Finite Group Databasebeta

← #13141138

Subgroups — #13141138

Order 1728.

990 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×#611C211yesmaximal, characteristic
2C22×Dic3×D9C211yesmaximal, characteristic, complemented
3C2×#88551C211yesmaximal, characteristic, complemented
4C2×C9×#217C211yesmaximal, characteristic, complemented
5C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
6#11720331C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591721—31nomaximal
9#10588636—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C23×C3×D9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C2×C3×#611—22no—
15C2×C3×#611—22no—
16C2×C3×#611C414yescomplemented
17C22×C9×Dic3C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×Dic3×D9—22no—
21C2×Dic3×D9—22no—
22C2×Dic3×D9—22no—
23#88551—22no—
24C9×#217—22no—
25(C2×D9)⋊Dic3—22no—
26C22×C3×#67—31no—
27C22×D3×Dic3—31no—
28C2×#3757—31no—
29C2×C3×#217D311yescharacteristic
30C2×(C2×D3)⋊Dic3—31no—
31C22×#611D311yescharacteristic
32C22×C4×D9—31no—
33#63579—31no—
34#63579—31no—
35C2×C9×C22⋊C4—31no—
36C2×#3676—31no—
37C2×#3676—61no—
38C2×(C2×D9)⋊C4—31no—
39C9⋊#95—31no—
40C9⋊#95—31no—
41C23×C3×C9—21no—
42C23×C3×C9—21no—
43C23×C3×C9—21no—
44C23×C3×C9—21no—
45C23×C3×C9—21no—
46C23×C3×C9—21no—
47C23×C3×C9C2×C412yescomplemented
48C23×C3×C9C2311yescharacteristic
49C22×C3×D9—21no—
50C22×C3×D9—21no—
51C22×C3×D9—21no—
52C22×C3×D9—21no—
53C22×C3×D9C2×C412yes—
54C22×C3×D9D412yes—
55C22×C3×D9D412yes—
56C2×C3⋊Dic9—21no—
57C2×C3⋊Dic9—21no—
58C2×C3⋊Dic9—21no—
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9—21no—
61C2×C3⋊Dic9D412yescomplemented
62C3×#611—28no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3—21no—
65C2×C9×Dic3—21no—
66C2×C9×Dic3—21no—
67C2×C9×Dic3—21no—
68C2×C9×Dic3D412yescomplemented
69C2×C3×Dic9—21no—
70C2×C3×Dic9—21no—
71C2×C3×Dic9C2×C412yescomplemented
72Dic3×D9—28no—
73#5220—91no—
74#4956—91no—
75C24×C32C2×D311yescharacteristic
76C23×C3×D3—31no—
77C22×C3⋊Dic3—31no—
78C22×C3⋊Dic3—31no—
79C2×C3×#67—34no—
80C2×C3×#67—62no—
81C2×C3×#67—62no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3C2×D311yescharacteristic
84C22×C3×Dic3C2×D311yescharacteristic
85C2×D3×Dic3—62no—
86C2×D3×Dic3—62no—
87C2×D3×Dic3—62no—
88C24×C9—21no—
89C24×C9C2×D311yescharacteristic
90C23×D9C2×D311yescharacteristic
91#3757—62no—
92C3×#217—22no—
93(C2×D3)⋊Dic3—62no—
94C22×C4×C9—31no—
95C22×C4×C9—31no—
96C2×#611—22no—
97C2×#611—22no—
98C2×#611Dic314yescomplemented
99C22×Dic9—31no—
100C22×Dic9—31no—
101C22×Dic9—61no—
102C22×Dic9—61no—
103C22×Dic9C2×D311yescharacteristic
104C2×C4×D9—62no—
105C2×C4×D9—62no—
106C2×C4×D9—62no—
107C9×C22⋊C4—62no—
108#3676—62no—
109#3676—64no—
110(C2×D9)⋊C4—62no—
111C22×C3×C9—22no—
112C22×C3×C9—22no—
113C22×C3×C9—22no—
114C22×C3×C9—41no—
115C22×C3×C9—41no—
116C22×C3×C9—41no—
117C22×C3×C9—41no—
118C22×C3×C9C22×C411yescharacteristic
119C22×C3×C9C2×D411yescharacteristic
120C22×C3×C9C2×D411yescharacteristic
121C22×C3×C9C2×D411yescharacteristic
122C22×C3×C9C2×D411yescharacteristic
123C22×C3×C9#4111yescharacteristic
124C22×C3×C9#4111yescharacteristic
125C2×C3×D9—22no—
126C2×C3×D9—22no—
127C2×C3×D9—22no—
128C2×C3×D9—22no—
129C2×C3×D9—22no—
130C2×C3×D9—22no—
131C2×C3×D9C22⋊C414yescomplemented
132C3⋊Dic9—22no—
133C3⋊Dic9—41no—
134C9×Dic3—22no—
135C9×Dic3—41no—
136C3×Dic9—22no—
137C22×C3×D4—91no—
138C22×#67—31no—
139C23×Dic3—91no—
140C22×C4×D3—91no—
141C2×C3×C22⋊C4—31no—
142C2×#217—91no—
143C2×#217—91no—
144C2×#217—91no—
145C2×#217—181no—
146C2×#217D911yescharacteristic, complemented
147C2×(C2×D3)⋊C4—91no—
148C3⋊#95—91no—
149C3⋊#95—91no—
150C3⋊#95—91no—
151C3⋊#95—91no—
152C23×C32—21no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32C4×D312yes—
159C23×C32C22×D311yescharacteristic
160C22×C3×D3—32no—
161C22×C3×D3—32no—
162C22×C3×D3—32no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C3×#67—68no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3#6712yes—
183D3×Dic3—68no—
184C23×C9—22no—
185C23×C9—22no—
186C23×C9—21no—
187C23×C9—21no—
188C23×C9—21no—
189C23×C9—21no—
190C23×C9—21no—
191C23×C9—21no—
192C23×C9—22no—
193C23×C9—21no—
194C23×C9—41no—
195C23×C9—41no—
196C23×C9—41no—
197C23×C9—41no—
198C23×C9C2×Dic312yescomplemented
199C23×C9C22×D311yescharacteristic
200C22×D9—21no—
201C22×D9—21no—
202C22×D9—21no—
203C22×D9—21no—
204C22×D9C2×Dic312yes—
205C22×D9#6712yes—
206C22×D9#6712yes—
207C2×C4×C9—32no—
208C2×C4×C9—61no—
209C2×C4×C9—61no—
210C2×C4×C9—61no—
211C2×C4×C9—61no—
212C2×C4×C9—61no—
213#611—28no—
214C2×Dic9—21no—
215C2×Dic9—21no—
216C2×Dic9—32no—
217C2×Dic9—62no—
218C2×Dic9—61no—
219C2×Dic9—61no—
220C2×Dic9—62no—
221C2×Dic9—61no—
222C2×Dic9—61no—
223C2×Dic9—61no—
224C2×Dic9—121no—
225C2×Dic9—121no—
226C2×Dic9—121no—
227C2×Dic9—121no—
228C2×Dic9C2×Dic312yescomplemented
229C4×D9—68no—
230#386—271noSylow
231C2×C3×C9—42no—
232C2×C3×C9C2×C22⋊C411yescharacteristic
233C2×C3×C9C4×D411yescharacteristic
234C2×C3×C9C4×D411yescharacteristic
235C2×C3×C9#12011yescharacteristic
236C2×C3×C9#12111yescharacteristic
237C2×C3×C9#12111yescharacteristic
238C2×C3×C9#12311yescharacteristic
239C3×D9—24no—
240C24×C3—21no—
241C24×C3—91no—
242C24×C3C2×D911yescharacteristic
243C24×C3D3211yescharacteristic
244C23×D3—31no—
245C2×C3×D4—94no—
246C2×C3×D4—182no—
247C2×C3×D4—182no—
248C22×C3×C4—31no—
249C22×C3×C4—31no—
250C22×C3×C4—91no—
251C2×#67—34no—
252C2×#67—62no—
253C2×#67—62no—
254C22×Dic3—31no—
255C22×Dic3—91no—
256C22×Dic3—91no—
257C22×Dic3—91no—
258C22×Dic3—91no—
259C22×Dic3—181no—
260C22×Dic3—181no—
261C22×Dic3—182no—
262C22×Dic3—182no—
263C22×Dic3—182no—
264C22×Dic3C2×D911yescharacteristic
265C22×Dic3C2×D911yescharacteristic
266C2×C4×D3—182no—
267C2×C4×D3—182no—
268C2×C4×D3—182no—
269C3×C22⋊C4—62no—
270#217—22no—
271#217—182no—
272#217—182no—
273#217—182no—
274#217—184no—
275(C2×D3)⋊C4—182no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32C2×C4×D311yescharacteristic
284C22×C32#23511yescharacteristic
285C22×C32D3×D411yescharacteristic
286C22×C32D3×D411yescharacteristic
287C22×C32D3×D411yescharacteristic
288C22×C32#23711yescharacteristic
289C22×C32C2×#6711yescharacteristic
290C2×C3×D3—34no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C3⋊Dic3—62no—
298C3⋊Dic3—121no—
299C3×Dic3—22no—
300C3×Dic3—41no—
301C3×Dic3—62no—
302C22×C9—24no—
303C22×C9—22no—
304C22×C9—22no—
305C22×C9—22no—
306C22×C9—21no—
307C22×C9—21no—
308C22×C9—21no—
309C22×C9—21no—
310C22×C9—21no—
311C22×C9—21no—
312C22×C9—21no—
313C22×C9—42no—
314C22×C9—42no—
315C22×C9—42no—
316C22×C9—42no—
317C22×C9—42no—
318C22×C9—42no—
319C22×C9—41no—
320C22×C9—41no—
321C22×C9—41no—
322C22×C9—41no—
323C22×C9D3×D411yescharacteristic
324C22×C9D3×D411yescharacteristic
325C22×C9#23711yescharacteristic
326C22×C9#23711yescharacteristic
327C22×C9C22×Dic311yescharacteristic
328C22×C9C2×#6711yescharacteristic
329C22×C9C2×#6711yescharacteristic
330C2×D9—22no—
331C2×D9—22no—
332C2×D9—22no—
333C2×D9—22no—
334C2×D9—22no—
335C2×D9—22no—
336C2×D9#21714yescomplemented
337C4×C9—62no—
338C4×C9—121no—
339Dic9—22no—
340Dic9—62no—
341Dic9—122no—
342Dic9—122no—
343Dic9—121no—
344C22×D4—91no—
345C23×C4—271no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—271no—
348C2×C22⋊C4—271no—
349#95—271no—
350#95—271no—
351C3×C9#38611yescharacteristic, complemented, Sylow
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—22no—
367C23×C3—21no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—92no—
373C23×C3—92no—
374C23×C3—92no—
375C23×C3—181no—
376C23×C3—181no—
377C23×C3—181no—
378C23×C3—181no—
379C23×C3C4×D912yescomplemented
380C23×C3C22×D911yescharacteristic
381C23×C3D3×Dic312yes—
382C23×C3C2×D3211yescharacteristic
383C22×D3—32no—
384C22×D3—32no—
385C22×D3—32no—
386C22×D3—61no—
387C22×D3—61no—
388C22×D3—61no—
389C22×D3—61no—
390C3×D4—188no—
391C2×C3×C4—32no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—61no—
397C2×C3×C4—92no—
398C2×C3×C4—181no—
399C2×C3×C4—181no—
400#67—68no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—32no—
407C2×Dic3—61no—
408C2×Dic3—61no—
409C2×Dic3—92no—
410C2×Dic3—92no—
411C2×Dic3—182no—
412C2×Dic3—188no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—181no—
416C2×Dic3—181no—
417C2×Dic3—182no—
418C2×Dic3—181no—
419C2×Dic3—181no—
420C2×Dic3—181no—
421C2×Dic3—181no—
422C2×Dic3—181no—
423C2×Dic3—181no—
424C2×Dic3—361no—
425C2×Dic3—361no—
426C2×Dic3—361no—
427C2×Dic3—361no—
428C2×Dic3#61112yescomplemented
429C4×D3—188no—
430C2×C32—42no—
431C2×C32D3×C22⋊C411yescharacteristic
432C2×C32#67⋊C411yescharacteristic
433C2×C32#88311yescharacteristic
434C2×C32#88411yescharacteristic
435C2×C32C4×#6711yescharacteristic
436C2×C32#93711yescharacteristic
437C2×C32#93811yescharacteristic
438C3×D3—64no—
439C2×C9—21no—
440C2×C9—21no—
441C2×C9—21no—
442C2×C9—21no—
443C2×C9—21no—
444C2×C9—21no—
445C2×C9—21no—
446C2×C9—44no—
447C2×C9—42no—
448C2×C9D4×Dic311yescharacteristic
449C2×C9D4×Dic311yescharacteristic
450C2×C9#93511yescharacteristic
451C2×C9#93711yescharacteristic
452C2×C9#93911yescharacteristic
453C2×C9#93911yescharacteristic
454C2×C9C2×#21711yescharacteristic
455D9—24no—
456C24—91no—
457C24D3×D911yescharacteristic
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—24no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—94no—
501C22×C3—182no—
502C22×C3—182no—
503C22×C3—182no—
504C22×C3—182no—
505C22×C3—182no—
506C22×C3—182no—
507C22×C3C2×C4×D911yescharacteristic
508C22×C3#369711yescharacteristic
509C22×C3D4×D911yescharacteristic
510C22×C3D4×D911yescharacteristic
511C22×C3D4×D911yescharacteristic
512C22×C3#369911yescharacteristic
513C22×C3C2×#61111yescharacteristic
514C22×C3C2×D3×Dic311yescharacteristic
515C22×C3#380411yescharacteristic
516C22×C3#380511yescharacteristic
517C22×C3D3×#6711yescharacteristic
518C22×C3D3×#6711yescharacteristic
519C22×C3D3×#6711yescharacteristic
520C22×C3#381111yescharacteristic
521C2×D3—34no—
522C2×D3—62no—
523C2×D3—62no—
524C2×D3—62no—
525C2×D3—62no—
526C2×D3—62no—
527C2×D3—62no—
528C3×C4—62no—
529C3×C4—121no—
530C3×C4—182no—
531Dic3—22no—
532Dic3—41no—
533Dic3—62no—
534Dic3—182no—
535Dic3—182no—
536Dic3—362no—
537Dic3—362no—
538Dic3—361no—
539Dic3—361no—
540C32#495611yescharacteristic
541C9—21no—
542C9#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×D912yescomplemented
557C23C2×D3×D911yescharacteristic
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—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—44no—
583C2×C3—42no—
584C2×C3—42no—
585C2×C3—184no—
586C2×C3C22⋊C4×D911yescharacteristic
587C2×C3#611⋊C411yescharacteristic
588C2×C3#6344511yescharacteristic
589C2×C3#6344611yescharacteristic
590C2×C3C4×#61111yescharacteristic
591C2×C3#6349911yescharacteristic
592C2×C3#6350011yescharacteristic
593C2×C3D3×#21711yescharacteristic
594C2×C3#6396911yescharacteristic
595C2×C3#6397011yescharacteristic
596C2×C3#6397111yescharacteristic
597C2×C3Dic3×#6711yescharacteristic
598C2×C3#67⋊Dic311yescharacteristic
599C2×C3#6398311yescharacteristic
600D3—64no—
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—
615C22C2×Dic3×D911yescharacteristic
616C22#8865811yescharacteristic
617C22#8866311yescharacteristic
618C22D3×#61111yescharacteristic
619C22D9×#6711yescharacteristic
620C22D9×#6711yescharacteristic
621C22#8866811yescharacteristic
622C4—62no—
623C4—121no—
624C4—182no—
625C4—542no—
626C4—1081no—
627C3—21no—
628C3#1058863611yescharacteristic, complemented
629C3#1059172111yescharacteristic
630C2—42no—
631C2—184no—
632C2D9×#21711yescharacteristic
633C2#1172126911yescharacteristic
634C2#1172127011yescharacteristic
635C2#1172127111yescharacteristic
636C2Dic3×#61111yescharacteristic
637C2#611⋊Dic311yescharacteristic
638C2#1172129211yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×C3×#611C2
cosocleC23×C3×C9C23
hypercenterC24D3×D9
hypocenterC3×C9#386
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×Dic3×D9C2semidirect
3. C2×#88551C2semidirect
4. C2×C9×#217C2semidirect
5. C2×(C2×D9)⋊Dic3C2semidirect
6. #11720331C2semidirect
7. #11720331C2semidirect
16. C2×C3×#611C4semidirect
47. C23×C3×C9C2×C4semidirect
61. C2×C3⋊Dic9D4semidirect
68. C2×C9×Dic3D4semidirect
71. C2×C3×Dic9C2×C4semidirect
98. C2×#611Dic3semidirect
131. C2×C3×D9C22⋊C4semidirect
146. C2×#217D9semidirect
198. C23×C9C2×Dic3semidirect
228. C2×Dic9C2×Dic3semidirect
336. C2×D9#217semidirect
351. C3×C9#386semidirect
379. C23×C3C4×D9semidirect
428. C2×Dic3#611semidirect
542. C9#5220semidirect
556. C23Dic3×D9semidirect
628. C3#10588636semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6112
C22×Dic3×D92
C2×#885512
C2×C9×#2172
C2×(C2×D9)⋊Dic32
#117203312
#105917213
#105886363