Finite Group Databasebeta

← C3×#24112 #10642719

Subgroups — C3×#24112 #10642719

Order 768.

1518 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3490C211yesmaximal, characteristic
2C3×#2849C212yesmaximal, complemented
3C2×C3×#385C211yesmaximal, characteristic, complemented
4C3×#2062C212yesmaximal, complemented
5C3×#2044C212yesmaximal, complemented
6C3×#1898C212yesmaximal, complemented
7C3×#1867C214yesmaximal, complemented
8C3×#1819C211yesmaximal, characteristic, complemented
9#24112C311yesmaximal, characteristic, direct product factor, Sylow
10C24×C3×C4C2211yescharacteristic
11C3×#551—22no—
12C3×#551—22no—
13C3×#546—22no—
14C3×#546—22no—
15C3×#543C2211yescharacteristic
16C3×#538—21no—
17C3×#538—21no—
18C3×#534C2211yescharacteristic
19C3×#525—21no—
20C3×#525—21no—
21C2×C3×#123—22no—
22C2×C3×#120C2212yes—
23C3×#518C2212yes—
24C22×C3×C4⋊C4—21no—
25C22×C3×C22⋊C4C2211yescharacteristic
26C22×C3×C22⋊C4C2211yescharacteristic
27C2×C3×#99C2214yes—
28C3×#386—22no—
29C3×#386C2214yes—
30C3×#385—21no—
31C3×#385—21no—
32C3×#385—22no—
33C3×#385C2212yes—
34C3×#385C2214yes—
35C3×#385C2212yes—
36C3×#381—22no—
37C3×#379C2212yes—
38C2×C3×#95—21no—
39C3×#342C2212yes—
40C3×#342C2214yes—
41C3×#342C2212yes—
42#3490C2×C311yescharacteristic
43#2849C2×C312yescomplemented
44C2×#385C2×C311yescharacteristic, complemented
45#2062C2×C312yescomplemented
46#2044C2×C312yescomplemented
47#1898C2×C312yescomplemented
48#1867C2×C314yescomplemented
49#1819C2×C311yescharacteristic, complemented
50C25×C3C2311yescharacteristic
51C2×C3×#41—21no—
52C2×C3×#41—21no—
53C22×C3×D4D412yes—
54C22×C3×D4D412yes—
55C22×C3×D4C2312yes—
56C23×C3×C4—21no—
57C23×C3×C4—22no—
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C23×C3×C4—22no—
61C23×C3×C4—44no—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2312yes—
65C3×#126—22no—
66C3×#126—22no—
67C3×#124—22no—
68C3×#124—22no—
69C3×#123—22no—
70C3×#123—22no—
71C3×#123—22no—
72C3×#123—22no—
73C3×#123—22no—
74C3×#123—48no—
75C3×C22⋊Q8—21no—
76C3×C22⋊Q8—21no—
77C3×C22⋊Q8—21no—
78C3×C22⋊Q8—21no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—22no—
81C3×#121—21no—
82C3×#121—21no—
83C3×#121—22no—
84C3×#121—22no—
85C3×#121—21no—
86C3×#121—21no—
87C3×#120—22no—
88C3×#120—22no—
89C3×#120—22no—
90C3×#120—22no—
91C3×#120—22no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×#117—21no—
99C3×#117—21no—
100C3×#117—22no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4—24no—
104C2×C3×C4⋊C4—44no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—24no—
110C2×C3×C22⋊C4—22no—
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—24no—
113C2×C3×C22⋊C4D412yes—
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C22⋊C4D412yes—
117C2×C3×C22⋊C4C2314yes—
118C2×C3×C22⋊C4C2311yescharacteristic
119C2×C3×C22⋊C4C2311yescharacteristic
120C2×C3×C22⋊C4C2312yes—
121C2×C3×C22⋊C4C2312yes—
122C3×#99—28no—
123C3×#95—22no—
124C3×#95—22no—
125C3×#95—24no—
126C3×#95—24no—
127C24×C4C22×C311yescharacteristic
128#551—22no—
129#551—22no—
130#546—22no—
131#546—22no—
132#543C22×C311yescharacteristic
133#538—21no—
134#538—21no—
135#534C22×C311yescharacteristic
136#525—21no—
137#525—21no—
138C2×#123—22no—
139C2×#120C22×C312yes—
140#518C22×C312yes—
141C22×C4⋊C4—21no—
142C22×C22⋊C4C22×C311yescharacteristic
143C22×C22⋊C4C22×C311yescharacteristic
144C2×#99C22×C314yes—
145#386—22no—
146#386C22×C314yes—
147#385—22no—
148#385—21no—
149#385—21no—
150#385C22×C314yes—
151#385C22×C312yes—
152#385C22×C312yes—
153#381—22no—
154#379C22×C312yes—
155C2×#95—21no—
156#342C22×C312yes—
157#342C22×C314yes—
158#342C22×C312yes—
159C24×C3—22no—
160C24×C3—22no—
161C24×C3—42no—
162C24×C3—42no—
163C24×C3C2×D411yescharacteristic
164C24×C3C2×D412yes—
165C24×C3C2×D412yes—
166C24×C3C2×D412yes—
167C24×C3C2×D411yescharacteristic
168C24×C3C2411yescharacteristic
169C3×#41—82no—
170C3×#41—82no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—24no—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—28no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—24no—
198C22×C3×C4—44no—
199C22×C3×C4—48no—
200C22×C3×C4—44no—
201C22×C3×C4—42no—
202C22×C3×C4—42no—
203C22×C3×C4—42no—
204C22×C3×C4—44no—
205C22×C3×C4—42no—
206C22×C3×C4—41no—
207C22×C3×C4—41no—
208C22×C3×C4—42no—
209C22×C3×C4—42no—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—44no—
226C3×C4⋊C4—44no—
227C3×C4⋊C4—48no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—48no—
247C3×C22⋊C4—44no—
248C3×C22⋊C4—44no—
249C3×C22⋊C4#4114yescomplemented
250C3×C42—22no—
251C3×C42—22no—
252C25C23×C311yescharacteristic
253C2×#41—21no—
254C2×#41—21no—
255C22×D4C3×D412yes—
256C22×D4C3×D412yes—
257C22×D4C23×C312yes—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—44no—
264C23×C4C3×D411yescharacteristic
265C23×C4C3×D411yescharacteristic
266C23×C4C23×C312yes—
267#126—22no—
268#126—22no—
269#124—22no—
270#124—22no—
271#123—22no—
272#123—22no—
273#123—22no—
274#123—22no—
275#123—22no—
276#123—48no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283#121—21no—
284#121—21no—
285#121—22no—
286#121—22no—
287#121—21no—
288#121—21no—
289#120—22no—
290#120—22no—
291#120—22no—
292#120—22no—
293#120—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300#117—21no—
301#117—21no—
302#117—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—44no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4C3×D411yescharacteristic
316C2×C22⋊C4C3×D411yescharacteristic
317C2×C22⋊C4C3×D412yes—
318C2×C22⋊C4C3×D412yes—
319C2×C22⋊C4C23×C314yes—
320C2×C22⋊C4C23×C311yescharacteristic
321C2×C22⋊C4C23×C311yescharacteristic
322C2×C22⋊C4C23×C312yes—
323C2×C22⋊C4C23×C312yes—
324#99—28no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—24no—
336C23×C3—24no—
337C23×C3—24no—
338C23×C3—24no—
339C23×C3—22no—
340C23×C3—24no—
341C23×C3—41no—
342C23×C3—42no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—42no—
346C23×C3—41no—
347C23×C3—48no—
348C23×C3—48no—
349C23×C3—42no—
350C23×C3—42no—
351C23×C3#12012yes—
352C23×C3#12011yescharacteristic
353C23×C3#12011yescharacteristic
354C23×C3C22×D411yescharacteristic
355C23×C3C22×D412yes—
356C23×C3C2×#4112yes—
357C23×C32−1+412yes—
358C3×Q8—81no—
359C3×Q8—81no—
360C3×D4—42no—
361C3×D4—42no—
362C3×D4—44no—
363C3×D4—44no—
364C3×D4—44no—
365C3×D4—82no—
366C3×D4—82no—
367C3×D4—81no—
368C3×D4—81no—
369C2×C3×C4—28no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—24no—
379C2×C3×C4—24no—
380C2×C3×C4—42no—
381C2×C3×C4—48no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—44no—
385C2×C3×C4—44no—
386C2×C3×C4—48no—
387C2×C3×C4—42no—
388C2×C3×C4—44no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—42no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—42no—
397C2×C3×C4—44no—
398C2×C3×C4—44no—
399C2×C3×C4—44no—
400C2×C3×C4—81no—
401C2×C3×C4—81no—
402C2×C3×C4—82no—
403C2×C3×C4—82no—
404C24—22no—
405C24—22no—
406C24—42no—
407C24—42no—
408C24C2×C3×D411yescharacteristic
409C24C2×C3×D412yes—
410C24C2×C3×D412yes—
411C24C2×C3×D412yes—
412C24C2×C3×D411yescharacteristic
413C24C24×C311yescharacteristic
414#41—82no—
415#41—82no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—44no—
429C2×D4—44no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4C2×C3×D412yes—
456C22×C4C2×C3×D411yescharacteristic
457C22×C4C2×C3×D412yes—
458C22×C4C2×C3×D411yescharacteristic
459C22×C4C2×C3×D411yescharacteristic
460C22×C4C2×C3×D412yes—
461C22×C4C2×C3×D411yescharacteristic
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4C3×#4114yescomplemented
495C42—22no—
496C42—22no—
497C22×C3—24no—
498C22×C3—24no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—22no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—22no—
506C22×C3—21no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—48no—
513C22×C3—48no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—84no—
520C22×C3B(4,𝔽2)12yes—
521C22×C3#45812yes—
522C22×C3C2×#12011yescharacteristic
523C22×C3#52511yescharacteristic
524C22×C3#53611yescharacteristic
525C22×C3#54714yes—
526C3×C4—44no—
527C3×C4—42no—
528C3×C4—44no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4—42no—
532C3×C4—41no—
533C3×C4—41no—
534C3×C4—42no—
535C3×C4—84no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—21no—
540C23—24no—
541C23—24no—
542C23—21no—
543C23—21no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—24no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—48no—
555C23—48no—
556C23—42no—
557C23—42no—
558C23C3×#12012yes—
559C23C3×#12011yescharacteristic
560C23C3×#12011yescharacteristic
561C23C22×C3×D411yescharacteristic
562C23C22×C3×D412yes—
563C23C2×C3×#4112yes—
564C23C3×2−1+412yes—
565Q8—81no—
566Q8—81no—
567D4—44no—
568D4—44no—
569D4—42no—
570D4—42no—
571D4—44no—
572D4—82no—
573D4—82no—
574D4—81no—
575D4—81no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C3—21no—
612C2×C3—24no—
613C2×C3—21no—
614C2×C3—22no—
615C2×C3—22no—
616C2×C3—42no—
617C2×C3—82no—
618C2×C3#240611yescharacteristic
619C2×C3C2×B(4,𝔽2)11yescharacteristic
620C2×C3C2×#45811yescharacteristic
621C22—24no—
622C22—24no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—21no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—42no—
635C22—42no—
636C22—48no—
637C22—48no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—84no—
644C22C3×B(4,𝔽2)12yes—
645C22C3×#45812yes—
646C22C2×C3×#12011yescharacteristic
647C22C3×#52511yescharacteristic
648C22C3×#53611yescharacteristic
649C22C3×#54714yes—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—41no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—42no—
659C4—84no—
660C3#2411211yescharacteristic, direct product factor, Sylow
661C2—21no—
662C2—24no—
663C2—22no—
664C2—22no—
665C2—21no—
666C2—42no—
667C2—82no—
668C2C3×#240611yescharacteristic
669C2C2×C3×B(4,𝔽2)11yescharacteristic
670C2C2×C3×#45811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC3×#3490C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2849C2semidirect
3. C2×C3×#385C2semidirect
4. C3×#2062C2semidirect
5. C3×#2044C2semidirect
6. C3×#1898C2semidirect
7. C3×#1867C2semidirect
8. C3×#1819C2semidirect
9. #24112C3direct (N × H)
43. #2849C2×C3semidirect
44. C2×#385C2×C3semidirect
45. #2062C2×C3semidirect
46. #2044C2×C3semidirect
47. #1898C2×C3semidirect
48. #1867C2×C3semidirect
49. #1819C2×C3semidirect
249. C3×C22⋊C4#41semidirect
494. C22⋊C4C3×#41semidirect
660. C3#24112direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34902
C3×#28492
C2×C3×#3852
C3×#20622
C3×#20442
C3×#18982
C3×#18672
C3×#18192
#241123