Finite Group Databasebeta

← #13151148

Subgroups — #13151148

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×D4×C9C211yesmaximal, characteristic, complemented
2C2×#88639C211yesmaximal, characteristic, complemented
3#11721838C211yesmaximal, characteristic, complemented
4C3×#63498C211yesmaximal, characteristic, complemented
5#11721287C212yesmaximal, complemented
6#11721282C212yesmaximal, complemented
7#11721280C212yesmaximal, complemented
8D3×#3676C212yesmaximal, complemented
9C4×#6195C211yesmaximal, characteristic, complemented
10D12×Dic9C211yesmaximal, characteristic, complemented
11#11720960C211yesmaximal, characteristic, complemented
12#10595229—31nomaximal
13#10589908—31nomaximal
14C23×D3×C9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C2×#6233C2212yescomplemented
17C2×C4×C3⋊D9C2211yescharacteristic, complemented
18C2×C9×#67C2212yes—
19D3×D4×C9—24no—
20C2×C9×D12C2211yescharacteristic
21C2×C4×D3×C9C2211yescharacteristic
22C2×C3×Dic18C2211yescharacteristic, complemented
23C2×#6195C2211yescharacteristic
24C2×#6195C2212yescomplemented
25C2×D3×Dic9C2212yescomplemented
26C2×D3×Dic9C2212yescomplemented
27#88639—24no—
28#88551C2212yescomplemented
29C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
30C3×#3676C2212yescomplemented
31C3×#3676C2212yescomplemented
32C3×C4×Dic9C2211yescharacteristic, complemented
33(C2×D3)⋊Dic9C2212yes—
34(C2×D3)⋊Dic9C2211yescharacteristic
35(C2×D3)⋊Dic9C2212yescomplemented
36#88445C2211yescharacteristic, complemented
37#88442C2212yescomplemented
38Dic9⋊Dic3C2211yescharacteristic, complemented
39C2×C3×D3×D4D311yescharacteristic
40C2×#3796—31no—
41#64147—31no—
42C3×#936—31no—
43#63977—32no—
44#63971—32no—
45#63969—32no—
46D3×#217—32no—
47C4×#626—31no—
48Dic3×D12—31no—
49#63856—31no—
50C22×D4×C9—31no—
51C2×#3699—31no—
52#63515—32no—
53C2×#3676—32no—
54#63501—32no—
55#63500—31no—
56#63500—61no—
57#63498D311yescharacteristic, complemented
58#63497—32no—
59D4×Dic9—31no—
60C4×#611—31no—
61#63486—31no—
62C23×C3×C9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9—24no—
65C22×D3×C9—24no—
66C22×D3×C9—42no—
67C22×D3×C9C2312yes—
68C22×D3×C9C2311yescharacteristic
69C3×D4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71#6233—42no—
72C2×C3⋊Dic9C2311yescharacteristic
73C2×C3⋊Dic9C2312yes—
74C4×C3⋊D9—22no—
75C9×#67—24no—
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2312yes—
79C2×C3×Dic9C2311yescharacteristic
80C9×D12D414yescomplemented
81C4×D3×C9—22no—
82C3×Dic18—22no—
83#6195—24no—
84#6195—22no—
85D3×Dic9—24no—
86D3×Dic9—42no—
87#5806—91no—
88#5676—91no—
89C23×C3×D3C2×D312yes—
90C2×C32×D4C2×D311yescharacteristic
91C2×#638—32no—
92C2×C4×C3⋊D3—31no—
93C2×C3×#67C2×D312yes—
94C3×D3×D4—24no—
95C2×C3×D12C2×D311yescharacteristic
96C2×C3×C4×D3C2×D311yescharacteristic
97C2×C3×Dic6—31no—
98C2×#626—32no—
99C2×#626—31no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102#3796—64no—
103C24×C9—32no—
104#3757—32no—
105C4⋊(C3⋊Dic3)—31no—
106C3×#217—32no—
107C3×#217—32no—
108C3×C4×Dic3—31no—
109#3724—32no—
110Dic3⋊Dic3—31no—
111#3722—31no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—31no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—21no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C2×D4×C9—64no—
119C2×D4×C9C2×D311yescharacteristic
120C22×C4×C9—31no—
121C2×#611—32no—
122C2×#611—31no—
123C2×#611—32no—
124C2×#611—62no—
125C22×Dic9—32no—
126C22×Dic9—32no—
127#3699—64no—
128C2×C4×D9—31no—
129C2×C4×D9—61no—
130C2×Dic18C2×D311yescharacteristic, complemented
131#3676—32no—
132#3676—31no—
133#3676—32no—
134#3676—32no—
135#3676—62no—
136#3676C2×D312yescomplemented
137#3676C2×D312yescomplemented
138(C2×D9)⋊C4—31no—
139C4⋊Dic9—31no—
140C4⋊Dic9—61no—
141Dic9⋊C4—31no—
142Dic9⋊C4—32no—
143C4×Dic9C2×D311yescharacteristic, complemented
144C22×C3×C9—22no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9C2411yescharacteristic
148C2×C3⋊D9—21no—
149C2×C3⋊D9—21no—
150C2×C3⋊D9—21no—
151C2×D3×C9—21no—
152C2×D3×C9—22no—
153C2×D3×C9—22no—
154C2×D3×C9—21no—
155C2×D3×C9—48no—
156C2×D3×C9C2×D414yes—
157C2×D3×C9#4112yescomplemented
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—21no—
160C3⋊Dic9—22no—
161C9×Dic3—21no—
162C3×Dic9—22no—
163C3×Dic9—21no—
164C3×Dic9—21no—
165C22×C3×D4—31no—
166C2×#239—91no—
167C2×#237—91no—
168C2×D3×D4D911yescharacteristic, complemented
169C3×#124—91no—
170#953—92no—
171C2×#217—92no—
172#946—91no—
173#939—92no—
174#938—91no—
175#938—91no—
176#938—181no—
177#936—31no—
178#935—92no—
179D4×Dic3—91no—
180C4×#67—91no—
181#924—91no—
182#884—92no—
183#883—92no—
184#882—92no—
185D3×C22⋊C4—92no—
186C4×D12—91no—
187C4×D12—91no—
188C23×C32C22×D312yes—
189C22×C3⋊D3—31no—
190C22×C3×D3—24no—
191C22×C3×D3—24no—
192C22×C3×D3—42no—
193C22×C3×D3C22×D311yescharacteristic
194C22×C3×D3C22×D312yes—
195C32×D4—22no—
196C2×C32×C4C22×D311yescharacteristic
197#638—122no—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—31no—
200C4×C3⋊D3—62no—
201C3×#67—24no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3C22×D311yescharacteristic
206C3×D12#6714yes—
207C3×C4×D3—22no—
208C3×Dic6—62no—
209#626—62no—
210#626—64no—
211D3×Dic3—64no—
212D3×Dic3—122no—
213C23×C9—22no—
214C23×C9—31no—
215C23×C9—32no—
216C23×C9—64no—
217C23×C9—64no—
218C23×C9—122no—
219C23×C9C22×D312yes—
220C22×D9—31no—
221C22×D9—61no—
222D4×C9—22no—
223D4×C9—34no—
224D4×C9—42no—
225D4×C9—64no—
226C2×C4×C9—21no—
227C2×C4×C9—31no—
228C2×C4×C9—62no—
229C2×C4×C9C22×D311yescharacteristic
230#611—64no—
231#611—62no—
232#611—122no—
233#611—124no—
234C2×Dic9—32no—
235C2×Dic9—31no—
236C2×Dic9—62no—
237C2×Dic9—61no—
238C2×Dic9—64no—
239C2×Dic9—122no—
240C2×Dic9C22×D311yescharacteristic
241C2×Dic9C22×D311yescharacteristic
242C2×Dic9C22×D312yes—
243C4×D9—62no—
244C4×D9—122no—
245Dic18—22no—
246#546—271noSylow
247C2×C3×C9—42no—
248C2×C3×C9C22×D411yescharacteristic
249C2×C3×C9C2×#4111yescharacteristic
250C2×C3×C9H(5,𝔽2)11yescharacteristic
251C3⋊D9—41no—
252D3×C9—22no—
253D3×C9—24no—
254C24×C3—32no—
255C23×D3C2×D912yes—
256C2×C3×Q8—91no—
257C2×C3×D4—21no—
258C2×C3×D4—32no—
259C2×C3×D4—31no—
260C2×C3×D4—64no—
261C2×C3×D4C2×D911yescharacteristic
262C2×C3×D4D3211yescharacteristic
263C22×C3×C4—31no—
264C2×#67—91no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—92no—
268C2×#67—182no—
269C2×#67C2×D912yes—
270C22×Dic3—92no—
271C22×Dic3—92no—
272#239—184no—
273#237—184no—
274D3×D4—24no—
275C2×D12—91no—
276C2×D12—92no—
277C2×D12C2×D911yescharacteristic
278C2×C4×D3—92no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—92no—
282C2×C4×D3—181no—
283C2×C4×D3C2×D911yescharacteristic
284C2×Dic6—31no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4—92no—
287C3×C42—91no—
288#217—32no—
289#217—32no—
290#217—92no—
291#217—91no—
292#217—92no—
293#217—92no—
294#217—92no—
295#217—182no—
296(C2×D3)⋊C4—91no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—91no—
301C4⋊Dic3—91no—
302C4⋊Dic3—91no—
303C4⋊Dic3—91no—
304C4⋊Dic3—181no—
305Dic3⋊C4—91no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308C4×Dic3—31no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—61no—
314C2×C3⋊D3—61no—
315C2×C3⋊D3—61no—
316C2×C3×D3—22no—
317C2×C3×D3—22no—
318C2×C3×D3—21no—
319C2×C3×D3—21no—
320C2×C3×D3—48no—
321C2×C3×D3#23712yes—
322C2×C3×D3C2×#6714yes—
323C32×C4C2×#6712yes—
324C3⋊Dic3—62no—
325C3⋊Dic3—61no—
326C3×Dic3—21no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—62no—
330C22×C9—22no—
331C22×C9—22no—
332C22×C9—22no—
333C22×C9—21no—
334C22×C9—34no—
335C22×C9—32no—
336C22×C9—42no—
337C22×C9—42no—
338C22×C9—42no—
339C22×C9—61no—
340C22×C9—62no—
341C22×C9—62no—
342C22×C9—61no—
343C22×C9—128no—
344C22×C9C23×D311yescharacteristic
345C2×D9—62no—
346C2×D9—61no—
347C2×D9—61no—
348C2×D9—61no—
349C2×D9—121no—
350C2×D9—121no—
351C4×C9—22no—
352C4×C9—61no—
353C4×C9D3×D412yes—
354Dic9—22no—
355Dic9—21no—
356Dic9—21no—
357Dic9—61no—
358Dic9—62no—
359Dic9—121no—
360Dic9—122no—
361C2×#41—271no—
362C22×D4—31no—
363#124—91no—
364#123—272no—
365C22⋊Q8—271no—
366#121—272no—
367#121—271no—
368#120—272no—
369C4×D4—271no—
370C4×D4—271no—
371C2×C22⋊C4—272no—
372C3×C9#54611yescharacteristic, complemented, Sylow
373C23×C3—22no—
374C23×C3—31no—
375C23×C3—32no—
376C23×C3—64no—
377C23×C3—64no—
378C23×C3—122no—
379C23×C3C22×D912yes—
380C23×C3C2×D3212yes—
381C22×D3—24no—
382C22×D3—24no—
383C22×D3—42no—
384C22×D3—91no—
385C22×D3—91no—
386C22×D3—181no—
387C22×D3C22×D911yescharacteristic
388C22×D3C22×D912yes—
389C3×Q8—182no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—34no—
393C3×D4—42no—
394C3×D4—64no—
395C2×C3×C4—21no—
396C2×C3×C4—31no—
397C2×C3×C4—62no—
398C2×C3×C4—92no—
399C2×C3×C4—91no—
400C2×C3×C4—91no—
401C2×C3×C4C22×D911yescharacteristic
402C2×C3×C4C2×D3211yescharacteristic
403#67—24no—
404#67—184no—
405#67—182no—
406#67—362no—
407#67—362no—
408#67—364no—
409C2×Dic3—31no—
410C2×Dic3—31no—
411C2×Dic3—32no—
412C2×Dic3—92no—
413C2×Dic3—91no—
414C2×Dic3—92no—
415C2×Dic3—91no—
416C2×Dic3—182no—
417C2×Dic3—181no—
418C2×Dic3—184no—
419C2×Dic3—362no—
420C2×Dic3C22×D911yescharacteristic
421D12—182no—
422D12—184no—
423D12#61114yescomplemented
424C4×D3—22no—
425C4×D3—184no—
426C4×D3—182no—
427C4×D3—182no—
428C4×D3—362no—
429C4×D3—362no—
430Dic6—62no—
431C2×C32—42no—
432C2×C32C2×#23711yescharacteristic
433C2×C32#100411yescharacteristic
434C2×C32C22×#6711yescharacteristic
435C3⋊D3—121no—
436C3×D3—22no—
437C3×D3—24no—
438C2×C9—21no—
439C2×C9—21no—
440C2×C9—21no—
441C2×C9—42no—
442C2×C9—64no—
443C2×C9—62no—
444C2×C9—82no—
445C2×C9C2×D3×D411yescharacteristic
446C2×C9D3×#4111yescharacteristic
447C2×C9#100911yescharacteristic
448D9—122no—
449D9—121no—
450C24—32no—
451#41—544no—
452C2×Q8—91no—
453C2×D4—32no—
454C2×D4—31no—
455C2×D4—64no—
456C2×D4—271no—
457C2×D4—272no—
458C2×D4—272no—
459C2×D4D3×D911yescharacteristic
460C22×C4—31no—
461C22×C4—272no—
462C22×C4—272no—
463C22×C4—271no—
464C4⋊C4—271no—
465C4⋊C4—272no—
466C4⋊C4—271no—
467C22⋊C4—92no—
468C22⋊C4—92no—
469C22⋊C4—271no—
470C22⋊C4—271no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—91no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—32no—
483C22×C3—34no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—62no—
488C22×C3—62no—
489C22×C3—61no—
490C22×C3—61no—
491C22×C3—128no—
492C22×C3C23×D911yescharacteristic
493C22×C3C22×D3211yescharacteristic
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—21no—
497C2×D3—21no—
498C2×D3—48no—
499C2×D3—182no—
500C2×D3—181no—
501C2×D3—181no—
502C2×D3—181no—
503C2×D3—181no—
504C2×D3—181no—
505C2×D3—181no—
506C2×D3—361no—
507C2×D3—361no—
508C2×D3#369912yescomplemented
509C2×D3C2×#61114yes—
510C3×C4—22no—
511C3×C4—61no—
512C3×C4—181no—
513C3×C4—181no—
514C3×C4—182no—
515C3×C4C2×#61112yes—
516C3×C4D3×#6712yes—
517Dic3—21no—
518Dic3—61no—
519Dic3—61no—
520Dic3—62no—
521Dic3—182no—
522Dic3—181no—
523Dic3—182no—
524Dic3—181no—
525Dic3—361no—
526Dic3—362no—
527C32#580611yescharacteristic
528C9—21no—
529C9#567611yescharacteristic, complemented
530C23—31no—
531C23—32no—
532C23—64no—
533C23—64no—
534C23—122no—
535C23—271no—
536C23C2×D3×D912yes—
537Q8—182no—
538D4—22no—
539D4—34no—
540D4—64no—
541D4—544no—
542D4—542no—
543D4—1082no—
544C2×C4—31no—
545C2×C4—62no—
546C2×C4—91no—
547C2×C4—91no—
548C2×C4—92no—
549C2×C4—272no—
550C2×C4—271no—
551C2×C4—544no—
552C2×C4—542no—
553C2×C4—1082no—
554C2×C4C2×D3×D911yescharacteristic
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—42no—
559C2×C3—42no—
560C2×C3—62no—
561C2×C3—64no—
562C2×C3—82no—
563C2×C3C2×#369911yescharacteristic
564C2×C3#6371011yescharacteristic
565C2×C3C22×#61111yescharacteristic
566C2×C3D3×#23711yescharacteristic
567C2×C3#6431411yescharacteristic
568C2×C3C2×D3×#6711yescharacteristic
569D3—22no—
570D3—24no—
571D3—362no—
572D3—361no—
573D3—361no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—32no—
578C22—34no—
579C22—62no—
580C22—62no—
581C22—61no—
582C22—61no—
583C22—128no—
584C22—541no—
585C22—541no—
586C22—541no—
587C22C22×D3×D911yescharacteristic
588C4—61no—
589C4—181no—
590C4—181no—
591C4—182no—
592C4—542no—
593C4—541no—
594C4D3×#61112yes—
595C3—21no—
596C3#1058990811yescharacteristic, complemented
597C3#1059522911yescharacteristic
598C2—42no—
599C2—62no—
600C2—64no—
601C2—1081no—
602C2D3×#369911yescharacteristic
603C2#1172250011yescharacteristic
604C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#546
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D3×D4×C9C2semidirect
2. C2×#88639C2semidirect
3. #11721838C2semidirect
4. C3×#63498C2semidirect
5. #11721287C2semidirect
6. #11721282C2semidirect
7. #11721280C2semidirect
8. D3×#3676C2semidirect
9. C4×#6195C2semidirect
10. D12×Dic9C2semidirect
11. #11720960C2semidirect
16. C2×#6233C22semidirect
17. C2×C4×C3⋊D9C22semidirect
22. C2×C3×Dic18C22semidirect
24. C2×#6195C22semidirect
25. C2×D3×Dic9C22semidirect
26. C2×D3×Dic9C22semidirect
28. #88551C22semidirect
29. C4⋊(C3⋊Dic9)C22semidirect
30. C3×#3676C22semidirect
31. C3×#3676C22semidirect
32. C3×C4×Dic9C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
36. #88445C22semidirect
37. #88442C22semidirect
38. Dic9⋊Dic3C22semidirect
57. #63498D3semidirect
80. C9×D12D4semidirect
130. C2×Dic18C2×D3semidirect
136. #3676C2×D3semidirect
137. #3676C2×D3semidirect
143. C4×Dic9C2×D3semidirect
157. C2×D3×C9#41semidirect
168. C2×D3×D4D9semidirect
372. C3×C9#546semidirect
423. D12#611semidirect
508. C2×D3#3699semidirect
529. C9#5676semidirect
596. C3#10589908semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D4×C92
C2×#886392
#117218382
C3×#634982
#117212872
#117212822
#117212802
D3×#36762
C4×#61952
D12×Dic92
#117209602
#105952293
#105899083