Finite Group Databasebeta

← C3×#15001 #10633608

Subgroups — C3×#15001 #10633608

Order 768.

1846 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#387C212yesmaximal, complemented
3C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
4C3×#1446C211yesmaximal, characteristic, complemented
5C3×#1446C211yesmaximal, characteristic, complemented
6C3×#1444C211yesmaximal, characteristic, complemented
7C3×#1442C211yesmaximal, characteristic, complemented
8C3×#1440C212yesmaximal
9C3×#1439C212yesmaximal
10C3×#1436C212yesmaximal, complemented
11C3×C22⋊#95C211yesmaximal, characteristic
12#15001C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C24×C3×C4C2211yescharacteristic
15C2×C3×C22⋊Q8—24no—
16C2×C3×C22⋊Q8—28no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C22⋊C4C2211yescharacteristic
24C22×C3×C42C2211yescharacteristic
25C22×C3×C42C2211yescharacteristic
26C3×#387—28no—
27C3×C4×C22⋊C4—24no—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2211yescharacteristic
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C22×C22⋊Q8C2×C311yescharacteristic
41C2×#387C2×C312yescomplemented
42C2×C4×C22⋊C4C2×C311yescharacteristic
43#1446C2×C311yescharacteristic, complemented
44#1446C2×C311yescharacteristic, complemented
45#1444C2×C311yescharacteristic, complemented
46#1442C2×C311yescharacteristic, complemented
47#1440C2×C312yes—
48#1439C2×C312yes—
49#1436C2×C312yescomplemented
50C22⋊#95C2×C311yescharacteristic
51C25×C3C2311yescharacteristic
52C22×C3×Q8—24no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—42no—
57C23×C3×C4—24no—
58C23×C3×C4—22no—
59C23×C3×C4—22no—
60C23×C3×C4—42no—
61C23×C3×C4D414yes—
62C23×C3×C4C2312yes—
63C23×C3×C4C2312yes—
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C3×C22⋊Q8—232no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4D418yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—24no—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4—42no—
92C2×C3×C22⋊C4Q814yescomplemented
93C2×C3×C42—22no—
94C2×C3×C42—22no—
95C2×C3×C42—24no—
96C2×C3×C42—42no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—24no—
100C3×#95—24no—
101C3×#95—24no—
102C3×#95—24no—
103C3×#95—24no—
104C3×#95—22no—
105C3×#95—24no—
106C3×#95—22no—
107C3×#95—42no—
108C3×#95—42no—
109C23×Q8C22×C311yescharacteristic
110C24×C4C22×C311yescharacteristic
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—28no—
113C22×C4⋊C4C22×C311yescharacteristic
114C22×C4⋊C4C22×C312yes—
115C22×C4⋊C4C22×C312yes—
116C22×C22⋊C4C22×C312yes—
117C22×C22⋊C4C22×C311yescharacteristic
118C22×C22⋊C4C22×C312yes—
119C22×C22⋊C4C22×C311yescharacteristic
120C22×C42C22×C311yescharacteristic
121C22×C42C22×C311yescharacteristic
122#387—28no—
123C4×C22⋊C4—24no—
124C2×#95C22×C312yes—
125C2×#95C22×C311yescharacteristic
126C2×#95C22×C311yescharacteristic
127C2×#95C22×C312yes—
128C2×#95C22×C312yes—
129C2×#95C22×C312yes—
130C2×#95C22×C312yes—
131C2×#95C22×C312yes—
132C2×#95C22×C311yescharacteristic
133C2×#95C22×C311yescharacteristic
134C2×#95C22×C312yes—
135C2×#95C22×C312yes—
136C24×C3—22no—
137C24×C3—21no—
138C24×C3—21no—
139C24×C3—22no—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3C2×D412yes—
145C24×C3C2×Q812yes—
146C24×C3#4112yes—
147C24×C3C2411yescharacteristic
148C2×C3×Q8—28no—
149C2×C3×Q8—48no—
150C2×C3×Q8—48no—
151C2×C3×Q8—44no—
152C2×C3×Q8—44no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
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—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—28no—
193C22×C3×C4—28no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—44no—
202C22×C3×C4—48no—
203C22×C3×C4—48no—
204C22×C3×C4C2×D414yes—
205C22×C3×C4C2×D414yes—
206C22×C3×C4C2×D414yes—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×Q812yes—
210C22×C3×C4C2×Q812yes—
211C22×C3×C4#4112yes—
212C22×C3×C4#4114yes—
213C22×C3×C4#4112yes—
214C22×C3×C4#4112yes—
215C22×C3×C4#4112yes—
216C3×C4⋊C4—216no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—216no—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—216no—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—48no—
223C3×C42—44no—
224C3×C42—44no—
225C25C23×C311yescharacteristic
226C22×Q8—24no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—42no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—42no—
235C23×C4C3×D414yes—
236C23×C4C23×C312yes—
237C23×C4C23×C312yes—
238C23×C4C23×C312yes—
239C23×C4C23×C311yescharacteristic
240C23×C4C23×C311yescharacteristic
241C23×C4C23×C311yescharacteristic
242C23×C4C23×C312yes—
243C23×C4C23×C311yescharacteristic
244C23×C4C23×C311yescharacteristic
245C23×C4C23×C311yescharacteristic
246C22⋊Q8—232no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4C3×D418yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4C3×Q814yescomplemented
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—42no—
271#95—22no—
272#95—22no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—22no—
279#95—24no—
280#95—22no—
281#95—42no—
282#95—42no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3C22⋊Q814yes—
306C23×C3C22×D411yescharacteristic
307C23×C3C22×D412yes—
308C23×C3C22×Q811yescharacteristic
309C23×C3C2×#4111yescharacteristic
310C23×C3C2×#4111yescharacteristic
311C23×C3C2×#4111yescharacteristic
312C23×C3C2×#4111yescharacteristic
313C23×C3C2×#4111yescharacteristic
314C23×C3C2×#4112yes—
315C23×C3H(5,𝔽2)12yes—
316C23×C32−1+412yes—
317C3×Q8—416no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—416no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4#12014yes—
376C2×C3×C4#12314yes—
377C24—22no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—21no—
382C24—21no—
383C24—22no—
384C24—22no—
385C24C2×C3×D412yes—
386C24C2×C3×Q812yes—
387C24C3×#4112yes—
388C24C24×C311yescharacteristic
389C2×Q8—28no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—44no—
393C2×Q8—44no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
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—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4C2×C3×D414yes—
446C22×C4C2×C3×D414yes—
447C22×C4C2×C3×D414yes—
448C22×C4C2×C3×D412yes—
449C22×C4C2×C3×D412yes—
450C22×C4C2×C3×Q812yes—
451C22×C4C2×C3×Q812yes—
452C22×C4C3×#4112yes—
453C22×C4C3×#4114yes—
454C22×C4C3×#4112yes—
455C22×C4C3×#4112yes—
456C22×C4C3×#4112yes—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C22⋊C4—216no—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—48no—
464C42—44no—
465C42—44no—
466C22×C3—24no—
467C22×C3—24no—
468C22×C3—24no—
469C22×C3—44no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—42no—
476C22×C3—44no—
477C22×C3C2×#12011yescharacteristic
478C22×C3C2×C22⋊Q811yescharacteristic
479C22×C3C2×#12311yescharacteristic
480C22×C3#52511yescharacteristic
481C22×C3#52512yes—
482C22×C3#52911yescharacteristic
483C22×C3#52911yescharacteristic
484C22×C3#53212yes—
485C22×C3#53211yescharacteristic
486C22×C3#53311yescharacteristic
487C22×C3#53311yescharacteristic
488C22×C3#53411yescharacteristic
489C22×C3#53611yescharacteristic
490C22×C3#53912yes—
491C22×C3#54212yes—
492C22×C3#54612yes—
493C22×C3#54612yes—
494C22×C3#54812yes—
495C22×C3D4×Q812yes—
496C22×C3#55112yes—
497C22×C3#55212yes—
498C22×C3#55412yes—
499C22×C3#55612yes—
500C3×C4—24no—
501C3×C4—24no—
502C3×C4—42no—
503C3×C4—42no—
504C3×C4—44no—
505C3×C4—42no—
506C3×C4—44no—
507C3×C4—42no—
508C3×C4—44no—
509C3×C4—82no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C3×C22⋊Q814yes—
533C23C22×C3×D411yescharacteristic
534C23C22×C3×D412yes—
535C23C22×C3×Q811yescharacteristic
536C23C2×C3×#4111yescharacteristic
537C23C2×C3×#4111yescharacteristic
538C23C2×C3×#4111yescharacteristic
539C23C2×C3×#4111yescharacteristic
540C23C2×C3×#4111yescharacteristic
541C23C2×C3×#4112yes—
542C23C3×H(5,𝔽2)12yes—
543C23C3×2−1+412yes—
544Q8—416no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—416no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4C3×#12014yes—
603C2×C4C3×#12314yes—
604C2×C3—44no—
605C2×C3#238811yescharacteristic
606C2×C3#240511yescharacteristic
607C2×C3#240911yescharacteristic
608C2×C3#246512yes—
609C2×C3#247712yes—
610C2×C3#249911yescharacteristic
611C2×C3#255612yes—
612C2×C3#256012yes—
613C2×C3#256411yescharacteristic
614C2×C3#263811yescharacteristic
615C2×C3#285511yescharacteristic
616C22—24no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—44no—
626C22—44no—
627C22C2×C3×#12011yescharacteristic
628C22C2×C3×C22⋊Q811yescharacteristic
629C22C2×C3×#12311yescharacteristic
630C22C3×#52511yescharacteristic
631C22C3×#52512yes—
632C22C3×#52911yescharacteristic
633C22C3×#52911yescharacteristic
634C22C3×#53211yescharacteristic
635C22C3×#53212yes—
636C22C3×#53311yescharacteristic
637C22C3×#53311yescharacteristic
638C22C3×#53411yescharacteristic
639C22C3×#53611yescharacteristic
640C22C3×#53912yes—
641C22C3×#54212yes—
642C22C3×#54612yes—
643C22C3×#54612yes—
644C22C3×#54812yes—
645C22C3×D4×Q812yes—
646C22C3×#55112yes—
647C22C3×#55212yes—
648C22C3×#55412yes—
649C22C3×#55612yes—
650C4—24no—
651C4—24no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—44no—
658C4—42no—
659C4—82no—
660C3#1500111yescharacteristic, direct product factor, Sylow
661C2—44no—
662C2C3×#238811yescharacteristic
663C2C3×#240511yescharacteristic
664C2C3×#240911yescharacteristic
665C2C3×#246512yes—
666C2C3×#247712yes—
667C2C3×#249911yescharacteristic
668C2C3×#255612yes—
669C2C3×#256012yes—
670C2C3×#256411yescharacteristic
671C2C3×#263811yescharacteristic
672C2C3×#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
2. C2×C3×#387C2semidirect
4. C3×#1446C2semidirect
5. C3×#1446C2semidirect
6. C3×#1444C2semidirect
7. C3×#1442C2semidirect
10. C3×#1436C2semidirect
12. #15001C3direct (N × H)
41. C2×#387C2×C3semidirect
43. #1446C2×C3semidirect
44. #1446C2×C3semidirect
45. #1444C2×C3semidirect
46. #1442C2×C3semidirect
49. #1436C2×C3semidirect
92. C2×C3×C22⋊C4Q8semidirect
266. C2×C22⋊C4C3×Q8semidirect
660. C3#15001direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×C3×#3872
C2×C3×C4×C22⋊C42
C3×#14462
C3×#14442
C3×#14422
C3×#14402
C3×#14392
C3×#14362
C3×C22⋊#952
#150013