Finite Group Databasebeta

← C3×#10596057 #13179337

Subgroups — C3×#10596057 #13179337

Order 1728.

2500 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
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#3830C211yesmaximal, characteristic, complemented
3C33×#120C211yesmaximal, characteristic, complemented
4C3×#64162C211yesmaximal, characteristic, complemented
5C3×#64148C212yesmaximal, complemented
6C3×#64144C211yesmaximal, characteristic, complemented
7C3×D4×C3⋊Dic3C212yesmaximal, complemented
8C3×#64095C211yesmaximal, characteristic, complemented
9C3×#64090C212yesmaximal, complemented
10C3×#64087C212yesmaximal, complemented
11C3×#64086C211yesmaximal, characteristic, complemented
12#10596057C311yesmaximal, characteristic, direct product factor
13C3×#5586—34nomaximal
14C24×C33C2211yescharacteristic
15C2×C33×D4C2212yes—
16C2×C33×D4C2211yescharacteristic
17C2×C3×#638C2212yes—
18C2×C3×#638C2211yescharacteristic
19C2×C3×#638C2211yescharacteristic, complemented
20C22×C3×C3⋊Dic3—24no—
21C22×C3×C3⋊Dic3C2212yes—
22C22×C3×C3⋊Dic3C2212yes—
23C22×C3×C3⋊Dic3C2211yescharacteristic
24C22×C3×C3⋊Dic3C2211yescharacteristic
25C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
26C3×#3830—24no—
27C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
28C2×C3×#634C2211yescharacteristic, complemented
29C33×C22⋊C4C2211yescharacteristic
30C33×C22⋊C4C2212yes—
31C3×#3757C2211yescharacteristic
32C3×#3757C2212yes—
33C3×#3757C2212yescomplemented
34C3×#3752C2212yescomplemented
35C3×C4⋊(C3⋊Dic3)C2212yescomplemented
36C3×#3750C2212yescomplemented
37C3×#3750C2212yescomplemented
38C3×C4×C3⋊Dic3C2212yescomplemented
39C23×C3⋊Dic3C2×C311yescharacteristic, complemented
40C2×#3830C2×C311yescharacteristic, complemented
41C23×C3×Dic3—34no—
42C2×C3×#237—34no—
43C32×#120—24no—
44C32×#120D314yescomplemented
45C32×#120C2×C311yescharacteristic, complemented
46#64162C2×C311yescharacteristic, complemented
47#64148C2×C312yescomplemented
48#64144C2×C311yescharacteristic, complemented
49D4×C3⋊Dic3C2×C312yescomplemented
50#64095C2×C311yescharacteristic, complemented
51#64090C2×C312yescomplemented
52#64087C2×C312yescomplemented
53#64086C2×C311yescharacteristic, complemented
54C3×#953—34no—
55C3×#939—38no—
56C3×#935—34no—
57C3×D4×Dic3—38no—
58C3×#886—34no—
59C3×#67⋊C4—38no—
60C3×#878—38no—
61C3×#877—34no—
62C23×C33—22no—
63C23×C33—24no—
64C23×C33C2311yescharacteristic
65C23×C33C2311yescharacteristic
66C23×C33C2312yes—
67C22×C3×C3⋊D3C2311yescharacteristic
68C33×D4—22no—
69C33×D4—24no—
70C2×C33×C4C2312yes—
71C2×C33×C4C2311yescharacteristic
72C3×#638—22no—
73C3×#638—24no—
74C3×#638—22no—
75C2×C3×C3⋊Dic3—22no—
76C2×C3×C3⋊Dic3—22no—
77C2×C3×C3⋊Dic3—24no—
78C2×C3×C3⋊Dic3—24no—
79C2×C3×C3⋊Dic3D414yescomplemented
80C2×C3×C3⋊Dic3C2312yes—
81C2×C3×C3⋊Dic3C2312yes—
82C2×C3×C3⋊Dic3C2311yescharacteristic
83C2×C3×C3⋊Dic3C2311yescharacteristic
84C2×C3×C3⋊Dic3C2311yescharacteristic
85C3×C4×C3⋊D3—22no—
86C3×#634—22no—
87C3×#525—91no—
88#5586—34no—
89C24×C32—24no—
90C24×C32C2×D314yes—
91C24×C32C22×C311yescharacteristic
92C2×C32×D4—28no—
93C2×C32×D4—24no—
94C2×C32×D4C2×D318yes—
95C2×C32×D4C2×D314yes—
96C2×C32×D4C22×C311yescharacteristic
97C2×C32×D4C22×C312yes—
98C2×#638C22×C312yes—
99C2×#638C22×C311yescharacteristic
100C2×#638C22×C311yescharacteristic, complemented
101C22×C3⋊Dic3—24no—
102C22×C3⋊Dic3C22×C312yes—
103C22×C3⋊Dic3C22×C312yes—
104C22×C3⋊Dic3C22×C311yescharacteristic
105C22×C3⋊Dic3C22×C311yescharacteristic
106C22×C3⋊Dic3C22×C311yescharacteristic, complemented
107#3830—24no—
108C2×C4×C3⋊D3C22×C311yescharacteristic, complemented
109C2×#634C22×C311yescharacteristic, complemented
110C2×C3×#67—38no—
111C2×C3×#67—34no—
112C2×C3×#67—34no—
113C22×C3×Dic3—34no—
114C22×C3×Dic3—38no—
115C22×C3×Dic3—38no—
116C22×C3×Dic3—34no—
117C22×C3×Dic3—34no—
118C22×C3×Dic3—616no—
119C3×#237—616no—
120C2×C3×C4×D3—34no—
121C2×C3×Dic6—34no—
122C32×C22⋊C4—24no—
123C32×C22⋊C4—28no—
124C32×C22⋊C4C2×D314yes—
125C32×C22⋊C4C2×D318yes—
126C32×C22⋊C4C22×C311yescharacteristic
127C32×C22⋊C4C22×C312yes—
128#3757C22×C311yescharacteristic
129#3757C22×C312yes—
130#3757C22×C312yescomplemented
131#3752C22×C312yescomplemented
132C4⋊(C3⋊Dic3)C22×C312yescomplemented
133#3750C22×C312yescomplemented
134#3750C22×C312yescomplemented
135C4×C3⋊Dic3C22×C312yescomplemented
136C3×#217—34no—
137C3×#217—38no—
138C3×#217—38no—
139C3×(C2×D3)⋊C4—38no—
140C3×C4⋊Dic3—38no—
141C3×Dic3⋊C4—38no—
142C3×Dic3⋊C4—38no—
143C3×C4×Dic3—38no—
144C22×C33—28no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33—22no—
148C22×C33—21no—
149C22×C33—22no—
150C22×C33C2×D412yes—
151C22×C33#4114yescomplemented
152C22×C33C2411yescharacteristic
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3—22no—
155C33×C4—21no—
156C33×C4—22no—
157C3×C3⋊Dic3—21no—
158C3×C3⋊Dic3—22no—
159C3×C3⋊Dic3—22no—
160C3×C3⋊Dic3C2×D412yes—
161C3×C3⋊Dic3C2×D412yes—
162C2×C3×#41—91no—
163C23×C3×C4—91no—
164C23×Dic3—34no—
165C2×#237—34no—
166C3×#123—91no—
167C3×#123—91no—
168C3×C22⋊Q8—92no—
169C3×#121—92no—
170C3×#120—24no—
171C3×#120—91no—
172C3×#120C3×D314yescomplemented
173C3×#120C3⋊D311yescharacteristic, complemented
174C3×C4×D4—92no—
175C3×C4×D4—92no—
176C3×#117—91no—
177#953—34no—
178#939—38no—
179#935—34no—
180D4×Dic3—38no—
181#886—34no—
182#67⋊C4—38no—
183#878—38no—
184#877—34no—
185C23×C32—22no—
186C23×C32—24no—
187C23×C32—24no—
188C23×C32—28no—
189C23×C32—24no—
190C23×C32—28no—
191C23×C32—216no—
192C23×C32—48no—
193C23×C32—416no—
194C23×C32C22×D314yes—
195C23×C32C22×D314yes—
196C23×C32C22×D318yes—
197C23×C32C23×C311yescharacteristic
198C23×C32C23×C311yescharacteristic
199C23×C32C23×C312yes—
200C22×C3⋊D3C23×C311yescharacteristic
201C22×C3×D3—34no—
202C32×D4—22no—
203C32×D4—24no—
204C32×D4—28no—
205C32×D4—216no—
206C32×D4—48no—
207C32×D4—416no—
208C2×C32×C4—28no—
209C2×C32×C4—24no—
210C2×C32×C4C22×D318yes—
211C2×C32×C4C22×D314yes—
212C2×C32×C4C23×C312yes—
213C2×C32×C4C23×C311yescharacteristic
214#638—22no—
215#638—24no—
216#638—22no—
217C2×C3⋊Dic3—22no—
218C2×C3⋊Dic3—22no—
219C2×C3⋊Dic3—24no—
220C2×C3⋊Dic3—24no—
221C2×C3⋊Dic3C3×D414yescomplemented
222C2×C3⋊Dic3C23×C312yes—
223C2×C3⋊Dic3C23×C312yes—
224C2×C3⋊Dic3C23×C311yescharacteristic
225C2×C3⋊Dic3C23×C311yescharacteristic
226C2×C3⋊Dic3C23×C311yescharacteristic
227C4×C3⋊D3—22no—
228#634—22no—
229C3×#67—68no—
230C3×#67—68no—
231C3×#67—616no—
232C2×C3×Dic3—316no—
233C2×C3×Dic3—34no—
234C2×C3×Dic3—34no—
235C2×C3×Dic3—38no—
236C2×C3×Dic3—38no—
237C2×C3×Dic3—34no—
238C2×C3×Dic3—68no—
239C2×C3×Dic3—616no—
240C2×C3×Dic3—616no—
241C2×C3×Dic3—68no—
242C3×C4×D3—68no—
243C3×Dic6—68no—
244#525—91noSylow
245C2×C33—22no—
246C2×C33—24no—
247C2×C33—41no—
248C2×C33C22×D411yescharacteristic
249C2×C33C2×#4112yes—
250C3×C3⋊D3—41no—
251C24×C3—24no—
252C24×C3C2×C3×D314yes—
253C24×C3C2×C3⋊D311yescharacteristic
254C3×#41—184no—
255C2×C3×Q8—91no—
256C2×C3×D4—28no—
257C2×C3×D4—24no—
258C2×C3×D4—92no—
259C2×C3×D4—91no—
260C2×C3×D4—91no—
261C2×C3×D4C2×C3×D318yes—
262C2×C3×D4C2×C3×D314yes—
263C2×C3×D4C2×C3⋊D312yes—
264C2×C3×D4C2×C3⋊D311yescharacteristic
265C22×C3×C4—91no—
266C22×C3×C4—92no—
267C22×C3×C4—91no—
268C22×C3×C4—92no—
269C22×C3×C4—91no—
270C22×C3×C4—91no—
271C22×C3×C4—184no—
272C2×#67—38no—
273C2×#67—34no—
274C2×#67—34no—
275C22×Dic3—34no—
276C22×Dic3—38no—
277C22×Dic3—38no—
278C22×Dic3—34no—
279C22×Dic3—34no—
280C22×Dic3—616no—
281#237—616no—
282C2×C4×D3—34no—
283C2×Dic6—34no—
284C3×C4⋊C4—92no—
285C3×C4⋊C4—92no—
286C3×C4⋊C4—92no—
287C3×C22⋊C4—24no—
288C3×C22⋊C4—28no—
289C3×C22⋊C4—91no—
290C3×C22⋊C4—92no—
291C3×C22⋊C4—92no—
292C3×C22⋊C4—92no—
293C3×C22⋊C4C2×C3×D314yes—
294C3×C22⋊C4C2×C3×D318yes—
295C3×C22⋊C4C2×C3⋊D311yescharacteristic
296C3×C22⋊C4C2×C3⋊D312yes—
297C3×C42—92no—
298#217—34no—
299#217—38no—
300#217—38no—
301(C2×D3)⋊C4—38no—
302C4⋊Dic3—38no—
303Dic3⋊C4—38no—
304Dic3⋊C4—38no—
305C4×Dic3—38no—
306C22×C32—28no—
307C22×C32—22no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—21no—
311C22×C32—22no—
312C22×C32—24no—
313C22×C32—28no—
314C22×C32—216no—
315C22×C32—232no—
316C22×C32—28no—
317C22×C32—28no—
318C22×C32—28no—
319C22×C32—24no—
320C22×C32—28no—
321C22×C32—48no—
322C22×C32—48no—
323C22×C32—48no—
324C22×C32—44no—
325C22×C32—48no—
326C22×C32—432no—
327C22×C32D3×D418yes—
328C22×C32#237116yescomplemented
329C22×C32C2×C3×D412yes—
330C22×C32C3×#4114yescomplemented
331C22×C32C23×D314yes—
332C22×C32C24×C311yescharacteristic
333C2×C3⋊D3—21no—
334C2×C3⋊D3—22no—
335C2×C3×D3—64no—
336C2×C3×D3—68no—
337C32×C4—21no—
338C32×C4—22no—
339C32×C4—24no—
340C32×C4—28no—
341C32×C4—44no—
342C32×C4—48no—
343C3⋊Dic3—21no—
344C3⋊Dic3—22no—
345C3⋊Dic3—22no—
346C3⋊Dic3C2×C3×D412yes—
347C3⋊Dic3C2×C3×D412yes—
348C3×Dic3—38no—
349C3×Dic3—38no—
350C3×Dic3—64no—
351C3×Dic3—68no—
352C3×Dic3—68no—
353C2×#41—91no—
354C23×C4—91no—
355#123—91no—
356#123—91no—
357C22⋊Q8—92no—
358#121—92no—
359#120—91no—
360#120C3×C3⋊D311yescharacteristic, complemented
361C4×D4—92no—
362C4×D4—92no—
363#117—91no—
364C33#52511yescharacteristic, complemented, Sylow
365C23×C3—28no—
366C23×C3—216no—
367C23×C3—22no—
368C23×C3—24no—
369C23×C3—24no—
370C23×C3—28no—
371C23×C3—24no—
372C23×C3—48no—
373C23×C3—416no—
374C23×C3—91no—
375C23×C3C22×C3×D314yes—
376C23×C3C22×C3×D314yes—
377C23×C3C22×C3×D318yes—
378C23×C3C22×C3⋊D311yescharacteristic
379C23×C3C22×C3⋊D311yescharacteristic
380C23×C3C22×C3⋊D312yes—
381C22×D3—34no—
382C3×Q8—182no—
383C3×D4—28no—
384C3×D4—216no—
385C3×D4—22no—
386C3×D4—24no—
387C3×D4—48no—
388C3×D4—416no—
389C3×D4—182no—
390C3×D4—182no—
391C3×D4—184no—
392C2×C3×C4—28no—
393C2×C3×C4—24no—
394C2×C3×C4—91no—
395C2×C3×C4—91no—
396C2×C3×C4—94no—
397C2×C3×C4—92no—
398C2×C3×C4—92no—
399C2×C3×C4—91no—
400C2×C3×C4—182no—
401C2×C3×C4—184no—
402C2×C3×C4—184no—
403C2×C3×C4—182no—
404C2×C3×C4—182no—
405C2×C3×C4C22×C3×D318yes—
406C2×C3×C4C22×C3×D314yes—
407C2×C3×C4C22×C3⋊D312yes—
408C2×C3×C4C22×C3⋊D311yescharacteristic
409#67—68no—
410#67—68no—
411#67—616no—
412C2×Dic3—34no—
413C2×Dic3—34no—
414C2×Dic3—316no—
415C2×Dic3—38no—
416C2×Dic3—38no—
417C2×Dic3—34no—
418C2×Dic3—68no—
419C2×Dic3—616no—
420C2×Dic3—616no—
421C2×Dic3—68no—
422C4×D3—68no—
423Dic6—68no—
424C2×C32—22no—
425C2×C32—24no—
426C2×C32—24no—
427C2×C32—28no—
428C2×C32—28no—
429C2×C32—216no—
430C2×C32—48no—
431C2×C32—416no—
432C2×C32—41no—
433C2×C32—44no—
434C2×C32—84no—
435C2×C32C2×D3×D414yes—
436C2×C32C2×#23718yes—
437C2×C32C22×C3×D411yescharacteristic
438C2×C32C2×C3×#4112yes—
439C3⋊D3—41no—
440C3×D3—124no—
441C24C2×C3×C3⋊D311yescharacteristic
442#41—184no—
443C2×Q8—91no—
444C2×D4—92no—
445C2×D4—91no—
446C2×D4—91no—
447C2×D4C2×C3×C3⋊D312yes—
448C2×D4C2×C3×C3⋊D311yescharacteristic
449C22×C4—91no—
450C22×C4—92no—
451C22×C4—91no—
452C22×C4—92no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—184no—
456C4⋊C4—92no—
457C4⋊C4—92no—
458C4⋊C4—92no—
459C22⋊C4—91no—
460C22⋊C4—92no—
461C22⋊C4—92no—
462C22⋊C4—92no—
463C22⋊C4C2×C3×C3⋊D311yescharacteristic
464C22⋊C4C2×C3×C3⋊D312yes—
465C42—92no—
466C22×C3—28no—
467C22×C3—232no—
468C22×C3—28no—
469C22×C3—28no—
470C22×C3—28no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—28no—
476C22×C3—21no—
477C22×C3—22no—
478C22×C3—24no—
479C22×C3—28no—
480C22×C3—216no—
481C22×C3—48no—
482C22×C3—48no—
483C22×C3—48no—
484C22×C3—44no—
485C22×C3—48no—
486C22×C3—432no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3C3×D3×D418yes—
490C22×C3C3×#237116yescomplemented
491C22×C3D4×C3⋊D312yes—
492C22×C3#383014yescomplemented
493C22×C3C23×C3×D314yes—
494C22×C3C23×C3⋊D311yescharacteristic
495C2×D3—64no—
496C2×D3—68no—
497C3×C4—24no—
498C3×C4—28no—
499C3×C4—21no—
500C3×C4—22no—
501C3×C4—44no—
502C3×C4—48no—
503C3×C4—92no—
504C3×C4—92no—
505C3×C4—182no—
506C3×C4—181no—
507C3×C4—182no—
508Dic3—38no—
509Dic3—38no—
510Dic3—64no—
511Dic3—68no—
512Dic3—68no—
513C32—24no—
514C32#558614yescomplemented
515C32C3×#52511yescharacteristic, complemented
516C23—22no—
517C23—24no—
518C23—91no—
519C23C22×C3×C3⋊D311yescharacteristic
520C23C22×C3×C3⋊D311yescharacteristic
521C23C22×C3×C3⋊D312yes—
522Q8—182no—
523D4—22no—
524D4—24no—
525D4—182no—
526D4—182no—
527D4—184no—
528C2×C4—91no—
529C2×C4—91no—
530C2×C4—94no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—91no—
534C2×C4—182no—
535C2×C4—184no—
536C2×C4—184no—
537C2×C4—182no—
538C2×C4—182no—
539C2×C4C22×C3×C3⋊D312yes—
540C2×C4C22×C3×C3⋊D311yescharacteristic
541C2×C3—28no—
542C2×C3—216no—
543C2×C3—22no—
544C2×C3—24no—
545C2×C3—24no—
546C2×C3—28no—
547C2×C3—48no—
548C2×C3—416no—
549C2×C3—44no—
550C2×C3—41no—
551C2×C3—84no—
552C2×C3—361no—
553C2×C3C2×C3×D3×D414yes—
554C2×C3C2×C3×#23718yes—
555C2×C3C2×D4×C3⋊D311yescharacteristic
556C2×C3C2×#383012yes—
557D3—124no—
558C22—28no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—21no—
563C22—22no—
564C22—181no—
565C22—182no—
566C22C3×D4×C3⋊D312yes—
567C22C3×#383014yescomplemented
568C22C23×C3×C3⋊D311yescharacteristic
569C4—21no—
570C4—22no—
571C4—92no—
572C4—92no—
573C4—182no—
574C4—181no—
575C4—182no—
576C3—24no—
577C3C3×#558614yescomplemented
578C3#1059605711yescharacteristic, direct product factor
579C2—22no—
580C2—24no—
581C2—41no—
582C2—361no—
583C2C2×C3×D4×C3⋊D311yescharacteristic
584C2C2×C3×#383012yes—
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×#120C2
socleC22×C33C24
hypercenterC3×#120C3⋊D3
hypocenterC32C3×#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
1. C23×C3×C3⋊Dic3C2semidirect
2. C2×C3×#3830C2semidirect
3. C33×#120C2semidirect
4. C3×#64162C2semidirect
5. C3×#64148C2semidirect
6. C3×#64144C2semidirect
7. C3×D4×C3⋊Dic3C2semidirect
8. C3×#64095C2semidirect
9. C3×#64090C2semidirect
10. C3×#64087C2semidirect
11. C3×#64086C2semidirect
12. #10596057C3direct (N × H)
19. C2×C3×#638C22semidirect
25. C22×C3×C3⋊Dic3C22semidirect
27. C2×C3×C4×C3⋊D3C22semidirect
28. C2×C3×#634C22semidirect
33. C3×#3757C22semidirect
34. C3×#3752C22semidirect
35. C3×C4⋊(C3⋊Dic3)C22semidirect
36. C3×#3750C22semidirect
37. C3×#3750C22semidirect
38. C3×C4×C3⋊Dic3C22semidirect
39. C23×C3⋊Dic3C2×C3semidirect
40. C2×#3830C2×C3semidirect
44. C32×#120D3semidirect
45. C32×#120C2×C3semidirect
46. #64162C2×C3semidirect
47. #64148C2×C3semidirect
48. #64144C2×C3semidirect
49. D4×C3⋊Dic3C2×C3semidirect
50. #64095C2×C3semidirect
51. #64090C2×C3semidirect
52. #64087C2×C3semidirect
53. #64086C2×C3semidirect
79. C2×C3×C3⋊Dic3D4semidirect
100. C2×#638C22×C3semidirect
106. C22×C3⋊Dic3C22×C3semidirect
108. C2×C4×C3⋊D3C22×C3semidirect
109. C2×#634C22×C3semidirect
130. #3757C22×C3semidirect
131. #3752C22×C3semidirect
132. C4⋊(C3⋊Dic3)C22×C3semidirect
133. #3750C22×C3semidirect
134. #3750C22×C3semidirect
135. C4×C3⋊Dic3C22×C3semidirect
151. C22×C33#41semidirect
172. C3×#120C3×D3semidirect
173. C3×#120C3⋊D3semidirect
221. C2×C3⋊Dic3C3×D4semidirect
328. C22×C32#237semidirect
330. C22×C32C3×#41semidirect
360. #120C3×C3⋊D3semidirect
364. C33#525semidirect
490. C22×C3C3×#237semidirect
492. C22×C3#3830semidirect
514. C32#5586semidirect
515. C32C3×#525semidirect
567. C22C3×#3830semidirect
577. C3C3×#5586semidirect
578. C3#10596057direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×C3×#38302
C33×#1202
C3×#641622
C3×#641482
C3×#641442
C3×D4×C3⋊Dic32
C3×#640952
C3×#640902
C3×#640872
C3×#640862
#105960573
C3×#55863