Finite Group Databasebeta

← C3×#10591394 #13158404

Subgroups — C3×#10591394 #13158404

Order 1728.

1362 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
1C22×C3×#625C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×C32×(C2×D3)⋊C4C212yesmaximal, complemented
4C2×C3×(C2×D3)⋊Dic3C212yesmaximal, complemented
5C3×#63579C211yesmaximal, characteristic, complemented
6C3×#4961—32nomaximal
7#10591394C311yesmaximal, characteristic, direct product factor
8C23×C32×D3C2212yes—
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×C3×#625—24no—
14C2×C3×#625—22no—
15C3×C4⋊(C3⋊Dic3)—22no—
16C32×(C2×D3)⋊C4—24no—
17C3×(C2×D3)⋊Dic3—24no—
18C22×C3×#67—32no—
19C22×#625C2×C311yescharacteristic
20C2×C32×C22⋊C4—32no—
21C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
22C2×C3×#217—32no—
23C2×C3×(C2×D3)⋊C4—22no—
24C2×C3×(C2×D3)⋊C4—32no—
25C2×C3×(C2×D3)⋊C4D312yescomplemented
26C2×C3×(C2×D3)⋊C4C2×C312yescomplemented
27C2×C3×C4⋊Dic3—32no—
28C2×C3×C4⋊Dic3—61no—
29C2×(C2×D3)⋊Dic3C2×C312yescomplemented
30C3×C3⋊#95—32no—
31#63579C2×C311yescharacteristic, complemented
32C23×C33C2311yescharacteristic
33C22×C32×D3—22no—
34C22×C32×D3—22no—
35C22×C32×D3—22no—
36C22×C32×D3—22no—
37C22×C32×D3—22no—
38C22×C32×D3—22no—
39C22×C32×D3D414yes—
40C2×C33×C4—21no—
41C2×C33×C4—21no—
42C2×C33×C4D412yescomplemented
43C2×C3×C3⋊Dic3—22no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3D412yes—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3—22no—
50C3×#625—28no—
51C3×#394—91no—
52#4961—32no—
53C24×C32—32no—
54C23×C3×D3—22no—
55C23×C3×D3C2×D312yes—
56C23×C3×D3C22×C312yes—
57C22×C32×C4—21no—
58C22×C32×C4—22no—
59C22×C32×C4—32no—
60C22×C32×C4—41no—
61C22×C32×C4C2×D312yes—
62C22×C32×C4C22×C311yescharacteristic
63C22×C3⋊Dic3C22×C311yescharacteristic
64C22×C3⋊Dic3C22×C311yescharacteristic
65C2×C3×#67—64no—
66C2×C3×#67—64no—
67C2×C3×#67—64no—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3—61no—
73C22×C3×Dic3C2×D312yes—
74C22×C3×Dic3C22×C312yes—
75C2×#625—24no—
76C2×#625—22no—
77C32×C22⋊C4—64no—
78C4⋊(C3⋊Dic3)—22no—
79C3×#217—64no—
80C3×(C2×D3)⋊C4—28no—
81C3×(C2×D3)⋊C4—24no—
82C3×(C2×D3)⋊C4—24no—
83C3×(C2×D3)⋊C4—64no—
84C3×C4⋊Dic3—64no—
85C3×C4⋊Dic3—64no—
86(C2×D3)⋊Dic3—24no—
87C22×C33C2×D412yes—
88C22×C33C2×D411yescharacteristic
89C22×C33C2×D411yescharacteristic
90C22×C33#4111yescharacteristic
91C22×C33#4112yes—
92C2×C32×D3—24no—
93C2×C32×D3—24no—
94C2×C32×D3—24no—
95C2×C32×D3—42no—
96C2×C32×D3—42no—
97C2×C32×D3—42no—
98C2×C32×D3—42no—
99C33×C4—22no—
100C3×C3⋊Dic3—22no—
101C3×C3⋊Dic3—41no—
102C32×Dic3—42no—
103C22×C3×D4—91no—
104C22×#67—32no—
105C2×C3×C4⋊C4—91no—
106C2×C3×C22⋊C4—32no—
107C2×C3×C22⋊C4—32no—
108C2×C3×C22⋊C4—62no—
109C2×C3×C22⋊C4—92no—
110C2×#217—32no—
111C2×(C2×D3)⋊C4—32no—
112C2×(C2×D3)⋊C4C3×D312yescomplemented
113C2×C4⋊Dic3—32no—
114C2×C4⋊Dic3—61no—
115C3×#95—91no—
116C3⋊#95—32no—
117C23×C32—21no—
118C23×C32—22no—
119C23×C32—34no—
120C23×C32—41no—
121C23×C32—62no—
122C23×C32—62no—
123C23×C32—62no—
124C23×C32—62no—
125C23×C32—62no—
126C23×C32—62no—
127C23×C32C22×D312yes—
128C23×C32C23×C311yescharacteristic
129C22×C3×D3—24no—
130C22×C3×D3—24no—
131C22×C3×D3—22no—
132C22×C3×D3—22no—
133C22×C3×D3—22no—
134C22×C3×D3—22no—
135C22×C3×D3—22no—
136C22×C3×D3—22no—
137C22×C3×D3—22no—
138C22×C3×D3—22no—
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—22no—
143C22×C3×D3—24no—
144C22×C3×D3—42no—
145C22×C3×D3—42no—
146C22×C3×D3—42no—
147C22×C3×D3—42no—
148C22×C3×D3D1214yes—
149C22×C3×D3C3×D414yes—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—24no—
158C2×C32×C4—42no—
159C2×C32×C4—42no—
160C2×C32×C4—42no—
161C2×C32×C4—42no—
162C2×C32×C4—42no—
163C2×C32×C4—62no—
164C2×C32×C4—62no—
165C2×C32×C4—62no—
166C2×C32×C4#6714yescomplemented
167C2×C32×C4C3×D412yescomplemented
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—22no—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3C3×D412yes—
172C3×#67—616no—
173C2×C3×Dic3—24no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—34no—
181C2×C3×Dic3—42no—
182C2×C3×Dic3—42no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—122no—
191C2×C3×Dic3—122no—
192#625—28no—
193#394—91noSylow
194C2×C33#12011yescharacteristic
195C2×C33#12112yes—
196C2×C33#12111yescharacteristic
197C2×C33#12312yes—
198C2×C33#12411yescharacteristic
199C32×D3—44no—
200C24×C3—32no—
201C24×C3—32no—
202C24×C3—62no—
203C23×D3C2×C3×D312yes—
204C2×C3×D4—182no—
205C2×C3×D4—184no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—32no—
209C22×C3×C4—32no—
210C22×C3×C4—41no—
211C22×C3×C4—62no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4D3211yescharacteristic
215C22×C3×C4C2×C3×D312yes—
216C2×#67—64no—
217C2×#67—64no—
218C2×#67—64no—
219C22×Dic3—32no—
220C22×Dic3—32no—
221C22×Dic3—61no—
222C22×Dic3—61no—
223C22×Dic3C2×C3×D312yes—
224C3×C4⋊C4—182no—
225C3×C22⋊C4—68no—
226C3×C22⋊C4—64no—
227C3×C22⋊C4—64no—
228C3×C22⋊C4—184no—
229#217—64no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—64no—
232C4⋊Dic3—64no—
233C4⋊Dic3—64no—
234C22×C32—22no—
235C22×C32—22no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32—42no—
249C22×C32—41no—
250C22×C32—42no—
251C22×C32—64no—
252C22×C32—64no—
253C22×C32—64no—
254C22×C32—122no—
255C22×C32—122no—
256C22×C32—122no—
257C22×C32—122no—
258C22×C32C2×D1212yes—
259C22×C32#23512yes—
260C22×C32D3×D412yes—
261C22×C32D3×D412yes—
262C22×C32#23712yes—
263C22×C32#23712yes—
264C22×C32C2×#6712yes—
265C22×C32C2×C3×D412yes—
266C22×C32C2×C3×D411yescharacteristic
267C22×C32C2×C3×D411yescharacteristic
268C22×C32C3×#4112yes—
269C22×C32C3×#4111yescharacteristic
270C2×C3×D3—28no—
271C2×C3×D3—24no—
272C2×C3×D3—24no—
273C2×C3×D3—24no—
274C2×C3×D3—24no—
275C2×C3×D3—24no—
276C2×C3×D3—24no—
277C2×C3×D3—44no—
278C2×C3×D3—44no—
279C2×C3×D3—44no—
280C2×C3×D3—44no—
281C2×C3×D3—44no—
282C2×C3×D3—44no—
283C2×C3×D3—42no—
284C2×C3×D3—42no—
285C2×C3×D3—42no—
286C2×C3×D3—42no—
287C2×C3×D3—42no—
288C2×C3×D3—42no—
289C2×C3×D3—42no—
290C2×C3×D3—42no—
291C32×C4—22no—
292C32×C4—24no—
293C32×C4—24no—
294C32×C4—44no—
295C32×C4—44no—
296C32×C4—122no—
297C3⋊Dic3—22no—
298C3⋊Dic3—41no—
299C3×Dic3—44no—
300C3×Dic3—42no—
301C3×Dic3—42no—
302C3×Dic3—64no—
303C3×Dic3—122no—
304C3×Dic3—122no—
305C3×Dic3—122no—
306C22×D4—91no—
307C2×C4⋊C4—91no—
308C2×C22⋊C4—32no—
309C2×C22⋊C4—92no—
310#95—91no—
311C33#39411yescharacteristic, complemented, Sylow
312C23×C3—22no—
313C23×C3—21no—
314C23×C3—34no—
315C23×C3—34no—
316C23×C3—41no—
317C23×C3—64no—
318C23×C3—64no—
319C23×C3—64no—
320C23×C3—62no—
321C23×C3—62no—
322C23×C3—62no—
323C23×C3—62no—
324C23×C3—62no—
325C23×C3—62no—
326C23×C3—62no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—62no—
330C23×C3—62no—
331C23×C3—62no—
332C23×C3—122no—
333C23×C3—122no—
334C23×C3—122no—
335C23×C3—122no—
336C23×C3C2×D3211yescharacteristic
337C23×C3C22×C3×D312yes—
338C22×D3—22no—
339C22×D3—22no—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—22no—
343C22×D3—22no—
344C22×D3C3×D1214yes—
345C3×D4—188no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—64no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—92no—
367C2×C3×C4—122no—
368C2×C3×C4—122no—
369C2×C3×C4—182no—
370C2×C3×C4—182no—
371C2×C3×C4—181no—
372C2×C3×C4#62512yescomplemented
373C2×C3×C4C3×#6714yescomplemented
374#67—616no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—34no—
379C2×Dic3—62no—
380C2×Dic3—62no—
381C2×Dic3—62no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—62no—
386C2×Dic3—122no—
387C2×Dic3—122no—
388C2×C32—22no—
389C2×C32—22no—
390C2×C32—22no—
391C2×C32—22no—
392C2×C32—22no—
393C2×C32—22no—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—22no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—42no—
403C2×C32—42no—
404C2×C32—41no—
405C2×C32—124no—
406C2×C32#88212yes—
407C2×C32#88312yes—
408C2×C32#88512yes—
409C2×C32#88612yes—
410C2×C32#93012yes—
411C2×C32#93812yes—
412C2×C32#93912yes—
413C2×C32C3×#12011yescharacteristic
414C2×C32C3×#12112yes—
415C2×C32C3×#12111yescharacteristic
416C2×C32C3×#12312yes—
417C2×C32C3×#12411yescharacteristic
418C3×D3—48no—
419C3×D3—44no—
420C3×D3—44no—
421C24—32no—
422C2×D4—182no—
423C2×D4—184no—
424C22×C4—32no—
425C22×C4—91no—
426C22×C4—91no—
427C22×C4C3×D3211yescharacteristic
428C4⋊C4—182no—
429C22⋊C4—64no—
430C22⋊C4—184no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—21no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—22no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—68no—
449C22×C3—64no—
450C22×C3—64no—
451C22×C3—64no—
452C22×C3—64no—
453C22×C3—64no—
454C22×C3—64no—
455C22×C3—124no—
456C22×C3—124no—
457C22×C3—124no—
458C22×C3—124no—
459C22×C3—124no—
460C22×C3—124no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3—122no—
464C22×C3—122no—
465C22×C3—122no—
466C22×C3—122no—
467C22×C3—122no—
468C22×C3—122no—
469C22×C3#379512yes—
470C22×C3D3×D1212yes—
471C22×C3#380511yescharacteristic
472C22×C3C2×#62511yescharacteristic
473C22×C3#381111yescharacteristic
474C22×C3C2×C3×D1212yes—
475C22×C3C3×#23512yes—
476C22×C3C3×D3×D412yes—
477C22×C3C3×D3×D412yes—
478C22×C3C3×#23712yes—
479C22×C3C3×#23712yes—
480C22×C3C2×C3×#6712yes—
481C2×D3—24no—
482C2×D3—24no—
483C2×D3—24no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—42no—
488C3×C4—24no—
489C3×C4—24no—
490C3×C4—22no—
491C3×C4—44no—
492C3×C4—44no—
493C3×C4—124no—
494C3×C4—122no—
495C3×C4—122no—
496C3×C4—182no—
497C3×C4—361no—
498Dic3—42no—
499Dic3—64no—
500Dic3—122no—
501Dic3—122no—
502Dic3—122no—
503C32—22no—
504C32—21no—
505C32—41no—
506C32#496112yescomplemented
507C32C3×#39411yescharacteristic, complemented
508C23—34no—
509C23—62no—
510C23—62no—
511C23—62no—
512C23—62no—
513C23—62no—
514C23—62no—
515C23C2×C3×D3211yescharacteristic
516D4—188no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—92no—
523C2×C4—182no—
524C2×C4—182no—
525C2×C4—181no—
526C2×C4C3×#62512yescomplemented
527C2×C3—21no—
528C2×C3—22no—
529C2×C3—22no—
530C2×C3—22no—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—22no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—22no—
539C2×C3—41no—
540C2×C3—42no—
541C2×C3—42no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—128no—
545C2×C3—124no—
546C2×C3—124no—
547C2×C3#6389112yes—
548C2×C3#6390812yes—
549C2×C3#6391311yescharacteristic
550C2×C3#6392311yescharacteristic
551C2×C3#6398011yescharacteristic
552C2×C3C3×#88212yes—
553C2×C3C3×#88312yes—
554C2×C3C3×#88512yes—
555C2×C3C3×#88612yes—
556C2×C3C3×#93012yes—
557C2×C3C3×#93812yes—
558C2×C3C3×#93912yes—
559D3—44no—
560C22—64no—
561C22—64no—
562C22—64no—
563C22—122no—
564C22—122no—
565C22—122no—
566C22—122no—
567C22C3×#379512yes—
568C22C3×D3×D1212yes—
569C22C3×#380511yescharacteristic
570C22C2×C3×#62511yescharacteristic
571C22C3×#381111yescharacteristic
572C4—22no—
573C4—122no—
574C4—182no—
575C4—361no—
576C3—22no—
577C3—21no—
578C3—41no—
579C3#1059139411yescharacteristic, direct product factor
580C3C3×#496112yescomplemented
581C2—124no—
582C2C3×#6389112yes—
583C2C3×#6390812yes—
584C2C3×#6391311yescharacteristic
585C2C3×#6392311yescharacteristic
586C2C3×#6398011yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC22×C3×#625C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#394
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
3. C2×C32×(C2×D3)⋊C4C2semidirect
4. C2×C3×(C2×D3)⋊Dic3C2semidirect
5. C3×#63579C2semidirect
7. #10591394C3direct (N × H)
21. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
25. C2×C3×(C2×D3)⋊C4D3semidirect
26. C2×C3×(C2×D3)⋊C4C2×C3semidirect
29. C2×(C2×D3)⋊Dic3C2×C3semidirect
31. #63579C2×C3semidirect
42. C2×C33×C4D4semidirect
112. C2×(C2×D3)⋊C4C3×D3semidirect
166. C2×C32×C4#67semidirect
167. C2×C32×C4C3×D4semidirect
311. C33#394semidirect
372. C2×C3×C4#625semidirect
373. C2×C3×C4C3×#67semidirect
506. C32#4961semidirect
507. C32C3×#394semidirect
526. C2×C4C3×#625semidirect
579. C3#10591394direct (N × H)
580. C3C3×#4961semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×(C2×D3)⋊C42
C2×C3×(C2×D3)⋊Dic32
C3×#635792
C3×#49613
#105913943