Finite Group Databasebeta

← C3×#25618 #10644225

Subgroups — C3×#25618 #10644225

Order 768.

1114 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic, complemented
2C3×#3153C211yesmaximal, characteristic, complemented
3C3×#3040C211yesmaximal, characteristic, complemented
4C3×#2640C211yesmaximal, characteristic, complemented
5C3×#2007C212yesmaximal, complemented
6C3×#2006C212yesmaximal, complemented
7C3×#1997C211yesmaximal, characteristic, complemented
8C3×#1963C211yesmaximal, characteristic, complemented
9C3×#1883C212yesmaximal, complemented
10C3×#1882C212yesmaximal, complemented
11C3×#1759C211yesmaximal, characteristic, complemented
12#25618C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C2×C3×#137C2211yescharacteristic
15C2×C3×AGL(1,ℤ/8)C2212yes—
16C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
17C3×#556—21no—
18C3×#555—21no—
19C3×#554—21no—
20C3×#553—22no—
21C3×#548—22no—
22C3×#547—21no—
23C3×#542—22no—
24C3×#537C2212yes—
25C3×#533—21no—
26C3×#532—21no—
27C3×#532C2211yescharacteristic
28C2×C3×C4⋊Q8C2211yescharacteristic
29C2×C3×#124C2211yescharacteristic
30C3×#520C2212yes—
31C3×C4×#41C2211yescharacteristic
32C3×#497—21no—
33C3×#496—21no—
34C3×#495—22no—
35C3×#464—21no—
36C3×#464—21no—
37C3×#463—21no—
38C3×#463—21no—
39C3×#423C2211yescharacteristic
40C2×C3×#104C2212yes—
41C2×C3×#104C2212yes—
42C2×C3×#104C2212yes—
43C3×#419C2211yescharacteristic
44C3×#418C2211yescharacteristic
45C3×C4×M16C2211yescharacteristic
46C3×#397—22no—
47C3×#393—22no—
48C3×#391C2212yes—
49C3×#390C2212yes—
50C3×#376C2211yescharacteristic
51C3×#343C2212yes—
52C3×#343C2212yes—
53C3×#339C2212yes—
54#3511C2×C311yescharacteristic, complemented
55#3153C2×C311yescharacteristic, complemented
56#3040C2×C311yescharacteristic, complemented
57#2640C2×C311yescharacteristic, complemented
58#2007C2×C312yescomplemented
59#2006C2×C312yescomplemented
60#1997C2×C311yescharacteristic, complemented
61#1963C2×C311yescharacteristic, complemented
62#1883C2×C312yescomplemented
63#1882C2×C312yescomplemented
64#1759C2×C311yescharacteristic, complemented
65C2×C3×#41—22no—
66C2×C3×#41C2311yescharacteristic
67C2×C3×#41C2312yes—
68C22×C3×Q8D412yes—
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4C2311yescharacteristic
71C3×#137—24no—
72C3×#137—42no—
73C3×#137—42no—
74C3×AGL(1,ℤ/8)—24no—
75C3×AGL(1,ℤ/8)—42no—
76C3×AGL(1,ℤ/8)—42no—
77C2×C3×Q16—21no—
78C2×C3×Q16—22no—
79C2×C3×SD16—21no—
80C2×C3×SD16—22no—
81C2×C3×SD16—21no—
82C2×C3×SD16—22no—
83C2×C3×D8—21no—
84C2×C3×D8—22no—
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2312yes—
88C3×C4⋊Q8—21no—
89C3×C4⋊Q8—21no—
90C3×C4⋊Q8—21no—
91C3×C4⋊Q8—21no—
92C3×C4⋊Q8D411yescharacteristic
93C3×#127D411yescharacteristic
94C3×#126—24no—
95C3×#125—22no—
96C3×#124—22no—
97C3×#124—24no—
98C3×#124—21no—
99C3×#124—21no—
100C3×#124—21no—
101C3×#124—21no—
102C3×#124D412yes—
103C3×#123—24no—
104C3×C22⋊Q8—22no—
105C3×C22⋊Q8—24no—
106C3×C22⋊Q8—22no—
107C3×C22⋊Q8—22no—
108C3×#121—22no—
109C3×#121—22no—
110C3×#121—22no—
111C3×#121—21no—
112C3×#121—21no—
113C3×#121—24no—
114C3×C4×Q8—22no—
115C3×C4×Q8—22no—
116C3×C4×Q8D411yescharacteristic
117C3×C4×Q8D411yescharacteristic
118C3×C4×D4—22no—
119C3×C4×D4—22no—
120C3×C4×D4—21no—
121C3×C4×D4—21no—
122C3×C4×D4—24no—
123C3×C4×D4—21no—
124C3×C4×D4—21no—
125C3×C4×D4D411yescharacteristic
126C3×C4×D4D411yescharacteristic
127C3×#117—22no—
128C3×#117—22no—
129C3×#117—21no—
130C3×#117C2311yescharacteristic
131C2×C3×C4⋊C4—22no—
132C2×C3×C4⋊C4D412yes—
133C2×C3×C4⋊C4C2312yes—
134C2×C3×C22⋊C4—21no—
135C2×C3×C22⋊C4—21no—
136C2×C3×C22⋊C4—24no—
137C2×C3×C42—21no—
138C2×C3×C42C2312yes—
139C2×C3×C42C2311yescharacteristic
140C3×C4⋊C8—21no—
141C3×C4⋊C8—21no—
142C3×#104—24no—
143C3×#104—42no—
144C3×#104—42no—
145C3×#103—21no—
146C3×#103—21no—
147C3×#102—21no—
148C3×#102—21no—
149C3×#97—21no—
150C3×C4×C8—21no—
151C3×#95—21no—
152C3×#95—21no—
153C3×#95—24no—
154C2×#137C22×C312yes—
155C2×#137C22×C311yescharacteristic
156C2×AGL(1,ℤ/8)C22×C312yes—
157C2×AGL(1,ℤ/8)C22×C311yescharacteristic
158#556—21no—
159#555—21no—
160#554—21no—
161#553—22no—
162#548—22no—
163#547—21no—
164#542—22no—
165#537C22×C312yes—
166#533—21no—
167#532—21no—
168#532C22×C311yescharacteristic
169C2×C4⋊Q8C22×C311yescharacteristic
170C2×#124C22×C311yescharacteristic
171#520C22×C312yes—
172C4×#41C22×C311yescharacteristic
173#497—21no—
174#496—21no—
175#495—22no—
176#464—21no—
177#464—21no—
178#463—21no—
179#463—21no—
180#423C22×C311yescharacteristic
181C2×#104C22×C312yes—
182C2×#104C22×C312yes—
183C2×#104C22×C312yes—
184#419C22×C311yescharacteristic
185#418C22×C311yescharacteristic
186C4×M16C22×C311yescharacteristic
187#397—22no—
188#393—22no—
189#391C22×C312yes—
190#390C22×C312yes—
191#376C22×C311yescharacteristic
192#343C22×C312yes—
193#343C22×C312yes—
194#339C22×C312yes—
195C24×C3—21no—
196C3×#41—22no—
197C3×#41—22no—
198C3×#41—22no—
199C3×#41—22no—
200C3×#41—22no—
201C3×#41—22no—
202C3×#41—84no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—22no—
208C2×C3×Q8—22no—
209C2×C3×Q8—82no—
210C2×C3×Q8C2×D412yes—
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×D4—21no—
213C2×C3×D4—22no—
214C2×C3×D4—24no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4C2×D412yes—
222C2×C3×D4C2×D411yescharacteristic
223C22×C3×C4—24no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—22no—
228C22×C3×C4—22no—
229C22×C3×C4—21no—
230C22×C3×C4C2×D414yes—
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C3×Q16—44no—
235C3×Q16—42no—
236C3×SD16—44no—
237C3×SD16—44no—
238C3×SD16—42no—
239C3×SD16—42no—
240C3×D8—44no—
241C3×D8—42no—
242C3×M16—22no—
243C3×M16—22no—
244C3×M16—22no—
245C3×M16#4114yescomplemented
246C2×C3×C8—22no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C22⋊C4—24no—
262C3×C22⋊C4—24no—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—21no—
265C3×C22⋊C4—21no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—41no—
270C3×C22⋊C4—41no—
271C3×C22⋊C4—41no—
272C3×C22⋊C4—41no—
273C3×C22⋊C4—84no—
274C3×C42—24no—
275C3×C42—21no—
276C3×C42—21no—
277C3×C42—21no—
278C3×C42—22no—
279C3×C42—22no—
280C3×C42C2×D411yescharacteristic
281C3×C42C2×D411yescharacteristic
282C3×C42C2×D411yescharacteristic
283C3×C42C2×D411yescharacteristic
284C2×#41—22no—
285C2×#41C23×C311yescharacteristic
286C2×#41C23×C312yes—
287C22×Q8C3×D412yes—
288C22×Q8C23×C311yescharacteristic
289C22×D4C23×C311yescharacteristic
290#137—24no—
291#137—42no—
292#137—42no—
293AGL(1,ℤ/8)—24no—
294AGL(1,ℤ/8)—42no—
295AGL(1,ℤ/8)—42no—
296C2×Q16—21no—
297C2×Q16—22no—
298C2×SD16—21no—
299C2×SD16—22no—
300C2×SD16—21no—
301C2×SD16—22no—
302C2×D8—21no—
303C2×D8—22no—
304C2×M16C23×C311yescharacteristic
305C2×M16C23×C311yescharacteristic
306C2×M16C23×C312yes—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8C3×D411yescharacteristic
312#127C3×D411yescharacteristic
313#126—24no—
314#125—22no—
315#124—22no—
316#124—24no—
317#124—21no—
318#124—21no—
319#124—21no—
320#124—21no—
321#124C3×D412yes—
322#123—24no—
323C22⋊Q8—22no—
324C22⋊Q8—24no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327#121—22no—
328#121—22no—
329#121—22no—
330#121—21no—
331#121—21no—
332#121—24no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8C3×D411yescharacteristic
336C4×Q8C3×D411yescharacteristic
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—21no—
340C4×D4—21no—
341C4×D4—24no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4C3×D411yescharacteristic
345C4×D4C3×D411yescharacteristic
346#117—22no—
347#117—22no—
348#117—21no—
349#117C23×C311yescharacteristic
350C2×C4⋊C4—22no—
351C2×C4⋊C4C3×D412yes—
352C2×C4⋊C4C23×C312yes—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C42—21no—
357C2×C42C23×C312yes—
358C2×C42C23×C311yescharacteristic
359C4⋊C8—21no—
360C4⋊C8—21no—
361#104—24no—
362#104—42no—
363#104—42no—
364#103—21no—
365#103—21no—
366#102—21no—
367#102—21no—
368#97—21no—
369C4×C8—21no—
370#95—21no—
371#95—21no—
372#95—24no—
373C23×C3—21no—
374C23×C3—22no—
375C23×C3—42no—
376C23×C3—42no—
377C23×C3—41no—
378C23×C3—42no—
379C23×C3C22×D411yescharacteristic
380C3×Q8—21no—
381C3×Q8—21no—
382C3×Q8—22no—
383C3×Q8—22no—
384C3×Q8—44no—
385C3×Q8—41no—
386C3×Q8—41no—
387C3×Q8—42no—
388C3×Q8—82no—
389C3×D4—21no—
390C3×D4—21no—
391C3×D4—22no—
392C3×D4—22no—
393C3×D4—42no—
394C3×D4—42no—
395C3×D4—42no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—41no—
399C3×D4—41no—
400C3×D4—42no—
401C3×D4—84no—
402C2×C3×C4—21no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—24no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—42no—
410C2×C3×C4—42no—
411C2×C3×C4—42no—
412C2×C3×C4—42no—
413C2×C3×C4—42no—
414C2×C3×C4—44no—
415C2×C3×C4—42no—
416C2×C3×C4—42no—
417C2×C3×C4—41no—
418C2×C3×C4—42no—
419C2×C3×C4—42no—
420C2×C3×C4—84no—
421C2×C3×C4#12011yescharacteristic
422C2×C3×C4#12011yescharacteristic
423C2×C3×C4#12011yescharacteristic
424C2×C3×C4#12011yescharacteristic
425C2×C3×C4C22×D411yescharacteristic
426C2×C3×C4C22×D411yescharacteristic
427C2×C3×C4C2×#4112yes—
428C2×C3×C4H(5,𝔽2)12yes—
429C3×C8—24no—
430C3×C8—41no—
431C3×C8—41no—
432C24—21no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—22no—
439#41—84no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—82no—
447C2×Q8C2×C3×D412yes—
448C2×Q8C2×C3×D411yescharacteristic
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4C2×C3×D412yes—
459C2×D4C2×C3×D411yescharacteristic
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4C2×C3×D414yes—
468C22×C4C2×C3×D411yescharacteristic
469C22×C4C2×C3×D411yescharacteristic
470C22×C4C24×C311yescharacteristic
471Q16—44no—
472Q16—42no—
473SD16—44no—
474SD16—44no—
475SD16—42no—
476SD16—42no—
477D8—44no—
478D8—42no—
479M16—22no—
480M16—22no—
481M16—22no—
482M16C3×#4114yescomplemented
483C2×C8—22no—
484C2×C8—21no—
485C2×C8—21no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4C2×C3×D411yescharacteristic
497C4⋊C4C2×C3×D411yescharacteristic
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—84no—
511C42—24no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—22no—
516C42—22no—
517C42C2×C3×D411yescharacteristic
518C42C2×C3×D411yescharacteristic
519C42C2×C3×D411yescharacteristic
520C42C2×C3×D411yescharacteristic
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—42no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—42no—
527C22×C3—41no—
528C22×C3—42no—
529C22×C3—84no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3C2×#12011yescharacteristic
534C22×C3#55012yes—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—42no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—44no—
543C3×C4—41no—
544C3×C4—82no—
545C3×C4—81no—
546C3×C4#53211yescharacteristic
547C3×C4#53511yescharacteristic
548C3×C4#54612yes—
549C23—21no—
550C23—22no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23C22×C3×D411yescharacteristic
556Q8—21no—
557Q8—21no—
558Q8—22no—
559Q8—22no—
560Q8—44no—
561Q8—41no—
562Q8—41no—
563Q8—42no—
564Q8—82no—
565D4—21no—
566D4—21no—
567D4—22no—
568D4—22no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—41no—
575D4—41no—
576D4—42no—
577D4—84no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—24no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4C3×#12011yescharacteristic
598C2×C4C3×#12011yescharacteristic
599C2×C4C3×#12011yescharacteristic
600C2×C4C3×#12011yescharacteristic
601C2×C4C22×C3×D411yescharacteristic
602C2×C4C22×C3×D411yescharacteristic
603C2×C4C2×C3×#4112yes—
604C2×C4C3×H(5,𝔽2)12yes—
605C8—24no—
606C8—41no—
607C8—41no—
608C2×C3—22no—
609C2×C3—81no—
610C2×C3—82no—
611C2×C3—161no—
612C2×C3#243111yescharacteristic
613C2×C3#301911yescharacteristic
614C2×C3#302011yescharacteristic
615C22—21no—
616C22—21no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—41no—
622C22—42no—
623C22—84no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22C2×C3×#12011yescharacteristic
628C22C3×#55012yes—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—44no—
637C4—41no—
638C4—82no—
639C4—81no—
640C4C3×#53211yescharacteristic
641C4C3×#53511yescharacteristic
642C4C3×#54612yes—
643C3#2561811yescharacteristic, direct product factor, Sylow
644C2—22no—
645C2—81no—
646C2—82no—
647C2—161no—
648C2C3×#243111yescharacteristic
649C2C3×#301911yescharacteristic
650C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3511C2semidirect
2. C3×#3153C2semidirect
3. C3×#3040C2semidirect
4. C3×#2640C2semidirect
5. C3×#2007C2semidirect
6. C3×#2006C2semidirect
7. C3×#1997C2semidirect
8. C3×#1963C2semidirect
9. C3×#1883C2semidirect
10. C3×#1882C2semidirect
11. C3×#1759C2semidirect
12. #25618C3direct (N × H)
54. #3511C2×C3semidirect
55. #3153C2×C3semidirect
56. #3040C2×C3semidirect
57. #2640C2×C3semidirect
58. #2007C2×C3semidirect
59. #2006C2×C3semidirect
60. #1997C2×C3semidirect
61. #1963C2×C3semidirect
62. #1883C2×C3semidirect
63. #1882C2×C3semidirect
64. #1759C2×C3semidirect
245. C3×M16#41semidirect
482. M16C3×#41semidirect
643. C3#25618direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35112
C3×#31532
C3×#30402
C3×#26402
C3×#20072
C3×#20062
C3×#19972
C3×#19632
C3×#18832
C3×#18822
C3×#17592
#256183