Finite Group Databasebeta

← #13149317

Subgroups — #13149317

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×D3×D36C211yesmaximal, characteristic, complemented
2C2×#88640C211yesmaximal, characteristic, complemented
3#11721751C211yesmaximal, characteristic, complemented
4C9×#892C211yesmaximal, characteristic, complemented
5C3×#63457C211yesmaximal, characteristic, complemented
6#11721051C212yesmaximal, complemented
7#11721049C212yesmaximal, complemented
8#11721007C211yesmaximal, characteristic, complemented
9#11721006C212yesmaximal, complemented
10#11721004C211yesmaximal, characteristic, complemented
11#11720940C212yesmaximal, complemented
12#10594628—31nomaximal
13#10589627—31nomaximal
14C22×D3×D9C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4⋊(C3⋊D9)C2212yescomplemented
17C2×C4×C3⋊D9C2211yescharacteristic, complemented
18C2×C4×D3×C9C2211yescharacteristic
19C2×C3×D36C2211yescharacteristic
20C2×C3×D36C2212yescomplemented
21C2×C3×C4×D9C2211yescharacteristic, complemented
22C2×#6195C2211yescharacteristic
23C2×#6194C2211yescharacteristic
24C2×#6192C2212yes—
25C2×#6192C2211yescharacteristic, complemented
26C2×#6192C2212yescomplemented
27C2×#6192C2212yescomplemented
28C2×#6189C2211yescharacteristic, complemented
29D3×D36—24no—
30#88640—24no—
31C3×C9×C4⋊C4C2211yescharacteristic, complemented
32#88536C2212yescomplemented
33C9×(C2×D3)⋊C4C2212yes—
34C3×(C2×D9)⋊C4C2212yescomplemented
35C9×C4⋊Dic3C2211yescharacteristic, complemented
36C4×C9×Dic3C2212yescomplemented
37#88445C2212yescomplemented
38(C2×D9)⋊Dic3C2212yescomplemented
39C2×D3×D12—31no—
40C2×#3798—31no—
41#64104—31no—
42C3×#895—31no—
43C3×#892D311yescharacteristic
44#63923—32no—
45#63922—32no—
46#63916—31no—
47#63911—32no—
48#63909—31no—
49#63852—32no—
50C2×#3697—31no—
51C22×D36—31no—
52C9×#117—31no—
53#63457—31no—
54#63457—31no—
55#63457—31no—
56#63457—61no—
57#63457D311yescharacteristic, complemented
58#63444—32no—
59#63444—32no—
60#63437—32no—
61#63436—32no—
62C22×C3⋊D9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9C2312yes—
66C22×C3×D9C2311yescharacteristic
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×D3×D9—42no—
70C2×C3×C4×C9C2312yes—
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C4⋊(C3⋊D9)—22no—
74C4⋊(C3⋊D9)—42no—
75C4×C3⋊D9—22no—
76C2×C9×Dic3C2312yes—
77C2×C9×Dic3C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C4×D3×C9D414yescomplemented
80C3×D36—22no—
81C3×D36—42no—
82C3×C4×D9—22no—
83#6195—22no—
84#6194—22no—
85#6192—22no—
86#6192—24no—
87#6192—42no—
88#6192—42no—
89#6189—22no—
90#5607—91no—
91#5525—91no—
92C22×D32—32no—
93C2×C4⋊(C3⋊D3)—31no—
94C2×C4⋊(C3⋊D3)—32no—
95C2×C4×C3⋊D3—31no—
96C2×C3×D12—31no—
97C2×C3×D12—32no—
98C2×C3×C4×D3—31no—
99C2×C3×C4×D3C2×D311yescharacteristic
100C2×#627—31no—
101C2×#626—31no—
102C2×#626—31no—
103C2×#626—32no—
104C2×#626—32no—
105C2×#626—32no—
106C2×#625—31no—
107D3×D12—64no—
108#3798—64no—
109C23×D9—32no—
110C32×C4⋊C4C2×D311yescharacteristic
111#3752—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4C2×D312yes—
114C3×C4⋊Dic3C2×D311yescharacteristic
115C3×C4×Dic3C2×D312yes—
116#3722—32no—
117(C2×D3)⋊Dic3—32no—
118C22×C4×C9—31no—
119C2×#611—31no—
120C2×#611—31no—
121#3697—64no—
122C2×D36—32no—
123C2×D36—31no—
124C2×D36—31no—
125C2×D36—32no—
126C2×D36—32no—
127C2×D36—32no—
128C2×D36—64no—
129C2×D36—61no—
130C2×D36—62no—
131C2×D36C2×D311yescharacteristic
132C2×D36C2×D312yescomplemented
133C2×C4×D9—31no—
134C2×C4×D9—61no—
135C2×C4×D9C2×D311yescharacteristic, complemented
136C2×Dic18—31no—
137C9×C4⋊C4—21no—
138C9×C4⋊C4—31no—
139C9×C4⋊C4C2×D311yescharacteristic, complemented
140C9×C22⋊C4—32no—
141C42×C9—32no—
142(C2×D9)⋊C4—32no—
143(C2×D9)⋊C4—32no—
144(C2×D9)⋊C4—32no—
145(C2×D9)⋊C4—62no—
146(C2×D9)⋊C4C2×D312yescomplemented
147C22×C3×C9C2411yescharacteristic
148C2×C3⋊D9—21no—
149C2×C3⋊D9—22no—
150C2×C3⋊D9—21no—
151C2×C3⋊D9—21no—
152C2×C3⋊D9—22no—
153C2×C3⋊D9—22no—
154C2×D3×C9—21no—
155C2×D3×C9—21no—
156C2×D3×C9C2×D412yes—
157C2×C3×D9—21no—
158C2×C3×D9—22no—
159C2×C3×D9—21no—
160C2×C3×D9—22no—
161C2×C3×D9—21no—
162C2×C3×D9—22no—
163D3×D9—48no—
164C3×C4×C9—22no—
165C3×C4×C9C2×D412yes—
166C3⋊Dic9—21no—
167C9×Dic3—22no—
168C9×Dic3C2×D412yes—
169C3×Dic9—21no—
170C2×D3×D4—91no—
171C2×#235—91no—
172C2×#235—91no—
173C22×D12—91no—
174C3×#121—91no—
175C3×#117—31no—
176#940—92no—
177#938—91no—
178#937—92no—
179#930—91no—
180#895—31no—
181#895—91no—
182#895—91no—
183#895—91no—
184#895—91no—
185#895—181no—
186#892D911yescharacteristic, complemented
187#885—92no—
188#882—92no—
189#882—92no—
190#882—92no—
191#875—92no—
192#874—92no—
193C22×C3⋊D3—32no—
194C22×C3⋊D3—31no—
195C22×C3×D3—32no—
196C22×C3×D3—31no—
197C22×C3×D3C22×D311yescharacteristic
198C2×D32—64no—
199C2×D32—122no—
200C2×D32—122no—
201C2×C32×C4C22×D312yes—
202C2×C32×C4C22×D311yescharacteristic
203C2×C3⋊Dic3—31no—
204C4⋊(C3⋊D3)—62no—
205C4⋊(C3⋊D3)—122no—
206C4×C3⋊D3—62no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3C22×D312yes—
209C2×C3×Dic3C22×D311yescharacteristic
210C3×D12—62no—
211C3×D12—122no—
212C3×C4×D3—62no—
213C3×C4×D3D1214yes—
214#627—62no—
215#626—62no—
216#626—62no—
217#626—64no—
218#626—122no—
219#626—122no—
220#625—62no—
221C23×C9—31no—
222C22×D9—32no—
223C22×D9—31no—
224C22×D9—61no—
225C22×D9—62no—
226C22×D9—64no—
227C22×D9—122no—
228C22×D9—122no—
229C22×D9C22×D312yes—
230C22×D9C22×D311yescharacteristic
231C2×C4×C9—21no—
232C2×C4×C9—22no—
233C2×C4×C9—31no—
234C2×C4×C9—34no—
235C2×C4×C9—32no—
236C2×C4×C9C22×D312yes—
237C2×C4×C9C22×D311yescharacteristic
238#611—62no—
239#611—62no—
240C2×Dic9—31no—
241C2×Dic9—61no—
242C2×Dic9C22×D311yescharacteristic
243D36—22no—
244D36—42no—
245D36—64no—
246D36—62no—
247D36—62no—
248D36—122no—
249D36—122no—
250D36—124no—
251D36—122no—
252D36—122no—
253C4×D9—22no—
254C4×D9—62no—
255C4×D9—122no—
256Dic18—62no—
257#535—271noSylow
258C2×C3×C9C22×D411yescharacteristic
259C2×C3×C9H(5,𝔽2)11yescharacteristic
260C2×C3×C9H(5,𝔽2)11yescharacteristic
261C3⋊D9—41no—
262C3⋊D9—42no—
263D3×C9—22no—
264C3×D9—42no—
265C3×D9—41no—
266C23×D3—92no—
267C23×D3—92no—
268C2×C3×D4—91no—
269C2×C3×D4—92no—
270C22×C3×C4—31no—
271C22×C3×C4—91no—
272C2×#67—91no—
273C2×#67—91no—
274C2×#67—91no—
275C2×#67—92no—
276C2×#67—91no—
277C2×#67—92no—
278C2×#67—92no—
279D3×D4—184no—
280#235—184no—
281#235—184no—
282C2×D12—31no—
283C2×D12—32no—
284C2×D12—91no—
285C2×D12—92no—
286C2×D12—91no—
287C2×D12—91no—
288C2×D12—91no—
289C2×D12—92no—
290C2×D12—92no—
291C2×D12—92no—
292C2×D12—92no—
293C2×D12—181no—
294C2×D12—182no—
295C2×D12—184no—
296C2×C4×D3—31no—
297C2×C4×D3—91no—
298C2×C4×D3—91no—
299C2×C4×D3—181no—
300C2×C4×D3C2×D911yescharacteristic
301C2×Dic6—91no—
302C2×Dic6—91no—
303C3×C4⋊C4—21no—
304C3×C4⋊C4—31no—
305C3×C4⋊C4C2×D911yescharacteristic, complemented
306C3×C4⋊C4D3211yescharacteristic
307C3×C22⋊C4—32no—
308C3×C22⋊C4—92no—
309C3×C42—32no—
310#217—92no—
311(C2×D3)⋊C4—32no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315(C2×D3)⋊C4—92no—
316(C2×D3)⋊C4—92no—
317(C2×D3)⋊C4—182no—
318(C2×D3)⋊C4C2×D912yes—
319C4⋊Dic3C2×D911yescharacteristic, complemented
320C4×Dic3C2×D912yescomplemented
321C22×C32C23×D311yescharacteristic
322C2×C3⋊D3—61no—
323C2×C3⋊D3—61no—
324C2×C3⋊D3—62no—
325C2×C3⋊D3—61no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—62no—
328C2×C3×D3—21no—
329C2×C3×D3—21no—
330C2×C3×D3—61no—
331C2×C3×D3—62no—
332C2×C3×D3—61no—
333C2×C3×D3—62no—
334C2×C3×D3—61no—
335C2×C3×D3—62no—
336C2×C3×D3C2×D1212yes—
337D32—128no—
338C32×C4—22no—
339C32×C4C2×D1212yes—
340C3⋊Dic3—61no—
341C3×Dic3—22no—
342C3×Dic3—61no—
343C3×Dic3C2×D1212yes—
344C22×C9—21no—
345C22×C9—32no—
346C22×C9—61no—
347C22×C9—61no—
348C22×C9C23×D311yescharacteristic
349C2×D9—21no—
350C2×D9—22no—
351C2×D9—21no—
352C2×D9—22no—
353C2×D9—21no—
354C2×D9—22no—
355C2×D9—62no—
356C2×D9—61no—
357C2×D9—62no—
358C2×D9—61no—
359C2×D9—62no—
360C2×D9—61no—
361C2×D9—62no—
362C2×D9—121no—
363C2×D9—121no—
364C2×D9—122no—
365C2×D9—122no—
366C2×D9—122no—
367C2×D9—128no—
368C4×C9—22no—
369C4×C9—22no—
370C4×C9—32no—
371C4×C9—42no—
372C4×C9—62no—
373C4×C9D3×D412yes—
374Dic9—21no—
375Dic9—61no—
376Dic9—121no—
377C2×#41—271no—
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—31no—
388C3×C9#53511yescharacteristic, complemented, Sylow
389C23×C3—31no—
390C23×C3—92no—
391C23×C3—91no—
392C22×D3—32no—
393C22×D3—31no—
394C22×D3—92no—
395C22×D3—91no—
396C22×D3—92no—
397C22×D3—91no—
398C22×D3—181no—
399C22×D3—182no—
400C22×D3—184no—
401C22×D3—184no—
402C22×D3—362no—
403C22×D3—362no—
404C22×D3—362no—
405C22×D3—362no—
406C22×D3C22×D911yescharacteristic
407C3×D4—182no—
408C3×D4—362no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—31no—
412C2×C3×C4—34no—
413C2×C3×C4—32no—
414C2×C3×C4—91no—
415C2×C3×C4—182no—
416C2×C3×C4C22×D912yes—
417C2×C3×C4C22×D911yescharacteristic
418C2×C3×C4C2×D3212yes—
419C2×C3×C4C2×D3211yescharacteristic
420#67—182no—
421#67—182no—
422#67—182no—
423#67—182no—
424#67—184no—
425#67—362no—
426#67—362no—
427C2×Dic3—31no—
428C2×Dic3—91no—
429C2×Dic3—91no—
430C2×Dic3—181no—
431C2×Dic3C22×D912yes—
432C2×Dic3C22×D911yescharacteristic
433D12—62no—
434D12—122no—
435D12—184no—
436D12—182no—
437D12—182no—
438D12—182no—
439D12—182no—
440D12—362no—
441D12—362no—
442D12—362no—
443D12—364no—
444D12—362no—
445D12—362no—
446C4×D3—62no—
447C4×D3—182no—
448C4×D3—182no—
449C4×D3—362no—
450C4×D3D3614yescomplemented
451Dic6—182no—
452Dic6—182no—
453C2×C32C22×D1211yescharacteristic
454C2×C32#100911yescharacteristic
455C2×C32#100911yescharacteristic
456C3⋊D3—121no—
457C3⋊D3—122no—
458C3×D3—22no—
459C3×D3—122no—
460C3×D3—121no—
461C2×C9—21no—
462C2×C9—21no—
463C2×C9—21no—
464C2×C9—62no—
465C2×C9C2×D3×D411yescharacteristic
466C2×C9#100411yescharacteristic
467C2×C9#100911yescharacteristic
468D9—41no—
469D9—42no—
470D9—122no—
471D9—121no—
472D9—122no—
473D9—242no—
474C24—272no—
475#41—544no—
476C2×Q8—271no—
477C2×D4—91no—
478C2×D4—92no—
479C2×D4—272no—
480C2×D4—271no—
481C2×D4—272no—
482C2×D4—272no—
483C2×D4—271no—
484C2×D4—271no—
485C2×D4—271no—
486C2×D4—272no—
487C2×D4—544no—
488C22×C4—31no—
489C22×C4—91no—
490C22×C4—271no—
491C4⋊C4—31no—
492C4⋊C4D3×D911yescharacteristic, complemented
493C22⋊C4—32no—
494C22⋊C4—92no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C22⋊C4—272no—
498C42—32no—
499C22×C3—21no—
500C22×C3—32no—
501C22×C3—61no—
502C22×C3—61no—
503C22×C3—181no—
504C22×C3—182no—
505C22×C3—181no—
506C22×C3—182no—
507C22×C3—181no—
508C22×C3—182no—
509C22×C3C23×D911yescharacteristic
510C22×C3C22×D3211yescharacteristic
511C2×D3—21no—
512C2×D3—21no—
513C2×D3—61no—
514C2×D3—62no—
515C2×D3—61no—
516C2×D3—62no—
517C2×D3—61no—
518C2×D3—62no—
519C2×D3—182no—
520C2×D3—181no—
521C2×D3—182no—
522C2×D3—181no—
523C2×D3—182no—
524C2×D3—181no—
525C2×D3—182no—
526C2×D3—181no—
527C2×D3—182no—
528C2×D3—181no—
529C2×D3—182no—
530C2×D3—181no—
531C2×D3—182no—
532C2×D3—361no—
533C2×D3—361no—
534C2×D3—362no—
535C2×D3—362no—
536C2×D3—362no—
537C2×D3—368no—
538C2×D3—368no—
539C2×D3C2×D3612yes—
540C3×C4—22no—
541C3×C4—22no—
542C3×C4—22no—
543C3×C4—32no—
544C3×C4—42no—
545C3×C4—62no—
546C3×C4—181no—
547C3×C4C2×D3612yes—
548C3×C4D3×D1212yes—
549Dic3—22no—
550Dic3—61no—
551Dic3—181no—
552Dic3—181no—
553Dic3—361no—
554Dic3C2×D3612yes—
555C32#552511yescharacteristic
556C9—21no—
557C9#560711yescharacteristic, complemented
558C23—31no—
559C23—92no—
560C23—91no—
561C23—272no—
562C23—271no—
563C23—544no—
564C23—1082no—
565C23—1082no—
566Q8—542no—
567D4—182no—
568D4—362no—
569D4—544no—
570D4—542no—
571D4—542no—
572D4—542no—
573D4—542no—
574D4—1082no—
575D4—1082no—
576D4—1082no—
577C2×C4—31no—
578C2×C4—34no—
579C2×C4—32no—
580C2×C4—91no—
581C2×C4—182no—
582C2×C4—271no—
583C2×C4—542no—
584C2×C4C2×D3×D912yes—
585C2×C4C2×D3×D911yescharacteristic
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—62no—
590C2×C3—362no—
591C2×C3—361no—
592C2×C3C22×D3611yescharacteristic
593C2×C3#6371511yescharacteristic
594C2×C3#6371511yescharacteristic
595C2×C3C2×D3×D1211yescharacteristic
596C2×C3#6431411yescharacteristic
597C2×C3#6432011yescharacteristic
598D3—22no—
599D3—122no—
600D3—121no—
601D3—362no—
602D3—362no—
603D3—361no—
604D3—362no—
605D3—361no—
606D3—722no—
607C22—32no—
608C22—61no—
609C22—61no—
610C22—181no—
611C22—182no—
612C22—181no—
613C22—182no—
614C22—181no—
615C22—182no—
616C22—541no—
617C22—542no—
618C22—541no—
619C22—542no—
620C22—541no—
621C22—542no—
622C22—1088no—
623C22C22×D3×D911yescharacteristic
624C4—22no—
625C4—32no—
626C4—62no—
627C4—181no—
628C4—541no—
629C4D3×D3612yes—
630C3—21no—
631C3#1058962711yescharacteristic, complemented
632C3#1059462811yescharacteristic
633C2—62no—
634C2—362no—
635C2—361no—
636C2—1082no—
637C2—1081no—
638C2C2×D3×D3611yescharacteristic
639C2#1172250111yescharacteristic
640C2#1172251511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#535
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×D36C2semidirect
2. C2×#88640C2semidirect
3. #11721751C2semidirect
4. C9×#892C2semidirect
5. C3×#63457C2semidirect
6. #11721051C2semidirect
7. #11721049C2semidirect
8. #11721007C2semidirect
9. #11721006C2semidirect
10. #11721004C2semidirect
11. #11720940C2semidirect
16. C2×C4⋊(C3⋊D9)C22semidirect
17. C2×C4×C3⋊D9C22semidirect
20. C2×C3×D36C22semidirect
21. C2×C3×C4×D9C22semidirect
25. C2×#6192C22semidirect
26. C2×#6192C22semidirect
27. C2×#6192C22semidirect
28. C2×#6189C22semidirect
31. C3×C9×C4⋊C4C22semidirect
32. #88536C22semidirect
34. C3×(C2×D9)⋊C4C22semidirect
35. C9×C4⋊Dic3C22semidirect
36. C4×C9×Dic3C22semidirect
37. #88445C22semidirect
38. (C2×D9)⋊Dic3C22semidirect
57. #63457D3semidirect
79. C4×D3×C9D4semidirect
132. C2×D36C2×D3semidirect
135. C2×C4×D9C2×D3semidirect
139. C9×C4⋊C4C2×D3semidirect
146. (C2×D9)⋊C4C2×D3semidirect
186. #892D9semidirect
305. C3×C4⋊C4C2×D9semidirect
319. C4⋊Dic3C2×D9semidirect
320. C4×Dic3C2×D9semidirect
388. C3×C9#535semidirect
450. C4×D3D36semidirect
492. C4⋊C4D3×D9semidirect
557. C9#5607semidirect
631. C3#10589627semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D362
C2×#886402
#117217512
C9×#8922
C3×#634572
#117210512
#117210492
#117210072
#117210062
#117210042
#117209402
#105946283
#105896273