Finite Group Databasebeta

← C3×#10596036 #13179316

Subgroups — C3×#10596036 #13179316

Order 1728.

1804 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic, complemented
2C3×#64148C212yesmaximal, complemented
3C3×#64139C211yesmaximal, characteristic, complemented
4C3×#64139C211yesmaximal, characteristic, complemented
5C3×#64135C211yesmaximal, characteristic, complemented
6C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7C3×#64107C212yesmaximal, complemented
8C3×#64092C212yesmaximal, complemented
9C3×#64090C212yesmaximal, complemented
10C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11C3×#64079C211yesmaximal, characteristic, complemented
12#10596036C311yesmaximal, characteristic, direct product factor
13C3×#5565—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C33×C4C2211yescharacteristic
16C22×C33×C4C2211yescharacteristic
17C2×C3×#638C2212yes—
18C2×C3×#638C2212yes—
19C22×C3×C3⋊Dic3C2212yes—
20C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C3×C4×C3⋊D3C2212yes—
22C33×C4⋊C4C2211yescharacteristic
23C33×C22⋊C4C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25C33×C42C2211yescharacteristic
26C3×#3757C2212yes—
27C3×#3752C2212yes—
28C3×#3752C2212yes—
29C3×#3752C2212yes—
30C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
33C3×C4⋊(C3⋊Dic3)C2212yes—
34C3×#3750C2212yes—
35C3×#3750C2212yes—
36C3×C4×C3⋊Dic3C2212yes—
37C32×C4×D4—24no—
38C32×C4×D4D314yescomplemented
39C32×C4×D4C2×C311yescharacteristic, complemented
40#64148C2×C312yescomplemented
41#64139C2×C311yescharacteristic, complemented
42#64139C2×C311yescharacteristic, complemented
43#64135C2×C311yescharacteristic, complemented
44C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
45#64107C2×C312yescomplemented
46#64092C2×C312yescomplemented
47#64090C2×C312yescomplemented
48C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
49#64079C2×C311yescharacteristic, complemented
50C3×#939—38no—
51C3×#930—34no—
52C3×#930—34no—
53C3×#926—34no—
54C2×C3×C4⋊Dic3—34no—
55C3×#898—38no—
56C3×#883—38no—
57C3×#67⋊C4—38no—
58C3×C4×D12—34no—
59C3×#870—34no—
60C23×C33C2311yescharacteristic
61C23×C33C2311yescharacteristic
62C22×C3×C3⋊D3C2312yes—
63C33×D4—22no—
64C2×C33×C4—22no—
65C2×C33×C4—22no—
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C2×C33×C4C2311yescharacteristic
70C2×C33×C4C2311yescharacteristic
71C3×#638—24no—
72C3×#638—42no—
73C2×C3×C3⋊Dic3—42no—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2312yes—
77C3×C4⋊(C3⋊D3)—22no—
78C3×C4×C3⋊D3—42no—
79C3×#553—91no—
80#5565—34no—
81C2×C32×D4—24no—
82C2×C32×D4C2×D314yes—
83C2×C32×D4C22×C311yescharacteristic
84C22×C32×C4—24no—
85C22×C32×C4—24no—
86C22×C32×C4C2×D314yes—
87C22×C32×C4C2×D314yes—
88C22×C32×C4C22×C311yescharacteristic
89C22×C32×C4C22×C311yescharacteristic
90C2×#638C22×C312yes—
91C2×#638C22×C312yes—
92C22×C3⋊Dic3C22×C312yes—
93C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
94C2×C4×C3⋊D3C22×C312yes—
95C2×C3×#67—38no—
96C2×C3×#67—38no—
97C22×C3×Dic3—38no—
98C2×C3×D12—34no—
99C2×C3×C4×D3—38no—
100C32×C4⋊C4—24no—
101C32×C4⋊C4C2×D314yes—
102C32×C4⋊C4C22×C311yescharacteristic
103C32×C22⋊C4—24no—
104C32×C22⋊C4—24no—
105C32×C22⋊C4C2×D314yes—
106C32×C22⋊C4C2×D314yes—
107C32×C22⋊C4C22×C311yescharacteristic
108C32×C22⋊C4C22×C311yescharacteristic
109C32×C42—24no—
110C32×C42C2×D314yes—
111C32×C42C22×C311yescharacteristic
112#3757C22×C312yes—
113#3752C22×C312yes—
114#3752C22×C312yes—
115#3752C22×C312yes—
116C4⋊(C3⋊Dic3)C22×C311yescharacteristic
117C4⋊(C3⋊Dic3)C22×C311yescharacteristic
118C4⋊(C3⋊Dic3)C22×C311yescharacteristic
119C4⋊(C3⋊Dic3)C22×C312yes—
120#3750C22×C312yes—
121#3750C22×C312yes—
122C4×C3⋊Dic3C22×C312yes—
123C3×#217—38no—
124C3×(C2×D3)⋊C4—38no—
125C3×(C2×D3)⋊C4—38no—
126C3×(C2×D3)⋊C4—38no—
127C3×C4⋊Dic3—38no—
128C3×C4⋊Dic3—34no—
129C3×C4⋊Dic3—34no—
130C3×C4⋊Dic3—34no—
131C3×Dic3⋊C4—38no—
132C3×Dic3⋊C4—38no—
133C3×C4×Dic3—38no—
134C22×C33—21no—
135C22×C33—21no—
136C22×C33—21no—
137C22×C33—21no—
138C22×C33—21no—
139C22×C33#4112yes—
140C22×C33C2411yescharacteristic
141C2×C3×C3⋊D3—22no—
142C2×C3×C3⋊D3—22no—
143C2×C3×C3⋊D3—22no—
144C33×C4—21no—
145C33×C4—21no—
146C33×C4—21no—
147C33×C4—21no—
148C33×C4#4112yes—
149C3×C3⋊Dic3—22no—
150C3×C3⋊Dic3—22no—
151C3×C3⋊Dic3—22no—
152C3×#126—92no—
153C3×#125—91no—
154C3×#123—92no—
155C3×#121—92no—
156C3×#121—91no—
157C3×#121—91no—
158C3×C4×D4—24no—
159C3×C4×D4—92no—
160C3×C4×D4—91no—
161C3×C4×D4C3×D314yescomplemented
162C3×C4×D4C3⋊D311yescharacteristic, complemented
163C3×#117—91no—
164C2×C3×C4⋊C4—91no—
165#939—38no—
166#930—34no—
167#930—34no—
168#926—34no—
169C2×C4⋊Dic3—34no—
170#898—38no—
171#883—38no—
172#67⋊C4—38no—
173C4×D12—34no—
174#870—34no—
175C23×C32—24no—
176C23×C32—24no—
177C23×C32C22×D314yes—
178C23×C32C22×D314yes—
179C23×C32C23×C311yescharacteristic
180C23×C32C23×C311yescharacteristic
181C22×C3⋊D3C23×C312yes—
182C22×C3×D3—38no—
183C32×D4—216no—
184C32×D4—28no—
185C32×D4—22no—
186C2×C32×C4—22no—
187C2×C32×C4—22no—
188C2×C32×C4—24no—
189C2×C32×C4—24no—
190C2×C32×C4—24no—
191C2×C32×C4—24no—
192C2×C32×C4—24no—
193C2×C32×C4—28no—
194C2×C32×C4—28no—
195C2×C32×C4—48no—
196C2×C32×C4—48no—
197C2×C32×C4C22×D314yes—
198C2×C32×C4C22×D314yes—
199C2×C32×C4C22×D314yes—
200C2×C32×C4C22×D314yes—
201C2×C32×C4C22×D314yes—
202C2×C32×C4C23×C311yescharacteristic
203C2×C32×C4C23×C311yescharacteristic
204C2×C32×C4C23×C311yescharacteristic
205C2×C32×C4C23×C311yescharacteristic
206C2×C32×C4C23×C311yescharacteristic
207#638—24no—
208#638—42no—
209C2×C3⋊Dic3—42no—
210C2×C3⋊Dic3C23×C312yes—
211C2×C3⋊Dic3C23×C312yes—
212C2×C3⋊Dic3C23×C312yes—
213C4⋊(C3⋊D3)—22no—
214C4×C3⋊D3—42no—
215C3×#67—616no—
216C3×#67—128no—
217C2×C3×Dic3—38no—
218C2×C3×Dic3—38no—
219C2×C3×Dic3—38no—
220C2×C3×Dic3—128no—
221C3×D12—68no—
222C3×C4×D3—128no—
223#553—91noSylow
224C2×C33—22no—
225C2×C33—41no—
226C2×C33C2×#4111yescharacteristic
227C2×C33C2×#4111yescharacteristic
228C2×C33H(5,𝔽2)11yescharacteristic
229C3×C3⋊D3—42no—
230C2×C3×D4—24no—
231C2×C3×D4—92no—
232C2×C3×D4—92no—
233C2×C3×D4—91no—
234C2×C3×D4C2×C3×D314yes—
235C2×C3×D4C2×C3⋊D311yescharacteristic
236C22×C3×C4—24no—
237C22×C3×C4—24no—
238C22×C3×C4—92no—
239C22×C3×C4—92no—
240C22×C3×C4C2×C3×D314yes—
241C22×C3×C4C2×C3×D314yes—
242C22×C3×C4C2×C3⋊D311yescharacteristic
243C22×C3×C4C2×C3⋊D311yescharacteristic
244C2×#67—38no—
245C2×#67—38no—
246C22×Dic3—38no—
247C2×D12—34no—
248C2×C4×D3—38no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—92no—
255C3×C4⋊C4—91no—
256C3×C4⋊C4C2×C3×D314yes—
257C3×C4⋊C4C2×C3⋊D311yescharacteristic
258C3×C22⋊C4—24no—
259C3×C22⋊C4—24no—
260C3×C22⋊C4—92no—
261C3×C22⋊C4—92no—
262C3×C22⋊C4—92no—
263C3×C22⋊C4—92no—
264C3×C22⋊C4C2×C3×D314yes—
265C3×C22⋊C4C2×C3×D314yes—
266C3×C22⋊C4C2×C3⋊D311yescharacteristic
267C3×C22⋊C4C2×C3⋊D311yescharacteristic
268C3×C42—24no—
269C3×C42—92no—
270C3×C42C2×C3×D314yes—
271C3×C42C2×C3⋊D311yescharacteristic
272#217—38no—
273(C2×D3)⋊C4—38no—
274(C2×D3)⋊C4—38no—
275(C2×D3)⋊C4—38no—
276C4⋊Dic3—34no—
277C4⋊Dic3—34no—
278C4⋊Dic3—34no—
279C4⋊Dic3—38no—
280Dic3⋊C4—38no—
281Dic3⋊C4—38no—
282C4×Dic3—38no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—28no—
286C22×C32—24no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—21no—
290C22×C32—24no—
291C22×C32—28no—
292C22×C32—24no—
293C22×C32—24no—
294C22×C32—24no—
295C22×C32—24no—
296C22×C32—44no—
297C22×C32—44no—
298C22×C32—44no—
299C22×C32—44no—
300C22×C32#23518yes—
301C22×C32C3×#4112yes—
302C22×C32C23×D314yes—
303C22×C32C24×C311yescharacteristic
304C2×C3⋊D3—22no—
305C2×C3⋊D3—22no—
306C2×C3⋊D3—22no—
307C2×C3×D3—68no—
308C2×C3×D3—68no—
309C2×C3×D3—68no—
310C32×C4—28no—
311C32×C4—24no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—28no—
317C32×C4—24no—
318C32×C4—24no—
319C32×C4—24no—
320C32×C4—44no—
321C32×C4—44no—
322C32×C4—44no—
323C32×C4#23718yes—
324C32×C4C3×#4112yes—
325C3⋊Dic3—22no—
326C3⋊Dic3—22no—
327C3⋊Dic3—22no—
328C3×Dic3—68no—
329C3×Dic3—68no—
330C3×Dic3—68no—
331#126—92no—
332#125—91no—
333#123—92no—
334#121—92no—
335#121—91no—
336#121—91no—
337C4×D4—92no—
338C4×D4—91no—
339C4×D4C3×C3⋊D311yescharacteristic, complemented
340#117—91no—
341C2×C4⋊C4—91no—
342C33#55311yescharacteristic, complemented, Sylow
343C23×C3—24no—
344C23×C3—24no—
345C23×C3—92no—
346C23×C3C22×C3×D314yes—
347C23×C3C22×C3×D314yes—
348C23×C3C22×C3⋊D311yescharacteristic
349C23×C3C22×C3⋊D311yescharacteristic
350C22×D3—38no—
351C3×D4—216no—
352C3×D4—28no—
353C3×D4—22no—
354C3×D4—184no—
355C3×D4—182no—
356C3×D4—362no—
357C2×C3×C4—28no—
358C2×C3×C4—28no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—48no—
367C2×C3×C4—48no—
368C2×C3×C4—92no—
369C2×C3×C4—92no—
370C2×C3×C4—92no—
371C2×C3×C4—362no—
372C2×C3×C4—362no—
373C2×C3×C4C22×C3×D314yes—
374C2×C3×C4C22×C3×D314yes—
375C2×C3×C4C22×C3×D314yes—
376C2×C3×C4C22×C3×D314yes—
377C2×C3×C4C22×C3×D314yes—
378C2×C3×C4C22×C3⋊D311yescharacteristic
379C2×C3×C4C22×C3⋊D311yescharacteristic
380C2×C3×C4C22×C3⋊D311yescharacteristic
381C2×C3×C4C22×C3⋊D311yescharacteristic
382C2×C3×C4C22×C3⋊D311yescharacteristic
383#67—616no—
384#67—128no—
385C2×Dic3—38no—
386C2×Dic3—38no—
387C2×Dic3—38no—
388C2×Dic3—128no—
389D12—68no—
390C4×D3—128no—
391C2×C32—22no—
392C2×C32—24no—
393C2×C32—24no—
394C2×C32—24no—
395C2×C32—28no—
396C2×C32—48no—
397C2×C32—48no—
398C2×C32—44no—
399C2×C32—41no—
400C2×C32C2×#23514yes—
401C2×C32C2×#23714yes—
402C2×C32#100914yes—
403C2×C32C2×C3×#4111yescharacteristic
404C2×C32C2×C3×#4111yescharacteristic
405C2×C32C3×H(5,𝔽2)11yescharacteristic
406C3⋊D3—42no—
407C3×D3—128no—
408C2×D4—92no—
409C2×D4—92no—
410C2×D4—91no—
411C2×D4C2×C3×C3⋊D311yescharacteristic
412C22×C4—92no—
413C22×C4—92no—
414C22×C4C2×C3×C3⋊D311yescharacteristic
415C22×C4C2×C3×C3⋊D311yescharacteristic
416C4⋊C4—91no—
417C4⋊C4—91no—
418C4⋊C4—92no—
419C4⋊C4—91no—
420C4⋊C4—92no—
421C4⋊C4—92no—
422C4⋊C4C2×C3×C3⋊D311yescharacteristic
423C22⋊C4—92no—
424C22⋊C4—92no—
425C22⋊C4—92no—
426C22⋊C4—92no—
427C22⋊C4C2×C3×C3⋊D311yescharacteristic
428C22⋊C4C2×C3×C3⋊D311yescharacteristic
429C42—92no—
430C42C2×C3×C3⋊D311yescharacteristic
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—28no—
436C22×C3—24no—
437C22×C3—24no—
438C22×C3—24no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—24no—
443C22×C3—28no—
444C22×C3—44no—
445C22×C3—44no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3—182no—
451C22×C3C3×#23518yes—
452C22×C3#382812yes—
453C22×C3C23×C3×D314yes—
454C22×C3C23×C3⋊D311yescharacteristic
455C2×D3—68no—
456C2×D3—68no—
457C2×D3—68no—
458C3×C4—28no—
459C3×C4—24no—
460C3×C4—24no—
461C3×C4—24no—
462C3×C4—24no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—28no—
468C3×C4—44no—
469C3×C4—44no—
470C3×C4—44no—
471C3×C4—182no—
472C3×C4—182no—
473C3×C4—182no—
474C3×C4C3×#23718yes—
475C3×C4#383012yes—
476Dic3—68no—
477Dic3—68no—
478Dic3—68no—
479C32—24no—
480C32#556514yescomplemented
481C32C3×#55311yescharacteristic, complemented
482C23—92no—
483C23C22×C3×C3⋊D311yescharacteristic
484C23C22×C3×C3⋊D311yescharacteristic
485D4—22no—
486D4—184no—
487D4—182no—
488D4—362no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—92no—
492C2×C4—92no—
493C2×C4—92no—
494C2×C4—362no—
495C2×C4—362no—
496C2×C4C22×C3×C3⋊D311yescharacteristic
497C2×C4C22×C3×C3⋊D311yescharacteristic
498C2×C4C22×C3×C3⋊D311yescharacteristic
499C2×C4C22×C3×C3⋊D311yescharacteristic
500C2×C4C22×C3×C3⋊D311yescharacteristic
501C2×C3—28no—
502C2×C3—22no—
503C2×C3—24no—
504C2×C3—24no—
505C2×C3—24no—
506C2×C3—48no—
507C2×C3—48no—
508C2×C3—44no—
509C2×C3—41no—
510C2×C3—362no—
511C2×C3C2×C3×#23514yes—
512C2×C3C2×C3×#23714yes—
513C2×C3C3×#100914yes—
514C2×C3C2×#382811yescharacteristic
515C2×C3C2×#383011yescharacteristic
516C2×C3#6436611yescharacteristic
517D3—128no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—182no—
524C22—182no—
525C22—182no—
526C22C3×#382812yes—
527C22C23×C3×C3⋊D311yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—182no—
533C4—182no—
534C4—182no—
535C4C3×#383012yes—
536C3—24no—
537C3C3×#556514yescomplemented
538C3#1059603611yescharacteristic, direct product factor
539C2—22no—
540C2—41no—
541C2—362no—
542C2C2×C3×#382811yescharacteristic
543C2C2×C3×#383011yescharacteristic
544C2C3×#6436611yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C24
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C33×C4×D4C2semidirect
2. C3×#64148C2semidirect
3. C3×#64139C2semidirect
4. C3×#64139C2semidirect
5. C3×#64135C2semidirect
6. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
7. C3×#64107C2semidirect
8. C3×#64092C2semidirect
9. C3×#64090C2semidirect
10. C3×C4×C4⋊(C3⋊D3)C2semidirect
11. C3×#64079C2semidirect
12. #10596036C3direct (N × H)
38. C32×C4×D4D3semidirect
39. C32×C4×D4C2×C3semidirect
40. #64148C2×C3semidirect
41. #64139C2×C3semidirect
42. #64139C2×C3semidirect
43. #64135C2×C3semidirect
44. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
45. #64107C2×C3semidirect
46. #64092C2×C3semidirect
47. #64090C2×C3semidirect
48. C4×C4⋊(C3⋊D3)C2×C3semidirect
49. #64079C2×C3semidirect
161. C3×C4×D4C3×D3semidirect
162. C3×C4×D4C3⋊D3semidirect
339. C4×D4C3×C3⋊D3semidirect
342. C33#553semidirect
480. C32#5565semidirect
481. C32C3×#553semidirect
537. C3C3×#5565semidirect
538. C3#10596036direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×D42
C3×#641482
C3×#641392
C3×#641352
C2×C3×C4⋊(C3⋊Dic3)2
C3×#641072
C3×#640922
C3×#640902
C3×C4×C4⋊(C3⋊D3)2
C3×#640792
#105960363
C3×#55653