Finite Group Databasebeta

← #13150539

Subgroups — #13150539

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic, complemented
2C2×D3×D36C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic, complemented
5#11721292C212yesmaximal, complemented
6#11721289C212yesmaximal, complemented
7#11721050C212yesmaximal, complemented
8#11721038C211yesmaximal, characteristic, complemented
9#11721037C211yesmaximal, characteristic, complemented
10#11721003C212yesmaximal, complemented
11#11720906C211yesmaximal, characteristic, complemented
12#10595064—31nomaximal
13#10589932—31nomaximal
14C22×D3×D9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic, complemented
16C2×#6233C2212yescomplemented
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C9×#67C2211yescharacteristic
19C2×C9×#67C2211yescharacteristic, complemented
20C2×C3×#611C2212yescomplemented
21C9×#235—24no—
22C2×C9×D12C2211yescharacteristic
23C2×C4×D3×C9C2211yescharacteristic
24C2×C4×D3×C9C2211yescharacteristic, complemented
25C2×C3×D36C2211yescharacteristic
26C2×C9×Dic6C2211yescharacteristic, complemented
27C2×#6195C2212yescomplemented
28C2×#6194C2212yescomplemented
29C2×#6192C2212yes—
30D3×D36—24no—
31#88536C2212yescomplemented
32C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
33C3×(C2×D9)⋊C4C2212yescomplemented
34C3×C4⋊Dic9C2211yescharacteristic, complemented
35(C2×D3)⋊Dic9C2212yes—
36#88445C2212yescomplemented
37(C2×D9)⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C2×C3×#235D311yescharacteristic
40C2×D3×D12—31no—
41#64139—31no—
42C3×#930—31no—
43#63983—32no—
44#63979—32no—
45#63922—32no—
46#63909—31no—
47#63908—31no—
48#63907—32no—
49#63841—31no—
50C2×C9×#41—31no—
51C22×D36—31no—
52#63509—32no—
53#63502—32no—
54#63499—32no—
55#63499—32no—
56#63492—31no—
57#63492—31no—
58#63492—31no—
59#63492—61no—
60#63492D311yescharacteristic, complemented
61#63488—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×D3×C9C2311yescharacteristic
65C22×D3×C9C2311yescharacteristic
66C22×C3×D9C2312yes—
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×D3×D9—42no—
70C2×C3×C4×C9—22no—
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3×C4×C9C2311yescharacteristic
73#6233—42no—
74C2×C3⋊Dic9C2312yes—
75C4⋊(C3⋊D9)—22no—
76C9×#67—22no—
77C9×#67—22no—
78C3×#611—42no—
79C2×C9×Dic3C2311yescharacteristic
80C2×C9×Dic3C2311yescharacteristic
81C2×C3×Dic9C2312yes—
82C9×D12—22no—
83C4×D3×C9—22no—
84C4×D3×C9D414yescomplemented
85C3×D36—22no—
86C9×Dic6—22no—
87#6195—42no—
88#6194—42no—
89#6192—24no—
90#5830—91no—
91#5607—91no—
92C22×D32—32no—
93C22×C32×C4C2×D311yescharacteristic
94C2×#638—32no—
95C2×C4⋊(C3⋊D3)—31no—
96C2×C3×#67—32no—
97C2×C3×#67C2×D311yescharacteristic
98C2×C3×#67C2×D311yescharacteristic
99C3×#235—24no—
100C2×C3×D12—31no—
101C2×C3×D12C2×D311yescharacteristic
102C2×C3×C4×D3C2×D311yescharacteristic
103C2×C3×C4×D3C2×D311yescharacteristic
104C2×C3×Dic6C2×D311yescharacteristic
105C2×#626—32no—
106C2×#626—32no—
107C2×#625—32no—
108D3×D12—64no—
109C23×D9—32no—
110#3752—32no—
111C4⋊(C3⋊Dic3)—31no—
112C3×(C2×D3)⋊C4—32no—
113C3×C4⋊Dic3—31no—
114#3722—32no—
115(C2×D3)⋊Dic3—32no—
116(C2×D3)⋊Dic3—32no—
117Dic32—32no—
118C9×#41—64no—
119C2×Q8×C9—31no—
120C2×D4×C9—31no—
121C2×D4×C9—31no—
122C2×D4×C9—31no—
123C22×C4×C9—21no—
124C22×C4×C9—31no—
125C22×C4×C9—31no—
126C22×C4×C9C2×D311yescharacteristic, complemented
127C2×#611—32no—
128C2×#611—32no—
129C2×#611—32no—
130C2×#611—62no—
131C2×#611C2×D312yescomplemented
132C2×D36—32no—
133C2×D36—31no—
134C2×D36—64no—
135C2×D36—61no—
136C2×D36C2×D311yescharacteristic
137#3676—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—32no—
141(C2×D9)⋊C4—62no—
142(C2×D9)⋊C4C2×D312yescomplemented
143C4⋊Dic9—31no—
144C4⋊Dic9—61no—
145C4⋊Dic9C2×D311yescharacteristic, complemented
146C4×Dic9—32no—
147C22×C3×C9—21no—
148C22×C3×C9—21no—
149C22×C3×C9—21no—
150C22×C3×C9C2411yescharacteristic
151C2×C3⋊D9—22no—
152C2×C3⋊D9—22no—
153C2×C3⋊D9—22no—
154C2×D3×C9—21no—
155C2×D3×C9—21no—
156C2×D3×C9—21no—
157C2×D3×C9—21no—
158C2×D3×C9—21no—
159C2×D3×C9C2×D412yes—
160C2×C3×D9—22no—
161C2×C3×D9—22no—
162C2×C3×D9—22no—
163D3×D9—48no—
164C3×C4×C9—21no—
165C3×C4×C9C2×D412yes—
166C3⋊Dic9—22no—
167C9×Dic3—21no—
168C9×Dic3C2×D412yes—
169C3×Dic9—22no—
170C2×C3×#41—31no—
171C2×D3×D4—91no—
172C2×#235D911yescharacteristic, complemented
173C22×D12—91no—
174C3×#121—91no—
175#947—92no—
176#940—92no—
177#940—92no—
178#938—91no—
179#937—92no—
180#937—92no—
181#937—92no—
182#930—31no—
183#930—91no—
184#930—91no—
185#930—91no—
186#930—91no—
187#930—181no—
188#926—91no—
189#895—91no—
190#892—91no—
191#885—92no—
192#882—92no—
193C23×C32C22×D311yescharacteristic
194C22×C3⋊D3—32no—
195C22×C3×D3—32no—
196C22×C3×D3C22×D311yescharacteristic
197C22×C3×D3C22×D311yescharacteristic
198C2×D32—64no—
199C2×D32—122no—
200C2×D32—122no—
201C2×C32×C4—22no—
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C22×D311yescharacteristic
204#638—122no—
205C2×C3⋊Dic3—32no—
206C4⋊(C3⋊D3)—62no—
207C3×#67—22no—
208C3×#67—22no—
209C3×#67—122no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3C22×D311yescharacteristic
212C2×C3×Dic3C22×D311yescharacteristic
213C3×D12—22no—
214C3×D12—62no—
215C3×C4×D3—22no—
216C3×C4×D3#6714yes—
217C3×Dic6—22no—
218#626—64no—
219#626—122no—
220#625—122no—
221C23×C9—21no—
222C23×C9—31no—
223C23×C9—31no—
224C23×C9C22×D311yescharacteristic
225C22×D9—32no—
226C22×D9—62no—
227C22×D9—64no—
228C22×D9—122no—
229C22×D9—122no—
230C22×D9C22×D312yes—
231Q8×C9—62no—
232D4×C9—62no—
233D4×C9—62no—
234D4×C9—62no—
235C2×C4×C9—22no—
236C2×C4×C9—21no—
237C2×C4×C9—21no—
238C2×C4×C9—31no—
239C2×C4×C9—34no—
240C2×C4×C9—31no—
241C2×C4×C9—42no—
242C2×C4×C9—62no—
243C2×C4×C9C22×D311yescharacteristic
244C2×C4×C9C22×D311yescharacteristic
245#611—42no—
246#611—122no—
247#611—122no—
248#611—122no—
249#611—124no—
250C2×Dic9—32no—
251C2×Dic9—62no—
252C2×Dic9C22×D312yes—
253D36—22no—
254D36—64no—
255D36—62no—
256D36—122no—
257#535—271noSylow
258C2×C3×C9—41no—
259C2×C3×C9C22×D411yescharacteristic
260C2×C3×C9H(5,𝔽2)11yescharacteristic
261C2×C3×C9H(5,𝔽2)11yescharacteristic
262C3⋊D9—42no—
263D3×C9—22no—
264D3×C9—41no—
265C3×D9—42no—
266C23×D3—92no—
267C23×D3—92no—
268C3×#41—64no—
269C2×C3×Q8—31no—
270C2×C3×D4—31no—
271C2×C3×D4—31no—
272C2×C3×D4—31no—
273C2×C3×D4—92no—
274C2×C3×D4—91no—
275C22×C3×C4—21no—
276C22×C3×C4—31no—
277C22×C3×C4—31no—
278C22×C3×C4C2×D911yescharacteristic, complemented
279C22×C3×C4D3211yescharacteristic
280C2×#67—32no—
281C2×#67—92no—
282C2×#67—92no—
283C2×#67—92no—
284C2×#67—92no—
285C2×#67—92no—
286C2×#67—92no—
287C2×#67—182no—
288C2×#67C2×D911yescharacteristic
289C2×#67C2×D911yescharacteristic, complemented
290D3×D4—184no—
291#235—24no—
292C2×D12—31no—
293C2×D12—92no—
294C2×D12—91no—
295C2×D12—91no—
296C2×D12—92no—
297C2×D12—181no—
298C2×D12—184no—
299C2×D12C2×D911yescharacteristic
300C2×C4×D3C2×D911yescharacteristic
301C2×C4×D3C2×D911yescharacteristic, complemented
302C2×Dic6C2×D911yescharacteristic, complemented
303C3×C4⋊C4—91no—
304C3×C22⋊C4—92no—
305#217—92no—
306#217—92no—
307(C2×D3)⋊C4—32no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4—92no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—182no—
315C4⋊Dic3—31no—
316C4⋊Dic3—91no—
317C4⋊Dic3—91no—
318C4⋊Dic3—181no—
319C4×Dic3—92no—
320C4×Dic3—92no—
321C22×C32—21no—
322C22×C32—21no—
323C22×C32—21no—
324C22×C32C23×D311yescharacteristic
325C2×C3⋊D3—62no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—62no—
328C2×C3×D3—21no—
329C2×C3×D3—21no—
330C2×C3×D3—21no—
331C2×C3×D3—21no—
332C2×C3×D3—21no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—62no—
336C2×C3×D3C2×#6712yes—
337D32—128no—
338C32×C4—21no—
339C32×C4C2×#6712yes—
340C3⋊Dic3—62no—
341C3×Dic3—21no—
342C3×Dic3—62no—
343C3×Dic3C2×#6712yes—
344C22×C9—22no—
345C22×C9—21no—
346C22×C9—21no—
347C22×C9—21no—
348C22×C9—21no—
349C22×C9—32no—
350C22×C9—41no—
351C22×C9—41no—
352C22×C9—61no—
353C22×C9—61no—
354C22×C9—61no—
355C22×C9—61no—
356C22×C9—61no—
357C22×C9C23×D311yescharacteristic
358C2×D9—22no—
359C2×D9—22no—
360C2×D9—22no—
361C2×D9—62no—
362C2×D9—62no—
363C2×D9—62no—
364C2×D9—122no—
365C2×D9—122no—
366C2×D9—122no—
367C2×D9—128no—
368C4×C9—21no—
369C4×C9—22no—
370C4×C9—32no—
371C4×C9—41no—
372C4×C9—61no—
373C4×C9D3×D412yes—
374Dic9—22no—
375Dic9—62no—
376Dic9—122no—
377C2×#41—31no—
378C22×D4—271no—
379#127—272no—
380#124—272no—
381#121—91no—
382#121—271no—
383#121—271no—
384#121—271no—
385#120—272no—
386#120—272no—
387#117—271no—
388C3×C9#53511yescharacteristic, complemented, Sylow
389C23×C3—21no—
390C23×C3—31no—
391C23×C3—31no—
392C23×C3—92no—
393C23×C3C22×D911yescharacteristic
394C23×C3C2×D3211yescharacteristic
395C22×D3—32no—
396C22×D3—92no—
397C22×D3—92no—
398C22×D3—182no—
399C22×D3—184no—
400C22×D3—184no—
401C22×D3—362no—
402C22×D3—362no—
403C22×D3—362no—
404C22×D3—362no—
405C22×D3C22×D911yescharacteristic
406C22×D3C22×D911yescharacteristic
407C3×Q8—62no—
408C3×D4—62no—
409C3×D4—62no—
410C3×D4—62no—
411C3×D4—182no—
412C3×D4—362no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—31no—
418C2×C3×C4—34no—
419C2×C3×C4—31no—
420C2×C3×C4—42no—
421C2×C3×C4—62no—
422C2×C3×C4—92no—
423C2×C3×C4C22×D911yescharacteristic
424C2×C3×C4C22×D911yescharacteristic
425C2×C3×C4C2×D3211yescharacteristic
426C2×C3×C4C2×D3211yescharacteristic
427#67—22no—
428#67—22no—
429#67—122no—
430#67—184no—
431#67—362no—
432#67—362no—
433#67—362no—
434#67—362no—
435#67—362no—
436#67—364no—
437C2×Dic3—32no—
438C2×Dic3—92no—
439C2×Dic3—92no—
440C2×Dic3—182no—
441C2×Dic3C22×D911yescharacteristic
442C2×Dic3C22×D911yescharacteristic
443D12—22no—
444D12—62no—
445D12—184no—
446D12—182no—
447D12—182no—
448D12—362no—
449D12—362no—
450C4×D3—22no—
451C4×D3#61114yescomplemented
452Dic6—22no—
453C2×C32—41no—
454C2×C32#100911yescharacteristic
455C2×C32#100911yescharacteristic
456C2×C32C22×#6711yescharacteristic
457C3⋊D3—122no—
458C3×D3—22no—
459C3×D3—41no—
460C3×D3—122no—
461C2×C9—21no—
462C2×C9—21no—
463C2×C9—21no—
464C2×C9—42no—
465C2×C9—41no—
466C2×C9—62no—
467C2×C9—121no—
468C2×C9C2×D3×D411yescharacteristic
469C2×C9#100411yescharacteristic
470C2×C9#100911yescharacteristic
471D9—42no—
472D9—122no—
473D9—242no—
474C24—272no—
475#41—64no—
476C2×Q8—31no—
477C2×D4—31no—
478C2×D4—31no—
479C2×D4—31no—
480C2×D4—92no—
481C2×D4—91no—
482C2×D4—272no—
483C2×D4—272no—
484C2×D4—272no—
485C2×D4—272no—
486C2×D4—271no—
487C2×D4—544no—
488C22×C4—31no—
489C22×C4—31no—
490C22×C4D3×D911yescharacteristic, complemented
491C4⋊C4—91no—
492C4⋊C4—271no—
493C22⋊C4—92no—
494C22⋊C4—272no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C22⋊C4—272no—
498C42—272no—
499C22×C3—22no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—32no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—61no—
511C22×C3—61no—
512C22×C3—61no—
513C22×C3—61no—
514C22×C3—61no—
515C22×C3—182no—
516C22×C3—182no—
517C22×C3—182no—
518C22×C3C23×D911yescharacteristic
519C22×C3C22×D3211yescharacteristic
520C2×D3—21no—
521C2×D3—21no—
522C2×D3—21no—
523C2×D3—21no—
524C2×D3—21no—
525C2×D3—62no—
526C2×D3—62no—
527C2×D3—62no—
528C2×D3—182no—
529C2×D3—182no—
530C2×D3—182no—
531C2×D3—182no—
532C2×D3—182no—
533C2×D3—182no—
534C2×D3—362no—
535C2×D3—362no—
536C2×D3—362no—
537C2×D3—368no—
538C2×D3—368no—
539C2×D3C2×#61112yes—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—22no—
543C3×C4—32no—
544C3×C4—41no—
545C3×C4—61no—
546C3×C4—182no—
547C3×C4C2×#61112yes—
548C3×C4D3×#6712yes—
549Dic3—21no—
550Dic3—62no—
551Dic3—182no—
552Dic3—182no—
553Dic3—362no—
554Dic3C2×#61112yes—
555C32#583011yescharacteristic
556C9—21no—
557C9#560711yescharacteristic, complemented
558C23—31no—
559C23—31no—
560C23—92no—
561C23—272no—
562C23—544no—
563C23—1082no—
564C23—1082no—
565C23C2×D3×D911yescharacteristic
566Q8—62no—
567D4—62no—
568D4—62no—
569D4—62no—
570D4—182no—
571D4—362no—
572D4—544no—
573D4—542no—
574D4—1082no—
575D4—1082no—
576D4—1082no—
577C2×C4—22no—
578C2×C4—31no—
579C2×C4—34no—
580C2×C4—31no—
581C2×C4—62no—
582C2×C4—92no—
583C2×C4—272no—
584C2×C4C2×D3×D911yescharacteristic
585C2×C4C2×D3×D911yescharacteristic
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—42no—
590C2×C3—41no—
591C2×C3—41no—
592C2×C3—62no—
593C2×C3—121no—
594C2×C3—362no—
595C2×C3#6371511yescharacteristic
596C2×C3#6371511yescharacteristic
597C2×C3C22×#61111yescharacteristic
598C2×C3#6430711yescharacteristic
599C2×C3#6430811yescharacteristic
600C2×C3C2×D3×#6711yescharacteristic
601D3—22no—
602D3—41no—
603D3—122no—
604D3—362no—
605D3—362no—
606D3—722no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—32no—
611C22—61no—
612C22—61no—
613C22—61no—
614C22—61no—
615C22—61no—
616C22—182no—
617C22—182no—
618C22—182no—
619C22—542no—
620C22—542no—
621C22—542no—
622C22—1088no—
623C22C22×D3×D911yescharacteristic
624C4—21no—
625C4—32no—
626C4—61no—
627C4—182no—
628C4—542no—
629C4D3×#61112yes—
630C3—21no—
631C3#1058993211yescharacteristic, complemented
632C3#1059506411yescharacteristic
633C2—41no—
634C2—62no—
635C2—121no—
636C2—362no—
637C2—1082no—
638C2#1172247711yescharacteristic
639C2#1172247811yescharacteristic
640C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#535
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C9×#235C2semidirect
2. C2×D3×D36C2semidirect
3. #11721819C2semidirect
4. C3×#63492C2semidirect
5. #11721292C2semidirect
6. #11721289C2semidirect
7. #11721050C2semidirect
8. #11721038C2semidirect
9. #11721037C2semidirect
10. #11721003C2semidirect
11. #11720906C2semidirect
15. C22×C3×C4×C9C22semidirect
16. C2×#6233C22semidirect
19. C2×C9×#67C22semidirect
20. C2×C3×#611C22semidirect
24. C2×C4×D3×C9C22semidirect
26. C2×C9×Dic6C22semidirect
27. C2×#6195C22semidirect
28. C2×#6194C22semidirect
31. #88536C22semidirect
32. C4⋊(C3⋊Dic9)C22semidirect
33. C3×(C2×D9)⋊C4C22semidirect
34. C3×C4⋊Dic9C22semidirect
36. #88445C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
60. #63492D3semidirect
84. C4×D3×C9D4semidirect
126. C22×C4×C9C2×D3semidirect
131. C2×#611C2×D3semidirect
142. (C2×D9)⋊C4C2×D3semidirect
145. C4⋊Dic9C2×D3semidirect
172. C2×#235D9semidirect
278. C22×C3×C4C2×D9semidirect
289. C2×#67C2×D9semidirect
301. C2×C4×D3C2×D9semidirect
302. C2×Dic6C2×D9semidirect
388. C3×C9#535semidirect
451. C4×D3#611semidirect
490. C22×C4D3×D9semidirect
557. C9#5607semidirect
631. C3#10589932semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C2×D3×D362
#117218192
C3×#634922
#117212922
#117212892
#117210502
#117210382
#117210372
#117210032
#117209062
#105950643
#105899323