Finite Group Databasebeta

← C3×#24653 #10643260

Subgroups — C3×#24653 #10643260

Order 768.

1036 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#460C211yesmaximal, characteristic, complemented
2C2×C3×#431C211yesmaximal, characteristic, complemented
3C3×#2856C211yesmaximal, characteristic
4C3×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#2000C212yesmaximal, complemented
7C3×#1982C211yesmaximal, characteristic, complemented
8C3×#1968C212yesmaximal, complemented
9C3×#1948C211yesmaximal, characteristic, complemented
10C3×#1934C212yesmaximal, complemented
11C3×#1803C211yesmaximal, characteristic, complemented
12#24653C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×#121—21no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C3×#460—24no—
22C3×#431—24no—
23C2×C3×#106C2212yes—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2212yes—
31C2×C3×C4×C8C2212yes—
32C3×#391C2211yescharacteristic
33C3×#390—21no—
34C3×#390—21no—
35C3×#390C2212yes—
36C3×#389C2211yescharacteristic
37C3×#387—21no—
38C3×#385—21no—
39C3×#384C2211yescharacteristic
40C3×#380C2211yescharacteristic
41C3×C4×C4⋊C4—21no—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×C4×C22⋊C4—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#460C2×C311yescharacteristic, complemented
47C2×#431C2×C311yescharacteristic, complemented
48#2856C2×C311yescharacteristic
49C4×C22⋊Q8C2×C311yescharacteristic, complemented
50#2034C2×C312yescomplemented
51#2000C2×C312yescomplemented
52#1982C2×C311yescharacteristic, complemented
53#1968C2×C312yescomplemented
54#1948C2×C311yescharacteristic, complemented
55#1934C2×C312yescomplemented
56#1803C2×C311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×SD16—24no—
66C2×C3×SD16—24no—
67C2×C3×SD16—42no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×#127—22no—
71C3×#127—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×#121—42no—
75C3×#121—42no—
76C3×#121—44no—
77C3×#121—44no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C42—22no—
95C2×C3×C42—22no—
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C3×#106—42no—
103C3×C4⋊C8—22no—
104C3×C4⋊C8D414yescomplemented
105C3×#103—42no—
106C3×#102—24no—
107C3×#102—24no—
108C3×#102—42no—
109C3×C22⋊C8—24no—
110C3×C4×C8—24no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C22×SD16C22×C312yes—
117C2×#127C22×C311yescharacteristic
118C2×C22⋊Q8C22×C311yescharacteristic
119C2×#121—21no—
120C2×#121—21no—
121C2×#121C22×C312yes—
122C2×C4×Q8C22×C311yescharacteristic
123C22×C42C22×C311yescharacteristic
124#460—24no—
125#431—24no—
126C2×#106C22×C312yes—
127C2×C4⋊C8C22×C311yescharacteristic
128C2×C4⋊C8C22×C311yescharacteristic
129C2×#103C22×C312yes—
130C2×#102C22×C312yes—
131C2×#102C22×C312yes—
132C2×#102C22×C312yes—
133C2×C22⋊C8C22×C312yes—
134C2×C4×C8C22×C312yes—
135#391C22×C311yescharacteristic
136#390—21no—
137#390—21no—
138#390C22×C312yes—
139#389C22×C311yescharacteristic
140#387—21no—
141#385—21no—
142#384C22×C311yescharacteristic
143#380C22×C311yescharacteristic
144C4×C4⋊C4—21no—
145C4×C4⋊C4C22×C311yescharacteristic
146C4×C22⋊C4—21no—
147#340C22×C312yes—
148#340C22×C312yes—
149C24×C3—22no—
150C24×C3C2×D411yescharacteristic
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8#4112yes—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—44no—
166C2×C3×D4—44no—
167C2×C3×D4—44no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—41no—
188C22×C3×C4SD1614yescomplemented
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C3×SD16—48no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8C2×D414yes—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4—42no—
212C3×C4⋊C4—42no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4#4112yes—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—44no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—44no—
219C3×C22⋊C4—44no—
220C3×C42—22no—
221C3×C42—22no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—42no—
225C3×C42—42no—
226C3×C42C2×D412yes—
227C3×C42#4112yes—
228C22×Q8C23×C311yescharacteristic
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C23×C312yes—
233C23×C4C3×D411yescharacteristic
234C23×C4C3×D411yescharacteristic
235C23×C4C23×C311yescharacteristic
236C2×SD16—24no—
237C2×SD16—24no—
238C2×SD16—42no—
239C22×C8C23×C312yes—
240C22×C8C23×C312yes—
241#127—22no—
242#127—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245#121—42no—
246#121—42no—
247#121—44no—
248#121—44no—
249C4×Q8—22no—
250C4×Q8—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C4⋊C4C23×C312yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42C3×D411yescharacteristic
270C2×C42C3×D411yescharacteristic
271C2×C42C23×C311yescharacteristic
272C2×C42C23×C311yescharacteristic
273#106—42no—
274C4⋊C8—22no—
275C4⋊C8C3×D414yescomplemented
276#103—42no—
277#102—24no—
278#102—24no—
279#102—42no—
280C22⋊C8—24no—
281C4×C8—24no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3C2×SD1612yes—
301C23×C3C22×D411yescharacteristic
302C3×Q8—22no—
303C3×Q8—22no—
304C3×Q8—41no—
305C3×Q8—41no—
306C3×D4—42no—
307C3×D4—42no—
308C3×D4—42no—
309C3×D4—42no—
310C3×D4—84no—
311C3×D4—88no—
312C3×D4—84no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—28no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—44no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4#12112yes—
344C2×C3×C4#12112yes—
345C2×C3×C4C2×SD1612yes—
346C2×C3×C4C2×SD1612yes—
347C2×C3×C4#13512yes—
348C2×C3×C4AGL(1,ℤ/8)12yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4H(5,𝔽2)11yescharacteristic
354C2×C3×C4H(5,𝔽2)11yescharacteristic
355C3×C8—24no—
356C3×C8—42no—
357C24—22no—
358C24C2×C3×D411yescharacteristic
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8C3×#4112yes—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—41no—
396C22×C4C3×SD1614yescomplemented
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C24×C311yescharacteristic
403SD16—48no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8C2×C3×D414yes—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4C3×#4112yes—
423C22⋊C4—24no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C42—22no—
429C42—22no—
430C42—21no—
431C42—21no—
432C42—42no—
433C42—42no—
434C42C2×C3×D412yes—
435C42C3×#4112yes—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3C2×#12111yescharacteristic
451C22×C3#52511yescharacteristic
452C22×C3#53511yescharacteristic
453C22×C3C22×SD1611yescharacteristic
454C22×C3C2×#13511yescharacteristic
455C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
456C22×C3#57511yescharacteristic
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—42no—
461C3×C4—41no—
462C3×C4—42no—
463C3×C4—82no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4#46514yes—
467C3×C4#54611yescharacteristic
468C3×C4#55011yescharacteristic
469C3×C4#55311yescharacteristic
470C3×C4#55511yescharacteristic
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23C2×C3×SD1612yes—
485C23C22×C3×D411yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—84no—
495D4—88no—
496D4—84no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—28no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4C3×#12112yes—
528C2×C4C3×#12112yes—
529C2×C4C2×C3×SD1612yes—
530C2×C4C2×C3×SD1612yes—
531C2×C4C3×#13512yes—
532C2×C4C3×AGL(1,ℤ/8)12yes—
533C2×C4C22×C3×D411yescharacteristic
534C2×C4C2×C3×#4111yescharacteristic
535C2×C4C2×C3×#4111yescharacteristic
536C2×C4C2×C3×#4111yescharacteristic
537C2×C4C3×H(5,𝔽2)11yescharacteristic
538C2×C4C3×H(5,𝔽2)11yescharacteristic
539C8—24no—
540C8—42no—
541C2×C3—42no—
542C2×C3—162no—
543C2×C3#241411yescharacteristic
544C2×C3C2×#46511yescharacteristic
545C2×C3#305511yescharacteristic
546C2×C3#310311yescharacteristic
547C2×C3#310611yescharacteristic
548C2×C3#320811yescharacteristic
549C2×C3#321211yescharacteristic
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22C2×C3×#12111yescharacteristic
565C22C3×#52511yescharacteristic
566C22C3×#53511yescharacteristic
567C22C22×C3×SD1611yescharacteristic
568C22C2×C3×#13511yescharacteristic
569C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
570C22C3×#57511yescharacteristic
571C4—22no—
572C4—22no—
573C4—22no—
574C4—41no—
575C4—42no—
576C4—42no—
577C4—82no—
578C4—81no—
579C4—81no—
580C4C3×#46514yes—
581C4C3×#54611yescharacteristic
582C4C3×#55011yescharacteristic
583C4C3×#55311yescharacteristic
584C4C3×#55511yescharacteristic
585C3#2465311yescharacteristic, direct product factor, Sylow
586C2—42no—
587C2—162no—
588C2C3×#241411yescharacteristic
589C2C2×C3×#46511yescharacteristic
590C2C3×#305511yescharacteristic
591C2C3×#310311yescharacteristic
592C2C3×#310611yescharacteristic
593C2C3×#320811yescharacteristic
594C2C3×#321211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#460C2semidirect
2. C2×C3×#431C2semidirect
4. C3×C4×C22⋊Q8C2semidirect
5. C3×#2034C2semidirect
6. C3×#2000C2semidirect
7. C3×#1982C2semidirect
8. C3×#1968C2semidirect
9. C3×#1948C2semidirect
10. C3×#1934C2semidirect
11. C3×#1803C2semidirect
12. #24653C3direct (N × H)
46. C2×#460C2×C3semidirect
47. C2×#431C2×C3semidirect
49. C4×C22⋊Q8C2×C3semidirect
50. #2034C2×C3semidirect
51. #2000C2×C3semidirect
52. #1982C2×C3semidirect
53. #1968C2×C3semidirect
54. #1948C2×C3semidirect
55. #1934C2×C3semidirect
56. #1803C2×C3semidirect
104. C3×C4⋊C8D4semidirect
188. C22×C3×C4SD16semidirect
275. C4⋊C8C3×D4semidirect
396. C22×C4C3×SD16semidirect
585. C3#24653direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4602
C2×C3×#4312
C3×#28562
C3×C4×C22⋊Q82
C3×#20342
C3×#20002
C3×#19822
C3×#19682
C3×#19482
C3×#19342
C3×#18032
#246533