2×D411yescharacteristic
122C22×C33C2×D411yescharacteristic
123C22×C33#4111yescharacteristic
124C22×C33#4111yescharacteristic
125C2×C32×D3—22no—
126C2×C32×D3—22no—
127C2×C32×D3—22no—
128C2×C32×D3—22no—
129C2×C32×D3—22no—
130C2×C32×D3—22no—
131C2×C32×D3C22⋊C414yescomplemented
132C3⋊(C3⋊Dic3)—22no—
133C3⋊(C3⋊Dic3)—41no—
134C3×C3⋊Dic3—22no—
135C3×C3⋊Dic3—41no—
136C32×Dic3—22no—
137C22×C3×D4—34no—
138C22×#67C3⋊D311yescharacteristic
139C23×Dic3—94no—
140C22×C4×D3—91no—
141C2×C3×C22⋊C4—91no—
142C2×#217—34no—
143C2×#217—94no—
144C2×#217—94no—
145C2×#217—91no—
146C2×#217—184no—
147C2×(C2×D3)⋊C4—91no—
148C3⋊#95—94no—
149C3⋊#95—94no—
150C3⋊#95—91no—
151C3⋊#95—91no—
152C23×C32—28no—
153C23×C32—28no—
154C23×C32—21no—
155C23×C32—24no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—24no—
160C23×C32—21no—
161C23×C32—21no—
162C23×C32—28no—
163C23×C32—24no—
164C23×C32—24no—
165C23×C32—24no—
166C23×C32—24no—
167C23×C32—24no—
168C23×C32—32no—
169C23×C32—32no—
170C23×C32—32no—
171C23×C32—44no—
172C23×C32—44no—
173C23×C32—44no—
174C23×C32—44no—
175C23×C32—61no—
176C23×C32—61no—
177C23×C32—61no—
178C23×C32—61no—
179C23×C32C4×D312yescomplemented
180C23×C32C2×Dic318yescomplemented
181C23×C32C22×D314yes—
182C23×C32C22×D311yescharacteristic
183C22×C3×D3—24no—
184C22×C3×D3—24no—
185C22×C3×D3—24no—
186C22×C3×D3—24no—
187C22×C3×D3C2×Dic318yes—
188C22×C3×D3#6718yes—
189C22×C3×D3#6718yes—
190C32×D4—68no—
191C2×C32×C4—32no—
192C2×C32×C4—61no—
193C2×C32×C4—61no—
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—21no—
196C2×C3⋊Dic3—21no—
197C2×C3⋊Dic3—21no—
198C2×C3⋊Dic3—21no—
199C2×C3⋊Dic3—32no—
200C2×C3⋊Dic3—38no—
201C2×C3⋊Dic3—68no—
202C2×C3⋊Dic3—68no—
203C2×C3⋊Dic3—61no—
204C2×C3⋊Dic3—61no—
205C2×C3⋊Dic3—68no—
206C2×C3⋊Dic3—61no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3—61no—
209C2×C3⋊Dic3—61no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3—64no—
213C2×C3⋊Dic3—64no—
214C2×C3⋊Dic3—124no—
215C2×C3⋊Dic3—124no—
216C2×C3⋊Dic3—124no—
217C2×C3⋊Dic3—124no—
218C2×C3⋊Dic3#6712yescomplemented
219C3×#67—232no—
220C2×C3×Dic3—24no—
221C2×C3×Dic3—24no—
222C2×C3×Dic3—38no—
223C2×C3×Dic3—64no—
224C2×C3×Dic3—64no—
225C2×C3×Dic3—64no—
226C2×C3×Dic3—64no—
227C2×C3×Dic3—64no—
228C2×C3×Dic3C2×Dic318yescomplemented
229D3×Dic3—632no—
230#386—271noSylow
231C2×C33—42no—
232C2×C33C2×C22⋊C411yescharacteristic
233C2×C33C4×D411yescharacteristic
234C2×C33C4×D411yescharacteristic
235C2×C33#12011yescharacteristic
236C2×C33#12111yescharacteristic
237C2×C33#12111yescharacteristic
238C2×C33#12311yescharacteristic
239C32×D3—24no—
240C24×C3—24no—
241C24×C3—34no—
242C24×C3D3214yes—
243C24×C3C2×C3⋊D311yescharacteristic
244C23×D3C2×C3⋊D311yescharacteristic
245C2×C3×D4—316no—
246C2×C3×D4—68no—
247C2×C3×D4—68no—
248C22×C3×C4—34no—
249C22×C3×C4—91no—
250C22×C3×C4—91no—
251C2×#67—22no—
252C2×#67—22no—
253C2×#67C3⋊Dic314yescomplemented
254C22×Dic3—34no—
255C22×Dic3—34no—
256C22×Dic3—94no—
257C22×Dic3—91no—
258C22×Dic3—94no—
259C22×Dic3—91no—
260C22×Dic3—184no—
261C22×Dic3—184no—
262C22×Dic3—188no—
263C22×Dic3—188no—
264C22×Dic3—188no—
265C22×Dic3C2×C3⋊D311yescharacteristic
266C2×C4×D3—182no—
267C2×C4×D3—182no—
268C2×C4×D3—182no—
269C3×C22⋊C4—182no—
270#217—68no—
271#217—188no—
272#217—188no—
273#217—182no—
274#217—1816no—
275(C2×D3)⋊C4—182no—
276C22×C32—216no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—28no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32—24no—
285C22×C32—24no—
286C22×C32—24no—
287C22×C32—24no—
288C22×C32—28no—
289C22×C32—28no—
290C22×C32—34no—
291C22×C32—48no—
292C22×C32—48no—
293C22×C32—48no—
294C22×C32—48no—
295C22×C32—48no—
296C22×C32—48no—
297C22×C32—41no—
298C22×C32—41no—
299C22×C32—44no—
300C22×C32—44no—
301C22×C32—41no—
302C22×C32—41no—
303C22×C32—44no—
304C22×C32—44no—
305C22×C32—62no—
306C22×C32—62no—
307C22×C32—62no—
308C22×C32—62no—
309C22×C32—62no—
310C22×C32—62no—
311C22×C32C2×C4×D311yescharacteristic
312C22×C32#23511yescharacteristic
313C22×C32D3×D414yes—
314C22×C32D3×D414yes—
315C22×C32D3×D411yescharacteristic
316C22×C32D3×D411yescharacteristic
317C22×C32D3×D411yescharacteristic
318C22×C32#23714yes—
319C22×C32#23714yes—
320C22×C32#23711yescharacteristic
321C22×C32C22×Dic314yes—
322C22×C32C2×#6714yes—
323C22×C32C2×#6714yes—
324C22×C32C2×#6711yescharacteristic
325C2×C3×D3—28no—
326C2×C3×D3—28no—
327C2×C3×D3—28no—
328C2×C3×D3—28no—
329C2×C3×D3—28no—
330C2×C3×D3—28no—
331C2×C3×D3#217116yescomplemented
332C32×C4—62no—
333C3⋊Dic3—22no—
334C3⋊Dic3—41no—
335C3⋊Dic3—62no—
336C3⋊Dic3—68no—
337C3⋊Dic3—128no—
338C3⋊Dic3—128no—
339C3⋊Dic3—121no—
340C3⋊Dic3—124no—
341C3×Dic3—28no—
342C3×Dic3—68no—
343C3×Dic3—124no—
344C22×D4—31no—
345C23×C4—271no—
346C2×C22⋊C4—91no—
347C2×C22⋊C4—271no—
348C2×C22⋊C4—271no—
349#95—271no—
350#95—271no—
351C33#38611yescharacteristic, complemented, Sylow
352C23×C3—28no—
353C23×C3—28no—
354C23×C3—24no—
355C23×C3—24no—
356C23×C3—24no—
357C23×C3—24no—
358C23×C3—24no—
359C23×C3—24no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—28no—
367C23×C3—24no—
368C23×C3—38no—
369C23×C3—38no—
370C23×C3—38no—
371C23×C3—44no—
372C23×C3—44no—
373C23×C3—44no—
374C23×C3—44no—
375C23×C3—64no—
376C23×C3—64no—
377C23×C3—64no—
378C23×C3—64no—
379C23×C3D3×Dic318yescomplemented
380C23×C3C2×C3⋊Dic312yescomplemented
381C23×C3C2×D3214yes—
382C23×C3C22×C3⋊D311yescharacteristic
383C22×D3—21no—
384C22×D3—21no—
385C22×D3—21no—
386C22×D3—21no—
387C22×D3C2×C3⋊Dic312yes—
388C22×D3#63812yes—
389C22×D3#63812yes—
390C3×D4—632no—
391C2×C3×C4—38no—
392C2×C3×C4—64no—
393C2×C3×C4—64no—
394C2×C3×C4—92no—
395C2×C3×C4—181no—
396C2×C3×C4—181no—
397C2×C3×C4—181no—
398C2×C3×C4—181no—
399C2×C3×C4—181no—
400#67—28no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—38no—
404C2×Dic3—64no—
405C2×Dic3—64no—
406C2×Dic3—64no—
407C2×Dic3—64no—
408C2×Dic3—64no—
409C2×Dic3—98no—
410C2×Dic3—92no—
411C2×Dic3—188no—
412C2×Dic3—1832no—
413C2×Dic3—184no—
414C2×Dic3—184no—
415C2×Dic3—181no—
416C2×Dic3—181no—
417C2×Dic3—188no—
418C2×Dic3—184no—
419C2×Dic3—184no—
420C2×Dic3—184no—
421C2×Dic3—181no—
422C2×Dic3—181no—
423C2×Dic3—181no—
424C2×Dic3—364no—
425C2×Dic3—364no—
426C2×Dic3—364no—
427C2×Dic3—364no—
428C2×Dic3C2×C3⋊Dic312yescomplemented
429C4×D3—188no—
430C2×C32—24no—
431C2×C32—24no—
432C2×C32—24no—
433C2×C32—24no—
434C2×C32—24no—
435C2×C32—24no—
436C2×C32—24no—
437C2×C32—416no—
438C2×C32—42no—
439C2×C32—48no—
440C2×C32—64no—
441C2×C32D3×C22⋊C411yescharacteristic
442C2×C32#67⋊C411yescharacteristic
443C2×C32#88311yescharacteristic
444C2×C32#88411yescharacteristic
445C2×C32C4×#6711yescharacteristic
446C2×C32D4×Dic314yes—
447C2×C32D4×Dic314yes—
448C2×C32#93514yes—
449C2×C32#93714yes—
450C2×C32#93711yescharacteristic
451C2×C32#93811yescharacteristic
452C2×C32#93914yes—
453C2×C32#93914yes—
454C2×C32C2×#21714yes—
455C3×D3—216no—
456C24—31no—
457C24D3×C3⋊D311yescharacteristic
458C2×D4—34no—
459C2×D4—62no—
460C2×D4—62no—
461C22×C4—31no—
462C22×C4—91no—
463C22×C4—91no—
464C22×C4—271no—
465C22×C4—271no—
466C22×C4—542no—
467C22×C4—542no—
468C22×C4—542no—
469C22⋊C4—182no—
470C22⋊C4—542no—
471C22⋊C4—542no—
472C22×C3—216no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—28no—
477C22×C3—28no—
478C22×C3—28no—
479C22×C3—24no—
480C22×C3—24no—
481C22×C3—24no—
482C22×C3—24no—
483C22×C3—24no—
484C22×C3—24no—
485C22×C3—24no—
486C22×C3—316no—
487C22×C3—48no—
488C22×C3—48no—
489C22×C3—48no—
490C22×C3—48no—
491C22×C3—48no—
492C22×C3—48no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—44no—
498C22×C3—44no—
499C22×C3—44no—
500C22×C3—44no—
501C22×C3—68no—
502C22×C3—68no—
503C22×C3—68no—
504C22×C3—68no—
505C22×C3—68no—
506C22×C3—68no—
507C22×C3C2×D3×Dic314yes—
508C22×C3#380414yes—
509C22×C3#380514yes—
510C22×C3D3×#6714yes—
511C22×C3D3×#6714yes—
512C22×C3D3×#6714yes—
513C22×C3#381114yes—
514C22×C3D4×C3⋊D311yescharacteristic
515C22×C3D4×C3⋊D311yescharacteristic
516C22×C3#383011yescharacteristic
517C22×C3#383011yescharacteristic
518C22×C3C22×C3⋊Dic311yescharacteristic
519C22×C3C2×#63811yescharacteristic
520C22×C3C2×#63811yescharacteristic
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—22no—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—22no—
527C2×D3#375714yescomplemented
528C3×C4—68no—
529C3×C4—182no—
530C3×C4—361no—
531Dic3—22no—
532Dic3—68no—
533Dic3—124no—
534Dic3—182no—
535Dic3—188no—
536Dic3—368no—
537Dic3—368no—
538Dic3—361no—
539Dic3—364no—
540C32—24no—
541C32#495611yescharacteristic, complemented
542C32#522014yescomplemented
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—32no—
550C23—32no—
551C23—32no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23D3×C3⋊Dic312yescomplemented
557C23C2×D3×C3⋊D311yescharacteristic
558D4—68no—
559C2×C4—32no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—92no—
563C2×C4—181no—
564C2×C4—181no—
565C2×C4—181no—
566C2×C4—181no—
567C2×C4—181no—
568C2×C4—272no—
569C2×C4—548no—
570C2×C4—541no—
571C2×C4—541no—
572C2×C4—541no—
573C2×C4—541no—
574C2×C4—541no—
575C2×C3—24no—
576C2×C3—24no—
577C2×C3—24no—
578C2×C3—24no—
579C2×C3—24no—
580C2×C3—24no—
581C2×C3—24no—
582C2×C3—416no—
583C2×C3—42no—
584C2×C3—48no—
585C2×C3—616no—
586C2×C3D3×#21714yes—
587C2×C3#6396914yes—
588C2×C3#6397014yes—
589C2×C3#6397114yes—
590C2×C3Dic3×#6714yes—
591C2×C3#67⋊Dic314yes—
592C2×C3#6398314yes—
593C2×C3D4×C3⋊Dic311yescharacteristic
594C2×C3D4×C3⋊Dic311yescharacteristic
595C2×C3#6414411yescharacteristic
596C2×C3#6414611yescharacteristic
597C2×C3#6414811yescharacteristic
598C2×C3#6414811yescharacteristic
599C2×C3C2×#375711yescharacteristic
600D3—24no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—34no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—62no—
610C22—62no—
611C22—62no—
612C22—62no—
613C22—62no—
614C22—62no—
615C22C2×D3×C3⋊Dic311yescharacteristic
616C22#8902811yescharacteristic
617C22#8902911yescharacteristic
618C22D3×#63811yescharacteristic
619C22D3×#63811yescharacteristic
620C22C3⋊D3×#6711yescharacteristic
621C22#8903911yescharacteristic
622C4—62no—
623C4—182no—
624C4—361no—
625C4—542no—
626C4—1081no—
627C3—24no—
628C3#1059172114yescomplemented
629C3#1059291911yescharacteristic, complemented
630C2—42no—
631C2—64no—
632C2D3×#375711yescharacteristic
633C2#1172361411yescharacteristic
634C2#1172361511yescharacteristic
635C2#1172361611yescharacteristic
636C2#67×C3⋊Dic311yescharacteristic
637C2#67⋊(C3⋊Dic3)11yescharacteristic
638C2#1172364311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC24×C33C22
socle, cosocleC23×C33C23
prime radicalC22×C32×#67C2
hypercenterC24D3×C3⋊D3
hypocenterC33#386
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×D3×C3⋊Dic3C2semidirect
3. C2×#88864C2semidirect
4. C2×C3×#3757C2semidirect
5. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
6. #11722061C2semidirect
7. #11722061C2semidirect
18. C2×C32×#67C4semidirect
34. C2×#3757D3semidirect
47. C23×C33C2×C4semidirect
61. C2×C3⋊(C3⋊Dic3)D4semidirect
67. C2×C3×C3⋊Dic3D4semidirect
71. C2×C32×Dic3C2×C4semidirect
97. C2×C3×#67Dic3semidirect
131. C2×C32×D3C22⋊C4semidirect
179. C23×C32C4×D3semidirect
180. C23×C32C2×Dic3semidirect
218. C2×C3⋊Dic3#67semidirect
228. C2×C3×Dic3C2×Dic3semidirect
253. C2×#67C3⋊Dic3semidirect
331. C2×C3×D3#217semidirect
351. C33#386semidirect
379. C23×C3D3×Dic3semidirect
380. C23×C3C2×C3⋊Dic3semidirect
428. C2×Dic3C2×C3⋊Dic3semidirect
527. C2×D3#3757semidirect
541. C32#4956semidirect
542. C32#5220semidirect
556. C23D3×C3⋊Dic3semidirect
628. C3#10591721semidirect
629. C3#10592919semidirect

Maximal subgroups

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

subgroupindex
C22×C32×#672
C22×D3×C3⋊Dic32
C2×#888642
C2×C3×#37572
C2×(C2×D3)⋊(C3⋊Dic3)2
#117220612
#105929193
#105917213