Finite Group Databasebeta

← C3×#32924 #10651531

Subgroups — C3×#32924 #10651531

Order 768.

1468 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#3040C212yesmaximal, complemented
3C2×C3×#455C211yesmaximal, characteristic, complemented
4C2×C3×#453C211yesmaximal, characteristic
5C3×#2544C211yesmaximal, characteristic, complemented
6C3×#1887C211yesmaximal, characteristic, complemented
7C3×#1885C211yesmaximal, characteristic, complemented
8C3×#1883C212yesmaximal, complemented
9C3×#1882C212yesmaximal, complemented
10C3×#1806C212yesmaximal, complemented
11C3×#1800C211yesmaximal, characteristic, complemented
12#32924C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×H(5,𝔽2)C2211yescharacteristic
14C2×C3×#137C2212yes—
15C2×C3×AGL(1,ℤ/8)C2212yes—
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic, complemented
19C3×#520—22no—
20C3×#520C2212yes—
21C3×#518—22no—
22C3×#518—22no—
23C3×#518C2211yescharacteristic
24C3×C4×#41—22no—
25C3×C4×#41C2212yes—
26C2×C3×C4×D4—22no—
27C2×C3×C4×D4—22no—
28C2×C3×C4×D4C2211yescharacteristic
29C3×#464—22no—
30C3×#464—22no—
31C3×#463—22no—
32C3×#463—22no—
33C3×#455—24no—
34C3×#453—24no—
35C3×#423C2212yescomplemented
36C2×C3×#104C2212yes—
37C2×C3×#104C2212yes—
38C3×#419C2212yescomplemented
39C3×#418C2212yescomplemented
40C2×C3×#100C2211yescharacteristic
41C2×C3×#100C2211yescharacteristic
42C3×#394—22no—
43C3×#391C2211yescharacteristic, complemented
44C3×#390C2211yescharacteristic, complemented
45C3×#386—22no—
46C3×C4×C4⋊C4C2211yescharacteristic, complemented
47C3×#343C2212yescomplemented
48C3×#339C2212yescomplemented
49C3×#339C2212yescomplemented
50C4×H(5,𝔽2)C2×C311yescharacteristic, complemented
51#3040C2×C312yescomplemented
52C2×#455C2×C311yescharacteristic, complemented
53C2×#453C2×C311yescharacteristic
54#2544C2×C311yescharacteristic, complemented
55#1887C2×C311yescharacteristic, complemented
56#1885C2×C311yescharacteristic, complemented
57#1883C2×C312yescomplemented
58#1882C2×C312yescomplemented
59#1806C2×C312yescomplemented
60#1800C2×C311yescharacteristic, complemented
61C3×H(5,𝔽2)—24no—
62C3×H(5,𝔽2)—22no—
63C3×H(5,𝔽2)D414yes—
64C2×C3×#41—22no—
65C2×C3×#41C2312yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4C2311yescharacteristic
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—22no—
73C23×C3×C4D412yes—
74C3×#137—24no—
75C3×#137—24no—
76C3×AGL(1,ℤ/8)—24no—
77C3×AGL(1,ℤ/8)—24no—
78C2×C3×Q16—22no—
79C2×C3×SD16—22no—
80C2×C3×SD16—22no—
81C2×C3×D8—22no—
82C2×C3×M16C2312yes—
83C2×C3×M16C2312yes—
84C3×#127—22no—
85C3×#127—22no—
86C3×#124—22no—
87C3×#124—22no—
88C3×#124—22no—
89C3×#124—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8D412yescomplemented
92C3×C4×Q8D412yescomplemented
93C3×C4×D4—24no—
94C3×C4×D4—24no—
95C3×C4×D4—24no—
96C3×C4×D4—24no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4—24no—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—22no—
105C3×C4×D4D412yescomplemented
106C3×C4×D4D412yescomplemented
107C3×#117—24no—
108C3×#117—22no—
109C3×#117—22no—
110C3×#117C2312yes—
111C2×C3×C4⋊C4—22no—
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—24no—
117C2×C3×C22⋊C4—22no—
118C2×C3×C22⋊C4—22no—
119C2×C3×C22⋊C4—21no—
120C2×C3×C22⋊C4—21no—
121C2×C3×C22⋊C4—22no—
122C2×C3×C22⋊C4D412yes—
123C2×C3×C42—22no—
124C2×C3×C42—22no—
125C2×C3×C42C2311yescharacteristic
126C2×C3×C42C2311yescharacteristic
127C2×C3×C42C2311yescharacteristic
128C3×C4⋊C8—22no—
129C3×C4⋊C8—22no—
130C3×#104—24no—
131C3×#104—24no—
132C3×#103—22no—
133C3×#103—22no—
134C3×#102—22no—
135C3×#102—22no—
136C3×#100—22no—
137C3×#100—22no—
138C3×#95—21no—
139C3×#95—22no—
140C2×H(5,𝔽2)C22×C311yescharacteristic
141C2×#137C22×C312yes—
142C2×AGL(1,ℤ/8)C22×C312yes—
143C2×#127C22×C311yescharacteristic
144C2×#124C22×C311yescharacteristic
145C2×#124C22×C311yescharacteristic, complemented
146#520—22no—
147#520C22×C312yes—
148#518—22no—
149#518—22no—
150#518C22×C311yescharacteristic
151C4×#41—22no—
152C4×#41C22×C312yes—
153C2×C4×D4—22no—
154C2×C4×D4—22no—
155C2×C4×D4C22×C311yescharacteristic
156#464—22no—
157#464—22no—
158#463—22no—
159#463—22no—
160#455—24no—
161#453—24no—
162#423C22×C312yescomplemented
163C2×#104C22×C312yes—
164C2×#104C22×C312yes—
165#419C22×C312yescomplemented
166#418C22×C312yescomplemented
167C2×#100C22×C311yescharacteristic
168C2×#100C22×C311yescharacteristic
169#394—22no—
170#391C22×C311yescharacteristic, complemented
171#390C22×C311yescharacteristic, complemented
172#386—22no—
173C4×C4⋊C4C22×C311yescharacteristic, complemented
174#343C22×C312yescomplemented
175#339C22×C312yescomplemented
176#339C22×C312yescomplemented
177C24×C3—22no—
178C24×C3—21no—
179C24×C3C2×D412yes—
180C3×#41—28no—
181C3×#41—28no—
182C3×#41—22no—
183C3×#41—22no—
184C3×#41C2×D414yes—
185C3×#41#4114yes—
186C2×C3×Q8—22no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—22no—
190C2×C3×Q8C2×D412yes—
191C2×C3×D4—24no—
192C2×C3×D4—28no—
193C2×C3×D4—28no—
194C2×C3×D4—28no—
195C2×C3×D4—22no—
196C2×C3×D4—28no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—22no—
201C2×C3×D4—21no—
202C2×C3×D4—22no—
203C2×C3×D4—44no—
204C2×C3×D4—44no—
205C2×C3×D4—82no—
206C2×C3×D4C2×D412yes—
207C2×C3×D4C2×D412yes—
208C2×C3×D4#4112yes—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—21no—
212C22×C3×C4—22no—
213C22×C3×C4—24no—
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—22no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—82no—
222C22×C3×C4—84no—
223C22×C3×C4C2×D412yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C3×Q16—44no—
228C3×SD16—44no—
229C3×SD16—44no—
230C3×D8—44no—
231C3×M16—22no—
232C3×M16—22no—
233C3×M16—22no—
234C3×M16—22no—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4C2×D412yes—
243C3×C4⋊C4C2×D412yes—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—44no—
251C3×C22⋊C4—41no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—41no—
254C3×C22⋊C4—82no—
255C3×C42—24no—
256C3×C42—22no—
257C3×C42—22no—
258C3×C42—22no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42C2×D412yes—
264C3×C42C2×D412yes—
265H(5,𝔽2)—24no—
266H(5,𝔽2)—22no—
267H(5,𝔽2)C3×D414yes—
268C2×#41—22no—
269C2×#41C23×C312yes—
270C22×Q8C23×C311yescharacteristic
271C22×D4—22no—
272C22×D4—22no—
273C22×D4—22no—
274C22×D4C23×C311yescharacteristic
275C22×D4C23×C311yescharacteristic
276C23×C4—22no—
277C23×C4C3×D412yes—
278#137—24no—
279#137—24no—
280AGL(1,ℤ/8)—24no—
281AGL(1,ℤ/8)—24no—
282C2×Q16—22no—
283C2×SD16—22no—
284C2×SD16—22no—
285C2×D8—22no—
286C2×M16C23×C312yes—
287C2×M16C23×C312yes—
288#127—22no—
289#127—22no—
290#124—22no—
291#124—22no—
292#124—22no—
293#124—22no—
294C4×Q8—22no—
295C4×Q8C3×D412yescomplemented
296C4×Q8C3×D412yescomplemented
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—24no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4C3×D412yescomplemented
310C4×D4C3×D412yescomplemented
311#117—24no—
312#117—22no—
313#117—22no—
314#117C23×C312yes—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C23×C311yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4C3×D412yes—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42C23×C311yescharacteristic
330C2×C42C23×C311yescharacteristic
331C2×C42C23×C311yescharacteristic
332C4⋊C8—22no—
333C4⋊C8—22no—
334#104—24no—
335#104—24no—
336#103—22no—
337#103—22no—
338#102—22no—
339#102—22no—
340#100—22no—
341#100—22no—
342#95—21no—
343#95—22no—
344C23×C3—28no—
345C23×C3—22no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—82no—
356C23×C3—82no—
357C23×C3#12114yes—
358C23×C3C22×D411yescharacteristic
359C3×Q8—22no—
360C3×Q8—42no—
361C3×Q8—42no—
362C3×Q8#12114yes—
363C3×D4—216no—
364C3×D4—24no—
365C3×D4—24no—
366C3×D4—24no—
367C3×D4—22no—
368C3×D4—24no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—88no—
372C3×D4#12114yes—
373C2×C3×C4—28no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—44no—
384C2×C3×C4—44no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—42no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—82no—
395C2×C3×C4—82no—
396C2×C3×C4—82no—
397C2×C3×C4—84no—
398C2×C3×C4—82no—
399C2×C3×C4#12011yescharacteristic
400C2×C3×C4#12012yes—
401C2×C3×C4#12011yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D412yes—
404C2×C3×C4C2×#4112yes—
405C2×C3×C4C2×#4111yescharacteristic
406C3×C8—42no—
407C3×C8—42no—
408C24—22no—
409C24—21no—
410C24C2×C3×D412yes—
411#41—28no—
412#41—28no—
413#41—22no—
414#41—22no—
415#41C2×C3×D414yes—
416#41C3×#4114yes—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8C2×C3×D412yes—
422C2×D4—24no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—22no—
427C2×D4—28no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—82no—
437C2×D4C2×C3×D412yes—
438C2×D4C2×C3×D412yes—
439C2×D4C3×#4112yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—82no—
453C22×C4—84no—
454C22×C4C2×C3×D412yes—
455C22×C4C2×C3×D411yescharacteristic
456C22×C4C2×C3×D411yescharacteristic
457C22×C4C24×C311yescharacteristic
458Q16—44no—
459SD16—44no—
460SD16—44no—
461D8—44no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466C2×C8—22no—
467C2×C8—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4C2×C3×D412yes—
474C4⋊C4C2×C3×D412yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—24no—
481C22⋊C4—44no—
482C22⋊C4—41no—
483C22⋊C4—42no—
484C22⋊C4—41no—
485C22⋊C4—82no—
486C42—24no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42C2×C3×D412yes—
495C42C2×C3×D412yes—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—24no—
499C22×C3—24no—
500C22×C3—44no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—82no—
506C22×C3—82no—
507C22×C3—82no—
508C22×C3—88no—
509C22×C3—81no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3C2×#12011yescharacteristic
513C22×C3C2×#12111yescharacteristic
514C22×C3#52511yescharacteristic
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—42no—
523C3×C4—81no—
524C3×C4—81no—
525C3×C4C2×#12112yes—
526C3×C4#52512yes—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—82no—
539C23—82no—
540C23C3×#12114yes—
541C23C22×C3×D411yescharacteristic
542Q8—22no—
543Q8—42no—
544Q8—42no—
545Q8C3×#12114yes—
546D4—216no—
547D4—24no—
548D4—24no—
549D4—24no—
550D4—22no—
551D4—24no—
552D4—42no—
553D4—42no—
554D4—88no—
555D4C3×#12114yes—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—84no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4C3×#12011yescharacteristic
583C2×C4C3×#12012yes—
584C2×C4C3×#12011yescharacteristic
585C2×C4C22×C3×D411yescharacteristic
586C2×C4C22×C3×D412yes—
587C2×C4C2×C3×#4112yes—
588C2×C4C2×C3×#4111yescharacteristic
589C8—42no—
590C8—42no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—44no—
594C2×C3—44no—
595C2×C3—161no—
596C2×C3#284711yescharacteristic
597C2×C3#301811yescharacteristic
598C2×C3#301911yescharacteristic
599C22—21no—
600C22—21no—
601C22—24no—
602C22—24no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—88no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22C2×C3×#12011yescharacteristic
616C22C2×C3×#12111yescharacteristic
617C22C3×#52511yescharacteristic
618C4—22no—
619C4—21no—
620C4—21no—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4C2×C3×#12112yes—
629C4C3×#52512yes—
630C3#3292411yescharacteristic, direct product factor, Sylow
631C2—21no—
632C2—21no—
633C2—44no—
634C2—44no—
635C2—161no—
636C2C3×#284711yescharacteristic
637C2C3×#301811yescharacteristic
638C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#453C2
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×C4×H(5,𝔽2)C2semidirect
2. C3×#3040C2semidirect
3. C2×C3×#455C2semidirect
5. C3×#2544C2semidirect
6. C3×#1887C2semidirect
7. C3×#1885C2semidirect
8. C3×#1883C2semidirect
9. C3×#1882C2semidirect
10. C3×#1806C2semidirect
11. C3×#1800C2semidirect
12. #32924C3direct (N × H)
18. C2×C3×#124C22semidirect
35. C3×#423C22semidirect
38. C3×#419C22semidirect
39. C3×#418C22semidirect
43. C3×#391C22semidirect
44. C3×#390C22semidirect
46. C3×C4×C4⋊C4C22semidirect
47. C3×#343C22semidirect
48. C3×#339C22semidirect
49. C3×#339C22semidirect
50. C4×H(5,𝔽2)C2×C3semidirect
51. #3040C2×C3semidirect
52. C2×#455C2×C3semidirect
54. #2544C2×C3semidirect
55. #1887C2×C3semidirect
56. #1885C2×C3semidirect
57. #1883C2×C3semidirect
58. #1882C2×C3semidirect
59. #1806C2×C3semidirect
60. #1800C2×C3semidirect
91. C3×C4×Q8D4semidirect
92. C3×C4×Q8D4semidirect
105. C3×C4×D4D4semidirect
106. C3×C4×D4D4semidirect
145. C2×#124C22×C3semidirect
162. #423C22×C3semidirect
165. #419C22×C3semidirect
166. #418C22×C3semidirect
170. #391C22×C3semidirect
171. #390C22×C3semidirect
173. C4×C4⋊C4C22×C3semidirect
174. #343C22×C3semidirect
175. #339C22×C3semidirect
176. #339C22×C3semidirect
295. C4×Q8C3×D4semidirect
296. C4×Q8C3×D4semidirect
309. C4×D4C3×D4semidirect
310. C4×D4C3×D4semidirect
630. C3#32924direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C3×#30402
C2×C3×#4552
C2×C3×#4532
C3×#25442
C3×#18872
C3×#18852
C3×#18832
C3×#18822
C3×#18062
C3×#18002
#329243