Finite Group Databasebeta

← C3×#24400 #10643007

Subgroups — C3×#24400 #10643007

Order 768.

1322 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#137C211yesmaximal, characteristic
2C2×C3×#473C212yesmaximal, complemented
3C3×#3007C211yesmaximal, characteristic
4C2×C3×#451C211yesmaximal, characteristic, complemented
5C3×#2880C211yesmaximal, characteristic
6C3×#2052C212yesmaximal, complemented
7C3×#2020C212yesmaximal, complemented
8C3×#1908C212yesmaximal, complemented
9C3×#1901C211yesmaximal, characteristic, complemented
10C3×#1855C211yesmaximal, characteristic, complemented
11C3×#1784C211yesmaximal, characteristic, complemented
12#24400C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C2×C3×#137—21no—
16C2×C3×#137—21no—
17C2×C3×#137—28no—
18C2×C3×#137C2212yes—
19C2×C3×#137C2211yescharacteristic
20C2×C3×#137C2212yes—
21C2×C3×#137C2211yescharacteristic
22C22×C3×Q16—21no—
23C22×C3×Q16C2212yescomplemented
24C22×C3×SD16—21no—
25C2×C3×#131C2212yes—
26C22×C3×M16C2211yescharacteristic
27C2×C3×C22⋊Q8C2211yescharacteristic
28C3×#473—28no—
29C3×#452—21no—
30C3×#452—21no—
31C3×#451—21no—
32C3×#451—21no—
33C3×#451—21no—
34C3×#451—22no—
35C3×#450—21no—
36C2×C3×#108C2212yescomplemented
37C3×#417C2211yescharacteristic
38C3×#417C2211yescharacteristic
39C2×C3×#103C2212yescomplemented
40C2×C3×#101C2212yescomplemented
41C2×C3×#101C2212yescomplemented
42C2×C3×#101C2212yescomplemented
43C3×#408—22no—
44C3×#408C2212yes—
45C3×#407—21no—
46C3×#407C2211yescharacteristic
47C3×#407C2211yescharacteristic
48C2×C3×C22⋊C8—21no—
49C2×C3×C22⋊C8C2211yescharacteristic
50C3×#343C2212yescomplemented
51C3×#341C2212yescomplemented
52C22×#137C2×C311yescharacteristic
53C2×#473C2×C312yescomplemented
54#3007C2×C311yescharacteristic
55C2×#451C2×C311yescharacteristic, complemented
56#2880C2×C311yescharacteristic
57#2052C2×C312yescomplemented
58#2020C2×C312yescomplemented
59#1908C2×C312yescomplemented
60#1901C2×C311yescharacteristic, complemented
61#1855C2×C311yescharacteristic, complemented
62#1784C2×C311yescharacteristic, complemented
63C2×C3×#41—21no—
64C2×C3×#41—21no—
65C2×C3×#41—24no—
66C2×C3×#41—42no—
67C2×C3×#41D411yescharacteristic
68C2×C3×#41D411yescharacteristic
69C2×C3×#41C2311yescharacteristic
70C2×C3×#41C2311yescharacteristic
71C22×C3×Q8—21no—
72C22×C3×Q8—24no—
73C22×C3×Q8—21no—
74C22×C3×Q8—21no—
75C22×C3×Q8—21no—
76C22×C3×Q8—21no—
77C22×C3×Q8—42no—
78C22×C3×Q8D411yescharacteristic
79C22×C3×Q8C2311yescharacteristic
80C22×C3×Q8C2312yes—
81C22×C3×D4—21no—
82C22×C3×D4D411yescharacteristic
83C23×C3×C4—21no—
84C23×C3×C4—21no—
85C23×C3×C4C2311yescharacteristic
86C3×#137—28no—
87C3×#137—28no—
88C3×#137—24no—
89C3×#137—24no—
90C3×#137—24no—
91C3×#137—24no—
92C3×#137—42no—
93C3×#137D418yes—
94C2×C3×Q16—28no—
95C2×C3×Q16—22no—
96C2×C3×Q16—22no—
97C2×C3×Q16—24no—
98C2×C3×Q16—21no—
99C2×C3×Q16—21no—
100C2×C3×Q16—22no—
101C2×C3×Q16—22no—
102C2×C3×Q16—42no—
103C2×C3×SD16—28no—
104C2×C3×SD16—22no—
105C2×C3×SD16—22no—
106C2×C3×SD16—21no—
107C2×C3×SD16—21no—
108C3×#131—24no—
109C3×#131—42no—
110C2×C3×M16—21no—
111C2×C3×M16—24no—
112C2×C3×M16—22no—
113C2×C3×M16C2312yes—
114C2×C3×M16C2311yescharacteristic
115C2×C3×M16C2311yescharacteristic
116C2×C3×M16C2312yes—
117C22×C3×C8—21no—
118C22×C3×C8—22no—
119C22×C3×C8C2312yes—
120C3×C22⋊Q8—21no—
121C3×C22⋊Q8—21no—
122C3×C22⋊Q8—21no—
123C3×C22⋊Q8—21no—
124C2×C3×C4⋊C4—21no—
125C2×C3×C4⋊C4C2311yescharacteristic
126C2×C3×C22⋊C4—21no—
127C3×#108—24no—
128C3×#103—21no—
129C3×#103—21no—
130C3×#103—22no—
131C3×#103—22no—
132C3×#102—21no—
133C3×#102—21no—
134C3×#101—24no—
135C3×#101—24no—
136C3×#101—42no—
137C3×C22⋊C8—21no—
138C3×C22⋊C8—21no—
139C3×C22⋊C8—21no—
140C3×C22⋊C8—21no—
141C3×C22⋊C8—22no—
142C3×C22⋊C8—22no—
143C3×C22⋊C8D411yescharacteristic
144C3×C22⋊C8D411yescharacteristic
145C3×C22⋊C8D412yes—
146C22×#41C22×C311yescharacteristic
147C23×Q8C22×C311yescharacteristic
148C2×#137—21no—
149C2×#137—21no—
150C2×#137—28no—
151C2×#137C22×C312yes—
152C2×#137C22×C311yescharacteristic
153C2×#137C22×C312yes—
154C2×#137C22×C311yescharacteristic
155C22×Q16—21no—
156C22×Q16C22×C312yescomplemented
157C22×SD16—21no—
158C2×#131C22×C312yes—
159C22×M16C22×C311yescharacteristic
160C2×C22⋊Q8C22×C311yescharacteristic
161#473—28no—
162#452—21no—
163#452—21no—
164#451—21no—
165#451—21no—
166#451—21no—
167#451—22no—
168#450—21no—
169C2×#108C22×C312yescomplemented
170#417C22×C311yescharacteristic
171#417C22×C311yescharacteristic
172C2×#103C22×C312yescomplemented
173C2×#101C22×C312yescomplemented
174C2×#101C22×C312yescomplemented
175C2×#101C22×C312yescomplemented
176#408—22no—
177#408C22×C312yes—
178#407—21no—
179#407C22×C311yescharacteristic
180#407C22×C311yescharacteristic
181C2×C22⋊C8—21no—
182C2×C22⋊C8C22×C311yescharacteristic
183#343C22×C312yescomplemented
184#341C22×C312yescomplemented
185C24×C3—21no—
186C24×C3#4111yescharacteristic
187C3×#41—22no—
188C3×#41—22no—
189C3×#41—22no—
190C3×#41—22no—
191C3×#41—22no—
192C3×#41—44no—
193C3×#41—41no—
194C3×#41—41no—
195C3×#41—44no—
196C3×#41C2×D414yes—
197C2×C3×Q8—21no—
198C2×C3×Q8—22no—
199C2×C3×Q8—24no—
200C2×C3×Q8—24no—
201C2×C3×Q8—22no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—22no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—22no—
209C2×C3×Q8—22no—
210C2×C3×Q8—44no—
211C2×C3×Q8—41no—
212C2×C3×Q8—44no—
213C2×C3×Q8—44no—
214C2×C3×Q8—42no—
215C2×C3×Q8—42no—
216C2×C3×Q8—41no—
217C2×C3×Q8—41no—
218C2×C3×Q8C2×D414yes—
219C2×C3×Q8C2×D411yescharacteristic
220C2×C3×Q8C2×D411yescharacteristic
221C2×C3×D4—21no—
222C2×C3×D4—22no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—41no—
228C2×C3×D4—44no—
229C2×C3×D4C2×D411yescharacteristic
230C2×C3×D4C2×D411yescharacteristic
231C22×C3×C4—21no—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4—21no—
235C22×C3×C4—22no—
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—44no—
242C22×C3×C4—44no—
243C22×C3×C4—41no—
244C22×C3×C4—41no—
245C22×C3×C4C2×D411yescharacteristic
246C22×C3×C4C2×D411yescharacteristic
247C22×C3×C4C2×D411yescharacteristic
248C22×C3×C4C2×D411yescharacteristic
249C22×C3×C4#4111yescharacteristic
250C22×C3×C4C2411yescharacteristic
251C3×Q16—28no—
252C3×Q16—28no—
253C3×Q16—44no—
254C3×Q16—44no—
255C3×Q16—42no—
256C3×Q16—42no—
257C3×SD16—28no—
258C3×SD16—28no—
259C3×SD16—42no—
260C3×SD16—42no—
261C3×M16—24no—
262C3×M16—22no—
263C3×M16—22no—
264C3×M16—22no—
265C3×M16—22no—
266C3×M16—22no—
267C3×M16—44no—
268C3×M16C2×D414yes—
269C2×C3×C8—24no—
270C2×C3×C8—22no—
271C2×C3×C8—22no—
272C2×C3×C8—22no—
273C2×C3×C8—22no—
274C2×C3×C8—21no—
275C2×C3×C8—21no—
276C2×C3×C8—44no—
277C2×C3×C8C2×D412yes—
278C2×C3×C8C2×D412yes—
279C3×C4⋊C4—21no—
280C3×C4⋊C4—21no—
281C3×C4⋊C4—41no—
282C3×C4⋊C4—41no—
283C3×C22⋊C4—41no—
284C3×C22⋊C4—41no—
285C2×#41—21no—
286C2×#41—21no—
287C2×#41—24no—
288C2×#41—42no—
289C2×#41C3×D411yescharacteristic
290C2×#41C3×D411yescharacteristic
291C2×#41C23×C311yescharacteristic
292C2×#41C23×C311yescharacteristic
293C22×Q8—21no—
294C22×Q8—24no—
295C22×Q8—21no—
296C22×Q8—21no—
297C22×Q8—21no—
298C22×Q8—21no—
299C22×Q8—42no—
300C22×Q8C3×D411yescharacteristic
301C22×Q8C23×C311yescharacteristic
302C22×Q8C23×C312yes—
303C22×D4—21no—
304C22×D4C3×D411yescharacteristic
305C23×C4—21no—
306C23×C4—21no—
307C23×C4C23×C311yescharacteristic
308#137—28no—
309#137—28no—
310#137—24no—
311#137—24no—
312#137—24no—
313#137—24no—
314#137—42no—
315#137C3×D418yes—
316C2×Q16—28no—
317C2×Q16—22no—
318C2×Q16—22no—
319C2×Q16—24no—
320C2×Q16—21no—
321C2×Q16—21no—
322C2×Q16—22no—
323C2×Q16—22no—
324C2×Q16—42no—
325C2×SD16—28no—
326C2×SD16—22no—
327C2×SD16—22no—
328C2×SD16—21no—
329C2×SD16—21no—
330#131—24no—
331#131—42no—
332C2×M16—21no—
333C2×M16—24no—
334C2×M16—22no—
335C2×M16C23×C312yes—
336C2×M16C23×C311yescharacteristic
337C2×M16C23×C311yescharacteristic
338C2×M16C23×C312yes—
339C22×C8—21no—
340C22×C8—22no—
341C22×C8C23×C312yes—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4C23×C311yescharacteristic
348C2×C22⋊C4—21no—
349#108—24no—
350#103—21no—
351#103—21no—
352#103—22no—
353#103—22no—
354#102—21no—
355#102—21no—
356#101—24no—
357#101—24no—
358#101—42no—
359C22⋊C8—21no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C22⋊C8—21no—
363C22⋊C8—22no—
364C22⋊C8—22no—
365C22⋊C8C3×D411yescharacteristic
366C22⋊C8C3×D411yescharacteristic
367C22⋊C8C3×D412yes—
368C23×C3—21no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—21no—
372C23×C3—41no—
373C23×C3—42no—
374C23×C3—42no—
375C23×C3—81no—
376C23×C3#12111yescharacteristic
377C23×C3#12111yescharacteristic
378C23×C3C2×#4111yescharacteristic
379C3×Q8—21no—
380C3×Q8—24no—
381C3×Q8—21no—
382C3×Q8—21no—
383C3×Q8—24no—
384C3×Q8—42no—
385C3×Q8—44no—
386C3×Q8—44no—
387C3×Q8—42no—
388C3×Q8—42no—
389C3×Q8—41no—
390C3×Q8—41no—
391C3×Q8—42no—
392C3×Q8#12012yes—
393C3×D4—24no—
394C3×D4—21no—
395C3×D4—21no—
396C3×D4—21no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—42no—
400C3×D4—42no—
401C3×D4#12012yes—
402C2×C3×C4—21no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—24no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—24no—
413C2×C3×C4—42no—
414C2×C3×C4—42no—
415C2×C3×C4—42no—
416C2×C3×C4—44no—
417C2×C3×C4—44no—
418C2×C3×C4—42no—
419C2×C3×C4—42no—
420C2×C3×C4—42no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—42no—
424C2×C3×C4—81no—
425C2×C3×C4#12111yescharacteristic
426C2×C3×C4#12111yescharacteristic
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D412yes—
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4H(5,𝔽2)12yes—
431C3×C8—24no—
432C3×C8—22no—
433C3×C8—22no—
434C3×C8—41no—
435C3×C8—41no—
436C3×C8—42no—
437C24—21no—
438C24C3×#4111yescharacteristic
439#41—22no—
440#41—22no—
441#41—22no—
442#41—22no—
443#41—22no—
444#41—44no—
445#41—41no—
446#41—41no—
447#41—44no—
448#41C2×C3×D414yes—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—44no—
463C2×Q8—41no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—41no—
469C2×Q8—41no—
470C2×Q8C2×C3×D414yes—
471C2×Q8C2×C3×D411yescharacteristic
472C2×Q8C2×C3×D411yescharacteristic
473C2×D4—21no—
474C2×D4—22no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—41no—
480C2×D4—44no—
481C2×D4C2×C3×D411yescharacteristic
482C2×D4C2×C3×D411yescharacteristic
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4C2×C3×D411yescharacteristic
498C22×C4C2×C3×D411yescharacteristic
499C22×C4C2×C3×D411yescharacteristic
500C22×C4C2×C3×D411yescharacteristic
501C22×C4C3×#4111yescharacteristic
502C22×C4C24×C311yescharacteristic
503Q16—28no—
504Q16—28no—
505Q16—44no—
506Q16—44no—
507Q16—42no—
508Q16—42no—
509SD16—28no—
510SD16—28no—
511SD16—42no—
512SD16—42no—
513M16—24no—
514M16—22no—
515M16—22no—
516M16—22no—
517M16—22no—
518M16—22no—
519M16—44no—
520M16C2×C3×D414yes—
521C2×C8—24no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—44no—
529C2×C8C2×C3×D412yes—
530C2×C8C2×C3×D412yes—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—22no—
543C22×C3—21no—
544C22×C3—21no—
545C22×C3—44no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3—84no—
552C22×C3#47312yescomplemented
553C22×C3C2×#12111yescharacteristic
554C22×C3#54612yes—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—22no—
558C3×C4—41no—
559C3×C4—44no—
560C3×C4—81no—
561C3×C4—81no—
562C3×C4C2×#12011yescharacteristic
563C3×C4#53411yescharacteristic
564C3×C4D4212yes—
565C23—21no—
566C23—22no—
567C23—22no—
568C23—21no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—81no—
573C23C3×#12111yescharacteristic
574C23C3×#12111yescharacteristic
575C23C2×C3×#4111yescharacteristic
576Q8—21no—
577Q8—24no—
578Q8—21no—
579Q8—21no—
580Q8—24no—
581Q8—42no—
582Q8—44no—
583Q8—44no—
584Q8—42no—
585Q8—42no—
586Q8—41no—
587Q8—41no—
588Q8—42no—
589Q8C3×#12012yes—
590D4—24no—
591D4—21no—
592D4—21no—
593D4—21no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4C3×#12012yes—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—24no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—81no—
622C2×C4C3×#12111yescharacteristic
623C2×C4C3×#12111yescharacteristic
624C2×C4C22×C3×D411yescharacteristic
625C2×C4C22×C3×D412yes—
626C2×C4C22×C3×D411yescharacteristic
627C2×C4C3×H(5,𝔽2)12yes—
628C8—24no—
629C8—22no—
630C8—22no—
631C8—41no—
632C8—41no—
633C8—42no—
634C2×C3—22no—
635C2×C3—22no—
636C2×C3—41no—
637C2×C3—42no—
638C2×C3—81no—
639C2×C3#241011yescharacteristic
640C2×C3C2×#47311yescharacteristic
641C2×C3#306911yescharacteristic
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—22no—
648C22—21no—
649C22—21no—
650C22—44no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—84no—
657C22C3×#47312yescomplemented
658C22C2×C3×#12111yescharacteristic
659C22C3×#54612yes—
660C4—21no—
661C4—21no—
662C4—22no—
663C4—41no—
664C4—44no—
665C4—81no—
666C4—81no—
667C4C2×C3×#12011yescharacteristic
668C4C3×#53411yescharacteristic
669C4C3×D4212yes—
670C3#2440011yescharacteristic, direct product factor, Sylow
671C2—22no—
672C2—22no—
673C2—41no—
674C2—42no—
675C2—81no—
676C2C3×#241011yescharacteristic
677C2C2×C3×#47311yescharacteristic
678C2C3×#306911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#473C2semidirect
4. C2×C3×#451C2semidirect
6. C3×#2052C2semidirect
7. C3×#2020C2semidirect
8. C3×#1908C2semidirect
9. C3×#1901C2semidirect
10. C3×#1855C2semidirect
11. C3×#1784C2semidirect
12. #24400C3direct (N × H)
23. C22×C3×Q16C22semidirect
36. C2×C3×#108C22semidirect
39. C2×C3×#103C22semidirect
40. C2×C3×#101C22semidirect
41. C2×C3×#101C22semidirect
42. C2×C3×#101C22semidirect
50. C3×#343C22semidirect
51. C3×#341C22semidirect
53. C2×#473C2×C3semidirect
55. C2×#451C2×C3semidirect
57. #2052C2×C3semidirect
58. #2020C2×C3semidirect
59. #1908C2×C3semidirect
60. #1901C2×C3semidirect
61. #1855C2×C3semidirect
62. #1784C2×C3semidirect
156. C22×Q16C22×C3semidirect
169. C2×#108C22×C3semidirect
172. C2×#103C22×C3semidirect
173. C2×#101C22×C3semidirect
174. C2×#101C22×C3semidirect
175. C2×#101C22×C3semidirect
183. #343C22×C3semidirect
184. #341C22×C3semidirect
552. C22×C3#473semidirect
657. C22C3×#473semidirect
670. C3#24400direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1372
C2×C3×#4732
C3×#30072
C2×C3×#4512
C3×#28802
C3×#20522
C3×#20202
C3×#19082
C3×#19012
C3×#18552
C3×#17842
#244003