Finite Group Databasebeta

← #13151993

Subgroups — #13151993

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C2×#88663C211yesmaximal, characteristic, complemented
3C9×#953C211yesmaximal, characteristic, complemented
4C3×#63515C211yesmaximal, characteristic, complemented
5#11721290C212yesmaximal, complemented
6#67⋊Dic9C212yesmaximal, complemented
7#611⋊Dic3C212yesmaximal, complemented
8#11721267C212yesmaximal, complemented
9#11721266C211yesmaximal, characteristic, complemented
10#11721260C211yesmaximal, characteristic, complemented
11#11721248C211yesmaximal, characteristic, complemented
12#10595501—31nomaximal
13#10589688—31nomaximal
14C24×C3×C9C2211yescharacteristic
15C22×C3⋊Dic9—24no—
16C22×C3⋊Dic9C2211yescharacteristic
17C22×C3⋊Dic9C2212yes—
18C22×C3⋊Dic9C2212yes—
19C22×C3⋊Dic9C2211yescharacteristic
20C2×C9×#67C2211yescharacteristic
21C2×C9×#67C2212yes—
22C2×C3×#611C2211yescharacteristic
23C2×C3×#611C2212yes—
24C2×#6194C2211yescharacteristic, complemented
25#88663—24no—
26C2×D3×Dic9C2211yescharacteristic, complemented
27C2×Dic3×D9C2211yescharacteristic, complemented
28C2×#6189C2211yescharacteristic, complemented
29C9×#217C2211yescharacteristic
30C9×#217C2212yes—
31C3×#3676C2211yescharacteristic
32C3×#3676C2212yes—
33(C2×D3)⋊Dic9C2212yescomplemented
34(C2×D9)⋊Dic3C2212yescomplemented
35Dic3⋊Dic9C2212yescomplemented
36#88442C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C23×C3⋊Dic3—31no—
40C2×#3805—31no—
41C3×#953—31no—
42C3×#953D311yescharacteristic
43#63982—32no—
44#63980—32no—
45#67⋊Dic3—32no—
46#67⋊Dic3—32no—
47#63962—31no—
48#63962—31no—
49#63957—31no—
50C23×Dic9—31no—
51C23×Dic9—61no—
52C2×#3699—31no—
53C9×#120—31no—
54#63515D311yescharacteristic, complemented
55#63501—32no—
56#63497—31no—
57D4×Dic9—32no—
58#63448—31no—
59#611⋊C4—32no—
60#63440—32no—
61#63439—31no—
62C23×C3×C9—22no—
63C23×C3×C9—24no—
64C23×C3×C9C2311yescharacteristic
65C23×C3×C9C2312yes—
66C22×D3×C9C2311yescharacteristic
67C22×C3×D9C2311yescharacteristic
68C2×C3⋊Dic9—22no—
69C2×C3⋊Dic9—24no—
70C2×C3⋊Dic9—24no—
71C2×C3⋊Dic9D414yescomplemented
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2311yescharacteristic
74C2×C3⋊Dic9C2311yescharacteristic
75C9×#67—22no—
76C9×#67—24no—
77C3×#611—22no—
78C3×#611—24no—
79C2×C9×Dic3C2311yescharacteristic
80C2×C9×Dic3C2312yes—
81C2×C3×Dic9C2311yescharacteristic
82C2×C3×Dic9C2312yes—
83#6194—22no—
84D3×Dic9—22no—
85Dic3×D9—22no—
86#6189—22no—
87#5586—91no—
88#5586—91no—
89C24×C32C2×D311yescharacteristic
90C22×C3⋊Dic3—31no—
91C22×C3⋊Dic3—31no—
92C22×C3⋊Dic3—32no—
93C22×C3⋊Dic3—32no—
94C22×C3⋊Dic3—64no—
95C2×C3×#67—31no—
96C2×C3×#67—32no—
97C2×C3×#67C2×D311yescharacteristic
98C2×C3×#67C2×D312yes—
99C2×#627—31no—
100C2×#625—31no—
101#3805—64no—
102C2×D3×Dic3—31no—
103C2×D3×Dic3—31no—
104C24×C9—21no—
105C24×C9C2×D311yescharacteristic
106C3×#217—31no—
107C3×#217—32no—
108C3×#217C2×D311yescharacteristic
109C3×#217C2×D312yes—
110#3724—32no—
111Dic3⋊Dic3—32no—
112Dic3⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C2×#611—31no—
119C2×#611C2×D312yes—
120C2×#611C2×D311yescharacteristic
121C22×Dic9—31no—
122C22×Dic9—32no—
123C22×Dic9—31no—
124C22×Dic9—32no—
125C22×Dic9—31no—
126C22×Dic9—64no—
127C22×Dic9—61no—
128C22×Dic9—62no—
129C22×Dic9—62no—
130C22×Dic9—61no—
131C22×Dic9—68no—
132#3699—64no—
133C2×C4×D9—31no—
134C2×Dic18—31no—
135C9×C22⋊C4—31no—
136C9×C22⋊C4—32no—
137#3676—32no—
138#3676C2×D312yes—
139#3676C2×D311yescharacteristic
140(C2×D9)⋊C4—32no—
141C4⋊Dic9—32no—
142Dic9⋊C4—32no—
143Dic9⋊C4—32no—
144C4×Dic9—32no—
145C22×C3×C9—28no—
146C22×C3×C9—22no—
147C22×C3×C9—22no—
148C22×C3×C9—22no—
149C22×C3×C9C2×D412yes—
150C22×C3×C9#4114yescomplemented
151C22×C3×C9C2411yescharacteristic
152C2×D3×C9—21no—
153C2×D3×C9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—22no—
156C3⋊Dic9—22no—
157C3⋊Dic9C2×D412yes—
158C3⋊Dic9C2×D412yes—
159C9×Dic3—21no—
160C9×Dic3—22no—
161C3×Dic9—21no—
162C3×Dic9—22no—
163C23×Dic3—91no—
164C23×Dic3—91no—
165C23×Dic3—181no—
166C2×#237—91no—
167C2×#237—91no—
168C3×#120—31no—
169C3×#120—91no—
170#953—31no—
171#953D911yescharacteristic, complemented
172#939—92no—
173#939—92no—
174#935—91no—
175#935—91no—
176D4×Dic3—92no—
177D4×Dic3—92no—
178#886—91no—
179#886—91no—
180#67⋊C4—92no—
181#67⋊C4—92no—
182#878—92no—
183#878—92no—
184#877—91no—
185#877—91no—
186C23×C32—22no—
187C23×C32—24no—
188C23×C32C22×D311yescharacteristic
189C23×C32C22×D312yes—
190C22×C3×D3—31no—
191C22×C3×D3C22×D311yescharacteristic
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—32no—
196C2×C3⋊Dic3—64no—
197C2×C3⋊Dic3—64no—
198C2×C3⋊Dic3—62no—
199C3×#67—22no—
200C3×#67—24no—
201C3×#67—62no—
202C3×#67—64no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3C22×D312yes—
206C2×C3×Dic3C22×D311yescharacteristic
207#627—62no—
208#625—62no—
209D3×Dic3—62no—
210D3×Dic3—62no—
211C23×C9—24no—
212C23×C9—28no—
213C23×C9—22no—
214C23×C9—24no—
215C23×C9—22no—
216C23×C9—21no—
217C23×C9—31no—
218C23×C9C22×D311yescharacteristic
219C23×C9C22×D312yes—
220C22×D9C22×D311yescharacteristic
221D4×C9—62no—
222D4×C9—64no—
223C2×C4×C9—32no—
224C2×C4×C9—31no—
225#611—22no—
226#611—24no—
227#611—62no—
228C2×Dic9—34no—
229C2×Dic9—31no—
230C2×Dic9—31no—
231C2×Dic9—32no—
232C2×Dic9—64no—
233C2×Dic9—61no—
234C2×Dic9—61no—
235C2×Dic9—62no—
236C2×Dic9—62no—
237C2×Dic9—64no—
238C2×Dic9—64no—
239C2×Dic9—68no—
240C2×Dic9—68no—
241C2×Dic9—64no—
242C2×Dic9—62no—
243C2×Dic9C22×D312yes—
244C2×Dic9C22×D311yescharacteristic
245C4×D9—62no—
246Dic18—62no—
247#525—271noSylow
248C2×C3×C9—22no—
249C2×C3×C9—24no—
250C2×C3×C9C22×D411yescharacteristic
251C2×C3×C9C2×#4112yes—
252D3×C9—41no—
253C3×D9—41no—
254C24×C3—21no—
255C24×C3C2×D911yescharacteristic
256C24×C3D3211yescharacteristic
257C2×C3×D4—32no—
258C2×C3×D4—31no—
259C2×C3×D4—92no—
260C2×C3×D4—91no—
261C2×#67—32no—
262C2×#67—31no—
263C2×#67—91no—
264C2×#67—91no—
265C2×#67C2×D912yes—
266C2×#67C2×D911yescharacteristic
267C22×Dic3—91no—
268C22×Dic3—91no—
269C22×Dic3—91no—
270C22×Dic3—92no—
271C22×Dic3—92no—
272C22×Dic3—91no—
273C22×Dic3—92no—
274C22×Dic3—92no—
275C22×Dic3—91no—
276C22×Dic3—91no—
277C22×Dic3—181no—
278C22×Dic3—182no—
279C22×Dic3—182no—
280C22×Dic3—181no—
281C22×Dic3—184no—
282C22×Dic3—184no—
283C22×Dic3—188no—
284#237—184no—
285#237—184no—
286C2×C4×D3—91no—
287C2×C4×D3—91no—
288C2×Dic6—91no—
289C2×Dic6—91no—
290C3×C22⋊C4—31no—
291C3×C22⋊C4—32no—
292C3×C22⋊C4—91no—
293C3×C22⋊C4—92no—
294#217—31no—
295#217—32no—
296#217—92no—
297#217—92no—
298#217C2×D911yescharacteristic
299#217C2×D912yes—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302C4⋊Dic3—92no—
303C4⋊Dic3—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308C4×Dic3—92no—
309C4×Dic3—92no—
310C22×C32—28no—
311C22×C32—22no—
312C22×C32—22no—
313C22×C32—22no—
314C22×C32D3×D412yes—
315C22×C32#23714yes—
316C22×C32C23×D311yescharacteristic
317C2×C3×D3—21no—
318C2×C3×D3—22no—
319C2×C3×D3—61no—
320C2×C3×D3—62no—
321C3⋊Dic3—32no—
322C3⋊Dic3—32no—
323C3⋊Dic3—62no—
324C3×Dic3—21no—
325C3×Dic3—22no—
326C3×Dic3—61no—
327C3×Dic3—62no—
328C22×C9—24no—
329C22×C9—216no—
330C22×C9—28no—
331C22×C9—24no—
332C22×C9—24no—
333C22×C9—22no—
334C22×C9—22no—
335C22×C9—22no—
336C22×C9—21no—
337C22×C9—24no—
338C22×C9—22no—
339C22×C9—61no—
340C22×C9—62no—
341C22×C9D3×D412yes—
342C22×C9#23714yescomplemented
343C22×C9C23×D311yescharacteristic
344C2×D9—21no—
345C2×D9—22no—
346C4×C9—61no—
347C4×C9—62no—
348Dic9—21no—
349Dic9—22no—
350Dic9—32no—
351Dic9—32no—
352Dic9—62no—
353Dic9—62no—
354Dic9—64no—
355Dic9—62no—
356C2×#41—271no—
357C23×C4—271no—
358#123—271no—
359#123—271no—
360C22⋊Q8—272no—
361#121—272no—
362#120—31no—
363#120—91no—
364C4×D4—272no—
365C4×D4—272no—
366#117—271no—
367C3×C9#52511yescharacteristic, complemented, Sylow
368C23×C3—24no—
369C23×C3—28no—
370C23×C3—22no—
371C23×C3—24no—
372C23×C3—24no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—21no—
376C23×C3—31no—
377C23×C3—91no—
378C23×C3C22×D911yescharacteristic
379C23×C3C22×D912yes—
380C23×C3C2×D3212yes—
381C23×C3C2×D3211yescharacteristic
382C22×D3—31no—
383C22×D3C22×D911yescharacteristic
384C3×D4—64no—
385C3×D4—62no—
386C3×D4—182no—
387C3×D4—184no—
388C2×C3×C4—32no—
389C2×C3×C4—31no—
390C2×C3×C4—92no—
391C2×C3×C4—91no—
392#67—22no—
393#67—24no—
394#67—64no—
395#67—62no—
396#67—182no—
397#67—182no—
398C2×Dic3—32no—
399C2×Dic3—31no—
400C2×Dic3—91no—
401C2×Dic3—91no—
402C2×Dic3—94no—
403C2×Dic3—91no—
404C2×Dic3—91no—
405C2×Dic3—94no—
406C2×Dic3—92no—
407C2×Dic3—92no—
408C2×Dic3—181no—
409C2×Dic3—181no—
410C2×Dic3—184no—
411C2×Dic3—182no—
412C2×Dic3—182no—
413C2×Dic3—182no—
414C2×Dic3—184no—
415C2×Dic3—184no—
416C2×Dic3—184no—
417C2×Dic3—184no—
418C2×Dic3—188no—
419C2×Dic3—188no—
420C2×Dic3—184no—
421C2×Dic3—182no—
422C2×Dic3—182no—
423C2×Dic3C22×D912yes—
424C2×Dic3C22×D911yescharacteristic
425C4×D3—182no—
426C4×D3—182no—
427Dic6—182no—
428Dic6—182no—
429C2×C32—22no—
430C2×C32—24no—
431C2×C32C2×D3×D411yescharacteristic
432C2×C32C2×#23712yes—
433C3×D3—41no—
434C3×D3—121no—
435C2×C9—28no—
436C2×C9—24no—
437C2×C9—22no—
438C2×C9—24no—
439C2×C9—22no—
440C2×C9—21no—
441C2×C9—121no—
442C2×C9C2×D3×D411yescharacteristic
443C2×C9C2×#23712yes—
444D9—41no—
445C24D3×D911yescharacteristic
446#41—544no—
447C2×Q8—271no—
448C2×D4—32no—
449C2×D4—31no—
450C2×D4—92no—
451C2×D4—91no—
452C2×D4—271no—
453C22×C4—271no—
454C22×C4—271no—
455C22×C4—271no—
456C22×C4—272no—
457C22×C4—272no—
458C22×C4—271no—
459C22×C4—544no—
460C4⋊C4—272no—
461C4⋊C4—272no—
462C4⋊C4—272no—
463C22⋊C4—31no—
464C22⋊C4—32no—
465C22⋊C4—91no—
466C22⋊C4—92no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—272no—
470C22×C3—24no—
471C22×C3—28no—
472C22×C3—216no—
473C22×C3—28no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—21no—
483C22×C3—24no—
484C22×C3—22no—
485C22×C3—62no—
486C22×C3—61no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3D4×D912yes—
490C22×C3#369914yescomplemented
491C22×C3C23×D911yescharacteristic
492C22×C3#380514yes—
493C22×C3#381112yes—
494C22×C3C22×D3211yescharacteristic
495C2×D3—21no—
496C2×D3—22no—
497C2×D3—62no—
498C2×D3—61no—
499C3×C4—62no—
500C3×C4—61no—
501C3×C4—181no—
502C3×C4—182no—
503Dic3—21no—
504Dic3—22no—
505Dic3—62no—
506Dic3—61no—
507Dic3—92no—
508Dic3—92no—
509Dic3—92no—
510Dic3—92no—
511Dic3—182no—
512Dic3—182no—
513Dic3—184no—
514Dic3—182no—
515Dic3—182no—
516C32#558611yescharacteristic
517C9—21no—
518C9#558611yescharacteristic, complemented
519C23—24no—
520C23—22no—
521C23—31no—
522C23—91no—
523C23C2×D3×D912yes—
524C23C2×D3×D911yescharacteristic
525Q8—542no—
526D4—64no—
527D4—62no—
528D4—184no—
529D4—182no—
530D4—542no—
531C2×C4—32no—
532C2×C4—31no—
533C2×C4—92no—
534C2×C4—91no—
535C2×C4—271no—
536C2×C4—271no—
537C2×C4—274no—
538C2×C4—272no—
539C2×C4—542no—
540C2×C4—542no—
541C2×C4—544no—
542C2×C4—544no—
543C2×C4—542no—
544C2×C3—28no—
545C2×C3—24no—
546C2×C3—24no—
547C2×C3—22no—
548C2×C3—22no—
549C2×C3—24no—
550C2×C3—22no—
551C2×C3—21no—
552C2×C3—121no—
553C2×C3—361no—
554C2×C3C2×D4×D911yescharacteristic
555C2×C3C2×#369912yes—
556C2×C3C2×#380512yes—
557C2×C3C2×#381111yescharacteristic
558D3—41no—
559D3—121no—
560C22—28no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—62no—
565C22—61no—
566C22—182no—
567C22—181no—
568C22#8866314yescomplemented
569C22#8866812yes—
570C22C22×D3×D911yescharacteristic
571C4—62no—
572C4—61no—
573C4—182no—
574C4—181no—
575C4—272no—
576C4—272no—
577C4—542no—
578C3—21no—
579C3#1058968811yescharacteristic, complemented
580C3#1059550111yescharacteristic
581C2—24no—
582C2—22no—
583C2—121no—
584C2—361no—
585C2C2×#8866312yes—
586C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC22×C32C23×D3
hypercenterC24D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊Dic9C2semidirect
2. C2×#88663C2semidirect
3. C9×#953C2semidirect
4. C3×#63515C2semidirect
5. #11721290C2semidirect
6. #67⋊Dic9C2semidirect
7. #611⋊Dic3C2semidirect
8. #11721267C2semidirect
9. #11721266C2semidirect
10. #11721260C2semidirect
11. #11721248C2semidirect
24. C2×#6194C22semidirect
26. C2×D3×Dic9C22semidirect
27. C2×Dic3×D9C22semidirect
28. C2×#6189C22semidirect
33. (C2×D3)⋊Dic9C22semidirect
34. (C2×D9)⋊Dic3C22semidirect
35. Dic3⋊Dic9C22semidirect
36. #88442C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
54. #63515D3semidirect
71. C2×C3⋊Dic9D4semidirect
150. C22×C3×C9#41semidirect
171. #953D9semidirect
342. C22×C9#237semidirect
367. C3×C9#525semidirect
490. C22×C3#3699semidirect
518. C9#5586semidirect
568. C22#88663semidirect
579. C3#10589688semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊Dic92
C2×#886632
C9×#9532
C3×#635152
#117212902
#67⋊Dic92
#611⋊Dic32
#117212672
#117212662
#117212602
#117212482
#105955013
#105896883