Finite Group Databasebeta

← #13150538

Subgroups — #13150538

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×C3×#3697C211yesmaximal, characteristic, complemented
2C2×D9×D12C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C9×#930C211yesmaximal, characteristic, complemented
5#11721292C212yesmaximal, complemented
6#11721289C212yesmaximal, complemented
7#11721049C212yesmaximal, complemented
8#11721036C211yesmaximal, characteristic, complemented
9#11721035C212yesmaximal, complemented
10#11721004C211yesmaximal, characteristic, complemented
11#11720907C211yesmaximal, characteristic, complemented
12#10595064—31nomaximal
13#10589709—31nomaximal
14C22×D3×D9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic, complemented
16C2×#6233C2212yescomplemented
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C9×#67C2212yescomplemented
19C2×C3×#611C2211yescharacteristic
20C2×C3×#611C2211yescharacteristic, complemented
21C2×C9×D12C2211yescharacteristic
22C3×#3697—24no—
23C2×C3×D36C2211yescharacteristic
24C2×C3×C4×D9C2211yescharacteristic
25C2×C3×C4×D9C2211yescharacteristic, complemented
26C2×C3×Dic18C2211yescharacteristic, complemented
27C2×#6195C2212yes—
28C2×#6194C2212yescomplemented
29C2×#6192C2212yescomplemented
30D9×D12—24no—
31#88536C2212yescomplemented
32C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
33C9×(C2×D3)⋊C4C2212yescomplemented
34C9×C4⋊Dic3C2211yescharacteristic, complemented
35(C2×D3)⋊Dic9C2212yescomplemented
36#88445C2212yescomplemented
37(C2×D9)⋊Dic3C2212yes—
38Dic3×Dic9C2212yescomplemented
39C2×C3×#235—31no—
40C2×D3×D12—31no—
41#64139—31no—
42C3×#930D311yescharacteristic
43#63983—32no—
44#63979—32no—
45#63922—32no—
46#63909—31no—
47#63908—31no—
48#63907—32no—
49#63841—31no—
50C2×D4×D9—31no—
51C2×#3697D311yescharacteristic, complemented
52C9×#121—31no—
53#63502—32no—
54#63500—31no—
55#63499—32no—
56#63492—31no—
57#63492—61no—
58#63457—31no—
59#63454—31no—
60#63447—32no—
61#63444—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2312yes—
64C22×D3×C9C2312yes—
65C22×C3×D9C2311yescharacteristic
66C22×C3×D9C2311yescharacteristic
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—42no—
77C3×#611—22no—
78C3×#611—22no—
79C2×C9×Dic3C2312yes—
80C2×C3×Dic9C2311yescharacteristic
81C2×C3×Dic9C2311yescharacteristic
82C9×D12—22no—
83C3×D36—22no—
84C3×C4×D9—22no—
85C3×C4×D9D414yescomplemented
86C3×Dic18—22no—
87#6195—24no—
88#6194—42no—
89#6192—42no—
90#5830—91no—
91#5607—91no—
92C22×D32—32no—
93C22×C32×C4C2×D311yescharacteristic
94C2×#638—32no—
95C2×C4⋊(C3⋊D3)—31no—
96C2×C3×#67—31no—
97C2×C3×#67—31no—
98C2×C3×#67C2×D312yes—
99C3×#235—64no—
100C2×C3×D12—31no—
101C2×C3×D12C2×D311yescharacteristic
102C2×C3×C4×D3—31no—
103C2×C3×C4×D3—31no—
104C2×C3×Dic6—31no—
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)⋊C4C2×D312yes—
113C3×C4⋊Dic3C2×D311yescharacteristic
114#3722—32no—
115(C2×D3)⋊Dic3—32no—
116(C2×D3)⋊Dic3—32no—
117Dic32—32no—
118C2×D4×C9—32no—
119C2×D4×C9—31no—
120C22×C4×C9—21no—
121C22×C4×C9C2×D311yescharacteristic, complemented
122C2×#611—32no—
123C2×#611—32no—
124C2×#611—32no—
125C2×#611—62no—
126C2×#611C2×D311yescharacteristic
127C2×#611C2×D311yescharacteristic, complemented
128D4×D9—64no—
129#3697—24no—
130C2×D36—31no—
131C2×D36—32no—
132C2×D36—61no—
133C2×D36C2×D311yescharacteristic
134C2×C4×D9C2×D311yescharacteristic
135C2×C4×D9C2×D311yescharacteristic, complemented
136C2×Dic18C2×D311yescharacteristic, complemented
137C9×C4⋊C4—31no—
138C9×C22⋊C4—32no—
139#3676—32no—
140(C2×D9)⋊C4—32no—
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4—32no—
143(C2×D9)⋊C4—62no—
144C4⋊Dic9—31no—
145C4⋊Dic9—61no—
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—22no—
155C2×D3×C9—22no—
156C2×D3×C9—22no—
157C2×C3×D9—21no—
158C2×C3×D9—21no—
159C2×C3×D9—21no—
160C2×C3×D9—21no—
161C2×C3×D9—21no—
162C2×C3×D9C2×D412yes—
163D3×D9—48no—
164C3×C4×C9—21no—
165C3×C4×C9C2×D412yes—
166C3⋊Dic9—22no—
167C9×Dic3—22no—
168C3×Dic9—21no—
169C3×Dic9C2×D412yes—
170C2×C3×#41—91no—
171C2×D3×D4—91no—
172C2×#235—31no—
173C22×D12—91no—
174C3×#121—31no—
175#947—92no—
176#940—92no—
177#940—92no—
178#938—91no—
179#937—92no—
180#937—92no—
181#937—92no—
182#930—91no—
183#930—91no—
184#930—91no—
185#930—91no—
186#930—181no—
187#930D911yescharacteristic, complemented
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—31no—
196C22×C3×D3—31no—
197C22×C3×D3C22×D312yes—
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—42no—
208C3×#67—62no—
209C3×#67—62no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3C22×D312yes—
213C3×D12—22no—
214C3×D12—62no—
215C3×C4×D3—34no—
216C3×C4×D3—62no—
217C3×Dic6—62no—
218#626—64no—
219#626—122no—
220#625—122no—
221C23×C9—21no—
222C23×C9—32no—
223C23×C9C22×D311yescharacteristic
224C22×D9—32no—
225C22×D9—62no—
226C22×D9—64no—
227C22×D9—122no—
228C22×D9—122no—
229C22×D9C22×D311yescharacteristic
230C22×D9C22×D311yescharacteristic
231D4×C9—62no—
232D4×C9—122no—
233C2×C4×C9—22no—
234C2×C4×C9—21no—
235C2×C4×C9—21no—
236C2×C4×C9—32no—
237C2×C4×C9—42no—
238C2×C4×C9C22×D311yescharacteristic
239C2×C4×C9C22×D311yescharacteristic
240#611—22no—
241#611—22no—
242#611—64no—
243#611—122no—
244#611—122no—
245#611—124no—
246C2×Dic9—32no—
247C2×Dic9—62no—
248C2×Dic9C22×D311yescharacteristic
249C2×Dic9C22×D311yescharacteristic
250D36—22no—
251D36—62no—
252D36—122no—
253D36—122no—
254C4×D9—22no—
255C4×D9#6714yescomplemented
256Dic18—22no—
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—42no—
264C3×D9—22no—
265C3×D9—41no—
266C23×D3—92no—
267C23×D3—92no—
268C3×#41—184no—
269C2×C3×Q8—91no—
270C2×C3×D4—32no—
271C2×C3×D4—31no—
272C2×C3×D4—91no—
273C2×C3×D4—91no—
274C2×C3×D4—91no—
275C22×C3×C4—21no—
276C22×C3×C4—91no—
277C22×C3×C4—91no—
278C22×C3×C4C2×D911yescharacteristic, complemented
279C22×C3×C4D3211yescharacteristic
280C2×#67—31no—
281C2×#67—31no—
282C2×#67—92no—
283C2×#67—92no—
284C2×#67—92no—
285C2×#67—92no—
286C2×#67—92no—
287C2×#67—92no—
288C2×#67—182no—
289C2×#67C2×D912yescomplemented
290D3×D4—184no—
291#235—64no—
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×D3—31no—
301C2×C4×D3—31no—
302C2×Dic6—31no—
303C3×C4⋊C4—31no—
304C3×C22⋊C4—32no—
305#217—92no—
306#217—92no—
307(C2×D3)⋊C4—92no—
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—182no—
314(C2×D3)⋊C4C2×D912yescomplemented
315C4⋊Dic3—91no—
316C4⋊Dic3—91no—
317C4⋊Dic3—181no—
318C4⋊Dic3C2×D911yescharacteristic, complemented
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—22no—
329C2×C3×D3—22no—
330C2×C3×D3—22no—
331C2×C3×D3—32no—
332C2×C3×D3—61no—
333C2×C3×D3—61no—
334C2×C3×D3—61no—
335C2×C3×D3—61no—
336C2×C3×D3—61no—
337D32—128no—
338C32×C4—21no—
339C32×C4D3×D412yes—
340C3⋊Dic3—62no—
341C3×Dic3—22no—
342C3×Dic3—32no—
343C3×Dic3—61no—
344C22×C9—22no—
345C22×C9—21no—
346C22×C9—21no—
347C22×C9—21no—
348C22×C9—21no—
349C22×C9—41no—
350C22×C9—41no—
351C22×C9—62no—
352C22×C9—62no—
353C22×C9—62no—
354C22×C9C23×D311yescharacteristic
355C2×D9—21no—
356C2×D9—21no—
357C2×D9—21no—
358C2×D9—21no—
359C2×D9—21no—
360C2×D9—62no—
361C2×D9—62no—
362C2×D9—62no—
363C2×D9—122no—
364C2×D9—122no—
365C2×D9—122no—
366C2×D9—128no—
367C2×D9C2×#6712yes—
368C4×C9—21no—
369C4×C9—22no—
370C4×C9—41no—
371C4×C9—62no—
372C4×C9C2×#6712yes—
373Dic9—21no—
374Dic9—62no—
375Dic9—122no—
376Dic9C2×#6712yes—
377C2×#41—91no—
378C22×D4—271no—
379#127—272no—
380#124—272no—
381#121—31no—
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—32no—
391C23×C3—91no—
392C23×C3—91no—
393C23×C3C22×D911yescharacteristic
394C23×C3C2×D3211yescharacteristic
395C22×D3—31no—
396C22×D3—31no—
397C22×D3—92no—
398C22×D3—92no—
399C22×D3—182no—
400C22×D3—184no—
401C22×D3—184no—
402C22×D3—362no—
403C22×D3—362no—
404C22×D3—362no—
405C22×D3—362no—
406C22×D3C22×D912yes—
407C3×Q8—182no—
408C3×D4—62no—
409C3×D4—122no—
410C3×D4—182no—
411C3×D4—182no—
412C3×D4—182no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—32no—
418C2×C3×C4—42no—
419C2×C3×C4—91no—
420C2×C3×C4—94no—
421C2×C3×C4—91no—
422C2×C3×C4—182no—
423C2×C3×C4C22×D911yescharacteristic
424C2×C3×C4C22×D911yescharacteristic
425C2×C3×C4C2×D3211yescharacteristic
426C2×C3×C4C2×D3211yescharacteristic
427#67—42no—
428#67—62no—
429#67—62no—
430#67—184no—
431#67—362no—
432#67—362no—
433#67—362no—
434#67—362no—
435#67—362no—
436#67—364no—
437C2×Dic3—31no—
438C2×Dic3—31no—
439C2×Dic3—92no—
440C2×Dic3—92no—
441C2×Dic3—182no—
442C2×Dic3C22×D912yes—
443D12—22no—
444D12—62no—
445D12—184no—
446D12—182no—
447D12—182no—
448D12—362no—
449D12—362no—
450C4×D3—34no—
451C4×D3—62no—
452Dic6—62no—
453C2×C32—41no—
454C2×C32C2×D3×D411yescharacteristic
455C2×C32#100411yescharacteristic
456C2×C32#100911yescharacteristic
457C3⋊D3—122no—
458C3×D3—42no—
459C3×D3—62no—
460C3×D3—121no—
461C2×C9—21no—
462C2×C9—21no—
463C2×C9—21no—
464C2×C9—42no—
465C2×C9—41no—
466C2×C9—122no—
467C2×C9#100911yescharacteristic
468C2×C9#100911yescharacteristic
469C2×C9C22×#6711yescharacteristic
470D9—22no—
471D9—41no—
472D9—122no—
473D9—242no—
474C24—272no—
475#41—184no—
476C2×Q8—91no—
477C2×D4—32no—
478C2×D4—31no—
479C2×D4—91no—
480C2×D4—91no—
481C2×D4—91no—
482C2×D4—272no—
483C2×D4—272no—
484C2×D4—272no—
485C2×D4—272no—
486C2×D4—271no—
487C2×D4—544no—
488C22×C4—91no—
489C22×C4—91no—
490C22×C4D3×D911yescharacteristic, complemented
491C4⋊C4—31no—
492C4⋊C4—271no—
493C22⋊C4—32no—
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—41no—
508C22×C3—41no—
509C22×C3—62no—
510C22×C3—62no—
511C22×C3—62no—
512C22×C3—92no—
513C22×C3—181no—
514C22×C3—181no—
515C22×C3—181no—
516C22×C3—181no—
517C22×C3—181no—
518C22×C3C23×D911yescharacteristic
519C22×C3C22×D3211yescharacteristic
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—32no—
524C2×D3—61no—
525C2×D3—61no—
526C2×D3—61no—
527C2×D3—61no—
528C2×D3—61no—
529C2×D3—182no—
530C2×D3—182no—
531C2×D3—182no—
532C2×D3—182no—
533C2×D3—182no—
534C2×D3—182no—
535C2×D3—362no—
536C2×D3—362no—
537C2×D3—362no—
538C2×D3—368no—
539C2×D3—368no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—22no—
543C3×C4—41no—
544C3×C4—62no—
545C3×C4—92no—
546C3×C4—181no—
547C3×C4D4×D912yes—
548C3×C4D3×#6712yes—
549Dic3—22no—
550Dic3—32no—
551Dic3—61no—
552Dic3—182no—
553Dic3—182no—
554Dic3—362no—
555C32#560711yescharacteristic
556C9—21no—
557C9#583011yescharacteristic, complemented
558C23—32no—
559C23—91no—
560C23—91no—
561C23—272no—
562C23—544no—
563C23—1082no—
564C23—1082no—
565C23C2×D3×D911yescharacteristic
566Q8—182no—
567D4—62no—
568D4—122no—
569D4—182no—
570D4—182no—
571D4—182no—
572D4—544no—
573D4—542no—
574D4—1082no—
575D4—1082no—
576D4—1082no—
577C2×C4—22no—
578C2×C4—32no—
579C2×C4—91no—
580C2×C4—94no—
581C2×C4—91no—
582C2×C4—182no—
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—122no—
593C2×C3—182no—
594C2×C3—361no—
595C2×C3C2×D4×D911yescharacteristic
596C2×C3#6371011yescharacteristic
597C2×C3#6371511yescharacteristic
598C2×C3#6430711yescharacteristic
599C2×C3#6430811yescharacteristic
600C2×C3C2×D3×#6711yescharacteristic
601D3—42no—
602D3—62no—
603D3—121no—
604D3—362no—
605D3—362no—
606D3—722no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—62no—
611C22—62no—
612C22—62no—
613C22—92no—
614C22—181no—
615C22—181no—
616C22—181no—
617C22—181no—
618C22—181no—
619C22—542no—
620C22—542no—
621C22—542no—
622C22—1088no—
623C22C22×D3×D911yescharacteristic
624C4—21no—
625C4—62no—
626C4—92no—
627C4—181no—
628C4—542no—
629C4D9×#6712yes—
630C3—21no—
631C3#1058970911yescharacteristic, complemented
632C3#1059506411yescharacteristic
633C2—41no—
634C2—122no—
635C2—182no—
636C2—361no—
637C2—1082no—
638C2#1172247611yescharacteristic
639C2#1172247811yescharacteristic
640C2C2×D9×#6711yescharacteristic
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×C3×#3697C2semidirect
2. C2×D9×D12C2semidirect
3. #11721819C2semidirect
4. C9×#930C2semidirect
5. #11721292C2semidirect
6. #11721289C2semidirect
7. #11721049C2semidirect
8. #11721036C2semidirect
9. #11721035C2semidirect
10. #11721004C2semidirect
11. #11720907C2semidirect
15. C22×C3×C4×C9C22semidirect
16. C2×#6233C22semidirect
18. C2×C9×#67C22semidirect
20. C2×C3×#611C22semidirect
25. C2×C3×C4×D9C22semidirect
26. C2×C3×Dic18C22semidirect
28. C2×#6194C22semidirect
29. C2×#6192C22semidirect
31. #88536C22semidirect
32. C4⋊(C3⋊Dic9)C22semidirect
33. C9×(C2×D3)⋊C4C22semidirect
34. C9×C4⋊Dic3C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
36. #88445C22semidirect
38. Dic3×Dic9C22semidirect
51. C2×#3697D3semidirect
85. C3×C4×D9D4semidirect
121. C22×C4×C9C2×D3semidirect
127. C2×#611C2×D3semidirect
135. C2×C4×D9C2×D3semidirect
136. C2×Dic18C2×D3semidirect
187. #930D9semidirect
255. C4×D9#67semidirect
278. C22×C3×C4C2×D9semidirect
289. C2×#67C2×D9semidirect
314. (C2×D3)⋊C4C2×D9semidirect
318. C4⋊Dic3C2×D9semidirect
388. C3×C9#535semidirect
490. C22×C4D3×D9semidirect
557. C9#5830semidirect
631. C3#10589709semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×D9×D122
#117218192
C9×#9302
#117212922
#117212892
#117210492
#117210362
#117210352
#117210042
#117209072
#105950643
#105897093