Finite Group Databasebeta

← #13141127

Subgroups — #13141127

Order 1728.

1010 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6233C211yesmaximal, characteristic
2C2×C9×#217C211yesmaximal, characteristic, complemented
3C2×C3×#3676C211yesmaximal, characteristic, complemented
4C2×#88445C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6C2×#88442C211yesmaximal, characteristic, complemented
7C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
8#10591704—31nomaximal
9#10588637—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C23×C3⋊D9C2211yescharacteristic
12C2×#6233—22no—
13C2×#6233—22no—
14C2×#6233C414yescomplemented
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C9×#217—22no—
21C3×#3676—22no—
22#88445—22no—
23#88445—22no—
24#88442—22no—
25Dic3×Dic9—22no—
26C22×#638—31no—
27C2×C3×#217—31no—
28C2×C3×#217D311yescharacteristic
29C2×#3724—31no—
30C2×#3722—31no—
31C2×#3722—31no—
32C2×Dic32—31no—
33C22×#611—31no—
34C22×#611—61no—
35C2×C9×C22⋊C4—31no—
36C2×#3676D311yescharacteristic, complemented
37C2×(C2×D9)⋊C4—31no—
38C2×(C2×D9)⋊C4—31no—
39C2×Dic9⋊C4—31no—
40C2×C4×Dic9—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×C412yes—
48C23×C3×C9C2311yescharacteristic
49C22×C3⋊D9—21no—
50C22×C3⋊D9—21no—
51C22×C3⋊D9—21no—
52C22×C3⋊D9—21no—
53C22×C3⋊D9—21no—
54C22×C3⋊D9—21no—
55C22×C3⋊D9C2×C412yes—
56#6233—28no—
57C2×C3⋊Dic9C2×C412yes—
58C2×C3⋊Dic9D412yes—
59C2×C3⋊Dic9D412yes—
60C2×C9×Dic3—21no—
61C2×C9×Dic3—21no—
62C2×C9×Dic3—21no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3—21no—
65C2×C9×Dic3D412yescomplemented
66C2×C3×Dic9—21no—
67C2×C3×Dic9—21no—
68C2×C3×Dic9—21no—
69C2×C3×Dic9—21no—
70C2×C3×Dic9—21no—
71C2×C3×Dic9D412yescomplemented
72#4957—91no—
73#4957—91no—
74C24×C32C2×D311yescharacteristic
75C23×C3⋊D3—31no—
76C2×#638—34no—
77C2×#638—62no—
78C2×#638—62no—
79C22×C3⋊Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3C2×D311yescharacteristic
83C22×C3×Dic3C2×D311yescharacteristic
84C24×C9—21no—
85C24×C9C2×D311yescharacteristic
86C23×D9—31no—
87C23×D9—61no—
88C3×#217—22no—
89C3×#217—62no—
90#3724—62no—
91#3722—62no—
92#3722—62no—
93Dic32—62no—
94C22×C4×C9—31no—
95C22×C4×C9—31no—
96C2×#611—34no—
97C2×#611—62no—
98C2×#611—62no—
99C2×#611—64no—
100C2×#611—64no—
101C2×#611—64no—
102C22×Dic9—31no—
103C22×Dic9—61no—
104C22×Dic9C2×D311yescharacteristic
105C22×Dic9C2×D311yescharacteristic
106C9×C22⋊C4—62no—
107#3676—22no—
108(C2×D9)⋊C4—62no—
109(C2×D9)⋊C4—62no—
110Dic9⋊C4—62no—
111C4×Dic9—62no—
112C22×C3×C9—22no—
113C22×C3×C9—22no—
114C22×C3×C9—22no—
115C22×C3×C9—41no—
116C22×C3×C9—41no—
117C22×C3×C9—41no—
118C22×C3×C9—41no—
119C22×C3×C9C22×C411yescharacteristic
120C22×C3×C9C2×D411yescharacteristic
121C22×C3×C9C2×D411yescharacteristic
122C22×C3×C9C2×D411yescharacteristic
123C22×C3×C9C2×D411yescharacteristic
124C22×C3×C9#4111yescharacteristic
125C22×C3×C9#4111yescharacteristic
126C2×C3⋊D9—22no—
127C2×C3⋊D9—22no—
128C2×C3⋊D9—22no—
129C2×C3⋊D9—41no—
130C2×C3⋊D9—41no—
131C2×C3⋊D9—41no—
132C2×C3⋊D9—41no—
133C3⋊Dic9C22⋊C414yescomplemented
134C9×Dic3—22no—
135C9×Dic3—41no—
136C3×Dic9—22no—
137C3×Dic9—41no—
138C22×#67—91no—
139C22×#67—91no—
140C22×#67—181no—
141C2×C3×C22⋊C4—31no—
142C2×C3×C22⋊C4—91no—
143C2×#217—31no—
144C2×#217D911yescharacteristic, complemented
145C2×(C2×D3)⋊C4—91no—
146C2×(C2×D3)⋊C4—91no—
147C2×(C2×D3)⋊C4—91no—
148C2×(C2×D3)⋊C4—91no—
149C2×Dic3⋊C4—91no—
150C2×Dic3⋊C4—91no—
151C2×C4×Dic3—91no—
152C2×C4×Dic3—91no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32C4×D312yes—
160C23×C32C22×D311yescharacteristic
161C22×C3⋊D3—32no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—61no—
164C22×C3⋊D3—61no—
165C22×C3⋊D3—61no—
166C22×C3⋊D3—61no—
167C22×C3⋊D3—61no—
168#638—68no—
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—32no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3#6712yes—
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×C9C4×D312yes—
199C23×C9C22×D311yescharacteristic
200C22×D9—32no—
201C22×D9—62no—
202C22×D9—62no—
203C22×D9—62no—
204C22×D9—61no—
205C22×D9—61no—
206C22×D9—61no—
207C22×D9—61no—
208C22×D9—61no—
209C22×D9—61no—
210C22×D9—121no—
211C22×D9—121no—
212C22×D9—121no—
213C22×D9—121no—
214C2×C4×C9—32no—
215C2×C4×C9—61no—
216C2×C4×C9—61no—
217C2×C4×C9—61no—
218C2×C4×C9—61no—
219C2×C4×C9—61no—
220#611—68no—
221#611—616no—
222C2×Dic9—21no—
223C2×Dic9—21no—
224C2×Dic9—21no—
225C2×Dic9—21no—
226C2×Dic9—21no—
227C2×Dic9—32no—
228C2×Dic9—32no—
229C2×Dic9—32no—
230C2×Dic9—62no—
231C2×Dic9—62no—
232C2×Dic9—62no—
233C2×Dic9#6712yescomplemented
234#390—271noSylow
235C2×C3×C9—42no—
236C2×C3×C9C2×C22⋊C411yescharacteristic
237C2×C3×C9C4×D411yescharacteristic
238C2×C3×C9C4×D411yescharacteristic
239C2×C3×C9#12111yescharacteristic
240C2×C3×C9#12111yescharacteristic
241C2×C3×C9#12411yescharacteristic
242C2×C3×C9#12711yescharacteristic
243C3⋊D9—42no—
244C24×C3—21no—
245C24×C3C2×D911yescharacteristic
246C24×C3D3211yescharacteristic
247C23×D3—91no—
248C23×D3—91no—
249C23×D3—181no—
250C22×C3×C4—31no—
251C22×C3×C4—31no—
252C22×C3×C4—91no—
253C22×C3×C4—91no—
254C2×#67—94no—
255C2×#67—94no—
256C2×#67—184no—
257C2×#67—182no—
258C2×#67—182no—
259C2×#67—182no—
260C2×#67—182no—
261C2×#67—184no—
262C2×#67—184no—
263C22×Dic3—31no—
264C22×Dic3—31no—
265C22×Dic3—91no—
266C22×Dic3—91no—
267C22×Dic3—181no—
268C22×Dic3C2×D911yescharacteristic
269C22×Dic3C2×D911yescharacteristic
270C3×C22⋊C4—62no—
271C3×C22⋊C4—182no—
272#217—22no—
273#217—62no—
274(C2×D3)⋊C4—182no—
275(C2×D3)⋊C4—182no—
276(C2×D3)⋊C4—182no—
277(C2×D3)⋊C4—182no—
278Dic3⋊C4—182no—
279Dic3⋊C4—182no—
280C4×Dic3—182no—
281C4×Dic3—182no—
282C22×C32—22no—
283C22×C32—22no—
284C22×C32—22no—
285C22×C32—41no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—41no—
289C22×C32C2×C4×D311yescharacteristic
290C22×C32#23511yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32D3×D411yescharacteristic
293C22×C32D3×D411yescharacteristic
294C22×C32#23711yescharacteristic
295C22×C32C2×#6711yescharacteristic
296C2×C3⋊D3—62no—
297C2×C3⋊D3—62no—
298C2×C3⋊D3—62no—
299C2×C3⋊D3—121no—
300C2×C3⋊D3—121no—
301C2×C3⋊D3—121no—
302C2×C3⋊D3—121no—
303C3⋊Dic3—34no—
304C3×Dic3—22no—
305C3×Dic3—41no—
306C3×Dic3—62no—
307C3×Dic3—121no—
308C22×C9—24no—
309C22×C9—22no—
310C22×C9—22no—
311C22×C9—22no—
312C22×C9—21no—
313C22×C9—21no—
314C22×C9—21no—
315C22×C9—21no—
316C22×C9—21no—
317C22×C9—21no—
318C22×C9—21no—
319C22×C9—42no—
320C22×C9—42no—
321C22×C9—42no—
322C22×C9—42no—
323C22×C9—42no—
324C22×C9—42no—
325C22×C9—41no—
326C22×C9—41no—
327C22×C9—41no—
328C22×C9—41no—
329C22×C9C2×C4×D311yescharacteristic
330C22×C9#23511yescharacteristic
331C22×C9D3×D411yescharacteristic
332C22×C9D3×D411yescharacteristic
333C22×C9D3×D411yescharacteristic
334C22×C9#23711yescharacteristic
335C22×C9C2×#6711yescharacteristic
336C2×D9—64no—
337C2×D9—62no—
338C2×D9—62no—
339C2×D9—62no—
340C2×D9—122no—
341C2×D9—122no—
342C2×D9—122no—
343C2×D9—122no—
344C2×D9—122no—
345C2×D9—122no—
346C2×D9—121no—
347C2×D9—121no—
348C2×D9—121no—
349C2×D9—121no—
350C4×C9—62no—
351C4×C9—121no—
352Dic9—22no—
353Dic9—34no—
354Dic9—41no—
355Dic9—64no—
356C22×D4—271no—
357C2×C4⋊C4—271no—
358C2×C22⋊C4—31no—
359C2×C22⋊C4—91no—
360C2×C22⋊C4—271no—
361C2×C22⋊C4—271no—
362C2×C42—271no—
363C3×C9#39011yescharacteristic, complemented, Sylow
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—22no—
379C23×C3—21no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3C4×D912yes—
385C23×C3C22×D911yescharacteristic
386C23×C3#62412yes—
387C23×C3C2×D3211yescharacteristic
388C22×D3—92no—
389C22×D3—92no—
390C22×D3—182no—
391C22×D3—182no—
392C22×D3—182no—
393C22×D3—181no—
394C22×D3—181no—
395C22×D3—181no—
396C22×D3—181no—
397C22×D3—181no—
398C22×D3—181no—
399C22×D3—181no—
400C22×D3—181no—
401C22×D3—181no—
402C22×D3—181no—
403C22×D3—181no—
404C22×D3—181no—
405C22×D3—361no—
406C22×D3—361no—
407C22×D3—361no—
408C22×D3—361no—
409C2×C3×C4—32no—
410C2×C3×C4—61no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4—61no—
414C2×C3×C4—61no—
415C2×C3×C4—92no—
416C2×C3×C4—181no—
417C2×C3×C4—181no—
418C2×C3×C4—181no—
419C2×C3×C4—181no—
420C2×C3×C4—181no—
421#67—188no—
422#67—188no—
423#67—1816no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—32no—
430C2×Dic3—61no—
431C2×Dic3—61no—
432C2×Dic3—61no—
433C2×Dic3—61no—
434C2×Dic3—61no—
435C2×Dic3—92no—
436C2×Dic3—92no—
437C2×Dic3—92no—
438C2×Dic3—92no—
439C2×Dic3—92no—
440C2×Dic3—92no—
441C2×Dic3—182no—
442C2×Dic3—182no—
443C2×Dic3—182no—
444C2×Dic3#61112yescomplemented
445C2×C32—42no—
446C2×C32D3×C22⋊C411yescharacteristic
447C2×C32#67⋊C411yescharacteristic
448C2×C32#88411yescharacteristic
449C2×C32#88511yescharacteristic
450C2×C32C4×#6711yescharacteristic
451C2×C32#93911yescharacteristic
452C2×C32#94011yescharacteristic
453C3⋊D3—122no—
454C2×C9—21no—
455C2×C9—21no—
456C2×C9—21no—
457C2×C9—21no—
458C2×C9—21no—
459C2×C9—21no—
460C2×C9—21no—
461C2×C9—44no—
462C2×C9—42no—
463C2×C9D3×C22⋊C411yescharacteristic
464C2×C9#67⋊C411yescharacteristic
465C2×C9#88411yescharacteristic
466C2×C9#88511yescharacteristic
467C2×C9C4×#6711yescharacteristic
468C2×C9#93911yescharacteristic
469C2×C9#94011yescharacteristic
470D9—124no—
471D9—122no—
472C24—271no—
473C24D3×D911yescharacteristic
474C2×D4—274no—
475C2×D4—542no—
476C2×D4—542no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—91no—
480C22×C4—91no—
481C22×C4—271no—
482C4⋊C4—542no—
483C22⋊C4—62no—
484C22⋊C4—182no—
485C22⋊C4—542no—
486C22⋊C4—542no—
487C42—542no—
488C22×C3—24no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3C2×C4×D911yescharacteristic
517C22×C3#369711yescharacteristic
518C22×C3D4×D911yescharacteristic
519C22×C3D4×D911yescharacteristic
520C22×C3D4×D911yescharacteristic
521C22×C3#369911yescharacteristic
522C22×C3C2×#61111yescharacteristic
523C22×C3#380411yescharacteristic
524C22×C3#380411yescharacteristic
525C22×C3C2×#62411yescharacteristic
526C22×C3D3×#6711yescharacteristic
527C22×C3D3×#6711yescharacteristic
528C22×C3#381111yescharacteristic
529C22×C3#381111yescharacteristic
530C2×D3—184no—
531C2×D3—182no—
532C2×D3—182no—
533C2×D3—182no—
534C2×D3—182no—
535C2×D3—182no—
536C2×D3—182no—
537C2×D3—362no—
538C2×D3—362no—
539C2×D3—362no—
540C2×D3—362no—
541C2×D3—362no—
542C2×D3—362no—
543C2×D3—361no—
544C2×D3—361no—
545C2×D3—361no—
546C2×D3—361no—
547C2×D3—361no—
548C2×D3—361no—
549C2×D3—361no—
550C2×D3—361no—
551C3×C4—62no—
552C3×C4—121no—
553C3×C4—182no—
554C3×C4—361no—
555Dic3—22no—
556Dic3—41no—
557Dic3—62no—
558Dic3—94no—
559Dic3—94no—
560Dic3—121no—
561Dic3—184no—
562C32#495711yescharacteristic
563C9—21no—
564C9#495711yescharacteristic, complemented
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—272no—
572C23—541no—
573C23—541no—
574C23—541no—
575C23—541no—
576C23—541no—
577C23—541no—
578C23(C3⋊D9)⋊C412yes—
579C23C2×D3×D911yescharacteristic
580D4—548no—
581C2×C4—32no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—92no—
588C2×C4—181no—
589C2×C4—181no—
590C2×C4—181no—
591C2×C4—181no—
592C2×C4—181no—
593C2×C4—272no—
594C2×C4—272no—
595C2×C4—272no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—44no—
604C2×C3—42no—
605C2×C3—42no—
606C2×C3C22⋊C4×D911yescharacteristic
607C2×C3#611⋊C411yescharacteristic
608C2×C3#6344611yescharacteristic
609C2×C3#6344711yescharacteristic
610C2×C3C4×#61111yescharacteristic
611C2×C3#6350111yescharacteristic
612C2×C3#6350211yescharacteristic
613C2×C3#6395211yescharacteristic
614C2×C3#6395211yescharacteristic
615C2×C3#6395311yescharacteristic
616C2×C3#6397111yescharacteristic
617C2×C3#6397111yescharacteristic
618C2×C3#6397411yescharacteristic
619C2×C3#6397911yescharacteristic
620D3—364no—
621D3—362no—
622D3—362no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—542no—
631C22—542no—
632C22—542no—
633C22—1081no—
634C22—1081no—
635C22—1081no—
636C22—1081no—
637C22#8865811yescharacteristic
638C22C2×(C3⋊D9)⋊C411yescharacteristic
639C22#8866211yescharacteristic
640C22D3×#61111yescharacteristic
641C22D9×#6711yescharacteristic
642C22#8866811yescharacteristic
643C22#8866811yescharacteristic
644C4—62no—
645C4—121no—
646C4—182no—
647C4—274no—
648C4—361no—
649C3—21no—
650C3#1058863711yescharacteristic, complemented
651C3#1059170411yescharacteristic
652C2—42no—
653C2—1082no—
654C2#1172124011yescharacteristic
655C2#1172124111yescharacteristic
656C2#1172124211yescharacteristic
657C2#1172127111yescharacteristic
658C2#1172127411yescharacteristic
659C2#1172128211yescharacteristic
660C2#1172128911yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×#6233C2
cosocleC23×C3×C9C23
hypercenterC24D3×D9
hypocenterC3×C9#390
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×C9×#217C2semidirect
3. C2×C3×#3676C2semidirect
4. C2×#88445C2semidirect
5. C2×#88445C2semidirect
6. C2×#88442C2semidirect
7. C2×Dic3×Dic9C2semidirect
14. C2×#6233C4semidirect
36. C2×#3676D3semidirect
65. C2×C9×Dic3D4semidirect
71. C2×C3×Dic9D4semidirect
133. C3⋊Dic9C22⋊C4semidirect
144. C2×#217D9semidirect
233. C2×Dic9#67semidirect
363. C3×C9#390semidirect
444. C2×Dic3#611semidirect
564. C9#4957semidirect
650. C3#10588637semidirect

Maximal subgroups

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

subgroupindex
C22×#62332
C2×C9×#2172
C2×C3×#36762
C2×#884452
C2×#884422
C2×Dic3×Dic92
#105917043
#105886373