Finite Group Databasebeta

← C3×#25581 #10644188

Subgroups — C3×#25581 #10644188

Order 768.

1326 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#553C211yesmaximal, characteristic, complemented
2C2×C3×#490C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic, complemented
4C3×#2541C211yesmaximal, characteristic, complemented
5C3×#1999C212yesmaximal, complemented
6C3×#1999C212yesmaximal, complemented
7C3×#1989C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1875C212yesmaximal, complemented
10C3×#1874C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25581C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2212yes—
14C3×#553—24no—
15C3×#553—24no—
16C2×C3×#127—21no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#126—21no—
19C2×C3×#125C2211yescharacteristic
20C2×C3×#123—21no—
21C2×C3×#121—21no—
22C2×C3×#121C2211yescharacteristic
23C2×C3×#121C2211yescharacteristic
24C2×C3×#121C2212yes—
25C2×C3×#121C2212yes—
26C2×C3×C4×D4—21no—
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C22×C3×C4⋊C4C2211yescharacteristic
31C3×#490—24no—
32C3×#459—24no—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C2×C3×C22⋊C8C2212yes—
40C2×C3×#97C2211yescharacteristic
41C3×#392—21no—
42C3×#392—21no—
43C3×#390—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#340C2212yes—
48C2×#553C2×C311yescharacteristic, complemented
49C2×#490C2×C311yescharacteristic, complemented
50C2×#459C2×C311yescharacteristic, complemented
51#2541C2×C311yescharacteristic, complemented
52#1999C2×C312yescomplemented
53#1999C2×C312yescomplemented
54#1989C2×C311yescharacteristic, complemented
55#1955C2×C311yescharacteristic, complemented
56#1875C2×C312yescomplemented
57#1874C2×C312yescomplemented
58#1762C2×C311yescharacteristic, complemented
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4D411yescharacteristic
66C22×C3×D4C2312yes—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C2×C3×D8—24no—
73C2×C3×D8—42no—
74C2×C3×D8—42no—
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C3×#127—22no—
78C3×#127—22no—
79C3×#127—44no—
80C3×#126—28no—
81C3×#125—22no—
82C3×#125D414yescomplemented
83C3×#123—28no—
84C3×#121—28no—
85C3×#121—22no—
86C3×#121—22no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#121—24no—
90C3×#121—24no—
91C3×#121—42no—
92C3×#121—42no—
93C3×C4×D4—28no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4D414yescomplemented
98C3×#117—22no—
99C3×#117—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—22no—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4D411yescharacteristic
117C2×C3×C22⋊C4D411yescharacteristic
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2311yescharacteristic
121C3×C4⋊C8—22no—
122C3×#102—24no—
123C3×#102—24no—
124C3×#102—24no—
125C3×#102—24no—
126C3×C22⋊C8—42no—
127C3×C22⋊C8—42no—
128C3×#97—22no—
129C3×#95—21no—
130C22×D8C22×C312yes—
131#553—24no—
132#553—24no—
133C2×#127—21no—
134C2×#127C22×C311yescharacteristic
135C2×#126—21no—
136C2×#125C22×C311yescharacteristic
137C2×#123—21no—
138C2×#121—21no—
139C2×#121C22×C311yescharacteristic
140C2×#121C22×C311yescharacteristic
141C2×#121C22×C312yes—
142C2×#121C22×C312yes—
143C2×C4×D4—21no—
144C2×C4×D4C22×C311yescharacteristic
145C2×C4×D4C22×C311yescharacteristic
146C2×#117C22×C311yescharacteristic
147C22×C4⋊C4C22×C311yescharacteristic
148#490—24no—
149#459—24no—
150C2×C4⋊C8C22×C311yescharacteristic
151C2×#102C22×C312yes—
152C2×#102C22×C312yes—
153C2×#102C22×C312yes—
154C2×#102C22×C312yes—
155C2×C22⋊C8C22×C312yes—
156C2×C22⋊C8C22×C312yes—
157C2×#97C22×C311yescharacteristic
158#392—21no—
159#392—21no—
160#390—21no—
161C4×C4⋊C4C22×C311yescharacteristic
162#340C22×C312yes—
163#340C22×C312yes—
164#340C22×C312yes—
165C24×C3—21no—
166C24×C3—22no—
167C24×C3C2×D411yescharacteristic
168C24×C3C2×D411yescharacteristic
169C2×C3×D4—24no—
170C2×C3×D4—22no—
171C2×C3×D4—24no—
172C2×C3×D4—22no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—41no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—42no—
192C2×C3×D4—42no—
193C2×C3×D4—82no—
194C2×C3×D4—82no—
195C2×C3×D4—82no—
196C2×C3×D4—82no—
197C2×C3×D4C2×D412yes—
198C22×C3×C4—22no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—24no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—24no—
212C22×C3×C4—82no—
213C22×C3×C4—82no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C3×D8—48no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4C2×D412yes—
235C3×C4⋊C4C2×D412yes—
236C3×C4⋊C4C2×D412yes—
237C3×C4⋊C4#4114yes—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—44no—
245C3×C22⋊C4—44no—
246C3×C42—24no—
247C3×C42—42no—
248C3×C42C2×D412yes—
249C3×C42C2×D412yes—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4C3×D411yescharacteristic
257C22×D4C23×C312yes—
258C22×D4C23×C311yescharacteristic
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C23×C311yescharacteristic
262C23×C4C23×C311yescharacteristic
263C2×D8—24no—
264C2×D8—42no—
265C2×D8—42no—
266C22×C8C23×C312yes—
267C22×C8C23×C312yes—
268#127—22no—
269#127—22no—
270#127—44no—
271#126—28no—
272#125—22no—
273#125C3×D414yescomplemented
274#123—28no—
275#121—28no—
276#121—22no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—24no—
281#121—24no—
282#121—42no—
283#121—42no—
284C4×D4—28no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4C3×D414yescomplemented
289#117—22no—
290#117—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4C3×D411yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C4⋊C4C23×C311yescharacteristic
300C2×C4⋊C4C23×C312yes—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4C3×D411yescharacteristic
308C2×C22⋊C4C3×D411yescharacteristic
309C2×C42—21no—
310C2×C42—21no—
311C2×C42C23×C311yescharacteristic
312C4⋊C8—22no—
313#102—24no—
314#102—24no—
315#102—24no—
316#102—24no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319#97—22no—
320#95—21no—
321C23×C3—21no—
322C23×C3—22no—
323C23×C3—21no—
324C23×C3—22no—
325C23×C3—21no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—22no—
331C23×C3—42no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3AGL(1,ℤ/8)12yes—
343C23×C3C22×D411yescharacteristic
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—44no—
347C3×D4—48no—
348C3×D4—42no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—42no—
353C3×D4—84no—
354C3×D4—84no—
355C3×D4—88no—
356C3×D4—84no—
357C3×D4—88no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—84no—
382C2×C3×C4—84no—
383C2×C3×C4#12012yes—
384C2×C3×C4#12012yes—
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4112yes—
388C2×C3×C4H(5,𝔽2)12yes—
389C3×C8—42no—
390C3×C8—42no—
391C24—21no—
392C24—22no—
393C24C2×C3×D411yescharacteristic
394C24C2×C3×D411yescharacteristic
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—41no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4C2×C3×D412yes—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C2×C3×D411yescharacteristic
443C22×C4C2×C3×D411yescharacteristic
444C22×C4C24×C311yescharacteristic
445D8—48no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4C2×C3×D412yes—
461C4⋊C4C2×C3×D412yes—
462C4⋊C4C2×C3×D412yes—
463C4⋊C4C3×#4114yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C42—24no—
473C42—42no—
474C42C2×C3×D412yes—
475C42C2×C3×D412yes—
476C22×C3—22no—
477C22×C3—24no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—42no—
481C22×C3—41no—
482C22×C3—42no—
483C22×C3—41no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—41no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—82no—
490C22×C3—82no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3C2×#12011yescharacteristic
499C22×C3#54612yes—
500C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
501C22×C3#57511yescharacteristic
502C22×C3#57612yes—
503C3×C4—22no—
504C3×C4—22no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—42no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—44no—
511C3×C4#53211yescharacteristic
512C3×C4#53511yescharacteristic
513C3×C4#55012yes—
514C23—21no—
515C23—22no—
516C23—21no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C3×AGL(1,ℤ/8)12yes—
536C23C22×C3×D411yescharacteristic
537D4—22no—
538D4—22no—
539D4—44no—
540D4—48no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—84no—
547D4—84no—
548D4—88no—
549D4—84no—
550D4—88no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4C3×#12012yes—
577C2×C4C3×#12012yes—
578C2×C4C22×C3×D411yescharacteristic
579C2×C4C22×C3×D411yescharacteristic
580C2×C4C2×C3×#4112yes—
581C2×C4C3×H(5,𝔽2)12yes—
582C8—42no—
583C8—42no—
584C2×C3—42no—
585C2×C3—81no—
586C2×C3—82no—
587C2×C3—162no—
588C2×C3#243111yescharacteristic
589C2×C3#300611yescharacteristic
590C2×C3#300811yescharacteristic
591C2×C3#330912yes—
592C2×C3#331812yes—
593C22—22no—
594C22—24no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—41no—
599C22—42no—
600C22—41no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22C2×C3×#12011yescharacteristic
616C22C3×#54612yes—
617C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
618C22C3×#57511yescharacteristic
619C22C3×#57612yes—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4C3×#53211yescharacteristic
629C4C3×#53511yescharacteristic
630C4C3×#55012yes—
631C3#2558111yescharacteristic, direct product factor, Sylow
632C2—42no—
633C2—81no—
634C2—82no—
635C2—162no—
636C2C3×#243111yescharacteristic
637C2C3×#300611yescharacteristic
638C2C3×#300811yescharacteristic
639C2C3×#330912yes—
640C2C3×#331812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#553C2semidirect
2. C2×C3×#490C2semidirect
3. C2×C3×#459C2semidirect
4. C3×#2541C2semidirect
5. C3×#1999C2semidirect
6. C3×#1999C2semidirect
7. C3×#1989C2semidirect
8. C3×#1955C2semidirect
9. C3×#1875C2semidirect
10. C3×#1874C2semidirect
11. C3×#1762C2semidirect
12. #25581C3direct (N × H)
48. C2×#553C2×C3semidirect
49. C2×#490C2×C3semidirect
50. C2×#459C2×C3semidirect
51. #2541C2×C3semidirect
52. #1999C2×C3semidirect
53. #1999C2×C3semidirect
54. #1989C2×C3semidirect
55. #1955C2×C3semidirect
56. #1875C2×C3semidirect
57. #1874C2×C3semidirect
58. #1762C2×C3semidirect
82. C3×#125D4semidirect
97. C3×C4×D4D4semidirect
273. #125C3×D4semidirect
288. C4×D4C3×D4semidirect
631. C3#25581direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5532
C2×C3×#4902
C2×C3×#4592
C3×#25412
C3×#19992
C3×#19892
C3×#19552
C3×#18752
C3×#18742
C3×#17622
#255813