Finite Group Databasebeta

← C3×#24241 #10642848

Subgroups — C3×#24241 #10642848

Order 768.

1438 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C2×C3×#452C212yesmaximal, complemented
4C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
5C3×#2538C211yesmaximal, characteristic, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1954C211yesmaximal, characteristic, complemented
10C3×#1802C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal, complemented
12#24241C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2211yescharacteristic
16C3×#547—24no—
17C3×#547—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#123—22no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×#121—21no—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C22×C3×C4⋊C4—21no—
27C3×#462—24no—
28C3×#452—28no—
29C3×C4×SD16—24no—
30C2×C3×#106C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2211yescharacteristic
37C2×C3×#102C2212yes—
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×#547C2×C311yescharacteristic
48C2×#462C2×C311yescharacteristic, complemented
49C2×#452C2×C312yescomplemented
50C2×C4×SD16C2×C311yescharacteristic, complemented
51#2538C2×C311yescharacteristic, complemented
52#2035C2×C312yescomplemented
53#2001C2×C312yescomplemented
54#1956C2×C311yescharacteristic, complemented
55#1954C2×C311yescharacteristic, complemented
56#1802C2×C311yescharacteristic, complemented
57#1788C2×C312yescomplemented
58C2×C3×#41—24no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41—42no—
63C2×C3×#41—42no—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2312yes—
66C22×C3×D4—22no—
67C22×C3×D4D411yescharacteristic
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—22no—
70C23×C3×C4—21no—
71C23×C3×C4C2312yes—
72C2×C3×Q16—24no—
73C2×C3×Q16—24no—
74C2×C3×Q16—42no—
75C2×C3×SD16—22no—
76C2×C3×SD16—22no—
77C2×C3×SD16D414yescomplemented
78C22×C3×C8C2312yes—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8D414yes—
83C3×#123—216no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—42no—
88C3×#121—28no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4D414yes—
95C2×C3×C4⋊C4—28no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—21no—
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—21no—
109C2×C3×C22⋊C4—22no—
110C2×C3×C22⋊C4—22no—
111C2×C3×C22⋊C4D412yes—
112C2×C3×C42—21no—
113C2×C3×C42C2311yescharacteristic
114C3×#106—22no—
115C3×C4⋊C8—22no—
116C3×#103—24no—
117C3×#103—24no—
118C3×#103—22no—
119C3×#103—42no—
120C3×#102—22no—
121C3×#102—24no—
122C3×C22⋊C8—24no—
123C3×C22⋊C8—42no—
124C3×C4×C8—22no—
125C3×#95—22no—
126C22×#41C22×C312yes—
127C22×Q16C22×C312yes—
128C22×SD16C22×C311yescharacteristic
129#547—24no—
130#547—24no—
131C2×C4⋊Q8C22×C311yescharacteristic
132C2×#123—22no—
133C2×C22⋊Q8C22×C312yes—
134C2×C22⋊Q8C22×C312yes—
135C2×#121—21no—
136C2×C4×Q8C22×C311yescharacteristic
137C2×C4×D4C22×C311yescharacteristic
138C2×C4×D4C22×C311yescharacteristic
139C22×C4⋊C4—21no—
140#462—24no—
141#452—28no—
142C4×SD16—24no—
143C2×#106C22×C311yescharacteristic
144C2×C4⋊C8C22×C311yescharacteristic
145C2×#103C22×C311yescharacteristic
146C2×#103C22×C312yes—
147C2×#103C22×C312yes—
148C2×#103C22×C312yes—
149C2×#102C22×C312yes—
150C2×#102C22×C311yescharacteristic
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×C3—21no—
161C24×C3C2×D412yes—
162C3×#41—48no—
163C3×#41—48no—
164C3×#41—48no—
165C3×#41—48no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8C2×D414yes—
177C2×C3×Q8C2×D412yes—
178C2×C3×D4—28no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4C2×D412yes—
188C2×C3×D4C2×D412yes—
189C2×C3×D4#4112yes—
190C22×C3×C4—28no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—24no—
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—21no—
202C22×C3×C4—22no—
203C22×C3×C4—44no—
204C22×C3×C4—44no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C3×Q16—48no—
212C3×SD16—28no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—21no—
219C2×C3×C8—22no—
220C2×C3×C8—21no—
221C2×C3×C8C2×D412yes—
222C3×C4⋊C4—28no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—44no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—81no—
235C3×C4⋊C4C2×D412yes—
236C3×C22⋊C4—28no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—44no—
240C3×C22⋊C4—82no—
241C3×C42—42no—
242C3×C42C2×D412yes—
243C3×C42C2×D412yes—
244C2×#41—24no—
245C2×#41—24no—
246C2×#41—24no—
247C2×#41—24no—
248C2×#41—42no—
249C2×#41—42no—
250C22×Q8C23×C311yescharacteristic
251C22×Q8C23×C312yes—
252C22×D4—22no—
253C22×D4C3×D411yescharacteristic
254C22×D4C23×C311yescharacteristic
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C23×C312yes—
258C2×Q16—24no—
259C2×Q16—24no—
260C2×Q16—42no—
261C2×SD16—22no—
262C2×SD16—22no—
263C2×SD16C3×D414yescomplemented
264C22×C8C23×C312yes—
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C4⋊Q8—22no—
268C4⋊Q8C3×D414yes—
269#123—216no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—42no—
274#121—28no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4C3×D414yes—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C3×D411yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C312yes—
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4C3×D412yes—
298C2×C42—21no—
299C2×C42C23×C311yescharacteristic
300#106—22no—
301C4⋊C8—22no—
302#103—24no—
303#103—24no—
304#103—22no—
305#103—42no—
306#102—22no—
307#102—24no—
308C22⋊C8—24no—
309C22⋊C8—42no—
310C4×C8—22no—
311#95—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3C22×D411yescharacteristic
327C3×Q8—22no—
328C3×Q8—22no—
329C3×Q8—24no—
330C3×Q8—24no—
331C3×Q8—42no—
332C3×Q8—41no—
333C3×Q8—42no—
334C3×Q8—41no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—416no—
339C3×D4—44no—
340C3×D4#12114yes—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—416no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—48no—
361C2×C3×C4—41no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4#12012yes—
368C2×C3×C4#12012yes—
369C2×C3×C4#13512yes—
370C2×C3×C4#13712yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C3×C8—22no—
378C3×C8—41no—
379C3×C8—42no—
380C24—21no—
381C24C2×C3×D412yes—
382#41—48no—
383#41—48no—
384#41—48no—
385#41—48no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8C2×C3×D414yes—
397C2×Q8C2×C3×D412yes—
398C2×D4—28no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4C2×C3×D412yes—
408C2×D4C2×C3×D412yes—
409C2×D4C3×#4112yes—
410C22×C4—28no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4C2×C3×D412yes—
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C24×C311yescharacteristic
431Q16—48no—
432SD16—28no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—21no—
441C2×C8C2×C3×D412yes—
442C4⋊C4—28no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4C2×C3×D412yes—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—44no—
460C22⋊C4—82no—
461C42—42no—
462C42C2×C3×D412yes—
463C42C2×C3×D412yes—
464C22×C3—24no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3C2×#12011yescharacteristic
479C22×C3D4211yescharacteristic
480C22×C3#54611yescharacteristic
481C22×C3C2×#13511yescharacteristic
482C22×C3C2×#13711yescharacteristic
483C22×C3#57711yescharacteristic
484C22×C3#57811yescharacteristic
485C3×C4—22no—
486C3×C4—42no—
487C3×C4—44no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—81no—
491C3×C4—82no—
492C3×C4#45214yescomplemented
493C3×C4C2×#12111yescharacteristic
494C3×C4#53511yescharacteristic
495C3×C4D4211yescharacteristic
496C3×C4#54611yescharacteristic
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C22×C3×D411yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—24no—
515Q8—24no—
516Q8—42no—
517Q8—42no—
518Q8—41no—
519Q8—41no—
520D4—22no—
521D4—22no—
522D4—22no—
523D4—416no—
524D4—44no—
525D4C3×#12114yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—416no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4C3×#12012yes—
553C2×C4C3×#12012yes—
554C2×C4C3×#13512yes—
555C2×C4C3×#13712yes—
556C2×C4C22×C3×D411yescharacteristic
557C2×C4C22×C3×D411yescharacteristic
558C2×C4C22×C3×D411yescharacteristic
559C2×C4C2×C3×#4111yescharacteristic
560C2×C4C3×H(5,𝔽2)11yescharacteristic
561C2×C4C3×H(5,𝔽2)11yescharacteristic
562C8—22no—
563C8—41no—
564C8—42no—
565C2×C3—44no—
566C2×C3—82no—
567C2×C3#240811yescharacteristic
568C2×C3C2×#45211yescharacteristic
569C2×C3#301311yescharacteristic
570C2×C3#327611yescharacteristic
571C2×C3#328411yescharacteristic
572C2×C3#330211yescharacteristic
573C2×C3#333111yescharacteristic
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22C2×C3×#12011yescharacteristic
589C22C3×D4211yescharacteristic
590C22C3×#54611yescharacteristic
591C22C2×C3×#13511yescharacteristic
592C22C2×C3×#13711yescharacteristic
593C22C3×#57711yescharacteristic
594C22C3×#57811yescharacteristic
595C4—22no—
596C4—42no—
597C4—44no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—82no—
602C4C3×#45214yescomplemented
603C4C2×C3×#12111yescharacteristic
604C4C3×#53511yescharacteristic
605C4C3×D4211yescharacteristic
606C4C3×#54611yescharacteristic
607C3#2424111yescharacteristic, direct product factor, Sylow
608C2—44no—
609C2—82no—
610C2C3×#240811yescharacteristic
611C2C2×C3×#45211yescharacteristic
612C2C3×#301311yescharacteristic
613C2C3×#327611yescharacteristic
614C2C3×#328411yescharacteristic
615C2C3×#330211yescharacteristic
616C2C3×#333111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#547C2
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×#452C2semidirect
4. C2×C3×C4×SD16C2semidirect
5. C3×#2538C2semidirect
6. C3×#2035C2semidirect
7. C3×#2001C2semidirect
8. C3×#1956C2semidirect
9. C3×#1954C2semidirect
10. C3×#1802C2semidirect
11. C3×#1788C2semidirect
12. #24241C3direct (N × H)
48. C2×#462C2×C3semidirect
49. C2×#452C2×C3semidirect
50. C2×C4×SD16C2×C3semidirect
51. #2538C2×C3semidirect
52. #2035C2×C3semidirect
53. #2001C2×C3semidirect
54. #1956C2×C3semidirect
55. #1954C2×C3semidirect
56. #1802C2×C3semidirect
57. #1788C2×C3semidirect
77. C2×C3×SD16D4semidirect
263. C2×SD16C3×D4semidirect
492. C3×C4#452semidirect
602. C4C3×#452semidirect
607. C3#24241direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#4622
C2×C3×#4522
C2×C3×C4×SD162
C3×#25382
C3×#20352
C3×#20012
C3×#19562
C3×#19542
C3×#18022
C3×#17882
#242413