Finite Group Databasebeta

← #13151154

Subgroups — #13151154

Order 1728.

1154 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88742C211yesmaximal, characteristic, complemented
2C22×D3×Dic9C211yesmaximal, characteristic, complemented
3C9×#937C211yesmaximal, characteristic, complemented
4C3×#63497C211yesmaximal, characteristic, complemented
5#11721287C211yesmaximal, characteristic, complemented
6#67⋊Dic9C212yesmaximal, complemented
7#11721282C212yesmaximal, complemented
8#11721255C211yesmaximal, characteristic, complemented
9#11721248C211yesmaximal, characteristic, complemented
10#11721024C212yesmaximal, complemented
11#11721014C212yesmaximal, complemented
12#10595160—31nomaximal
13#10589688—31nomaximal
14C23×D3×C9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic
16C2×#6233C2211yescharacteristic
17C2×#6233C2211yescharacteristic, complemented
18C22×C3⋊Dic9C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic, complemented
20#88742—24no—
21C2×C4×C3⋊D9C2211yescharacteristic, complemented
22C2×#6226C2211yescharacteristic, complemented
23C2×C9×#67C2212yes—
24C22×C3×Dic9C2211yescharacteristic
25C2×#6195C2212yes—
26C2×D3×Dic9—24no—
27C2×D3×Dic9C2212yes—
28C2×D3×Dic9C2212yes—
29C9×#217C2211yescharacteristic
30C3×#3676C2211yescharacteristic
31C3×#3676C2212yescomplemented
32C9×(C2×D3)⋊C4C2212yes—
33C3×Dic9⋊C4C2212yescomplemented
34(C2×D3)⋊Dic9C2212yes—
35#88445C2212yescomplemented
36Dic3⋊Dic9C2212yescomplemented
37#88442C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C2×#3830—31no—
40C22×D3×Dic3—31no—
41C3×#937D311yescharacteristic
42C3×#935—31no—
43#63977—31no—
44#67⋊Dic3—32no—
45#63971—32no—
46#63966—31no—
47#63957—31no—
48#63896—32no—
49#63881—32no—
50C23×Dic9—31no—
51C2×#3699—31no—
52C2×#3699—61no—
53C9×#120—31no—
54#63515—31no—
55#63501—32no—
56#63497D311yescharacteristic, complemented
57D4×Dic9—32no—
58#63448—31no—
59#611⋊C4—32no—
60#63440—32no—
61#63439—31no—
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2311yescharacteristic
64C22×C3⋊D9C2311yescharacteristic
65C22×D3×C9—22no—
66C22×D3×C9—24no—
67C22×D3×C9C2312yes—
68C3×D4×C9—22no—
69C2×C3×C4×C9C2311yescharacteristic
70#6233—22no—
71#6233—22no—
72C2×C3⋊Dic9—22no—
73C2×C3⋊Dic9D414yescomplemented
74C2×C3⋊Dic9C2311yescharacteristic
75C2×C3⋊Dic9C2311yescharacteristic
76C2×C3⋊Dic9C2311yescharacteristic
77C4×C3⋊D9—22no—
78#6226—22no—
79C9×#67—24no—
80C2×C9×Dic3C2312yes—
81C2×C3×Dic9—22no—
82C2×C3×Dic9C2312yes—
83C2×C3×Dic9C2312yes—
84#6195—24no—
85D3×Dic9—24no—
86D3×Dic9—24no—
87#5656—91no—
88#5586—91no—
89C23×C3×D3C2×D311yescharacteristic
90C2×C32×D4C2×D311yescharacteristic
91C2×#638—31no—
92C2×#638—31no—
93C22×C3⋊Dic3—31no—
94C22×C3⋊Dic3—31no—
95#3830—64no—
96C2×C4×C3⋊D3—31no—
97C2×#634—31no—
98C2×C3×#67C2×D312yes—
99C22×C3×Dic3—31no—
100C2×#626—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—64no—
104C24×C9—31no—
105C3×#217—31no—
106C3×#217—32no—
107C3×#217C2×D311yescharacteristic
108C3×(C2×D3)⋊C4C2×D312yes—
109C3×Dic3⋊C4—32no—
110#3724—32no—
111Dic3⋊Dic3—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114Dic32—32no—
115C2×D4×C9—21no—
116C2×D4×C9—32no—
117C2×D4×C9C2×D311yescharacteristic
118C2×#611—31no—
119C2×#611—31no—
120C2×#611—32no—
121C2×#611—61no—
122C2×#611—61no—
123C22×Dic9—31no—
124C22×Dic9—31no—
125C22×Dic9—32no—
126C22×Dic9—32no—
127C22×Dic9—64no—
128C22×Dic9—61no—
129C22×Dic9—61no—
130C22×Dic9C2×D311yescharacteristic
131#3699—64no—
132#3699—68no—
133C2×C4×D9—31no—
134C2×C4×D9—61no—
135C2×Dic18—31no—
136C2×Dic18—61no—
137C9×C22⋊C4—31no—
138C9×C22⋊C4—32no—
139#3676—32no—
140#3676C2×D311yescharacteristic
141#3676C2×D312yescomplemented
142(C2×D9)⋊C4—32no—
143C4⋊Dic9—32no—
144Dic9⋊C4—32no—
145Dic9⋊C4C2×D312yescomplemented
146C4×Dic9—32no—
147C22×C3×C9—22no—
148C22×C3×C9—21no—
149C22×C3×C9—22no—
150C22×C3×C9C2×D412yes—
151C22×C3×C9C2411yescharacteristic
152C2×C3⋊D9—21no—
153C2×C3⋊D9—22no—
154C2×D3×C9—28no—
155C2×D3×C9—22no—
156C2×D3×C9—22no—
157C2×D3×C9#4114yescomplemented
158C3×C4×C9—21no—
159C3⋊Dic9—21no—
160C3⋊Dic9C2×D412yes—
161C3⋊Dic9C2×D412yes—
162C9×Dic3—22no—
163C3×Dic9—22no—
164C3×Dic9—22no—
165C23×Dic3—91no—
166C2×#237—91no—
167C2×#237—91no—
168C2×#237—181no—
169C22×C4×D3—91no—
170C3×#123—91no—
171C3×#120—31no—
172#953—91no—
173#939—92no—
174#937D911yescharacteristic, complemented
175#935—31no—
176D4×Dic3—92no—
177#929—91no—
178#896—92no—
179#893—92no—
180#886—91no—
181#884—92no—
182#882—91no—
183#67⋊C4—92no—
184#67⋊C4—92no—
185#878—92no—
186#877—91no—
187#877—91no—
188C23×C32C22×D311yescharacteristic
189C23×C32C22×D311yescharacteristic
190C22×C3⋊D3—31no—
191C22×C3×D3—22no—
192C22×C3×D3—24no—
193C22×C3×D3C22×D312yes—
194C32×D4—22no—
195C2×C32×C4C22×D311yescharacteristic
196#638—62no—
197#638—62no—
198C2×C3⋊Dic3—34no—
199C2×C3⋊Dic3—31no—
200C2×C3⋊Dic3—31no—
201C2×C3⋊Dic3—31no—
202C2×C3⋊Dic3—62no—
203C4×C3⋊D3—62no—
204#634—62no—
205C3×#67—24no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3C22×D312yes—
210#626—64no—
211D3×Dic3—64no—
212D3×Dic3—64no—
213C23×C9—21no—
214C23×C9—21no—
215C23×C9—32no—
216C23×C9—64no—
217C23×C9—62no—
218C23×C9C22×D311yescharacteristic
219C23×C9C22×D311yescharacteristic
220C22×D9—31no—
221C22×D9—61no—
222D4×C9—24no—
223D4×C9—22no—
224D4×C9—64no—
225C2×C4×C9—21no—
226C2×C4×C9—32no—
227C2×C4×C9C22×D311yescharacteristic
228#611—64no—
229#611—62no—
230#611—64no—
231#611—64no—
232#611—62no—
233C2×Dic9—22no—
234C2×Dic9—34no—
235C2×Dic9—31no—
236C2×Dic9—31no—
237C2×Dic9—31no—
238C2×Dic9—64no—
239C2×Dic9—61no—
240C2×Dic9—61no—
241C2×Dic9—61no—
242C2×Dic9—64no—
243C2×Dic9—64no—
244C2×Dic9—62no—
245C2×Dic9—64no—
246C2×Dic9C22×D312yes—
247C2×Dic9C22×D312yes—
248C4×D9—62no—
249C4×D9—64no—
250Dic18—62no—
251Dic18—64no—
252#525—271noSylow
253C2×C3×C9—22no—
254C2×C3×C9—41no—
255C2×C3×C9C22×D411yescharacteristic
256C2×C3×C9C2×#4112yes—
257C3⋊D9—41no—
258D3×C9—24no—
259C24×C3—31no—
260C23×D3C2×D911yescharacteristic
261C2×C3×D4—21no—
262C2×C3×D4—32no—
263C2×C3×D4C2×D911yescharacteristic
264C2×C3×D4D3211yescharacteristic
265C22×C3×C4—91no—
266C2×#67—92no—
267C2×#67—91no—
268C2×#67—91no—
269C2×#67—91no—
270C2×#67—91no—
271C2×#67—181no—
272C2×#67—181no—
273C2×#67C2×D912yes—
274C22×Dic3—31no—
275C22×Dic3—92no—
276C22×Dic3—92no—
277C22×Dic3—91no—
278C22×Dic3—91no—
279C22×Dic3—91no—
280C22×Dic3—91no—
281C22×Dic3—181no—
282C22×Dic3—181no—
283C22×Dic3—184no—
284#237—184no—
285#237—184no—
286#237—188no—
287C2×D12—92no—
288C2×C4×D3—92no—
289C2×C4×D3—92no—
290C2×C4×D3—91no—
291C2×C4×D3—91no—
292C2×C4×D3—181no—
293C2×C4×D3—184no—
294C2×Dic6—91no—
295C2×Dic6—91no—
296C2×Dic6—181no—
297C3×C4⋊C4—92no—
298C3×C22⋊C4—31no—
299C3×C22⋊C4—32no—
300C3×C22⋊C4—91no—
301C3×C22⋊C4—92no—
302#217—31no—
303#217—32no—
304#217—92no—
305#217C2×D911yescharacteristic
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4C2×D912yes—
310C4⋊Dic3—92no—
311Dic3⋊C4—32no—
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315C4×Dic3—92no—
316C4×Dic3—92no—
317C22×C32—21no—
318C22×C32—22no—
319C22×C32—22no—
320C22×C32D3×D412yes—
321C22×C32C23×D311yescharacteristic
322C2×C3⋊D3—61no—
323C2×C3⋊D3—62no—
324C2×C3×D3—28no—
325C2×C3×D3—22no—
326C2×C3×D3—22no—
327C2×C3×D3#23714yes—
328C32×C4—21no—
329C3⋊Dic3—32no—
330C3⋊Dic3—32no—
331C3⋊Dic3—61no—
332C3×Dic3—22no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C22×C9—22no—
336C22×C9—22no—
337C22×C9—21no—
338C22×C9—22no—
339C22×C9—21no—
340C22×C9—22no—
341C22×C9—34no—
342C22×C9—42no—
343C22×C9—42no—
344C22×C9—68no—
345C22×C9—62no—
346C22×C9—62no—
347C22×C9D3×D412yes—
348C22×C9C23×D311yescharacteristic
349C2×D9—62no—
350C2×D9—61no—
351C2×D9—62no—
352C2×D9—122no—
353C4×C9—22no—
354C4×C9—21no—
355C4×C9—62no—
356Dic9—22no—
357Dic9—22no—
358Dic9—32no—
359Dic9—32no—
360Dic9—62no—
361Dic9—62no—
362Dic9—62no—
363Dic9—61no—
364C2×#41—271no—
365C23×C4—271no—
366#123—91no—
367#123—271no—
368C22⋊Q8—272no—
369#121—272no—
370#120—31no—
371#120—271no—
372C4×D4—272no—
373C4×D4—272no—
374#117—271no—
375C3×C9#52511yescharacteristic, complemented, Sylow
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—32no—
379C23×C3—64no—
380C23×C3—62no—
381C23×C3C22×D911yescharacteristic
382C23×C3C22×D911yescharacteristic
383C23×C3C2×D3211yescharacteristic
384C23×C3C2×D3211yescharacteristic
385C22×D3—22no—
386C22×D3—24no—
387C22×D3—91no—
388C22×D3—91no—
389C22×D3—181no—
390C22×D3C22×D912yes—
391C3×D4—24no—
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—64no—
395C2×C3×C4—21no—
396C2×C3×C4—32no—
397C2×C3×C4—92no—
398C2×C3×C4—92no—
399C2×C3×C4—182no—
400C2×C3×C4C22×D911yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402#67—24no—
403#67—184no—
404#67—182no—
405#67—182no—
406#67—184no—
407#67—184no—
408#67—182no—
409#67—182no—
410C2×Dic3—32no—
411C2×Dic3—32no—
412C2×Dic3—62no—
413C2×Dic3—91no—
414C2×Dic3—91no—
415C2×Dic3—94no—
416C2×Dic3—91no—
417C2×Dic3—91no—
418C2×Dic3—94no—
419C2×Dic3—91no—
420C2×Dic3—91no—
421C2×Dic3—181no—
422C2×Dic3—181no—
423C2×Dic3—181no—
424C2×Dic3—184no—
425C2×Dic3—184no—
426C2×Dic3—184no—
427C2×Dic3—182no—
428C2×Dic3—182no—
429C2×Dic3—184no—
430C2×Dic3C22×D912yes—
431D12—184no—
432C4×D3—182no—
433C4×D3—182no—
434C4×D3—184no—
435C4×D3—184no—
436C4×D3—184no—
437Dic6—182no—
438Dic6—182no—
439Dic6—184no—
440C2×C32—22no—
441C2×C32—41no—
442C2×C32C2×D3×D411yescharacteristic
443C2×C32C2×#23712yes—
444C3⋊D3—121no—
445C3×D3—24no—
446C2×C9—22no—
447C2×C9—22no—
448C2×C9—21no—
449C2×C9—42no—
450C2×C9—42no—
451C2×C9—41no—
452C2×C9—64no—
453C2×C9C2×D3×D411yescharacteristic
454C2×C9D3×#4112yes—
455D9—122no—
456D9—121no—
457C24—31no—
458#41—544no—
459C2×Q8—271no—
460C2×D4—32no—
461C2×D4—272no—
462C2×D4—271no—
463C2×D4—271no—
464C2×D4D3×D911yescharacteristic
465C22×C4—91no—
466C22×C4—272no—
467C22×C4—272no—
468C22×C4—271no—
469C22×C4—271no—
470C22×C4—271no—
471C22×C4—544no—
472C4⋊C4—92no—
473C4⋊C4—272no—
474C4⋊C4—272no—
475C22⋊C4—31no—
476C22⋊C4—32no—
477C22⋊C4—91no—
478C22⋊C4—92no—
479C22⋊C4—272no—
480C22⋊C4—272no—
481C42—272no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—22no—
489C22×C3—21no—
490C22×C3—22no—
491C22×C3—34no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—68no—
495C22×C3—62no—
496C22×C3—62no—
497C22×C3D4×D912yes—
498C22×C3C23×D911yescharacteristic
499C22×C3#381112yes—
500C22×C3C22×D3211yescharacteristic
501C2×D3—28no—
502C2×D3—22no—
503C2×D3—22no—
504C2×D3—182no—
505C2×D3—182no—
506C2×D3—181no—
507C2×D3—181no—
508C2×D3—182no—
509C2×D3—362no—
510C2×D3#369914yescomplemented
511C3×C4—22no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—62no—
515C3×C4—182no—
516C3×C4—182no—
517Dic3—22no—
518Dic3—62no—
519Dic3—62no—
520Dic3—92no—
521Dic3—92no—
522Dic3—92no—
523Dic3—92no—
524Dic3—182no—
525Dic3—182no—
526Dic3—182no—
527Dic3—181no—
528Dic3—181no—
529C32#558611yescharacteristic
530C9—21no—
531C9#565611yescharacteristic, complemented
532C23—32no—
533C23—64no—
534C23—62no—
535C23—271no—
536C23C2×D3×D911yescharacteristic
537C23C2×D3×D911yescharacteristic
538Q8—542no—
539D4—22no—
540D4—64no—
541D4—544no—
542D4—542no—
543D4—542no—
544C2×C4—32no—
545C2×C4—92no—
546C2×C4—92no—
547C2×C4—182no—
548C2×C4—271no—
549C2×C4—271no—
550C2×C4—274no—
551C2×C4—271no—
552C2×C4—542no—
553C2×C4—544no—
554C2×C4—544no—
555C2×C4—542no—
556C2×C4C2×D3×D911yescharacteristic
557C2×C3—22no—
558C2×C3—22no—
559C2×C3—22no—
560C2×C3—21no—
561C2×C3—42no—
562C2×C3—42no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—64no—
566C2×C3C2×D4×D911yescharacteristic
567C2×C3C2×#369912yes—
568C2×C3D3×#23712yes—
569C2×C3C2×#381111yescharacteristic
570D3—24no—
571D3—362no—
572D3—361no—
573D3—361no—
574C22—22no—
575C22—22no—
576C22—21no—
577C22—34no—
578C22—68no—
579C22—62no—
580C22—62no—
581C22—542no—
582C22—541no—
583C22#8866812yes—
584C22C22×D3×D911yescharacteristic
585C4—21no—
586C4—62no—
587C4—182no—
588C4—182no—
589C4—272no—
590C4—272no—
591C4—541no—
592C3—21no—
593C3#1058968811yescharacteristic, complemented
594C3#1059516011yescharacteristic
595C2—22no—
596C2—41no—
597C2—64no—
598C2—1081no—
599C2D3×#369912yes—
600C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#525
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×#88742C2semidirect
2. C22×D3×Dic9C2semidirect
3. C9×#937C2semidirect
4. C3×#63497C2semidirect
5. #11721287C2semidirect
6. #67⋊Dic9C2semidirect
7. #11721282C2semidirect
8. #11721255C2semidirect
9. #11721248C2semidirect
10. #11721024C2semidirect
11. #11721014C2semidirect
17. C2×#6233C22semidirect
19. C22×C3⋊Dic9C22semidirect
21. C2×C4×C3⋊D9C22semidirect
22. C2×#6226C22semidirect
31. C3×#3676C22semidirect
33. C3×Dic9⋊C4C22semidirect
35. #88445C22semidirect
36. Dic3⋊Dic9C22semidirect
37. #88442C22semidirect
38. Dic3×Dic9C22semidirect
56. #63497D3semidirect
73. C2×C3⋊Dic9D4semidirect
141. #3676C2×D3semidirect
145. Dic9⋊C4C2×D3semidirect
157. C2×D3×C9#41semidirect
174. #937D9semidirect
375. C3×C9#525semidirect
510. C2×D3#3699semidirect
531. C9#5656semidirect
593. C3#10589688semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
C22×D3×Dic92
C9×#9372
C3×#634972
#117212872
#67⋊Dic92
#117212822
#117212552
#117212482
#117210242
#117210142
#105951603
#105896883