Finite Group Databasebeta

← D3×#64144 #13177410

Subgroups — D3×#64144 #13177410

Order 1728.

2364 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×C32×D3×D4C211yesmaximal, characteristic
2C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#11723976C211yesmaximal, characteristic, complemented
4C3×#64144C211yesmaximal, characteristic, complemented
5#11723614C212yesmaximal, complemented
6D3×#3757C211yesmaximal, characteristic, complemented
7D3×#3757C212yesmaximal, complemented
8#11723603C211yesmaximal, characteristic, complemented
9#11723602C211yesmaximal, characteristic, complemented
10#11723473C212yesmaximal, complemented
11D3×#3750C212yesmaximal, complemented
12C2×#64144—31nomaximal
13D3×#935—34nomaximal
14C23×C32×D3C2211yescharacteristic
15C23×C32×D3C2211yescharacteristic
16C2×C33×D4C2211yescharacteristic
17C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
19C2×C32×#67C2211yescharacteristic
20C2×C32×#67C2211yescharacteristic
21C32×D3×D4—24no—
22C2×C32×D12C2211yescharacteristic
23C2×C32×C4×D3C2211yescharacteristic
24C2×D3×C3⋊Dic3—24no—
25C2×D3×C3⋊Dic3C2212yes—
26C2×D3×C3⋊Dic3C2212yes—
27C2×D3×C3⋊Dic3C2212yescomplemented
28#88864C2211yescharacteristic, complemented
29#88864C2212yescomplemented
30#88857C2212yescomplemented
31C3×#3757C2211yescharacteristic, complemented
32C3×#3757C2212yescomplemented
33C3×#3750C2212yescomplemented
34#88807C2212yescomplemented
35#88806C2212yescomplemented
36(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C22×C32×D4—31no—
40C23×C3⋊Dic3—31no—
41C2×C3×D3×D4D314yes—
42C22×D3×Dic3—34no—
43C2×#3757—31no—
44C2×#3757—32no—
45#64144—31no—
46#64144—31no—
47#64144—31no—
48#64144—32no—
49#64144—34no—
50#64144—32no—
51#64144—64no—
52#64144D311yescharacteristic, direct product factor
53C2×#3750—32no—
54C3×#935—34no—
55#63969—38no—
56D3×#217—38no—
57D3×#217—34no—
58#63962—34no—
59#63961—34no—
60#63884—38no—
61D3×Dic3⋊C4—38no—
62C23×C33C2311yescharacteristic
63C23×C33C2311yescharacteristic
64C22×C32×D3—24no—
65C22×C32×D3—22no—
66C22×C32×D3—24no—
67C22×C32×D3D414yes—
68C22×C32×D3C2311yescharacteristic
69C22×C32×D3C2311yescharacteristic
70C22×C32×D3C2311yescharacteristic
71C33×D4—22no—
72C2×C33×C4C2311yescharacteristic
73C2×C3⋊(C3⋊Dic3)—22no—
74C2×C3⋊(C3⋊Dic3)C2312yes—
75C2×C3⋊(C3⋊Dic3)C2312yes—
76C2×C3×C3⋊Dic3—22no—
77C2×C3×C3⋊Dic3C2312yes—
78C2×C3×C3⋊Dic3C2312yes—
79C32×#67—22no—
80C32×#67—22no—
81C2×C32×Dic3C2311yescharacteristic
82C32×D12—22no—
83C32×C4×D3—22no—
84D3×C3⋊Dic3—24no—
85D3×C3⋊Dic3—24no—
86D3×C3⋊Dic3—24no—
87C2×#935—94no—
88D3×#123—91no—
89C24×C32—31no—
90C24×C32—31no—
91C23×C3×D3C2×D314yes—
92C23×C3×D3C2×D314yes—
93C2×C32×D4—24no—
94C2×C32×D4—31no—
95C2×C32×D4—31no—
96C2×C32×D4—31no—
97C2×C32×D4—64no—
98C2×C32×D4C2×D314yes—
99C2×C32×D4C2×D311yescharacteristic
100C22×C32×C4—31no—
101C22×C3⋊Dic3—31no—
102C22×C3⋊Dic3—34no—
103C22×C3⋊Dic3—32no—
104C22×C3⋊Dic3—32no—
105C22×C3⋊Dic3—32no—
106C22×C3⋊Dic3—64no—
107C22×C3⋊Dic3—64no—
108C22×C3⋊Dic3C2×D311yescharacteristic, complemented
109C2×C3×#67C2×D314yes—
110C2×C3×#67C2×D314yes—
111C22×C3×Dic3—34no—
112C3×D3×D4—216no—
113C2×C3×D12C2×D314yes—
114C2×C3×C4×D3C2×D314yes—
115C2×D3×Dic3—38no—
116C2×D3×Dic3—38no—
117C2×D3×Dic3—38no—
118C2×D3×Dic3—616no—
119#3757—31no—
120#3757—32no—
121#3757—34no—
122#3757—38no—
123#3757—32no—
124#3757—32no—
125#3757—32no—
126#3757—64no—
127#3757—68no—
128#3757C2×D311yescharacteristic, complemented
129#3757C2×D312yescomplemented
130#3750—32no—
131#3750—38no—
132#3750—32no—
133#3750—32no—
134#3750—68no—
135#3750C2×D312yescomplemented
136C3×#217—34no—
137C3×#217—38no—
138C3×Dic3⋊C4—38no—
139#3724—38no—
140Dic3⋊Dic3—38no—
141(C2×D3)⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33—22no—
145C22×C33—21no—
146C22×C33—22no—
147C22×C33C2×D412yes—
148C22×C33C2411yescharacteristic
149C2×C32×D3—22no—
150C2×C32×D3—28no—
151C2×C32×D3—21no—
152C2×C32×D3—22no—
153C2×C32×D3—44no—
154C2×C32×D3C2×D412yes—
155C2×C32×D3C2×D412yes—
156C2×C32×D3#4114yes—
157C33×C4—21no—
158C3⋊(C3⋊Dic3)—22no—
159C3⋊(C3⋊Dic3)—22no—
160C3×C3⋊Dic3—22no—
161C3×C3⋊Dic3—22no—
162C32×Dic3—21no—
163C22×C3×D4—34no—
164C23×Dic3—94no—
165C2×D3×D4C3⋊D311yescharacteristic
166C22×C4×D3—91no—
167C3×#123—91no—
168C2×#217—94no—
169C2×#217—98no—
170#935—34no—
171#935—98no—
172#935—94no—
173#935—98no—
174#935—94no—
175#935—94no—
176#935—91no—
177#935—184no—
178#929—91no—
179C2×Dic3⋊C4—98no—
180#894—92no—
181D3×C4⋊C4—92no—
182#886—91no—
183#883—92no—
184D3×C22⋊C4—91no—
185D3×C22⋊C4—92no—
186C23×C32—24no—
187C23×C32—24no—
188C23×C32—31no—
189C23×C32—34no—
190C23×C32—31no—
191C23×C32—31no—
192C23×C32—64no—
193C23×C32—64no—
194C23×C32—62no—
195C23×C32C22×D314yes—
196C23×C32C22×D314yes—
197C23×C32C22×D311yescharacteristic
198C23×C32C22×D311yescharacteristic
199C22×C3×D3—216no—
200C22×C3×D3—216no—
201C22×C3×D3—28no—
202C22×C3×D3#67116yes—
203C22×C3×D3C22×D314yes—
204C22×C3×D3C22×D314yes—
205C22×C3×D3C22×D314yes—
206C32×D4—22no—
207C32×D4—28no—
208C32×D4—48no—
209C32×D4—62no—
210C32×D4—62no—
211C32×D4—62no—
212C2×C32×C4—24no—
213C2×C32×C4—31no—
214C2×C32×C4—62no—
215C2×C32×C4C22×D314yes—
216C2×C32×C4C22×D311yescharacteristic
217C2×C3⋊Dic3—22no—
218C2×C3⋊Dic3—32no—
219C2×C3⋊Dic3—32no—
220C2×C3⋊Dic3—38no—
221C2×C3⋊Dic3—38no—
222C2×C3⋊Dic3—68no—
223C2×C3⋊Dic3—68no—
224C2×C3⋊Dic3—64no—
225C2×C3⋊Dic3—64no—
226C2×C3⋊Dic3—64no—
227C2×C3⋊Dic3—62no—
228C2×C3⋊Dic3—68no—
229C2×C3⋊Dic3—128no—
230C2×C3⋊Dic3C22×D312yes—
231C2×C3⋊Dic3C22×D312yes—
232C3×#67—28no—
233C3×#67—28no—
234C2×C3×Dic3—38no—
235C2×C3×Dic3—38no—
236C2×C3×Dic3—68no—
237C2×C3×Dic3C22×D314yes—
238C3×D12—28no—
239C3×C4×D3—28no—
240D3×Dic3—616no—
241D3×Dic3—616no—
242D3×Dic3—616no—
243C2×#123—271noSylow
244C2×C33—22no—
245C2×C33—41no—
246C2×C33C22×D411yescharacteristic
247C2×C33C2×#4112yes—
248C32×D3—22no—
249C32×D3—41no—
250C32×D3#12314yescomplemented
251C24×C3—34no—
252C24×C3—34no—
253C23×D3C2×C3⋊D311yescharacteristic
254C23×D3C2×C3⋊D311yescharacteristic
255C2×C3×D4—24no—
256C2×C3×D4—34no—
257C2×C3×D4—34no—
258C2×C3×D4—34no—
259C2×C3×D4—616no—
260C2×C3×D4D3214yes—
261C2×C3×D4C2×C3⋊D311yescharacteristic
262C22×C3×C4—34no—
263C22×C3×C4—91no—
264C2×#67C2×C3⋊D311yescharacteristic
265C2×#67C2×C3⋊D311yescharacteristic
266C22×Dic3—34no—
267C22×Dic3—98no—
268C22×Dic3—98no—
269C22×Dic3—98no—
270C22×Dic3—94no—
271C22×Dic3—91no—
272C22×Dic3—184no—
273C22×Dic3—1816no—
274D3×D4—24no—
275C2×D12C2×C3⋊D311yescharacteristic
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—184no—
280C2×C4×D3C2×C3⋊D311yescharacteristic
281C3×C4⋊C4—92no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4—91no—
284#217—34no—
285#217—38no—
286#217—98no—
287#217—98no—
288#217—98no—
289#217—94no—
290#217—98no—
291#217—92no—
292#217—91no—
293#217—184no—
294#217—188no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298C4⋊Dic3—92no—
299Dic3⋊C4—38no—
300Dic3⋊C4—98no—
301Dic3⋊C4—98no—
302Dic3⋊C4—98no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—188no—
306C22×C32—22no—
307C22×C32—21no—
308C22×C32—22no—
309C22×C32—24no—
310C22×C32—28no—
311C22×C32—28no—
312C22×C32—28no—
313C22×C32—24no—
314C22×C32—32no—
315C22×C32—34no—
316C22×C32—32no—
317C22×C32—48no—
318C22×C32—44no—
319C22×C32—48no—
320C22×C32—68no—
321C22×C32—62no—
322C22×C32—61no—
323C22×C32—62no—
324C22×C32—124no—
325C22×C32D3×D412yes—
326C22×C32C2×#6718yes—
327C22×C32C23×D314yes—
328C22×C32C23×D311yescharacteristic
329C2×C3×D3—28no—
330C2×C3×D3—232no—
331C2×C3×D3—28no—
332C2×C3×D3—24no—
333C2×C3×D3—416no—
334C2×C3×D3#237116yes—
335C2×C3×D3C2×#6718yes—
336C2×C3×D3C2×#6718yes—
337C32×C4—21no—
338C32×C4—24no—
339C32×C4—44no—
340C32×C4—61no—
341C3⋊Dic3—22no—
342C3⋊Dic3—22no—
343C3⋊Dic3—62no—
344C3⋊Dic3—62no—
345C3⋊Dic3—68no—
346C3⋊Dic3—68no—
347C3⋊Dic3—128no—
348C3⋊Dic3—128no—
349C3×Dic3—24no—
350C3×Dic3—68no—
351C3×Dic3—68no—
352C22×D4—31no—
353C23×C4—271no—
354#123—91no—
355#123—272no—
356#123—271no—
357#123—272no—
358#123—271no—
359#123—271no—
360C2×C4⋊C4—272no—
361C2×C22⋊C4—271no—
362C2×C22⋊C4—272no—
363C33C2×#12311yescharacteristic, complemented, Sylow
364C23×C3—24no—
365C23×C3—24no—
366C23×C3—34no—
367C23×C3—34no—
368C23×C3—316no—
369C23×C3—34no—
370C23×C3—616no—
371C23×C3—68no—
372C23×C3—616no—
373C23×C3C2×D3214yes—
374C23×C3C2×D3214yes—
375C23×C3C22×C3⋊D311yescharacteristic
376C23×C3C22×C3⋊D311yescharacteristic
377C22×D3—24no—
378C22×D3—24no—
379C22×D3—22no—
380C22×D3#63814yes—
381C22×D3C22×C3⋊D311yescharacteristic
382C22×D3C22×C3⋊D311yescharacteristic
383C22×D3C22×C3⋊D311yescharacteristic
384C3×D4—28no—
385C3×D4—22no—
386C3×D4—48no—
387C3×D4—68no—
388C3×D4—68no—
389C3×D4—68no—
390C2×C3×C4—24no—
391C2×C3×C4—34no—
392C2×C3×C4—68no—
393C2×C3×C4—92no—
394C2×C3×C4—92no—
395C2×C3×C4—182no—
396C2×C3×C4C2×D3214yes—
397C2×C3×C4C22×C3⋊D311yescharacteristic
398#67—22no—
399#67—22no—
400C2×Dic3—38no—
401C2×Dic3—38no—
402C2×Dic3—68no—
403C2×Dic3—98no—
404C2×Dic3—98no—
405C2×Dic3—92no—
406C2×Dic3—92no—
407C2×Dic3—188no—
408C2×Dic3—188no—
409C2×Dic3—1816no—
410C2×Dic3—1816no—
411C2×Dic3—1816no—
412C2×Dic3—188no—
413C2×Dic3—182no—
414C2×Dic3—368no—
415C2×Dic3C22×C3⋊D311yescharacteristic
416D12—22no—
417C4×D3—22no—
418C4×D3—184no—
419C4×D3—184no—
420C4×D3—184no—
421C2×C32—22no—
422C2×C32—28no—
423C2×C32—24no—
424C2×C32—28no—
425C2×C32—34no—
426C2×C32—48no—
427C2×C32—41no—
428C2×C32—44no—
429C2×C32—62no—
430C2×C32—84no—
431C2×C32—121no—
432C2×C32C2×D3×D411yescharacteristic
433C2×C32C2×#23718yes—
434C2×C32D3×#4112yes—
435C2×C32C22×#6714yes—
436C3×D3—28no—
437C3×D3—44no—
438C3×D3#935116yescomplemented
439C24—31no—
440C24—31no—
441C2×D4—31no—
442C2×D4—31no—
443C2×D4—31no—
444C2×D4—64no—
445C2×D4D3×C3⋊D311yescharacteristic
446C22×C4—31no—
447C22×C4—91no—
448C22×C4—272no—
449C22×C4—272no—
450C22×C4—272no—
451C22×C4—271no—
452C22×C4—544no—
453C4⋊C4—92no—
454C4⋊C4—272no—
455C4⋊C4—272no—
456C4⋊C4—272no—
457C22⋊C4—91no—
458C22⋊C4—92no—
459C22⋊C4—272no—
460C22⋊C4—272no—
461C22⋊C4—272no—
462C22⋊C4—271no—
463C22⋊C4—272no—
464C22×C3—28no—
465C22×C3—22no—
466C22×C3—28no—
467C22×C3—24no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—24no—
471C22×C3—28no—
472C22×C3—316no—
473C22×C3—38no—
474C22×C3—38no—
475C22×C3—48no—
476C22×C3—44no—
477C22×C3—48no—
478C22×C3—632no—
479C22×C3—68no—
480C22×C3—68no—
481C22×C3—64no—
482C22×C3—1216no—
483C22×C3D3×#6718yes—
484C22×C3C2×#63812yes—
485C22×C3C22×D3214yes—
486C22×C3C23×C3⋊D311yescharacteristic
487C2×D3—22no—
488C2×D3—28no—
489C2×D3—22no—
490C2×D3—21no—
491C2×D3—44no—
492C2×D3#383014yes—
493C2×D3C2×#63812yes—
494C2×D3C2×#63812yes—
495C3×C4—24no—
496C3×C4—21no—
497C3×C4—44no—
498C3×C4—64no—
499C3×C4—182no—
500C3×C4—182no—
501Dic3—21no—
502Dic3—68no—
503Dic3—68no—
504Dic3—188no—
505Dic3—188no—
506Dic3—182no—
507Dic3—182no—
508Dic3—368no—
509Dic3—368no—
510C32—24no—
511C32D3×#12311yescharacteristic, complemented
512C32C2×#93514yescomplemented
513C23—31no—
514C23—31no—
515C23—34no—
516C23—31no—
517C23—64no—
518C23—62no—
519C23—64no—
520C23C2×D3×C3⋊D311yescharacteristic
521C23C2×D3×C3⋊D311yescharacteristic
522D4—22no—
523D4—62no—
524D4—62no—
525D4—62no—
526C2×C4—31no—
527C2×C4—62no—
528C2×C4—92no—
529C2×C4—92no—
530C2×C4—182no—
531C2×C4—272no—
532C2×C4—272no—
533C2×C4—544no—
534C2×C4—544no—
535C2×C4—544no—
536C2×C4—542no—
537C2×C4C2×D3×C3⋊D311yescharacteristic
538C2×C3—28no—
539C2×C3—22no—
540C2×C3—28no—
541C2×C3—24no—
542C2×C3—316no—
543C2×C3—48no—
544C2×C3—44no—
545C2×C3—41no—
546C2×C3—68no—
547C2×C3—84no—
548C2×C3—124no—
549C2×C3D3×#23718yes—
550C2×C3C2×D3×#6714yes—
551C2×C3C2×#383012yes—
552C2×C3C22×#63811yescharacteristic
553D3—22no—
554D3—41no—
555D3#6414414yesdirect product factor
556C22—22no—
557C22—22no—
558C22—21no—
559C22—34no—
560C22—32no—
561C22—32no—
562C22—68no—
563C22—62no—
564C22—62no—
565C22—61no—
566C22—124no—
567C22D3×#63812yes—
568C22C22×D3×C3⋊D311yescharacteristic
569C4—21no—
570C4—61no—
571C4—182no—
572C4—182no—
573C4—542no—
574C4—542no—
575C3—24no—
576C3D3×#93514yescomplemented
577C3C2×#6414411yescharacteristic, complemented
578C2—22no—
579C2—34no—
580C2—41no—
581C2—62no—
582C2—121no—
583C2D3×#383012yes—
584C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC2×C32×D3×D4C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×D3×C3⋊Dic3C2semidirect
3. #11723976C2semidirect
4. C3×#64144C2semidirect
5. #11723614C2semidirect
6. D3×#3757C2semidirect
7. D3×#3757C2semidirect
8. #11723603C2semidirect
9. #11723602C2semidirect
10. #11723473C2semidirect
11. D3×#3750C2semidirect
17. C22×C3⋊(C3⋊Dic3)C22semidirect
18. C22×C3×C3⋊Dic3C22semidirect
27. C2×D3×C3⋊Dic3C22semidirect
28. #88864C22semidirect
29. #88864C22semidirect
30. #88857C22semidirect
31. C3×#3757C22semidirect
32. C3×#3757C22semidirect
33. C3×#3750C22semidirect
34. #88807C22semidirect
35. #88806C22semidirect
36. (C2×D3)⋊(C3⋊Dic3)C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
52. #64144D3direct (N × H)
108. C22×C3⋊Dic3C2×D3semidirect
128. #3757C2×D3semidirect
129. #3757C2×D3semidirect
135. #3750C2×D3semidirect
250. C32×D3#123semidirect
363. C33C2×#123semidirect
438. C3×D3#935semidirect
511. C32D3×#123semidirect
512. C32C2×#935semidirect
555. D3#64144direct (N × H)
576. C3D3×#935semidirect
577. C3C2×#64144semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×D42
C22×D3×C3⋊Dic32
#117239762
C3×#641442
#117236142
D3×#37572
#117236032
#117236022
#117234732
D3×#37502
C2×#641443
D3×#9353