Finite Group Databasebeta

← C3×#15065 #10633672

Subgroups — C3×#15065 #10633672

Order 768.

1594 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#400C211yesmaximal, characteristic, complemented
2C2×C3×#399C211yesmaximal, characteristic
3C2×C3×#395C211yesmaximal, characteristic, complemented
4C2×C3×#393C211yesmaximal, characteristic
5C2×C3×#391C211yesmaximal, characteristic, complemented
6C2×C3×#388C211yesmaximal, characteristic
7C3×#1445C212yesmaximal, complemented
8C3×#1439C212yesmaximal
9C3×#1437C212yesmaximal
10C3×C4⋊#95C212yesmaximal, complemented
11C3×C4×#95C211yesmaximal, characteristic, complemented
12#15065C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2211yescharacteristic
23C22×C3×C42C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×#400—24no—
26C3×#399—24no—
27C3×#395—24no—
28C3×#393—24no—
29C3×#391—24no—
30C3×#388—24no—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2211yescharacteristic
36C2×C3×#95C2211yescharacteristic
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×C3×#95C2211yescharacteristic
41C2×C3×#95C2211yescharacteristic
42C2×#400C2×C311yescharacteristic, complemented
43C2×#399C2×C311yescharacteristic
44C2×#395C2×C311yescharacteristic, complemented
45C2×#393C2×C311yescharacteristic
46C2×#391C2×C311yescharacteristic, complemented
47C2×#388C2×C311yescharacteristic
48#1445C2×C312yescomplemented
49#1439C2×C312yes—
50#1437C2×C312yes—
51C4⋊#95C2×C312yescomplemented
52C4×#95C2×C311yescharacteristic, complemented
53C25×C3C2311yescharacteristic
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—24no—
58C22×C3×Q8—42no—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—42no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—24no—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4—24no—
92C2×C3×C22⋊C4—42no—
93C2×C3×C22⋊C4—42no—
94C2×C3×C22⋊C4Q814yescomplemented
95C2×C3×C42—24no—
96C2×C3×C42—24no—
97C2×C3×C42—42no—
98C2×C3×C42—42no—
99C2×C3×C42D414yes—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—24no—
103C3×#95—24no—
104C3×#95—24no—
105C3×#95—24no—
106C3×#95—24no—
107C3×#95—24no—
108C3×#95—24no—
109C3×#95D414yes—
110C3×#95D414yes—
111C23×Q8C22×C311yescharacteristic
112C22×C4⋊C4C22×C311yescharacteristic
113C22×C4⋊C4C22×C311yescharacteristic
114C22×C4⋊C4C22×C312yes—
115C22×C4⋊C4C22×C312yes—
116C22×C22⋊C4C22×C311yescharacteristic
117C22×C22⋊C4C22×C312yes—
118C22×C22⋊C4C22×C312yes—
119C22×C22⋊C4C22×C311yescharacteristic
120C22×C22⋊C4C22×C311yescharacteristic
121C22×C42C22×C312yes—
122C22×C42C22×C311yescharacteristic
123#400—24no—
124#399—24no—
125#395—24no—
126#393—24no—
127#391—24no—
128#388—24no—
129C2×#95C22×C312yes—
130C2×#95C22×C312yes—
131C2×#95C22×C312yes—
132C2×#95C22×C312yes—
133C2×#95C22×C312yes—
134C2×#95C22×C311yescharacteristic
135C2×#95C22×C312yes—
136C2×#95C22×C312yes—
137C2×#95C22×C311yescharacteristic
138C2×#95C22×C311yescharacteristic
139C2×#95C22×C311yescharacteristic
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3—21no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3C2×Q812yes—
151C24×C3C2411yescharacteristic
152C2×C3×Q8—28no—
153C2×C3×Q8—48no—
154C2×C3×Q8—48no—
155C2×C3×Q8—44no—
156C2×C3×Q8—44no—
157C22×C3×C4—22no—
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—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
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—21no—
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—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4C2×D414yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D414yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×Q812yes—
212C22×C3×C4C2×Q812yes—
213C22×C3×C4#4114yes—
214C22×C3×C4#4112yes—
215C22×C3×C4#4114yes—
216C22×C3×C4#4114yes—
217C3×C4⋊C4—48no—
218C3×C4⋊C4—48no—
219C3×C4⋊C4—44no—
220C3×C4⋊C4—44no—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—48no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—48no—
225C3×C22⋊C4—44no—
226C3×C42—44no—
227C3×C42—48no—
228C25C23×C311yescharacteristic
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—24no—
233C22×Q8—42no—
234C23×C4C23×C311yescharacteristic
235C23×C4C23×C311yescharacteristic
236C23×C4C23×C312yes—
237C23×C4C23×C311yescharacteristic
238C23×C4C23×C312yes—
239C23×C4C23×C311yescharacteristic
240C23×C4C23×C312yes—
241C23×C4C23×C311yescharacteristic
242C23×C4C23×C311yescharacteristic
243C23×C4C23×C312yes—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4C3×Q814yescomplemented
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42C3×D414yes—
275#95—22no—
276#95—22no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95C3×D414yes—
285#95C3×D414yes—
286C23×C3—22no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—24no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3C22×D411yescharacteristic
310C23×C3C22×D411yescharacteristic
311C23×C3C22×D411yescharacteristic
312C23×C3C22×Q811yescharacteristic
313C23×C3C2×#4111yescharacteristic
314C23×C3C2×#4112yes—
315C23×C3C2×#4112yes—
316C23×C3C2×#4112yes—
317C23×C3H(5,𝔽2)12yes—
318C23×C32−1+411yescharacteristic
319C23×C32−1+411yescharacteristic
320C3×Q8—416no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4C22⋊Q814yes—
379C2×C3×C4#12418yes—
380C2×C3×C4#12714yes—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—21no—
390C24—21no—
391C24C2×C3×Q812yes—
392C24C24×C311yescharacteristic
393C2×Q8—28no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—44no—
397C2×Q8—44no—
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—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4C2×C3×D414yes—
446C22×C4C2×C3×D412yes—
447C22×C4C2×C3×D412yes—
448C22×C4C2×C3×D414yes—
449C22×C4C2×C3×D412yes—
450C22×C4C2×C3×D412yes—
451C22×C4C2×C3×D412yes—
452C22×C4C2×C3×Q812yes—
453C22×C4C2×C3×Q812yes—
454C22×C4C3×#4114yes—
455C22×C4C3×#4112yes—
456C22×C4C3×#4114yes—
457C22×C4C3×#4114yes—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C22⋊C4—28no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C42—44no—
468C42—48no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—44no—
472C22×C3—42no—
473C22×C3—44no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—82no—
478C22×C3—82no—
479C22×C3—81no—
480C22×C3C2×C22⋊Q811yescharacteristic
481C22×C3#52511yescharacteristic
482C22×C3C2×#12412yes—
483C22×C3#52912yes—
484C22×C3C2×#12711yescharacteristic
485C22×C3#53211yescharacteristic
486C22×C3#53211yescharacteristic
487C22×C3#53212yes—
488C22×C3#53312yes—
489C22×C3#53411yescharacteristic
490C22×C3#53611yescharacteristic
491C22×C3#53912yes—
492C22×C3#54212yes—
493C22×C3#54612yes—
494C22×C3#54612yes—
495C22×C3#54711yescharacteristic
496C22×C3#54711yescharacteristic
497C22×C3D4×Q811yescharacteristic
498C22×C3D4×Q811yescharacteristic
499C22×C3#55112yes—
500C22×C3#55212yes—
501C22×C3#55412yes—
502C22×C3#55612yes—
503C3×C4—24no—
504C3×C4—28no—
505C3×C4—24no—
506C3×C4—42no—
507C3×C4—44no—
508C3×C4—44no—
509C3×C4—42no—
510C3×C4—82no—
511C3×C4—81no—
512C3×C4—81no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C22×C3×D411yescharacteristic
537C23C22×C3×D411yescharacteristic
538C23C22×C3×D411yescharacteristic
539C23C22×C3×Q811yescharacteristic
540C23C2×C3×#4111yescharacteristic
541C23C2×C3×#4112yes—
542C23C2×C3×#4112yes—
543C23C2×C3×#4112yes—
544C23C3×H(5,𝔽2)12yes—
545C23C3×2−1+411yescharacteristic
546C23C3×2−1+411yescharacteristic
547Q8—416no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4C3×C22⋊Q814yes—
606C2×C4C3×#12418yes—
607C2×C4C3×#12714yes—
608C2×C3—82no—
609C2×C3#239611yescharacteristic
610C2×C3#248812yes—
611C2×C3#249212yes—
612C2×C3#252112yes—
613C2×C3#253911yescharacteristic
614C2×C3#254211yescharacteristic
615C2×C3#255612yes—
616C2×C3#256411yescharacteristic
617C2×C3#260411yescharacteristic
618C2×C3#266311yescharacteristic
619C2×C3#285811yescharacteristic
620C22—42no—
621C22—44no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—81no—
626C22—82no—
627C22—82no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×C3×C22⋊Q811yescharacteristic
632C22C3×#52511yescharacteristic
633C22C2×C3×#12412yes—
634C22C3×#52912yes—
635C22C2×C3×#12711yescharacteristic
636C22C3×#53211yescharacteristic
637C22C3×#53212yes—
638C22C3×#53211yescharacteristic
639C22C3×#53312yes—
640C22C3×#53411yescharacteristic
641C22C3×#53611yescharacteristic
642C22C3×#53912yes—
643C22C3×#54212yes—
644C22C3×#54612yes—
645C22C3×#54612yes—
646C22C3×#54711yescharacteristic
647C22C3×#54711yescharacteristic
648C22C3×D4×Q811yescharacteristic
649C22C3×D4×Q811yescharacteristic
650C22C3×#55112yes—
651C22C3×#55212yes—
652C22C3×#55412yes—
653C22C3×#55612yes—
654C4—28no—
655C4—24no—
656C4—24no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—81no—
662C4—82no—
663C4—81no—
664C3#1506511yescharacteristic, direct product factor, Sylow
665C2—82no—
666C2C3×#239611yescharacteristic
667C2C3×#248812yes—
668C2C3×#249212yes—
669C2C3×#252112yes—
670C2C3×#253911yescharacteristic
671C2C3×#254211yescharacteristic
672C2C3×#255612yes—
673C2C3×#256411yescharacteristic
674C2C3×#260411yescharacteristic
675C2C3×#266311yescharacteristic
676C2C3×#285811yescharacteristic
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
1. C2×C3×#400C2semidirect
3. C2×C3×#395C2semidirect
5. C2×C3×#391C2semidirect
7. C3×#1445C2semidirect
10. C3×C4⋊#95C2semidirect
11. C3×C4×#95C2semidirect
12. #15065C3direct (N × H)
42. C2×#400C2×C3semidirect
44. C2×#395C2×C3semidirect
46. C2×#391C2×C3semidirect
48. #1445C2×C3semidirect
51. C4⋊#95C2×C3semidirect
52. C4×#95C2×C3semidirect
94. C2×C3×C22⋊C4Q8semidirect
269. C2×C22⋊C4C3×Q8semidirect
664. C3#15065direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4002
C2×C3×#3992
C2×C3×#3952
C2×C3×#3932
C2×C3×#3912
C2×C3×#3882
C3×#14452
C3×#14392
C3×#14372
C3×C4⋊#952
C3×C4×#952
#150653