Finite Group Databasebeta

← C3×#24355 #10642962

Subgroups — C3×#24355 #10642962

Order 768.

1742 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D8C211yesmaximal, characteristic
2C2×C3×#463C212yesmaximal, complemented
3C2×C3×#450C211yesmaximal, characteristic, complemented
4C3×#2435C211yesmaximal, characteristic, complemented
5C3×D4×C22⋊C4C211yesmaximal, characteristic, complemented
6C3×#2054C212yesmaximal, complemented
7C3×#2012C212yesmaximal, complemented
8C3×#1892C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1850C211yesmaximal, characteristic, complemented
11C3×#1786C211yesmaximal, characteristic, complemented
12#24355C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×SD16C2211yescharacteristic, complemented
16C22×C3×SD16C2211yescharacteristic, complemented
17C22×C3×D8—24no—
18C22×C3×D8—28no—
19C22×C3×D8C2212yes—
20C22×C3×D8C2212yes—
21C23×C3×C8C2211yescharacteristic
22C2×C3×#124C2212yescomplemented
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×C4×D4C2212yescomplemented
25C22×C3×C22⋊C4—21no—
26C3×#463—28no—
27C3×#450—24no—
28C2×C3×C4⋊C8C2212yescomplemented
29C2×C3×#103C2212yescomplemented
30C2×C3×#102C2212yescomplemented
31C2×C3×#102C2211yescharacteristic, complemented
32C2×C3×#102C2211yescharacteristic, complemented
33C2×C3×#102C2212yescomplemented
34C2×C3×C22⋊C8C2211yescharacteristic
35C3×#394—22no—
36C3×#390C2212yescomplemented
37C3×#390C2212yescomplemented
38C3×#386—22no—
39C3×#380C2211yescharacteristic
40C3×#379—21no—
41C3×#379—21no—
42C3×#379—21no—
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×#340C2212yescomplemented
45C3×#336C2212yescomplemented
46C23×D8C2×C311yescharacteristic
47C2×#463C2×C312yescomplemented
48C2×#450C2×C311yescharacteristic, complemented
49#2435C2×C311yescharacteristic, complemented
50D4×C22⋊C4C2×C311yescharacteristic, complemented
51#2054C2×C312yescomplemented
52#2012C2×C312yescomplemented
53#1892C2×C311yescharacteristic, complemented
54#1879C2×C312yescomplemented
55#1850C2×C311yescharacteristic, complemented
56#1786C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C25×C3—21no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—24no—
61C22×C3×D4—24no—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4—42no—
68C22×C3×D4—42no—
69C22×C3×D4D414yes—
70C22×C3×D4C2311yescharacteristic
71C22×C3×D4C2312yes—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—22no—
74C23×C3×C4C2311yescharacteristic
75C2×C3×SD16—22no—
76C2×C3×SD16—22no—
77C2×C3×SD16—24no—
78C2×C3×SD16—42no—
79C2×C3×D8—232no—
80C2×C3×D8—24no—
81C2×C3×D8—24no—
82C2×C3×D8—24no—
83C2×C3×D8—24no—
84C2×C3×D8—24no—
85C2×C3×D8D418yes—
86C22×C3×C8—22no—
87C22×C3×C8—24no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2312yes—
91C3×#124—24no—
92C3×#124—42no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—22no—
95C3×C4×D4—24no—
96C3×C4×D4—42no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—44no—
110C2×C3×C22⋊C4D411yescharacteristic
111C2×C3×C22⋊C4D411yescharacteristic
112C2×C3×C42C2312yes—
113C3×C4⋊C8—24no—
114C3×#103—24no—
115C3×#102—22no—
116C3×#102—22no—
117C3×#102—24no—
118C3×#102—42no—
119C3×C22⋊C8—22no—
120C3×#95—21no—
121C3×#95D412yes—
122C23×D4C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C22×SD16C22×C311yescharacteristic, complemented
125C22×SD16C22×C311yescharacteristic, complemented
126C22×D8—24no—
127C22×D8—28no—
128C22×D8C22×C312yes—
129C22×D8C22×C312yes—
130C23×C8C22×C311yescharacteristic
131C2×#124C22×C312yescomplemented
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×C4×D4C22×C312yescomplemented
134C22×C22⋊C4—21no—
135#463—28no—
136#450—24no—
137C2×C4⋊C8C22×C312yescomplemented
138C2×#103C22×C312yescomplemented
139C2×#102C22×C311yescharacteristic, complemented
140C2×#102C22×C312yescomplemented
141C2×#102C22×C311yescharacteristic, complemented
142C2×#102C22×C312yescomplemented
143C2×C22⋊C8C22×C311yescharacteristic
144#394—22no—
145#390C22×C312yescomplemented
146#390C22×C312yescomplemented
147#386—22no—
148#380C22×C311yescharacteristic
149#379—21no—
150#379—21no—
151#379—21no—
152C4×C22⋊C4C22×C311yescharacteristic
153#340C22×C312yescomplemented
154#336C22×C312yescomplemented
155C24×C3—21no—
156C24×C3—21no—
157C24×C3—22no—
158C24×C3—24no—
159C24×C3—44no—
160C24×C3—44no—
161C24×C3—42no—
162C24×C3—42no—
163C24×C3—42no—
164C24×C3—42no—
165C24×C3—42no—
166C24×C3—82no—
167C24×C3—82no—
168C24×C3C2×D411yescharacteristic
169C2×C3×Q8—22no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—22no—
173C2×C3×D4—28no—
174C2×C3×D4—28no—
175C2×C3×D4—24no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—44no—
187C2×C3×D4—48no—
188C2×C3×D4—48no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—44no—
192C2×C3×D4—42no—
193C2×C3×D4C2×D414yes—
194C2×C3×D4C2×D412yes—
195C2×C3×D4C2×D412yes—
196C2×C3×D4C2×D414yes—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—44no—
203C22×C3×C4—44no—
204C22×C3×C4—82no—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D412yes—
209C22×C3×C4#4112yes—
210C22×C3×C4C2411yescharacteristic
211C3×SD16—44no—
212C3×SD16—44no—
213C3×D8—216no—
214C3×D8—216no—
215C2×C3×C8—28no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8C2×D414yes—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—42no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—44no—
231C3×C22⋊C4—44no—
232C3×C22⋊C4—44no—
233C3×C22⋊C4—44no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—82no—
237C3×C22⋊C4—82no—
238C3×C22⋊C4—82no—
239C3×C42—22no—
240C3×C42—22no—
241C25—21no—
242C25—21no—
243C22×Q8C23×C311yescharacteristic
244C22×D4—24no—
245C22×D4—24no—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—42no—
252C22×D4—42no—
253C22×D4C3×D414yes—
254C22×D4C23×C312yes—
255C22×D4C23×C311yescharacteristic
256C22×D4C23×C311yescharacteristic
257C23×C4—22no—
258C23×C4C23×C311yescharacteristic
259C2×SD16—22no—
260C2×SD16—22no—
261C2×SD16—24no—
262C2×SD16—42no—
263C2×D8—232no—
264C2×D8—24no—
265C2×D8—24no—
266C2×D8—24no—
267C2×D8—24no—
268C2×D8—24no—
269C2×D8C3×D418yes—
270C22×C8—22no—
271C22×C8—24no—
272C22×C8C23×C311yescharacteristic
273C22×C8C23×C311yescharacteristic
274C22×C8C23×C312yes—
275#124—24no—
276#124—42no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C4×D4—24no—
280C4×D4—42no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C23×C312yes—
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4C3×D411yescharacteristic
295C2×C22⋊C4C3×D411yescharacteristic
296C2×C42C23×C312yes—
297C4⋊C8—24no—
298#103—24no—
299#102—22no—
300#102—22no—
301#102—24no—
302#102—42no—
303C22⋊C8—22no—
304#95—21no—
305#95C3×D412yes—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—24no—
310C23×C3—22no—
311C23×C3—24no—
312C23×C3—48no—
313C23×C3—48no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—48no—
328C23×C3—84no—
329C23×C3—88no—
330C23×C3—88no—
331C23×C3—88no—
332C23×C3—88no—
333C23×C3—84no—
334C23×C3—84no—
335C23×C3#12112yes—
336C23×C3#13512yes—
337C23×C3AGL(1,ℤ/8)12yes—
338C23×C3C22×D411yescharacteristic
339C3×Q8—41no—
340C3×Q8—41no—
341C3×Q8—42no—
342C3×D4—22no—
343C3×D4—24no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—24no—
347C3×D4—24no—
348C3×D4—48no—
349C3×D4—48no—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—42no—
353C3×D4#12014yes—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—88no—
372C2×C3×C4#12112yes—
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C22×D412yes—
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)12yes—
377C3×C8—24no—
378C3×C8—41no—
379C3×C8—41no—
380C24—21no—
381C24—21no—
382C24—22no—
383C24—24no—
384C24—44no—
385C24—44no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—82no—
392C24—82no—
393C24C2×C3×D411yescharacteristic
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4C2×C3×D414yes—
419C2×D4C2×C3×D412yes—
420C2×D4C2×C3×D412yes—
421C2×D4C2×C3×D414yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4C2×C3×D412yes—
431C22×C4C2×C3×D412yes—
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D412yes—
434C22×C4C3×#4112yes—
435C22×C4C24×C311yescharacteristic
436SD16—44no—
437SD16—44no—
438D8—216no—
439D8—216no—
440C2×C8—28no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8C2×C3×D414yes—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C42—22no—
465C42—22no—
466C22×C3—22no—
467C22×C3—24no—
468C22×C3—24no—
469C22×C3—22no—
470C22×C3—24no—
471C22×C3—44no—
472C22×C3—44no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—44no—
477C22×C3—44no—
478C22×C3—44no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—816no—
482C22×C3—816no—
483C22×C3—82no—
484C22×C3—82no—
485C22×C3—82no—
486C22×C3—82no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3#46314yescomplemented
490C22×C3C2×#12111yescharacteristic
491C22×C3D4212yes—
492C22×C3C2×#13511yescharacteristic
493C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
494C22×C3#57612yes—
495C3×C4—22no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—81no—
499C3×C4—82no—
500C3×C4—81no—
501C3×C4C2×#12011yescharacteristic
502C3×C4#53411yescharacteristic
503C3×C4#54612yes—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—48no—
511C23—48no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—48no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—84no—
527C23—88no—
528C23—88no—
529C23—88no—
530C23—88no—
531C23—84no—
532C23—84no—
533C23C3×#12112yes—
534C23C3×#13512yes—
535C23C3×AGL(1,ℤ/8)12yes—
536C23C22×C3×D411yescharacteristic
537Q8—41no—
538Q8—41no—
539Q8—42no—
540D4—22no—
541D4—24no—
542D4—22no—
543D4—22no—
544D4—24no—
545D4—24no—
546D4—48no—
547D4—48no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4C3×#12014yes—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—88no—
570C2×C4C3×#12112yes—
571C2×C4C22×C3×D411yescharacteristic
572C2×C4C22×C3×D412yes—
573C2×C4C2×C3×#4111yescharacteristic
574C2×C4C3×H(5,𝔽2)12yes—
575C8—24no—
576C8—41no—
577C8—41no—
578C2×C3—24no—
579C2×C3—44no—
580C2×C3—84no—
581C2×C3—82no—
582C2×C3#241011yescharacteristic
583C2×C3C2×#46311yescharacteristic
584C2×C3#304111yescharacteristic
585C2×C3#326412yes—
586C2×C3#328012yes—
587C22—22no—
588C22—24no—
589C22—24no—
590C22—22no—
591C22—24no—
592C22—44no—
593C22—44no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—816no—
603C22—816no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22C3×#46314yescomplemented
611C22C2×C3×#12111yescharacteristic
612C22C3×D4212yes—
613C22C2×C3×#13511yescharacteristic
614C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
615C22C3×#57612yes—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—82no—
621C4—81no—
622C4C2×C3×#12011yescharacteristic
623C4C3×#53411yescharacteristic
624C4C3×#54612yes—
625C3#2435511yescharacteristic, direct product factor, Sylow
626C2—24no—
627C2—44no—
628C2—84no—
629C2—82no—
630C2C3×#241011yescharacteristic
631C2C2×C3×#46311yescharacteristic
632C2C3×#304111yescharacteristic
633C2C3×#326412yes—
634C2C3×#328012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D8C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#463C2semidirect
3. C2×C3×#450C2semidirect
4. C3×#2435C2semidirect
5. C3×D4×C22⋊C4C2semidirect
6. C3×#2054C2semidirect
7. C3×#2012C2semidirect
8. C3×#1892C2semidirect
9. C3×#1879C2semidirect
10. C3×#1850C2semidirect
11. C3×#1786C2semidirect
12. #24355C3direct (N × H)
15. C22×C3×SD16C22semidirect
16. C22×C3×SD16C22semidirect
22. C2×C3×#124C22semidirect
24. C2×C3×C4×D4C22semidirect
28. C2×C3×C4⋊C8C22semidirect
29. C2×C3×#103C22semidirect
30. C2×C3×#102C22semidirect
31. C2×C3×#102C22semidirect
32. C2×C3×#102C22semidirect
33. C2×C3×#102C22semidirect
36. C3×#390C22semidirect
37. C3×#390C22semidirect
44. C3×#340C22semidirect
45. C3×#336C22semidirect
47. C2×#463C2×C3semidirect
48. C2×#450C2×C3semidirect
49. #2435C2×C3semidirect
50. D4×C22⋊C4C2×C3semidirect
51. #2054C2×C3semidirect
52. #2012C2×C3semidirect
53. #1892C2×C3semidirect
54. #1879C2×C3semidirect
55. #1850C2×C3semidirect
56. #1786C2×C3semidirect
124. C22×SD16C22×C3semidirect
125. C22×SD16C22×C3semidirect
131. C2×#124C22×C3semidirect
133. C2×C4×D4C22×C3semidirect
137. C2×C4⋊C8C22×C3semidirect
138. C2×#103C22×C3semidirect
139. C2×#102C22×C3semidirect
140. C2×#102C22×C3semidirect
141. C2×#102C22×C3semidirect
142. C2×#102C22×C3semidirect
145. #390C22×C3semidirect
146. #390C22×C3semidirect
153. #340C22×C3semidirect
154. #336C22×C3semidirect
489. C22×C3#463semidirect
610. C22C3×#463semidirect
625. C3#24355direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D82
C2×C3×#4632
C2×C3×#4502
C3×#24352
C3×D4×C22⋊C42
C3×#20542
C3×#20122
C3×#18922
C3×#18792
C3×#18502
C3×#17862
#243553