Finite Group Databasebeta

← C3×#24238 #10642845

Subgroups — C3×#24238 #10642845

Order 768.

1574 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic, complemented
3C2×C3×#449C212yesmaximal, complemented
4C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
5C3×#2537C211yesmaximal, characteristic, complemented
6C3×#2034C212yesmaximal, complemented
7C3×#1999C212yesmaximal, complemented
8C3×#1957C211yesmaximal, characteristic, complemented
9C3×#1954C211yesmaximal, characteristic, complemented
10C3×#1803C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal, complemented
12#24238C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×SD16C2211yescharacteristic
15C22×C3×D8C2212yes—
16C3×#550—24no—
17C3×#550—24no—
18C2×C3×#127—21no—
19C2×C3×#127C2211yescharacteristic
20C2×C3×#121—21no—
21C2×C3×#121—22no—
22C2×C3×#121C2212yes—
23C2×C3×#121C2212yes—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×C4×D4C2211yescharacteristic
28C3×#459—24no—
29C3×#449—28no—
30C3×C4×SD16—24no—
31C2×C3×#106C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#102C2211yescharacteristic
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×#102C2212yes—
39C2×C3×C22⋊C8C2212yes—
40C2×C3×C22⋊C8C2212yes—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#394—22no—
43C3×#384C2211yescharacteristic
44C3×#380—21no—
45C3×#380C2212yes—
46C3×#340C2212yes—
47C2×#550C2×C311yescharacteristic, complemented
48C2×#459C2×C311yescharacteristic, complemented
49C2×#449C2×C312yescomplemented
50C2×C4×SD16C2×C311yescharacteristic, complemented
51#2537C2×C311yescharacteristic, complemented
52#2034C2×C312yescomplemented
53#1999C2×C312yescomplemented
54#1957C2×C311yescharacteristic, complemented
55#1954C2×C311yescharacteristic, complemented
56#1803C2×C311yescharacteristic, complemented
57#1788C2×C312yescomplemented
58C2×C3×#41—24no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41—42no—
63C2×C3×#41—42no—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4—21no—
69C22×C3×D4—21no—
70C22×C3×D4D411yescharacteristic
71C22×C3×D4C2311yescharacteristic
72C22×C3×D4C2312yes—
73C23×C3×C4—22no—
74C23×C3×C4—21no—
75C23×C3×C4C2312yes—
76C2×C3×SD16—22no—
77C2×C3×SD16—22no—
78C2×C3×SD16D414yescomplemented
79C2×C3×D8—24no—
80C2×C3×D8—24no—
81C2×C3×D8—42no—
82C22×C3×C8C2312yes—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#127—28no—
86C3×#127—22no—
87C3×#127D414yescomplemented
88C3×#121—216no—
89C3×#121—24no—
90C3×#121—24no—
91C3×#121—24no—
92C3×#121—42no—
93C3×#121—44no—
94C3×C4×Q8—22no—
95C3×C4×Q8D414yescomplemented
96C3×C4×D4—28no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—22no—
110C2×C3×C22⋊C4—22no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4D412yes—
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115C3×#106—22no—
116C3×C4⋊C8—22no—
117C3×#103—22no—
118C3×#103—24no—
119C3×#102—24no—
120C3×#102—24no—
121C3×#102—22no—
122C3×#102—42no—
123C3×C22⋊C8—24no—
124C3×C22⋊C8—42no—
125C3×C4×C8—22no—
126C3×#95—22no—
127C22×#41C22×C312yes—
128C22×SD16C22×C311yescharacteristic
129C22×D8C22×C312yes—
130#550—24no—
131#550—24no—
132C2×#127—21no—
133C2×#127C22×C311yescharacteristic
134C2×#121—21no—
135C2×#121—22no—
136C2×#121C22×C312yes—
137C2×#121C22×C312yes—
138C2×C4×Q8C22×C311yescharacteristic
139C2×C4×D4—21no—
140C2×C4×D4C22×C311yescharacteristic
141C2×C4×D4C22×C311yescharacteristic
142#459—24no—
143#449—28no—
144C4×SD16—24no—
145C2×#106C22×C311yescharacteristic
146C2×C4⋊C8C22×C311yescharacteristic
147C2×#103C22×C312yes—
148C2×#103C22×C311yescharacteristic
149C2×#102C22×C311yescharacteristic
150C2×#102C22×C312yes—
151C2×#102C22×C312yes—
152C2×#102C22×C312yes—
153C2×C22⋊C8C22×C312yes—
154C2×C22⋊C8C22×C312yes—
155C2×C4×C8C22×C311yescharacteristic
156#394—22no—
157#384C22×C311yescharacteristic
158#380—21no—
159#380C22×C312yes—
160#340C22×C312yes—
161C24×C3—21no—
162C24×C3—22no—
163C24×C3C2×D412yes—
164C3×#41—48no—
165C3×#41—48no—
166C3×#41—48no—
167C3×#41—48no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8C2×D412yes—
172C2×C3×Q8C2×D412yes—
173C2×C3×Q8#4112yes—
174C2×C3×D4—28no—
175C2×C3×D4—28no—
176C2×C3×D4—22no—
177C2×C3×D4—24no—
178C2×C3×D4—24no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—21no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—21no—
191C2×C3×D4—44no—
192C2×C3×D4—44no—
193C2×C3×D4—44no—
194C2×C3×D4—44no—
195C2×C3×D4—42no—
196C2×C3×D4—42no—
197C2×C3×D4—81no—
198C2×C3×D4—81no—
199C2×C3×D4—82no—
200C2×C3×D4—82no—
201C2×C3×D4—82no—
202C2×C3×D4C2×D414yes—
203C2×C3×D4C2×D412yes—
204C22×C3×C4—21no—
205C22×C3×C4—28no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—21no—
214C22×C3×C4—22no—
215C22×C3×C4—44no—
216C22×C3×C4—44no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4—42no—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C3×SD16—28no—
225C3×D8—48no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—21no—
232C2×C3×C8—22no—
233C2×C3×C8—21no—
234C2×C3×C8C2×D412yes—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4C2×D412yes—
245C3×C22⋊C4—28no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—44no—
248C3×C22⋊C4—44no—
249C3×C22⋊C4—42no—
250C3×C42—24no—
251C3×C42C2×D412yes—
252C3×C42C2×D412yes—
253C2×#41—24no—
254C2×#41—24no—
255C2×#41—24no—
256C2×#41—24no—
257C2×#41—42no—
258C2×#41—42no—
259C22×Q8C23×C311yescharacteristic
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4C3×D411yescharacteristic
266C22×D4C23×C311yescharacteristic
267C22×D4C23×C312yes—
268C23×C4—22no—
269C23×C4—21no—
270C23×C4C23×C312yes—
271C2×SD16—22no—
272C2×SD16—22no—
273C2×SD16C3×D414yescomplemented
274C2×D8—24no—
275C2×D8—24no—
276C2×D8—42no—
277C22×C8C23×C312yes—
278C22×C8C23×C311yescharacteristic
279C22×C8C23×C311yescharacteristic
280#127—28no—
281#127—22no—
282#127C3×D414yescomplemented
283#121—216no—
284#121—24no—
285#121—24no—
286#121—24no—
287#121—42no—
288#121—44no—
289C4×Q8—22no—
290C4×Q8C3×D414yescomplemented
291C4×D4—28no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4C3×D411yescharacteristic
300C2×C4⋊C4C23×C311yescharacteristic
301C2×C4⋊C4C23×C312yes—
302C2×C4⋊C4C23×C311yescharacteristic
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4C3×D412yes—
308C2×C42—21no—
309C2×C42C23×C311yescharacteristic
310#106—22no—
311C4⋊C8—22no—
312#103—22no—
313#103—24no—
314#102—24no—
315#102—24no—
316#102—22no—
317#102—42no—
318C22⋊C8—24no—
319C22⋊C8—42no—
320C4×C8—22no—
321#95—22no—
322C23×C3—22no—
323C23×C3—24no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—42no—
337C23×C3—41no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—41no—
341C23×C3—42no—
342C23×C3—41no—
343C23×C3C22×D411yescharacteristic
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—22no—
347C3×Q8#12114yescomplemented
348C3×D4—22no—
349C3×D4—22no—
350C3×D4—24no—
351C3×D4—24no—
352C3×D4—416no—
353C3×D4—42no—
354C3×D4—41no—
355C3×D4—42no—
356C3×D4—41no—
357C3×D4—44no—
358C3×D4—48no—
359C3×D4—84no—
360C3×D4—88no—
361C3×D4—88no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—416no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—41no—
379C2×C3×C4—42no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—84no—
385C2×C3×C4#12012yes—
386C2×C3×C4#12012yes—
387C2×C3×C4#13512yes—
388C2×C3×C4AGL(1,ℤ/8)12yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C4H(5,𝔽2)11yescharacteristic
395C3×C8—22no—
396C3×C8—41no—
397C3×C8—42no—
398C24—21no—
399C24—22no—
400C24C2×C3×D412yes—
401#41—48no—
402#41—48no—
403#41—48no—
404#41—48no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×C3×D412yes—
409C2×Q8C2×C3×D412yes—
410C2×Q8C3×#4112yes—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—81no—
435C2×D4—81no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4C2×C3×D414yes—
440C2×D4C2×C3×D412yes—
441C22×C4—21no—
442C22×C4—28no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4C2×C3×D412yes—
458C22×C4C2×C3×D411yescharacteristic
459C22×C4C2×C3×D411yescharacteristic
460C22×C4C24×C311yescharacteristic
461SD16—28no—
462D8—48no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—21no—
471C2×C8C2×C3×D412yes—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4C2×C3×D412yes—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—42no—
487C42—24no—
488C42C2×C3×D412yes—
489C42C2×C3×D412yes—
490C22×C3—42no—
491C22×C3—44no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—44no—
496C22×C3—42no—
497C22×C3—44no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—82no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—82no—
508C22×C3—82no—
509C22×C3—82no—
510C22×C3—81no—
511C22×C3C2×#12011yescharacteristic
512C22×C3D4211yescharacteristic
513C22×C3#54611yescharacteristic
514C22×C3C2×#13511yescharacteristic
515C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C3×C4—24no—
519C3×C4—22no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—81no—
523C3×C4—82no—
524C3×C4#44914yescomplemented
525C3×C4C2×#12111yescharacteristic
526C3×C4#53511yescharacteristic
527C3×C4D4211yescharacteristic
528C3×C4#54611yescharacteristic
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23C22×C3×D411yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—22no—
554Q8C3×#12114yescomplemented
555D4—22no—
556D4—22no—
557D4—24no—
558D4—24no—
559D4—416no—
560D4—44no—
561D4—48no—
562D4—42no—
563D4—42no—
564D4—41no—
565D4—41no—
566D4—84no—
567D4—88no—
568D4—88no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—416no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4C3×#12012yes—
593C2×C4C3×#12012yes—
594C2×C4C3×#13512yes—
595C2×C4C3×AGL(1,ℤ/8)12yes—
596C2×C4C22×C3×D411yescharacteristic
597C2×C4C22×C3×D411yescharacteristic
598C2×C4C22×C3×D411yescharacteristic
599C2×C4C2×C3×#4111yescharacteristic
600C2×C4C3×H(5,𝔽2)11yescharacteristic
601C2×C4C3×H(5,𝔽2)11yescharacteristic
602C8—22no—
603C8—41no—
604C8—42no—
605C2×C3—82no—
606C2×C3—84no—
607C2×C3—82no—
608C2×C3#240811yescharacteristic
609C2×C3C2×#44911yescharacteristic
610C2×C3#301211yescharacteristic
611C2×C3#327411yescharacteristic
612C2×C3#328411yescharacteristic
613C2×C3#330511yescharacteristic
614C2×C3#333211yescharacteristic
615C22—44no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—81no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22—81no—
633C22—82no—
634C22—81no—
635C22—82no—
636C22C2×C3×#12011yescharacteristic
637C22C3×D4211yescharacteristic
638C22C3×#54611yescharacteristic
639C22C2×C3×#13511yescharacteristic
640C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
641C22C3×#57611yescharacteristic
642C22C3×#57711yescharacteristic
643C4—24no—
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—82no—
649C4C3×#44914yescomplemented
650C4C2×C3×#12111yescharacteristic
651C4C3×#53511yescharacteristic
652C4C3×D4211yescharacteristic
653C4C3×#54611yescharacteristic
654C3#2423811yescharacteristic, direct product factor, Sylow
655C2—82no—
656C2—84no—
657C2—82no—
658C2C3×#240811yescharacteristic
659C2C2×C3×#44911yescharacteristic
660C2C3×#301211yescharacteristic
661C2C3×#327411yescharacteristic
662C2C3×#328411yescharacteristic
663C2C3×#330511yescharacteristic
664C2C3×#333211yescharacteristic
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×#550C2semidirect
2. C2×C3×#459C2semidirect
3. C2×C3×#449C2semidirect
4. C2×C3×C4×SD16C2semidirect
5. C3×#2537C2semidirect
6. C3×#2034C2semidirect
7. C3×#1999C2semidirect
8. C3×#1957C2semidirect
9. C3×#1954C2semidirect
10. C3×#1803C2semidirect
11. C3×#1788C2semidirect
12. #24238C3direct (N × H)
47. C2×#550C2×C3semidirect
48. C2×#459C2×C3semidirect
49. C2×#449C2×C3semidirect
50. C2×C4×SD16C2×C3semidirect
51. #2537C2×C3semidirect
52. #2034C2×C3semidirect
53. #1999C2×C3semidirect
54. #1957C2×C3semidirect
55. #1954C2×C3semidirect
56. #1803C2×C3semidirect
57. #1788C2×C3semidirect
78. C2×C3×SD16D4semidirect
87. C3×#127D4semidirect
95. C3×C4×Q8D4semidirect
273. C2×SD16C3×D4semidirect
282. #127C3×D4semidirect
290. C4×Q8C3×D4semidirect
347. C3×Q8#121semidirect
524. C3×C4#449semidirect
554. Q8C3×#121semidirect
649. C4C3×#449semidirect
654. C3#24238direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4592
C2×C3×#4492
C2×C3×C4×SD162
C3×#25372
C3×#20342
C3×#19992
C3×#19572
C3×#19542
C3×#18032
C3×#17882
#242383