Finite Group Databasebeta

← C3×#15063 #10633670

Subgroups — C3×#15063 #10633670

Order 768.

1486 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#380C211yesmaximal, characteristic
2C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
3C3×#1445C211yesmaximal, characteristic, complemented
4C3×#1445C211yesmaximal, characteristic, complemented
5C3×#1444C211yesmaximal, characteristic, complemented
6C3×#1442C211yesmaximal, characteristic, complemented
7C3×#1440C212yesmaximal
8C3×#1439C212yesmaximal
9C3×#1436C212yesmaximal, complemented
10C3×#1434C212yesmaximal, complemented
11C3×C22⋊#95C211yesmaximal, characteristic
12#15063C311yesmaximal, characteristic, direct product factor, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C42C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C3×#380—24no—
24C3×C4×C22⋊C4—24no—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×#380C2×C311yescharacteristic
39C2×C4×C22⋊C4C2×C311yescharacteristic
40#1445C2×C311yescharacteristic, complemented
41#1445C2×C311yescharacteristic, complemented
42#1444C2×C311yescharacteristic, complemented
43#1442C2×C311yescharacteristic, complemented
44#1440C2×C312yes—
45#1439C2×C312yes—
46#1436C2×C312yescomplemented
47#1434C2×C312yescomplemented
48C22⋊#95C2×C311yescharacteristic
49C25×C3C2311yescharacteristic
50C23×C3×C4—24no—
51C23×C3×C4—22no—
52C23×C3×C4—22no—
53C23×C3×C4—42no—
54C23×C3×C4D414yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—42no—
71C2×C3×C4⋊C4—42no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—42no—
82C2×C3×C22⋊C4—42no—
83C2×C3×C22⋊C4—42no—
84C2×C3×C22⋊C4Q814yes—
85C2×C3×C42—22no—
86C2×C3×C42—22no—
87C2×C3×C42—24no—
88C2×C3×C42—42no—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—24no—
92C3×#95—22no—
93C3×#95—24no—
94C3×#95—24no—
95C3×#95—24no—
96C3×#95—24no—
97C3×#95—22no—
98C3×#95—42no—
99C3×#95—42no—
100C3×#95—42no—
101C3×#95—42no—
102C24×C4C22×C311yescharacteristic
103C22×C4⋊C4C22×C311yescharacteristic
104C22×C4⋊C4C22×C312yes—
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C22⋊C4C22×C311yescharacteristic
107C22×C22⋊C4C22×C312yes—
108C22×C22⋊C4C22×C311yescharacteristic
109C22×C22⋊C4C22×C312yes—
110C22×C42C22×C311yescharacteristic
111C22×C42C22×C311yescharacteristic
112#380—24no—
113C4×C22⋊C4—24no—
114C2×#95C22×C312yes—
115C2×#95C22×C311yescharacteristic
116C2×#95C22×C311yescharacteristic
117C2×#95C22×C312yes—
118C2×#95C22×C312yes—
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C311yescharacteristic
122C2×#95C22×C311yescharacteristic
123C2×#95C22×C312yes—
124C2×#95C22×C312yes—
125C2×#95C22×C312yes—
126C2×#95C22×C312yes—
127C24×C3—21no—
128C24×C3—21no—
129C24×C3—22no—
130C24×C3—21no—
131C24×C3—21no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3C2×D412yes—
136C24×C3C2×Q812yes—
137C24×C3#4112yes—
138C24×C3C2411yescharacteristic
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—28no—
167C22×C3×C4—28no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—44no—
190C22×C3×C4—48no—
191C22×C3×C4—48no—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×Q812yes—
195C22×C3×C4C2×Q812yes—
196C22×C3×C4#4112yes—
197C22×C3×C4#4114yes—
198C22×C3×C4#4112yes—
199C22×C3×C4#4114yes—
200C22×C3×C4#4112yes—
201C22×C3×C4#4112yes—
202C3×C4⋊C4—44no—
203C3×C4⋊C4—48no—
204C3×C4⋊C4—44no—
205C3×C22⋊C4—28no—
206C3×C22⋊C4—28no—
207C3×C22⋊C4—48no—
208C3×C22⋊C4—48no—
209C3×C42—44no—
210C3×C42—44no—
211C25C23×C311yescharacteristic
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—42no—
216C23×C4C3×D414yes—
217C23×C4C23×C312yes—
218C23×C4C23×C312yes—
219C23×C4C23×C312yes—
220C23×C4C23×C311yescharacteristic
221C23×C4C23×C311yescharacteristic
222C23×C4C23×C311yescharacteristic
223C23×C4C23×C312yes—
224C23×C4C23×C311yescharacteristic
225C23×C4C23×C311yescharacteristic
226C23×C4C23×C311yescharacteristic
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4C3×Q814yes—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—42no—
251#95—22no—
252#95—22no—
253#95—24no—
254#95—22no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—24no—
277C23×C3—24no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3C22⋊Q814yes—
287C23×C3C22×D411yescharacteristic
288C23×C3C22×Q811yescharacteristic
289C23×C3C2×#4111yescharacteristic
290C23×C3C2×#4112yes—
291C23×C3C2×#4111yescharacteristic
292C23×C3C2×#4111yescharacteristic
293C23×C3C2×#4111yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4112yes—
296C23×C3H(5,𝔽2)12yes—
297C23×C3H(5,𝔽2)12yes—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—416no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4#12314yes—
356C2×C3×C4#12314yes—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24C2×C3×D412yes—
366C24C2×C3×Q812yes—
367C24C3×#4112yes—
368C24C24×C311yescharacteristic
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4C2×C3×D412yes—
423C22×C4C2×C3×D412yes—
424C22×C4C2×C3×Q812yes—
425C22×C4C2×C3×Q812yes—
426C22×C4C3×#4112yes—
427C22×C4C3×#4114yes—
428C22×C4C3×#4112yes—
429C22×C4C3×#4114yes—
430C22×C4C3×#4112yes—
431C22×C4C3×#4112yes—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C22⋊C4—28no—
436C22⋊C4—28no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C42—44no—
440C42—44no—
441C22×C3—24no—
442C22×C3—24no—
443C22×C3—24no—
444C22×C3—44no—
445C22×C3—44no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3C2×C22⋊Q811yescharacteristic
453C22×C3C2×#12311yescharacteristic
454C22×C3C2×#12311yescharacteristic
455C22×C3#52511yescharacteristic
456C22×C3#52912yes—
457C22×C3#52911yescharacteristic
458C22×C3#52911yescharacteristic
459C22×C3#52912yes—
460C22×C3#53211yescharacteristic
461C22×C3#53311yescharacteristic
462C22×C3#53311yescharacteristic
463C22×C3#53411yescharacteristic
464C22×C3#53511yescharacteristic
465C22×C3#53812yes—
466C22×C3#54012yes—
467C22×C3#55112yes—
468C22×C3#55112yes—
469C22×C3#55112yes—
470C22×C3#55312yes—
471C22×C3#55312yes—
472C22×C3#55412yes—
473C22×C3#55412yes—
474C22×C3#55512yes—
475C3×C4—24no—
476C3×C4—24no—
477C3×C4—42no—
478C3×C4—44no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—82no—
483C3×C4—82no—
484C3×C4—82no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C3×C22⋊Q814yes—
508C23C22×C3×D411yescharacteristic
509C23C22×C3×Q811yescharacteristic
510C23C2×C3×#4111yescharacteristic
511C23C2×C3×#4112yes—
512C23C2×C3×#4111yescharacteristic
513C23C2×C3×#4111yescharacteristic
514C23C2×C3×#4111yescharacteristic
515C23C2×C3×#4111yescharacteristic
516C23C2×C3×#4112yes—
517C23C3×H(5,𝔽2)12yes—
518C23C3×H(5,𝔽2)12yes—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—416no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4C3×#12314yes—
577C2×C4C3×#12314yes—
578C2×C3—44no—
579C2×C3#239511yescharacteristic
580C2×C3#248311yescharacteristic
581C2×C3#248711yescharacteristic
582C2×C3#250011yescharacteristic
583C2×C3#252212yes—
584C2×C3#253112yes—
585C2×C3#256411yescharacteristic
586C2×C3#257412yes—
587C2×C3#257812yes—
588C2×C3#263511yescharacteristic
589C2×C3#285511yescharacteristic
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22C2×C3×C22⋊Q811yescharacteristic
602C22C2×C3×#12311yescharacteristic
603C22C2×C3×#12311yescharacteristic
604C22C3×#52511yescharacteristic
605C22C3×#52912yes—
606C22C3×#52912yes—
607C22C3×#52911yescharacteristic
608C22C3×#52911yescharacteristic
609C22C3×#53211yescharacteristic
610C22C3×#53311yescharacteristic
611C22C3×#53311yescharacteristic
612C22C3×#53411yescharacteristic
613C22C3×#53511yescharacteristic
614C22C3×#53812yes—
615C22C3×#54012yes—
616C22C3×#55112yes—
617C22C3×#55112yes—
618C22C3×#55112yes—
619C22C3×#55312yes—
620C22C3×#55312yes—
621C22C3×#55412yes—
622C22C3×#55412yes—
623C22C3×#55512yes—
624C4—24no—
625C4—24no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—82no—
634C3#1506311yescharacteristic, direct product factor, Sylow
635C2—44no—
636C2C3×#239511yescharacteristic
637C2C3×#248311yescharacteristic
638C2C3×#248711yescharacteristic
639C2C3×#250011yescharacteristic
640C2C3×#252212yes—
641C2C3×#253112yes—
642C2C3×#256411yescharacteristic
643C2C3×#257412yes—
644C2C3×#257812yes—
645C2C3×#263511yescharacteristic
646C2C3×#285511yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#1445C2semidirect
4. C3×#1445C2semidirect
5. C3×#1444C2semidirect
6. C3×#1442C2semidirect
9. C3×#1436C2semidirect
10. C3×#1434C2semidirect
12. #15063C3direct (N × H)
40. #1445C2×C3semidirect
41. #1445C2×C3semidirect
42. #1444C2×C3semidirect
43. #1442C2×C3semidirect
46. #1436C2×C3semidirect
47. #1434C2×C3semidirect
634. C3#15063direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3802
C2×C3×C4×C22⋊C42
C3×#14452
C3×#14442
C3×#14422
C3×#14402
C3×#14392
C3×#14362
C3×#14342
C3×C22⋊#952
#150633