Finite Group Databasebeta

← #13148416

Subgroups — #13148416

Order 1728.

1168 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
1C2×D9×#67C211yesmaximal, characteristic, complemented
2C22×#6192C211yesmaximal, characteristic, complemented
3#11721728C211yesmaximal, characteristic, complemented
4C9×#886C211yesmaximal, characteristic, complemented
5C3×C22⋊C4×D9C211yesmaximal, characteristic, complemented
6#11721291C211yesmaximal, characteristic, complemented
7#11721251C211yesmaximal, characteristic, complemented
8#11721051C212yesmaximal, complemented
9#11721033C212yesmaximal, complemented
10#11721004C212yesmaximal, complemented
11#11720988C212yesmaximal, complemented
12#10594291—31nomaximal
13#10589755—31nomaximal
14C23×C3⋊D9C2211yescharacteristic, complemented
15C23×C3×D9C2211yescharacteristic
16C22×D3×D9C2211yescharacteristic, complemented
17C2×#6233C2211yescharacteristic
18C2×C4⋊(C3⋊D9)C2212yescomplemented
19C2×C9×#67C2211yescharacteristic
20C22×C9×Dic3C2211yescharacteristic, complemented
21C2×C3×C4×D9C2212yescomplemented
22D9×#67—24no—
23C2×#6195C2212yescomplemented
24C2×#6194C2211yescharacteristic, complemented
25C2×#6192—24no—
26C2×#6192C2212yes—
27C2×#6192C2212yescomplemented
28C2×#6192C2211yescharacteristic, complemented
29C2×Dic3×D9C2211yescharacteristic, complemented
30C3×C9×C22⋊C4C2211yescharacteristic, complemented
31#88536C2212yescomplemented
32C3×#3676C2211yescharacteristic, complemented
33C9×(C2×D3)⋊C4C2212yescomplemented
34C3×(C2×D9)⋊C4C2212yescomplemented
35C9×C4⋊Dic3C2212yescomplemented
36#88445C2212yescomplemented
37#88445C2212yescomplemented
38Dic9⋊Dic3C2212yescomplemented
39C2×D3×#67—31no—
40C22×#626—31no—
41#64091—31no—
42C3×#886D311yescharacteristic
43C3×D3×C22⋊C4—31no—
44#63981—31no—
45#63966—31no—
46#63923—32no—
47#63909—32no—
48#63906—32no—
49#63890—32no—
50C2×D4×D9—31no—
51C22×D36—31no—
52C9×#123—31no—
53#63499—31no—
54#63491—31no—
55#63457—32no—
56#63456—32no—
57#63446—32no—
58#63444—31no—
59#63444—32no—
60#63444—61no—
61C22⋊C4×D9D311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—22no—
64C22×C3⋊D9—42no—
65C22×C3⋊D9C2311yescharacteristic
66C22×C3⋊D9C2312yes—
67C22×D3×C9C2311yescharacteristic
68C22×C3×D9—42no—
69C22×C3×D9D414yescomplemented
70C22×C3×D9C2311yescharacteristic
71C22×C3×D9C2311yescharacteristic
72C2×D3×D9—22no—
73C2×D3×D9—42no—
74C2×C3×C4×C9C2312yes—
75#6233—22no—
76C2×C3⋊Dic9C2311yescharacteristic
77C4⋊(C3⋊D9)—42no—
78C9×#67—22no—
79C2×C9×Dic3—22no—
80C2×C9×Dic3C2311yescharacteristic
81C2×C9×Dic3C2312yes—
82C2×C3×Dic9C2312yes—
83C3×C4×D9—42no—
84#6195—42no—
85#6194—22no—
86#6192—24no—
87#6192—24no—
88#6192—22no—
89Dic3×D9—22no—
90#5653—91no—
91#5493—91no—
92C23×C3⋊D3—31no—
93C23×C3×D3—31no—
94C22×D32—31no—
95C2×#638—31no—
96C2×C4⋊(C3⋊D3)—32no—
97C2×C3×#67C2×D311yescharacteristic
98C22×C3×Dic3C2×D311yescharacteristic
99C2×C3×C4×D3—32no—
100D3×#67—64no—
101C2×#626—32no—
102C2×#626—32no—
103C2×#626—31no—
104C2×#626—32no—
105C2×#626—64no—
106C2×#625—31no—
107C2×D3×Dic3—31no—
108C23×D9—31no—
109C23×D9—31no—
110C23×D9—61no—
111C23×D9C2×D311yescharacteristic
112C32×C22⋊C4C2×D311yescharacteristic
113#3752—32no—
114C3×#217—31no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4C2×D312yes—
117C3×C4⋊Dic3C2×D312yes—
118Dic3⋊Dic3—32no—
119#3722—32no—
120#3722—32no—
121C2×D4×C9—31no—
122C22×C4×C9—31no—
123C2×#611—31no—
124C2×#611—31no—
125C2×#611—32no—
126C2×#611—61no—
127D4×D9—64no—
128C2×D36—32no—
129C2×D36—32no—
130C2×D36—31no—
131C2×D36—32no—
132C2×D36—64no—
133C2×D36—62no—
134C2×C4×D9—31no—
135C2×C4×D9C2×D312yescomplemented
136C9×C4⋊C4—32no—
137C9×C22⋊C4—21no—
138C9×C22⋊C4—32no—
139C9×C22⋊C4C2×D311yescharacteristic, complemented
140#3676C2×D311yescharacteristic, complemented
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4—32no—
143(C2×D9)⋊C4—32no—
144(C2×D9)⋊C4—62no—
145(C2×D9)⋊C4C2×D312yescomplemented
146Dic9⋊C4—32no—
147C22×C3×C9—22no—
148C22×C3×C9C2×D412yes—
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—22no—
151C2×C3⋊D9—21no—
152C2×C3⋊D9—22no—
153C2×C3⋊D9—22no—
154C2×C3⋊D9—22no—
155C2×C3⋊D9—44no—
156C2×D3×C9—21no—
157C2×D3×C9—22no—
158C2×C3×D9—22no—
159C2×C3×D9—22no—
160C2×C3×D9—44no—
161C2×C3×D9C2×D412yes—
162C2×C3×D9C2×D412yes—
163D3×D9—44no—
164C3×C4×C9—22no—
165C3⋊Dic9—21no—
166C9×Dic3—21no—
167C9×Dic3—22no—
168C3×Dic9—22no—
169C22×#67—91no—
170C22×#67—91no—
171C2×D3×D4—91no—
172C22×D12—91no—
173C3×#123—31no—
174C2×C3×C22⋊C4—91no—
175#937—91no—
176#930—92no—
177#930—92no—
178#929—91no—
179#895—92no—
180#894—92no—
181#886—91no—
182#886—92no—
183#886D911yescharacteristic, complemented
184#884—92no—
185#882—92no—
186#882—91no—
187#882—91no—
188#882—91no—
189#882—92no—
190#882—181no—
191D3×C22⋊C4—31no—
192C23×C32C22×D311yescharacteristic
193C22×C3⋊D3—31no—
194C22×C3⋊D3—32no—
195C22×C3⋊D3—62no—
196C22×C3⋊D3—122no—
197C22×C3×D3—31no—
198C22×C3×D3—31no—
199C22×C3×D3—34no—
200C22×C3×D3—122no—
201C22×C3×D3C22×D311yescharacteristic
202C2×D32—62no—
203C2×D32—122no—
204C2×C32×C4C22×D312yes—
205#638—62no—
206C2×C3⋊Dic3—31no—
207C4⋊(C3⋊D3)—122no—
208C3×#67—22no—
209C2×C3×Dic3—22no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3C22×D311yescharacteristic
212C2×C3×Dic3C22×D312yes—
213C3×C4×D3—122no—
214#626—64no—
215#626—64no—
216#626—62no—
217#626—122no—
218#625—62no—
219D3×Dic3—62no—
220C23×C9—21no—
221C23×C9—31no—
222C23×C9C22×D311yescharacteristic
223C22×D9—32no—
224C22×D9—31no—
225C22×D9—42no—
226C22×D9—62no—
227C22×D9—61no—
228C22×D9—62no—
229C22×D9—62no—
230C22×D9—122no—
231C22×D9—122no—
232C22×D9—122no—
233C22×D9—124no—
234C22×D9D1214yescomplemented
235C22×D9C22×D311yescharacteristic
236C22×D9C22×D311yescharacteristic
237D4×C9—62no—
238C2×C4×C9—22no—
239C2×C4×C9—32no—
240C2×C4×C9—31no—
241C2×C4×C9—62no—
242C2×C4×C9C22×D312yes—
243#611—62no—
244#611—62no—
245#611—122no—
246#611—122no—
247C2×Dic9—31no—
248C2×Dic9—61no—
249C2×Dic9C22×D312yes—
250D36—64no—
251D36—64no—
252D36—62no—
253D36—122no—
254D36—124no—
255C4×D9—42no—
256C4×D9—62no—
257#534—271noSylow
258C2×C3×C9—22no—
259C2×C3×C9C22×D411yescharacteristic
260C2×C3×C9H(5,𝔽2)12yes—
261C3⋊D9—42no—
262C3⋊D9—41no—
263D3×C9—41no—
264C3×D9—22no—
265C3×D9—22no—
266C24×C3—91no—
267C23×D3—31no—
268C23×D3—91no—
269C23×D3—91no—
270C23×D3—91no—
271C23×D3—91no—
272C23×D3—181no—
273C2×C3×D4—31no—
274C22×C3×C4—31no—
275C22×C3×C4—92no—
276C2×#67—92no—
277C2×#67—91no—
278C2×#67—91no—
279C2×#67—91no—
280C2×#67—91no—
281C2×#67—92no—
282C2×#67—92no—
283C2×#67—91no—
284C2×#67—181no—
285C2×#67—184no—
286C2×#67—184no—
287C2×#67C2×D911yescharacteristic
288C22×Dic3—91no—
289C22×Dic3C2×D911yescharacteristic, complemented
290D3×D4—184no—
291C2×D12—92no—
292C2×D12—92no—
293C2×D12—91no—
294C2×D12—92no—
295C2×D12—92no—
296C2×D12—92no—
297C2×D12—182no—
298C2×D12—184no—
299C2×C4×D3—32no—
300C2×C4×D3—91no—
301C3×C4⋊C4—32no—
302C3×C22⋊C4—21no—
303C3×C22⋊C4—32no—
304C3×C22⋊C4—91no—
305C3×C22⋊C4—92no—
306C3×C22⋊C4C2×D911yescharacteristic, complemented
307C3×C22⋊C4D3211yescharacteristic
308#217—31no—
309(C2×D3)⋊C4—32no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315(C2×D3)⋊C4—92no—
316(C2×D3)⋊C4—182no—
317(C2×D3)⋊C4C2×D912yescomplemented
318C4⋊Dic3—92no—
319C4⋊Dic3C2×D912yescomplemented
320Dic3⋊C4—92no—
321C22×C32—22no—
322C22×C32D3×D412yes—
323C22×C32C23×D311yescharacteristic
324C2×C3⋊D3—62no—
325C2×C3⋊D3—62no—
326C2×C3⋊D3—61no—
327C2×C3⋊D3—62no—
328C2×C3⋊D3—62no—
329C2×C3⋊D3—124no—
330C2×C3×D3—21no—
331C2×C3×D3—22no—
332C2×C3×D3—32no—
333C2×C3×D3—32no—
334C2×C3×D3—62no—
335C2×C3×D3—62no—
336C2×C3×D3—124no—
337D32—124no—
338C32×C4—22no—
339C3⋊Dic3—61no—
340C3×Dic3—21no—
341C3×Dic3—22no—
342C3×Dic3—62no—
343C22×C9—22no—
344C22×C9—21no—
345C22×C9—22no—
346C22×C9—42no—
347C22×C9—61no—
348C22×C9—62no—
349C22×C9C2×D1212yes—
350C22×C9C23×D311yescharacteristic
351C2×D9—22no—
352C2×D9—22no—
353C2×D9—44no—
354C2×D9—64no—
355C2×D9—62no—
356C2×D9—61no—
357C2×D9—62no—
358C2×D9—62no—
359C2×D9—62no—
360C2×D9—122no—
361C2×D9—122no—
362C2×D9—122no—
363C2×D9—121no—
364C2×D9—124no—
365C2×D9—124no—
366C2×D9—128no—
367C2×D9C2×D1212yes—
368C2×D9C2×D1212yes—
369C4×C9—22no—
370C4×C9—42no—
371C4×C9—61no—
372C4×C9—62no—
373Dic9—22no—
374Dic9—61no—
375Dic9—121no—
376C22×D4—271no—
377C22×D4—271no—
378#123—31no—
379#123—272no—
380#123—271no—
381#121—272no—
382#121—272no—
383#120—272no—
384#120—271no—
385#120—271no—
386C2×C22⋊C4—91no—
387C3×C9#53411yescharacteristic, complemented, Sylow
388C23×C3—21no—
389C23×C3—31no—
390C23×C3—91no—
391C23×C3—91no—
392C23×C3—94no—
393C23×C3—362no—
394C23×C3C22×D911yescharacteristic
395C23×C3C2×D3211yescharacteristic
396C22×D3—31no—
397C22×D3—31no—
398C22×D3—34no—
399C22×D3—92no—
400C22×D3—91no—
401C22×D3—91no—
402C22×D3—92no—
403C22×D3—122no—
404C22×D3—182no—
405C22×D3—181no—
406C22×D3—182no—
407C22×D3—182no—
408C22×D3—182no—
409C22×D3—182no—
410C22×D3—362no—
411C22×D3—362no—
412C22×D3—362no—
413C22×D3—362no—
414C22×D3—362no—
415C22×D3—364no—
416C22×D3C22×D911yescharacteristic
417C3×D4—62no—
418C2×C3×C4—22no—
419C2×C3×C4—32no—
420C2×C3×C4—31no—
421C2×C3×C4—62no—
422C2×C3×C4—92no—
423C2×C3×C4—362no—
424C2×C3×C4C22×D912yes—
425C2×C3×C4C2×D3212yes—
426#67—22no—
427#67—184no—
428#67—184no—
429#67—182no—
430#67—182no—
431#67—182no—
432#67—182no—
433#67—182no—
434#67—362no—
435#67—362no—
436C2×Dic3—22no—
437C2×Dic3—32no—
438C2×Dic3—91no—
439C2×Dic3—91no—
440C2×Dic3—181no—
441C2×Dic3—182no—
442C2×Dic3C22×D912yes—
443C2×Dic3C22×D911yescharacteristic
444D12—184no—
445D12—184no—
446D12—182no—
447D12—362no—
448D12—362no—
449D12—364no—
450D12—362no—
451C4×D3—122no—
452C4×D3—182no—
453C2×C32—22no—
454C2×C32C2×D3×D411yescharacteristic
455C2×C32#100912yes—
456C3⋊D3—122no—
457C3⋊D3—121no—
458C3×D3—41no—
459C3×D3—62no—
460C3×D3—62no—
461C2×C9—22no—
462C2×C9—21no—
463C2×C9—22no—
464C2×C9—42no—
465C2×C9—121no—
466C2×C9C22×D1211yescharacteristic
467C2×C9#100412yes—
468D9—22no—
469D9—22no—
470D9—124no—
471D9—122no—
472D9—121no—
473D9—241no—
474C24—91no—
475C24—271no—
476C24—271no—
477C2×D4—31no—
478C2×D4—272no—
479C2×D4—272no—
480C2×D4—271no—
481C2×D4—272no—
482C2×D4—271no—
483C2×D4—271no—
484C2×D4—272no—
485C2×D4—544no—
486C2×D4—544no—
487C22×C4—31no—
488C22×C4—92no—
489C22×C4—271no—
490C4⋊C4—32no—
491C4⋊C4—272no—
492C22⋊C4—32no—
493C22⋊C4—91no—
494C22⋊C4—92no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C22⋊C4—272no—
498C22⋊C4D3×D911yescharacteristic, complemented
499C22×C3—22no—
500C22×C3—22no—
501C22×C3—21no—
502C22×C3—22no—
503C22×C3—42no—
504C22×C3—61no—
505C22×C3—62no—
506C22×C3—92no—
507C22×C3—92no—
508C22×C3—182no—
509C22×C3—182no—
510C22×C3—364no—
511C22×C3D4×D912yes—
512C22×C3C23×D911yescharacteristic
513C22×C3D3×D1212yes—
514C22×C3C22×D3211yescharacteristic
515C2×D3—21no—
516C2×D3—22no—
517C2×D3—32no—
518C2×D3—32no—
519C2×D3—62no—
520C2×D3—62no—
521C2×D3—124no—
522C2×D3—184no—
523C2×D3—182no—
524C2×D3—181no—
525C2×D3—182no—
526C2×D3—182no—
527C2×D3—182no—
528C2×D3—182no—
529C2×D3—181no—
530C2×D3—182no—
531C2×D3—182no—
532C2×D3—182no—
533C2×D3—362no—
534C2×D3—362no—
535C2×D3—362no—
536C2×D3—361no—
537C2×D3—364no—
538C2×D3—364no—
539C2×D3—364no—
540C2×D3—364no—
541C2×D3—368no—
542C3×C4—22no—
543C3×C4—22no—
544C3×C4—42no—
545C3×C4—61no—
546C3×C4—62no—
547C3×C4—182no—
548Dic3—22no—
549Dic3—21no—
550Dic3—62no—
551Dic3—181no—
552Dic3—181no—
553Dic3—361no—
554C32#565311yescharacteristic
555C9—21no—
556C9#549311yescharacteristic, complemented
557C23—31no—
558C23—91no—
559C23—91no—
560C23—94no—
561C23—272no—
562C23—271no—
563C23—362no—
564C23—542no—
565C23—542no—
566C23—1082no—
567C23—1082no—
568C23C2×D3×D911yescharacteristic
569D4—62no—
570D4—544no—
571D4—544no—
572D4—542no—
573D4—542no—
574D4—542no—
575D4—1082no—
576D4—1082no—
577C2×C4—32no—
578C2×C4—31no—
579C2×C4—62no—
580C2×C4—92no—
581C2×C4—271no—
582C2×C4—362no—
583C2×C4—542no—
584C2×C4C2×D3×D912yes—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—22no—
588C2×C3—21no—
589C2×C3—42no—
590C2×C3—121no—
591C2×C3—182no—
592C2×C3—182no—
593C2×C3C2×D4×D911yescharacteristic
594C2×C3#6371512yes—
595C2×C3C2×D3×D1211yescharacteristic
596C2×C3#6431412yes—
597D3—41no—
598D3—62no—
599D3—62no—
600D3—364no—
601D3—362no—
602D3—361no—
603D3—362no—
604D3—361no—
605D3—721no—
606C22—22no—
607C22—62no—
608C22—61no—
609C22—92no—
610C22—92no—
611C22—182no—
612C22—182no—
613C22—364no—
614C22—542no—
615C22—542no—
616C22—542no—
617C22—542no—
618C22—541no—
619C22—1084no—
620C22—1084no—
621C22D9×D1212yes—
622C22C22×D3×D911yescharacteristic
623C4—22no—
624C4—62no—
625C4—61no—
626C4—182no—
627C4—541no—
628C3—21no—
629C3#1058975511yescharacteristic, complemented
630C3#1059429111yescharacteristic
631C2—22no—
632C2—121no—
633C2—182no—
634C2—182no—
635C2—1082no—
636C2—1081no—
637C2C2×D9×D1211yescharacteristic
638C2#1172250112yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#534
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×D9×#67C2semidirect
2. C22×#6192C2semidirect
3. #11721728C2semidirect
4. C9×#886C2semidirect
5. C3×C22⋊C4×D9C2semidirect
6. #11721291C2semidirect
7. #11721251C2semidirect
8. #11721051C2semidirect
9. #11721033C2semidirect
10. #11721004C2semidirect
11. #11720988C2semidirect
14. C23×C3⋊D9C22semidirect
16. C22×D3×D9C22semidirect
18. C2×C4⋊(C3⋊D9)C22semidirect
20. C22×C9×Dic3C22semidirect
21. C2×C3×C4×D9C22semidirect
23. C2×#6195C22semidirect
24. C2×#6194C22semidirect
27. C2×#6192C22semidirect
28. C2×#6192C22semidirect
29. C2×Dic3×D9C22semidirect
30. C3×C9×C22⋊C4C22semidirect
31. #88536C22semidirect
32. C3×#3676C22semidirect
33. C9×(C2×D3)⋊C4C22semidirect
34. C3×(C2×D9)⋊C4C22semidirect
35. C9×C4⋊Dic3C22semidirect
36. #88445C22semidirect
37. #88445C22semidirect
38. Dic9⋊Dic3C22semidirect
61. C22⋊C4×D9D3semidirect
69. C22×C3×D9D4semidirect
135. C2×C4×D9C2×D3semidirect
139. C9×C22⋊C4C2×D3semidirect
140. #3676C2×D3semidirect
145. (C2×D9)⋊C4C2×D3semidirect
183. #886D9semidirect
234. C22×D9D12semidirect
289. C22×Dic3C2×D9semidirect
306. C3×C22⋊C4C2×D9semidirect
317. (C2×D3)⋊C4C2×D9semidirect
319. C4⋊Dic3C2×D9semidirect
387. C3×C9#534semidirect
498. C22⋊C4D3×D9semidirect
556. C9#5493semidirect
629. C3#10589755semidirect

Maximal subgroups

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

subgroupindex
C2×D9×#672
C22×#61922
#117217282
C9×#8862
C3×C22⋊C4×D92
#117212912
#117212512
#117210512
#117210332
#117210042
#117209882
#105942913
#105897553