Finite Group Databasebeta

← #13150992

Subgroups — #13150992

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic, complemented
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3#11721837C211yesmaximal, characteristic, complemented
4C3×#63497C211yesmaximal, characteristic, complemented
5#11721291C211yesmaximal, characteristic, complemented
6#11721265C211yesmaximal, characteristic, complemented
7#11721262C212yesmaximal, complemented
8#11721244C211yesmaximal, characteristic, complemented
9#11721240C212yesmaximal, complemented
10#11721030C212yesmaximal, complemented
11#11720986C212yesmaximal, complemented
12#10595159—31nomaximal
13#10589930—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic
16C2×#6233C2212yes—
17C2×C9×#67C2211yescharacteristic
18C2×C9×#67C2211yescharacteristic, complemented
19C22×C9×Dic3C2211yescharacteristic
20C22×C9×Dic3C2211yescharacteristic, complemented
21C22×C3×Dic9C2211yescharacteristic
22C9×#237—24no—
23C2×C4×D3×C9C2211yescharacteristic, complemented
24C2×C9×Dic6C2211yescharacteristic, complemented
25C2×#6195C2212yes—
26C2×(C3⋊D9)⋊C4—24no—
27C2×(C3⋊D9)⋊C4C2212yes—
28C2×(C3⋊D9)⋊C4C2212yes—
29#88551C2211yescharacteristic
30#88536C2212yes—
31C3×#3676C2211yescharacteristic
32C3×#3676C2212yescomplemented
33C3×Dic9⋊C4C2212yescomplemented
34(C2×D3)⋊Dic9C2212yescomplemented
35#88445C2212yes—
36Dic3⋊Dic9C2212yescomplemented
37#88442C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C2×C3×#237D311yescharacteristic
40C22×#624—31no—
41#64146—31no—
42C3×#935—31no—
43#63981—31no—
44#63961—31no—
45#63958—32no—
46#63955—31no—
47#63952—32no—
48#63904—32no—
49#63888—32no—
50C2×C9×#41—31no—
51C22×C4×D9—31no—
52#63508—32no—
53#63500—32no—
54#63499—31no—
55#63499—31no—
56#63499—61no—
57#63497—31no—
58#63497D311yescharacteristic, complemented
59C4×#611—32no—
60C4×#611—32no—
61#63488—31no—
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2311yescharacteristic
64C22×C3⋊D9—22no—
65C22×C3⋊D9—24no—
66C22×C3⋊D9C2312yes—
67C22×D3×C9C2311yescharacteristic
68C3×D4×C9—22no—
69C2×C3×C4×C9C2311yescharacteristic
70#6233—24no—
71C2×C3⋊Dic9C2312yes—
72C9×#67—22no—
73C9×#67—22no—
74C2×C9×Dic3—22no—
75C2×C9×Dic3D414yescomplemented
76C2×C9×Dic3C2311yescharacteristic
77C2×C9×Dic3C2311yescharacteristic
78C2×C9×Dic3C2311yescharacteristic
79C2×C3×Dic9—22no—
80C2×C3×Dic9C2312yes—
81C2×C3×Dic9C2312yes—
82C4×D3×C9—22no—
83C9×Dic6—22no—
84#6195—24no—
85(C3⋊D9)⋊C4—24no—
86(C3⋊D9)⋊C4—24no—
87#5828—91no—
88#5656—91no—
89C23×C3⋊D3—31no—
90C2×C32×D4C2×D311yescharacteristic
91C2×#638—32no—
92C2×C3×#67C2×D311yescharacteristic
93C2×C3×#67C2×D311yescharacteristic
94C22×C3×Dic3—31no—
95C22×C3×Dic3C2×D311yescharacteristic
96C22×C3×Dic3C2×D311yescharacteristic
97C3×#237—24no—
98C2×C3×C4×D3C2×D311yescharacteristic
99C2×C3×Dic6C2×D311yescharacteristic
100C2×#626—32no—
101C2×#624—32no—
102C2×#624—32no—
103C2×#624—64no—
104C23×D9—31no—
105C23×D9—61no—
106#3757—31no—
107#3752—32no—
108C3×#217—31no—
109C3×#217—32no—
110C3×Dic3⋊C4—32no—
111#3724—32no—
112Dic3⋊Dic3—32no—
113#3722—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C9×#41—64no—
117C2×Q8×C9—31no—
118C2×D4×C9—21no—
119C2×D4×C9—31no—
120C2×D4×C9—31no—
121C2×D4×C9C2×D311yescharacteristic
122C22×C4×C9—31no—
123C22×C4×C9—31no—
124C22×C4×C9—31no—
125C2×#611—32no—
126C2×#611—32no—
127C2×#611—62no—
128C22×Dic9C2×D311yescharacteristic
129C2×C4×D9—32no—
130C2×C4×D9—32no—
131C2×C4×D9—64no—
132#3676—31no—
133#3676—32no—
134#3676—61no—
135#3676C2×D311yescharacteristic
136#3676C2×D312yescomplemented
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—62no—
140C4⋊Dic9—32no—
141Dic9⋊C4—32no—
142Dic9⋊C4C2×D312yescomplemented
143C4×Dic9—32no—
144C22×C3×C9—22no—
145C22×C3×C9—21no—
146C22×C3×C9—22no—
147C22×C3×C9C2×D412yes—
148C22×C3×C9C2411yescharacteristic
149C2×C3⋊D9—28no—
150C2×C3⋊D9—22no—
151C2×C3⋊D9—22no—
152C2×C3⋊D9#4114yescomplemented
153C2×D3×C9—21no—
154C2×D3×C9—22no—
155C3×C4×C9—21no—
156C3⋊Dic9—22no—
157C9×Dic3—21no—
158C9×Dic3C2×D412yes—
159C9×Dic3C2×D412yes—
160C3×Dic9—22no—
161C3×Dic9—22no—
162C2×C3×#41—31no—
163C2×#237D911yescharacteristic, complemented
164C22×C4×D3—91no—
165C22×C4×D3—91no—
166C3×#123—91no—
167#946—92no—
168#938—92no—
169#937—91no—
170#937—91no—
171#937—91no—
172#937—181no—
173#935—31no—
174#935—91no—
175#929—91no—
176C4×#67—92no—
177C4×#67—92no—
178#926—91no—
179#896—92no—
180#893—92no—
181#884—92no—
182#882—91no—
183#67⋊C4—92no—
184#877—91no—
185C23×C32C22×D311yescharacteristic
186C23×C32C22×D311yescharacteristic
187C22×C3⋊D3—32no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3—62no—
190C22×C3×D3C22×D311yescharacteristic
191C32×D4—22no—
192C2×C32×C4C22×D311yescharacteristic
193#638—64no—
194C2×C3⋊Dic3—32no—
195C3×#67—22no—
196C3×#67—22no—
197C2×C3×Dic3—22no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3#6714yes—
202C2×C3×Dic3C22×D311yescharacteristic
203C2×C3×Dic3C22×D311yescharacteristic
204C2×C3×Dic3C22×D311yescharacteristic
205C3×C4×D3—22no—
206C3×Dic6—22no—
207#626—64no—
208#624—64no—
209#624—64no—
210C23×C9—21no—
211C23×C9—21no—
212C23×C9—31no—
213C23×C9C22×D311yescharacteristic
214C23×C9C22×D311yescharacteristic
215C22×D9—32no—
216C22×D9—62no—
217C22×D9—64no—
218C22×D9—62no—
219C22×D9—124no—
220C22×D9—122no—
221Q8×C9—62no—
222D4×C9—22no—
223D4×C9—42no—
224D4×C9—62no—
225D4×C9—62no—
226C2×C4×C9—21no—
227C2×C4×C9—34no—
228C2×C4×C9—31no—
229C2×C4×C9—31no—
230C2×C4×C9—31no—
231C2×C4×C9—62no—
232C2×C4×C9—62no—
233C2×C4×C9C22×D311yescharacteristic
234#611—64no—
235#611—64no—
236#611—124no—
237C2×Dic9—22no—
238C2×Dic9—32no—
239C2×Dic9—62no—
240C2×Dic9C22×D312yes—
241C2×Dic9C22×D312yes—
242C4×D9—64no—
243C4×D9—64no—
244#525—271noSylow
245C2×C3×C9—22no—
246C2×C3×C9—41no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4112yes—
249C3⋊D9—24no—
250D3×C9—41no—
251C23×D3—91no—
252C23×D3—91no—
253C23×D3—181no—
254C3×#41—64no—
255C2×C3×Q8—31no—
256C2×C3×D4—21no—
257C2×C3×D4—31no—
258C2×C3×D4—31no—
259C2×C3×D4C2×D911yescharacteristic
260C2×C3×D4D3211yescharacteristic
261C22×C3×C4—31no—
262C22×C3×C4—31no—
263C22×C3×C4—31no—
264C22×C3×C4—91no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—92no—
268C2×#67—182no—
269C2×#67C2×D911yescharacteristic
270C2×#67C2×D911yescharacteristic, complemented
271C22×Dic3—31no—
272C22×Dic3C2×D911yescharacteristic
273C22×Dic3C2×D911yescharacteristic, complemented
274#237—24no—
275C2×D12—92no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C2×C4×D3—184no—
281C2×C4×D3—184no—
282C2×C4×D3C2×D911yescharacteristic, complemented
283C2×Dic6C2×D911yescharacteristic, complemented
284C3×C4⋊C4—92no—
285C3×C22⋊C4—91no—
286C3×C22⋊C4—92no—
287#217—31no—
288#217—32no—
289#217—92no—
290#217—91no—
291#217—91no—
292#217—181no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—182no—
299C4⋊Dic3—92no—
300Dic3⋊C4—32no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304C4×Dic3—92no—
305C4×Dic3—92no—
306C22×C32—22no—
307C22×C32—22no—
308C22×C32—21no—
309C22×C32C2×#6712yes—
310C22×C32C23×D311yescharacteristic
311C2×C3⋊D3—34no—
312C2×C3⋊D3—68no—
313C2×C3⋊D3—62no—
314C2×C3⋊D3—62no—
315C2×C3×D3—22no—
316C2×C3×D3—21no—
317C32×C4—21no—
318C3⋊Dic3—62no—
319C3×Dic3—21no—
320C3×Dic3—62no—
321C3×Dic3—62no—
322C3×Dic3C2×#6712yes—
323C3×Dic3C2×#6712yes—
324C22×C9—22no—
325C22×C9—21no—
326C22×C9—22no—
327C22×C9—21no—
328C22×C9—22no—
329C22×C9—42no—
330C22×C9—42no—
331C22×C9—41no—
332C22×C9—61no—
333C22×C9—62no—
334C22×C9D3×D412yes—
335C22×C9C23×D311yescharacteristic
336C2×D9—34no—
337C2×D9—64no—
338C2×D9—68no—
339C2×D9—62no—
340C2×D9—62no—
341C2×D9—122no—
342C2×D9—122no—
343C2×D9—128no—
344C4×C9—21no—
345C4×C9—32no—
346C4×C9—32no—
347C4×C9—41no—
348C4×C9—61no—
349Dic9—22no—
350Dic9—22no—
351Dic9—62no—
352Dic9—122no—
353C2×#41—31no—
354C23×C4—271no—
355#123—91no—
356#123—271no—
357C22⋊Q8—272no—
358#121—272no—
359#120—271no—
360#120—271no—
361C4×D4—272no—
362C4×D4—272no—
363#117—271no—
364C3×C9#52511yescharacteristic, complemented, Sylow
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—31no—
368C23×C3C22×D911yescharacteristic
369C23×C3C22×D911yescharacteristic
370C23×C3C2×D3211yescharacteristic
371C23×C3C2×D3211yescharacteristic
372C22×D3—92no—
373C22×D3—92no—
374C22×D3—182no—
375C22×D3—184no—
376C22×D3—184no—
377C22×D3—182no—
378C22×D3—182no—
379C22×D3—364no—
380C22×D3—362no—
381C22×D3C22×D911yescharacteristic
382C3×Q8—62no—
383C3×D4—22no—
384C3×D4—22no—
385C3×D4—42no—
386C3×D4—62no—
387C3×D4—62no—
388C2×C3×C4—21no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—34no—
392C2×C3×C4—31no—
393C2×C3×C4—62no—
394C2×C3×C4—62no—
395C2×C3×C4—92no—
396C2×C3×C4—92no—
397C2×C3×C4—182no—
398C2×C3×C4C22×D911yescharacteristic
399C2×C3×C4C2×D3211yescharacteristic
400#67—22no—
401#67—22no—
402#67—184no—
403#67—184no—
404#67—184no—
405#67—364no—
406C2×Dic3—22no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—62no—
410C2×Dic3—92no—
411C2×Dic3—92no—
412C2×Dic3—182no—
413C2×Dic3#61114yescomplemented
414C2×Dic3C22×D911yescharacteristic
415C2×Dic3C22×D911yescharacteristic
416C2×Dic3C22×D911yescharacteristic
417D12—184no—
418C4×D3—22no—
419C4×D3—184no—
420C4×D3—184no—
421C4×D3—184no—
422C4×D3—184no—
423Dic6—22no—
424C2×C32—22no—
425C2×C32—41no—
426C2×C32D3×#4112yes—
427C2×C32C22×#6711yescharacteristic
428C3⋊D3—64no—
429C3×D3—41no—
430C2×C9—22no—
431C2×C9—22no—
432C2×C9—21no—
433C2×C9—42no—
434C2×C9—41no—
435C2×C9—81no—
436C2×C9—121no—
437C2×C9C2×D3×D411yescharacteristic
438C2×C9D3×#4112yes—
439D9—64no—
440D9—124no—
441C24—271no—
442#41—64no—
443C2×Q8—31no—
444C2×D4—31no—
445C2×D4—31no—
446C2×D4—272no—
447C2×D4—272no—
448C2×D4D3×D911yescharacteristic
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—91no—
453C22×C4—272no—
454C22×C4—272no—
455C22×C4—544no—
456C4⋊C4—92no—
457C4⋊C4—272no—
458C4⋊C4—272no—
459C22⋊C4—91no—
460C22⋊C4—92no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C42—272no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—22no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—22no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—62no—
478C22×C3—61no—
479C22×C3C2×#61112yes—
480C22×C3C23×D911yescharacteristic
481C22×C3D3×#6712yes—
482C22×C3C22×D3211yescharacteristic
483C2×D3—22no—
484C2×D3—21no—
485C2×D3—94no—
486C2×D3—94no—
487C2×D3—184no—
488C2×D3—188no—
489C2×D3—188no—
490C2×D3—182no—
491C2×D3—182no—
492C2×D3—182no—
493C2×D3—182no—
494C2×D3—362no—
495C2×D3—362no—
496C2×D3—368no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—32no—
500C3×C4—32no—
501C3×C4—41no—
502C3×C4—61no—
503C3×C4—182no—
504C3×C4—182no—
505Dic3—21no—
506Dic3—62no—
507Dic3—62no—
508Dic3—182no—
509Dic3—182no—
510Dic3—362no—
511Dic3C2×#61112yes—
512Dic3C2×#61112yes—
513C32#582811yescharacteristic
514C9—21no—
515C9#565611yescharacteristic, complemented
516C23—31no—
517C23—272no—
518C23—544no—
519C23—542no—
520C23C2×D3×D911yescharacteristic
521C23C2×D3×D911yescharacteristic
522Q8—62no—
523D4—22no—
524D4—62no—
525D4—62no—
526D4—544no—
527D4—544no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—34no—
531C2×C4—31no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—92no—
535C2×C4—92no—
536C2×C4—182no—
537C2×C4—272no—
538C2×C4—544no—
539C2×C4—544no—
540C2×C4C2×D3×D911yescharacteristic
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—22no—
544C2×C3—21no—
545C2×C3—42no—
546C2×C3—41no—
547C2×C3—41no—
548C2×C3—81no—
549C2×C3—121no—
550C2×C3#41×D912yes—
551C2×C3C22×#61111yescharacteristic
552C2×C3#6431212yes—
553C2×C3C2×D3×#6711yescharacteristic
554D3—41no—
555D3—184no—
556D3—184no—
557D3—364no—
558C22—22no—
559C22—22no—
560C22—21no—
561C22—62no—
562C22—61no—
563C22—274no—
564C22—548no—
565C22—542no—
566C22—542no—
567C22D3×#61112yes—
568C22C22×D3×D911yescharacteristic
569C4—21no—
570C4—32no—
571C4—32no—
572C4—61no—
573C4—182no—
574C4—182no—
575C4—542no—
576C3—21no—
577C3#1058993011yescharacteristic, complemented
578C3#1059515911yescharacteristic
579C2—22no—
580C2—41no—
581C2—121no—
582C2—544no—
583C2#1172249812yes—
584C2C2×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#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×C9×#237C2semidirect
2. C22×(C3⋊D9)⋊C4C2semidirect
3. #11721837C2semidirect
4. C3×#63497C2semidirect
5. #11721291C2semidirect
6. #11721265C2semidirect
7. #11721262C2semidirect
8. #11721244C2semidirect
9. #11721240C2semidirect
10. #11721030C2semidirect
11. #11720986C2semidirect
18. C2×C9×#67C22semidirect
20. C22×C9×Dic3C22semidirect
23. C2×C4×D3×C9C22semidirect
24. C2×C9×Dic6C22semidirect
32. C3×#3676C22semidirect
33. C3×Dic9⋊C4C22semidirect
34. (C2×D3)⋊Dic9C22semidirect
36. Dic3⋊Dic9C22semidirect
37. #88442C22semidirect
38. Dic3×Dic9C22semidirect
58. #63497D3semidirect
75. C2×C9×Dic3D4semidirect
136. #3676C2×D3semidirect
142. Dic9⋊C4C2×D3semidirect
152. C2×C3⋊D9#41semidirect
163. C2×#237D9semidirect
270. C2×#67C2×D9semidirect
273. C22×Dic3C2×D9semidirect
282. C2×C4×D3C2×D9semidirect
283. C2×Dic6C2×D9semidirect
364. C3×C9#525semidirect
413. C2×Dic3#611semidirect
515. C9#5656semidirect
577. C3#10589930semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
C22×(C3⋊D9)⋊C42
#117218372
C3×#634972
#117212912
#117212652
#117212622
#117212442
#117212402
#117210302
#117209862
#105951593
#105899303