Finite Group Databasebeta

← C3×#24243 #10642850

Subgroups — C3×#24243 #10642850

Order 768.

1406 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C2×C3×#451C212yesmaximal, complemented
4C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
5C3×#2538C211yesmaximal, characteristic
6C3×#2036C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1954C211yesmaximal, characteristic
10C3×#1803C211yesmaximal, characteristic, complemented
11C3×#1787C212yesmaximal, complemented
12#24243C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2212yes—
14C22×C3×Q16C2212yes—
15C22×C3×Q16C2211yescharacteristic
16C3×D4×Q8—24no—
17C3×D4×Q8—24no—
18C2×C3×C4⋊Q8—21no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8—22no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C3×#462—24no—
28C3×#451—28no—
29C3×C4×Q16—24no—
30C2×C3×#107C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2211yescharacteristic
36C2×C3×#103C2212yes—
37C2×C3×#103C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C2×C3×C22⋊C8C2212yes—
40C2×C3×C4×C8C2211yescharacteristic
41C3×#396—22no—
42C3×#390—21no—
43C3×#389—21no—
44C3×#384C2211yescharacteristic
45C3×#380C2212yes—
46C3×#340C2212yes—
47C2×D4×Q8C2×C311yescharacteristic
48C2×#462C2×C311yescharacteristic, complemented
49C2×#451C2×C312yescomplemented
50C2×C4×Q16C2×C311yescharacteristic, complemented
51#2538C2×C311yescharacteristic
52#2036C2×C312yescomplemented
53#2001C2×C312yescomplemented
54#1956C2×C311yescharacteristic, complemented
55#1954C2×C311yescharacteristic
56#1803C2×C311yescharacteristic, complemented
57#1787C2×C312yescomplemented
58C22×C3×Q8—24no—
59C22×C3×Q8—24no—
60C22×C3×Q8—22no—
61C22×C3×Q8—24no—
62C22×C3×Q8—24no—
63C22×C3×Q8—42no—
64C22×C3×Q8—42no—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×Q8C2312yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4D411yescharacteristic
69C23×C3×C4—22no—
70C23×C3×C4C2312yes—
71C2×C3×Q16—22no—
72C2×C3×Q16—22no—
73C2×C3×Q16—24no—
74C2×C3×Q16—24no—
75C2×C3×Q16—42no—
76C2×C3×Q16D414yescomplemented
77C22×C3×C8C2312yes—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C4⋊Q8—28no—
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8D414yes—
83C3×C22⋊Q8—216no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—42no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8D414yes—
92C3×C4×D4—28no—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—22no—
110C2×C3×C22⋊C4D412yes—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42C2311yescharacteristic
114C3×#107—22no—
115C3×C4⋊C8—22no—
116C3×#103—22no—
117C3×#103—24no—
118C3×#103—24no—
119C3×#103—24no—
120C3×#103—22no—
121C3×#103—42no—
122C3×C22⋊C8—24no—
123C3×C22⋊C8—42no—
124C3×C4×C8—22no—
125C3×#95—22no—
126C23×Q8C22×C312yes—
127C22×Q16C22×C312yes—
128C22×Q16C22×C311yescharacteristic
129D4×Q8—24no—
130D4×Q8—24no—
131C2×C4⋊Q8—21no—
132C2×C4⋊Q8C22×C311yescharacteristic
133C2×C22⋊Q8—22no—
134C2×C22⋊Q8C22×C312yes—
135C2×C22⋊Q8C22×C312yes—
136C2×C4×Q8C22×C311yescharacteristic
137C2×C4×Q8C22×C311yescharacteristic
138C2×C4×D4—21no—
139C2×C4×D4C22×C311yescharacteristic
140#462—24no—
141#451—28no—
142C4×Q16—24no—
143C2×#107C22×C311yescharacteristic
144C2×C4⋊C8C22×C311yescharacteristic
145C2×#103C22×C312yes—
146C2×#103C22×C311yescharacteristic
147C2×#103C22×C311yescharacteristic
148C2×#103C22×C312yes—
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×C22⋊C8C22×C312yes—
152C2×C22⋊C8C22×C312yes—
153C2×C4×C8C22×C311yescharacteristic
154#396—22no—
155#390—21no—
156#389—21no—
157#384C22×C311yescharacteristic
158#380C22×C312yes—
159#340C22×C312yes—
160C24×C3C2×D412yes—
161C2×C3×Q8—28no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—44no—
176C2×C3×Q8—44no—
177C2×C3×Q8—48no—
178C2×C3×Q8—48no—
179C2×C3×Q8—48no—
180C2×C3×Q8—48no—
181C2×C3×Q8C2×D414yes—
182C2×C3×Q8C2×D412yes—
183C2×C3×Q8C2×D412yes—
184C2×C3×Q8C2×D412yes—
185C2×C3×Q8#4112yes—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4Q1614yescomplemented
189C22×C3×C4—21no—
190C22×C3×C4—28no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
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—21no—
201C22×C3×C4—22no—
202C22×C3×C4—44no—
203C22×C3×C4—44no—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C3×Q16—28no—
209C3×Q16—48no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—21no—
216C2×C3×C8—22no—
217C2×C3×C8—21no—
218C2×C3×C8C2×D412yes—
219C3×C4⋊C4—28no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—44no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—81no—
233C3×C4⋊C4C2×D412yes—
234C3×C22⋊C4—28no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—44no—
237C3×C22⋊C4—82no—
238C3×C42—24no—
239C3×C42—42no—
240C3×C42C2×D412yes—
241C3×C42C2×D412yes—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—22no—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—42no—
248C22×Q8—42no—
249C22×Q8C23×C311yescharacteristic
250C22×Q8C23×C312yes—
251C22×Q8C23×C311yescharacteristic
252C22×D4C3×D411yescharacteristic
253C23×C4—22no—
254C23×C4C23×C312yes—
255C2×Q16—22no—
256C2×Q16—22no—
257C2×Q16—24no—
258C2×Q16—24no—
259C2×Q16—42no—
260C2×Q16C3×D414yescomplemented
261C22×C8C23×C312yes—
262C22×C8C23×C311yescharacteristic
263C22×C8C23×C311yescharacteristic
264C4⋊Q8—28no—
265C4⋊Q8—22no—
266C4⋊Q8C3×D414yes—
267C22⋊Q8—216no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—42no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8C3×D414yes—
276C4×D4—28no—
277C4×D4—22no—
278C4×D4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C3×D411yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C312yes—
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4C3×D412yes—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42C23×C311yescharacteristic
298#107—22no—
299C4⋊C8—22no—
300#103—22no—
301#103—24no—
302#103—24no—
303#103—24no—
304#103—22no—
305#103—42no—
306C22⋊C8—24no—
307C22⋊C8—42no—
308C4×C8—22no—
309#95—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3C2×Q1614yes—
317C23×C3C22×D411yescharacteristic
318C3×Q8—22no—
319C3×Q8—22no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—24no—
324C3×Q8—24no—
325C3×Q8—416no—
326C3×Q8—42no—
327C3×Q8—41no—
328C3×Q8—42no—
329C3×Q8—41no—
330C3×Q8#12114yes—
331C3×D4—28no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—416no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4#12012yes—
361C2×C3×C4#12012yes—
362C2×C3×C4C2×Q1612yes—
363C2×C3×C4#13712yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C3×C8—22no—
371C3×C8—41no—
372C3×C8—42no—
373C24C2×C3×D412yes—
374C2×Q8—28no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8C2×C3×D414yes—
395C2×Q8C2×C3×D412yes—
396C2×Q8C2×C3×D412yes—
397C2×Q8C2×C3×D412yes—
398C2×Q8C3×#4112yes—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4C3×Q1614yescomplemented
402C22×C4—21no—
403C22×C4—28no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4C2×C3×D412yes—
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C24×C311yescharacteristic
421Q16—28no—
422Q16—48no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—21no—
431C2×C8C2×C3×D412yes—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—81no—
446C4⋊C4C2×C3×D412yes—
447C22⋊C4—28no—
448C22⋊C4—24no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C42—24no—
452C42—42no—
453C42C2×C3×D412yes—
454C42C2×C3×D412yes—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—24no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3C2×#12011yescharacteristic
463C22×C3D4211yescharacteristic
464C22×C3#54611yescharacteristic
465C22×C3C22×Q1611yescharacteristic
466C22×C3C2×#13711yescharacteristic
467C22×C3#57711yescharacteristic
468C22×C3#57811yescharacteristic
469C3×C4—24no—
470C3×C4—22no—
471C3×C4—42no—
472C3×C4—44no—
473C3×C4—42no—
474C3×C4—42no—
475C3×C4—81no—
476C3×C4—82no—
477C3×C4#45114yescomplemented
478C3×C4C2×#12111yescharacteristic
479C3×C4#53511yescharacteristic
480C3×C4D4211yescharacteristic
481C3×C4#54611yescharacteristic
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23C2×C3×Q1614yes—
489C23C22×C3×D411yescharacteristic
490Q8—22no—
491Q8—22no—
492Q8—22no—
493Q8—22no—
494Q8—22no—
495Q8—24no—
496Q8—24no—
497Q8—416no—
498Q8—42no—
499Q8—42no—
500Q8—41no—
501Q8—41no—
502Q8C3×#12114yes—
503D4—28no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—416no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4C3×#12012yes—
533C2×C4C3×#12012yes—
534C2×C4C2×C3×Q1612yes—
535C2×C4C3×#13712yes—
536C2×C4C22×C3×D411yescharacteristic
537C2×C4C22×C3×D411yescharacteristic
538C2×C4C22×C3×D411yescharacteristic
539C2×C4C2×C3×#4111yescharacteristic
540C2×C4C3×H(5,𝔽2)11yescharacteristic
541C2×C4C3×H(5,𝔽2)11yescharacteristic
542C8—22no—
543C8—41no—
544C8—42no—
545C2×C3—44no—
546C2×C3#240811yescharacteristic
547C2×C3C2×#45111yescharacteristic
548C2×C3#301311yescharacteristic
549C2×C3#327811yescharacteristic
550C2×C3D4×Q1611yescharacteristic
551C2×C3#329911yescharacteristic
552C2×C3#332811yescharacteristic
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22C2×C3×#12011yescharacteristic
561C22C3×D4211yescharacteristic
562C22C3×#54611yescharacteristic
563C22C22×C3×Q1611yescharacteristic
564C22C2×C3×#13711yescharacteristic
565C22C3×#57711yescharacteristic
566C22C3×#57811yescharacteristic
567C4—24no—
568C4—22no—
569C4—42no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—81no—
574C4—82no—
575C4C3×#45114yescomplemented
576C4C2×C3×#12111yescharacteristic
577C4C3×#53511yescharacteristic
578C4C3×D4211yescharacteristic
579C4C3×#54611yescharacteristic
580C3#2424311yescharacteristic, direct product factor, Sylow
581C2—44no—
582C2C3×#240811yescharacteristic
583C2C2×C3×#45111yescharacteristic
584C2C3×#301311yescharacteristic
585C2C3×#327811yescharacteristic
586C2C3×D4×Q1611yescharacteristic
587C2C3×#329911yescharacteristic
588C2C3×#332811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
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
2. C2×C3×#462C2semidirect
3. C2×C3×#451C2semidirect
4. C2×C3×C4×Q16C2semidirect
6. C3×#2036C2semidirect
7. C3×#2001C2semidirect
8. C3×#1956C2semidirect
10. C3×#1803C2semidirect
11. C3×#1787C2semidirect
12. #24243C3direct (N × H)
48. C2×#462C2×C3semidirect
49. C2×#451C2×C3semidirect
50. C2×C4×Q16C2×C3semidirect
52. #2036C2×C3semidirect
53. #2001C2×C3semidirect
54. #1956C2×C3semidirect
56. #1803C2×C3semidirect
57. #1787C2×C3semidirect
76. C2×C3×Q16D4semidirect
188. C2×C3×D4Q16semidirect
260. C2×Q16C3×D4semidirect
401. C2×D4C3×Q16semidirect
477. C3×C4#451semidirect
575. C4C3×#451semidirect
580. C3#24243direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×C3×#4622
C2×C3×#4512
C2×C3×C4×Q162
C3×#25382
C3×#20362
C3×#20012
C3×#19562
C3×#19542
C3×#18032
C3×#17872
#242433