Finite Group Databasebeta

← #13161455

Subgroups — #13161455

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×C3×#638C211yesmaximal, characteristic
2C22×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×#88864C211yesmaximal, characteristic, complemented
4C2×C32×#217C211yesmaximal, characteristic, complemented
5C2×#88803C211yesmaximal, characteristic, complemented
6#11722061C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592655—31nomaximal
9#10591721—34nomaximal
10C24×C33C2211yescharacteristic
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×#638—22no—
15C2×C3×#638—22no—
16C2×C3×#638C414yescomplemented
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C2×Dic3×C3⋊D3—22no—
21C2×Dic3×C3⋊D3—22no—
22C2×Dic3×C3⋊D3—22no—
23#88864—22no—
24C32×#217—22no—
25#88803—22no—
26C22×#638D311yescharacteristic
27C22×C4×C3⋊D3—31no—
28C22×C3×#67—34no—
29C22×D3×Dic3—34no—
30C2×C32×C22⋊C4—31no—
31C2×#3757—31no—
32C2×#3757—34no—
33C2×#3757—64no—
34C2×#3752—31no—
35C2×C3×#217D314yescomplemented
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×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3—21no—
52C22×C3×C3⋊D3—21no—
53C22×C3×C3⋊D3C2×C412yes—
54C22×C3×C3⋊D3D412yes—
55C22×C3×C3⋊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
62C3×#638—28no—
63C2×C3×C3⋊Dic3—21no—
64C2×C3×C3⋊Dic3—21no—
65C2×C3×C3⋊Dic3C2×C412yescomplemented
66C2×C32×Dic3—21no—
67C2×C32×Dic3—21no—
68C2×C32×Dic3—21no—
69C2×C32×Dic3—21no—
70C2×C32×Dic3—21no—
71C2×C32×Dic3D412yescomplemented
72Dic3×C3⋊D3—28no—
73#5220—91no—
74#4956—94no—
75C24×C32—24no—
76C24×C32C2×D314yes—
77C24×C32C2×D311yescharacteristic
78C23×C3⋊D3C2×D311yescharacteristic
79C23×C3×D3—34no—
80C22×C32×C4—31no—
81C22×C32×C4—31no—
82C2×#638—22no—
83C2×#638—22no—
84C2×#638Dic314yescomplemented
85C22×C3⋊Dic3—31no—
86C22×C3⋊Dic3—34no—
87C22×C3⋊Dic3—31no—
88C22×C3⋊Dic3—34no—
89C22×C3⋊Dic3—64no—
90C22×C3⋊Dic3—64no—
91C22×C3⋊Dic3C2×D311yescharacteristic
92C2×C4×C3⋊D3—62no—
93C2×C4×C3⋊D3—62no—
94C2×C4×C3⋊D3—62no—
95C2×C3×#67—316no—
96C2×C3×#67—68no—
97C2×C3×#67—68no—
98C22×C3×Dic3—34no—
99C22×C3×Dic3C2×D314yes—
100C22×C3×Dic3C2×D314yes—
101C2×D3×Dic3—68no—
102C2×D3×Dic3—68no—
103C2×D3×Dic3—68no—
104C32×C22⋊C4—62no—
105#3757—62no—
106#3757—68no—
107#3757—616no—
108#3752—62no—
109C3×#217—28no—
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×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—22no—
128C2×C3×C3⋊D3—22no—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3C22⋊C414yescomplemented
132C3⋊(C3⋊Dic3)—22no—
133C3⋊(C3⋊Dic3)—41no—
134C3×C3⋊Dic3—22no—
135C32×Dic3—22no—
136C32×Dic3—41no—
137C22×C3×D4—91no—
138C22×#67—34no—
139C23×Dic3—91no—
140C22×C4×D3—94no—
141C2×C3×C22⋊C4—34no—
142C2×#217—94no—
143C2×#217—91no—
144C2×#217—91no—
145C2×#217—184no—
146C2×#217C3⋊D311yescharacteristic, complemented
147C2×(C2×D3)⋊C4—94no—
148C3⋊#95—94no—
149C3⋊#95—94no—
150C3⋊#95—91no—
151C3⋊#95—91no—
152C23×C32—28no—
153C23×C32—28no—
154C23×C32—24no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—24no—
158C23×C32—21no—
159C23×C32—21no—
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—44no—
169C23×C32—44no—
170C23×C32—44no—
171C23×C32—44no—
172C23×C32C4×D318yescomplemented
173C23×C32C2×Dic312yescomplemented
174C23×C32C22×D314yes—
175C23×C32C22×D311yescharacteristic
176C22×C3⋊D3—21no—
177C22×C3⋊D3—21no—
178C22×C3⋊D3—21no—
179C22×C3⋊D3—21no—
180C22×C3⋊D3C2×Dic312yes—
181C22×C3⋊D3#6712yes—
182C22×C3⋊D3#6712yes—
183C22×C3×D3—38no—
184C22×C3×D3—38no—
185C22×C3×D3—38no—
186C22×C3×D3—64no—
187C22×C3×D3—64no—
188C22×C3×D3—64no—
189C22×C3×D3—64no—
190C2×C32×C4—32no—
191C2×C32×C4—61no—
192C2×C32×C4—61no—
193C2×C32×C4—61no—
194C2×C32×C4—61no—
195C2×C32×C4—61no—
196#638—28no—
197C2×C3⋊Dic3—21no—
198C2×C3⋊Dic3—21no—
199C2×C3⋊Dic3—32no—
200C2×C3⋊Dic3—38no—
201C2×C3⋊Dic3—68no—
202C2×C3⋊Dic3—61no—
203C2×C3⋊Dic3—61no—
204C2×C3⋊Dic3—68no—
205C2×C3⋊Dic3—64no—
206C2×C3⋊Dic3—61no—
207C2×C3⋊Dic3—61no—
208C2×C3⋊Dic3—61no—
209C2×C3⋊Dic3—64no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3—64no—
213C2×C3⋊Dic3—124no—
214C2×C3⋊Dic3—124no—
215C2×C3⋊Dic3—124no—
216C2×C3⋊Dic3—124no—
217C2×C3⋊Dic3C2×Dic312yescomplemented
218C4×C3⋊D3—68no—
219C3×#67—632no—
220C2×C3×Dic3—24no—
221C2×C3×Dic3—24no—
222C2×C3×Dic3—24no—
223C2×C3×Dic3—24no—
224C2×C3×Dic3—24no—
225C2×C3×Dic3—38no—
226C2×C3×Dic3—64no—
227C2×C3×Dic3—64no—
228C2×C3×Dic3#6718yescomplemented
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
239C3×C3⋊D3—24no—
240C24×C3—24no—
241C24×C3—91no—
242C24×C3D3214yes—
243C24×C3C2×C3⋊D311yescharacteristic
244C23×D3—34no—
245C2×C3×D4—94no—
246C2×C3×D4—182no—
247C2×C3×D4—182no—
248C22×C3×C4—34no—
249C22×C3×C4—34no—
250C22×C3×C4—91no—
251C2×#67—316no—
252C2×#67—68no—
253C2×#67—68no—
254C22×Dic3—34no—
255C22×Dic3—94no—
256C22×Dic3—91no—
257C22×Dic3—94no—
258C22×Dic3—91no—
259C22×Dic3—184no—
260C22×Dic3—184no—
261C22×Dic3—182no—
262C22×Dic3—182no—
263C22×Dic3—182no—
264C22×Dic3C2×C3⋊D311yescharacteristic
265C22×Dic3C2×C3⋊D311yescharacteristic
266C2×C4×D3—188no—
267C2×C4×D3—188no—
268C2×C4×D3—188no—
269C3×C22⋊C4—68no—
270#217—22no—
271#217—182no—
272#217—188no—
273#217—182no—
274#217—1816no—
275(C2×D3)⋊C4—188no—
276C22×C32—216no—
277C22×C32—28no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—22no—
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—48no—
291C22×C32—48no—
292C22×C32—48no—
293C22×C32—48no—
294C22×C32—48no—
295C22×C32—48no—
296C22×C32—44no—
297C22×C32—44no—
298C22×C32—41no—
299C22×C32—44no—
300C22×C32—41no—
301C22×C32—44no—
302C22×C32—41no—
303C22×C32—41no—
304C22×C32C2×C4×D314yes—
305C22×C32#23514yes—
306C22×C32D3×D414yes—
307C22×C32D3×D414yes—
308C22×C32D3×D414yes—
309C22×C32D3×D411yescharacteristic
310C22×C32D3×D411yescharacteristic
311C22×C32#23714yes—
312C22×C32#23711yescharacteristic
313C22×C32#23711yescharacteristic
314C22×C32C22×Dic311yescharacteristic
315C22×C32C2×#6714yes—
316C22×C32C2×#6711yescharacteristic
317C22×C32C2×#6711yescharacteristic
318C2×C3⋊D3—22no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3—22no—
321C2×C3⋊D3—22no—
322C2×C3⋊D3—22no—
323C2×C3⋊D3—22no—
324C2×C3⋊D3#21714yescomplemented
325C2×C3×D3—316no—
326C2×C3×D3—68no—
327C2×C3×D3—68no—
328C2×C3×D3—68no—
329C2×C3×D3—68no—
330C2×C3×D3—68no—
331C2×C3×D3—68no—
332C32×C4—62no—
333C32×C4—121no—
334C3⋊Dic3—22no—
335C3⋊Dic3—62no—
336C3⋊Dic3—68no—
337C3⋊Dic3—128no—
338C3⋊Dic3—128no—
339C3⋊Dic3—124no—
340C3⋊Dic3—121no—
341C3×Dic3—28no—
342C3×Dic3—44no—
343C3×Dic3—68no—
344C22×D4—91no—
345C23×C4—271no—
346C2×C22⋊C4—31no—
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—44no—
369C23×C3—44no—
370C23×C3—44no—
371C23×C3—44no—
372C23×C3—92no—
373C23×C3—92no—
374C23×C3—92no—
375C23×C3—181no—
376C23×C3—181no—
377C23×C3—181no—
378C23×C3—181no—
379C23×C3D3×Dic318yescomplemented
380C23×C3C4×C3⋊D312yescomplemented
381C23×C3C2×D3214yes—
382C23×C3C22×C3⋊D311yescharacteristic
383C22×D3—38no—
384C22×D3—38no—
385C22×D3—38no—
386C22×D3—64no—
387C22×D3—64no—
388C22×D3—64no—
389C22×D3—64no—
390C3×D4—188no—
391C2×C3×C4—38no—
392C2×C3×C4—64no—
393C2×C3×C4—64no—
394C2×C3×C4—64no—
395C2×C3×C4—64no—
396C2×C3×C4—64no—
397C2×C3×C4—92no—
398C2×C3×C4—181no—
399C2×C3×C4—181no—
400#67—632no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—38no—
407C2×Dic3—64no—
408C2×Dic3—64no—
409C2×Dic3—98no—
410C2×Dic3—92no—
411C2×Dic3—188no—
412C2×Dic3—188no—
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×Dic3#63812yescomplemented
429C4×D3—1832no—
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—48no—
439C2×C32—42no—
440C2×C32D3×C22⋊C414yes—
441C2×C32#67⋊C414yes—
442C2×C32#88314yes—
443C2×C32#88414yes—
444C2×C32C4×#6714yes—
445C2×C32D4×Dic311yescharacteristic
446C2×C32D4×Dic311yescharacteristic
447C2×C32#93511yescharacteristic
448C2×C32#93714yes—
449C2×C32#93711yescharacteristic
450C2×C32#93814yes—
451C2×C32#93911yescharacteristic
452C2×C32#93911yescharacteristic
453C2×C32C2×#21711yescharacteristic
454C3⋊D3—24no—
455C3×D3—616no—
456C24—91no—
457C24D3×C3⋊D311yescharacteristic
458C2×D4—94no—
459C2×D4—182no—
460C2×D4—182no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—91no—
464C22×C4—271no—
465C22×C4—271no—
466C22×C4—542no—
467C22×C4—542no—
468C22×C4—542no—
469C22⋊C4—62no—
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—48no—
487C22×C3—48no—
488C22×C3—48no—
489C22×C3—48no—
490C22×C3—48no—
491C22×C3—48no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—44no—
497C22×C3—44no—
498C22×C3—44no—
499C22×C3—44no—
500C22×C3—94no—
501C22×C3—182no—
502C22×C3—182no—
503C22×C3—182no—
504C22×C3—182no—
505C22×C3—182no—
506C22×C3—182no—
507C22×C3C2×D3×Dic314yes—
508C22×C3#380414yes—
509C22×C3#380514yes—
510C22×C3D3×#6714yes—
511C22×C3D3×#6714yes—
512C22×C3D3×#6714yes—
513C22×C3#381114yes—
514C22×C3C2×C4×C3⋊D311yescharacteristic
515C22×C3#382811yescharacteristic
516C22×C3D4×C3⋊D311yescharacteristic
517C22×C3D4×C3⋊D311yescharacteristic
518C22×C3D4×C3⋊D311yescharacteristic
519C22×C3#383011yescharacteristic
520C22×C3C2×#63811yescharacteristic
521C2×D3—316no—
522C2×D3—68no—
523C2×D3—68no—
524C2×D3—68no—
525C2×D3—68no—
526C2×D3—68no—
527C2×D3—68no—
528C3×C4—68no—
529C3×C4—124no—
530C3×C4—182no—
531Dic3—22no—
532Dic3—41no—
533Dic3—68no—
534Dic3—182no—
535Dic3—188no—
536Dic3—368no—
537Dic3—368no—
538Dic3—361no—
539Dic3—364no—
540C32—24no—
541C32#495614yescomplemented
542C32#522011yescharacteristic, complemented
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—92no—
550C23—92no—
551C23—92no—
552C23—181no—
553C23—181no—
554C23—181no—
555C23—181no—
556C23Dic3×C3⋊D312yescomplemented
557C23C2×D3×C3⋊D311yescharacteristic
558D4—188no—
559C2×C4—32no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—92no—
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—184no—
586C2×C3D3×#21714yes—
587C2×C3#6396914yes—
588C2×C3#6397014yes—
589C2×C3#6397114yes—
590C2×C3Dic3×#6714yes—
591C2×C3#67⋊Dic314yes—
592C2×C3#6398314yes—
593C2×C3C22⋊C4×C3⋊D311yescharacteristic
594C2×C3#6409011yescharacteristic
595C2×C3#6409211yescharacteristic
596C2×C3#6409311yescharacteristic
597C2×C3C4×#63811yescharacteristic
598C2×C3#6414611yescharacteristic
599C2×C3#6414711yescharacteristic
600D3—616no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—94no—
609C22—182no—
610C22—182no—
611C22—182no—
612C22—182no—
613C22—182no—
614C22—182no—
615C22#8902911yescharacteristic
616C22C2×Dic3×C3⋊D311yescharacteristic
617C22#8903111yescharacteristic
618C22D3×#63811yescharacteristic
619C22C3⋊D3×#6711yescharacteristic
620C22C3⋊D3×#6711yescharacteristic
621C22#8903911yescharacteristic
622C4—62no—
623C4—121no—
624C4—182no—
625C4—542no—
626C4—1081no—
627C3—24no—
628C3#1059172114yescomplemented
629C3#1059265511yescharacteristic, complemented
630C2—42no—
631C2—184no—
632C2Dic3×#63811yescharacteristic
633C2C3⋊D3×#21711yescharacteristic
634C2#1172362111yescharacteristic
635C2#1172362411yescharacteristic
636C2#1172362611yescharacteristic
637C2#1172363111yescharacteristic
638C2#1172364311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC24×C33C22
socle, cosocleC23×C33C23
prime radicalC22×C3×#638C2
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×Dic3×C3⋊D3C2semidirect
3. C2×#88864C2semidirect
4. C2×C32×#217C2semidirect
5. C2×#88803C2semidirect
6. #11722061C2semidirect
7. #11722061C2semidirect
16. C2×C3×#638C4semidirect
35. C2×C3×#217D3semidirect
47. C23×C33C2×C4semidirect
61. C2×C3⋊(C3⋊Dic3)D4semidirect
65. C2×C3×C3⋊Dic3C2×C4semidirect
71. C2×C32×Dic3D4semidirect
84. C2×#638Dic3semidirect
131. C2×C3×C3⋊D3C22⋊C4semidirect
146. C2×#217C3⋊D3semidirect
172. C23×C32C4×D3semidirect
173. C23×C32C2×Dic3semidirect
217. C2×C3⋊Dic3C2×Dic3semidirect
228. C2×C3×Dic3#67semidirect
324. C2×C3⋊D3#217semidirect
351. C33#386semidirect
379. C23×C3D3×Dic3semidirect
380. C23×C3C4×C3⋊D3semidirect
428. C2×Dic3#638semidirect
541. C32#4956semidirect
542. C32#5220semidirect
556. C23Dic3×C3⋊D3semidirect
628. C3#10591721semidirect
629. C3#10592655semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C22×Dic3×C3⋊D32
C2×#888642
C2×C32×#2172
C2×#888032
#117220612
#105926553
#105917213