Finite Group Databasebeta

← C3×#24146 #10642753

Subgroups — C3×#24146 #10642753

Order 768.

1142 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3508C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic, complemented
3C3×C4×#137C211yesmaximal, characteristic, complemented
4C3×#2548C211yesmaximal, characteristic, complemented
5C3×#2037C212yesmaximal, complemented
6C3×#2004C212yesmaximal, complemented
7C3×#1986C211yesmaximal, characteristic, complemented
8C3×#1965C211yesmaximal, characteristic, complemented
9C3×#1885C212yesmaximal, complemented
10C3×#1856C212yesmaximal, complemented
11C3×#1806C211yesmaximal, characteristic, complemented
12#24146C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2211yescharacteristic
14C3×#559—21no—
15C3×#559—21no—
16C3×#550—21no—
17C3×#550—21no—
18C3×#546—22no—
19C3×#546—22no—
20C3×#540—22no—
21C3×#535C2212yes—
22C3×#533C2211yescharacteristic
23C3×#532—21no—
24C3×#532—21no—
25C2×C3×#127C2211yescharacteristic
26C2×C3×#124C2212yes—
27C3×#518C2212yes—
28C3×C4×#41C2211yescharacteristic
29C2×C3×C4×Q8C2211yescharacteristic
30C3×#479—22no—
31C3×#477—21no—
32C3×#475—21no—
33C3×#441—21no—
34C3×#440—21no—
35C3×C4×Q16—21no—
36C3×C4×SD16—21no—
37C3×#428C2212yes—
38C3×#428C2211yescharacteristic
39C3×#423C2211yescharacteristic
40C2×C3×#104C2212yes—
41C3×#419C2212yes—
42C3×#419C2211yescharacteristic
43C3×#417C2211yescharacteristic
44C2×C3×#100C2212yes—
45C2×C3×#100C2212yes—
46C3×C4×M16C2211yescharacteristic
47C3×#394—22no—
48C3×#390—22no—
49C3×#390C2212yes—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×#339C2212yes—
52C3×#339C2212yes—
53C3×#339C2212yes—
54#3508C2×C311yescharacteristic
55#3078C2×C311yescharacteristic, complemented
56C4×#137C2×C311yescharacteristic, complemented
57#2548C2×C311yescharacteristic, complemented
58#2037C2×C312yescomplemented
59#2004C2×C312yescomplemented
60#1986C2×C311yescharacteristic, complemented
61#1965C2×C311yescharacteristic, complemented
62#1885C2×C312yescomplemented
63#1856C2×C312yescomplemented
64#1806C2×C311yescharacteristic, complemented
65C2×C3×#41—22no—
66C2×C3×#41—22no—
67C2×C3×#41C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4D412yes—
70C22×C3×D4D412yes—
71C22×C3×D4C2312yes—
72C3×#137—22no—
73C3×#137—22no—
74C2×C3×Q16—21no—
75C2×C3×SD16—21no—
76C2×C3×M16C2311yescharacteristic
77C2×C3×M16C2312yes—
78C2×C3×M16C2311yescharacteristic
79C3×C4⋊Q8D411yescharacteristic
80C3×C4⋊Q8D411yescharacteristic
81C3×#127—22no—
82C3×#127—21no—
83C3×#127—21no—
84C3×#127—21no—
85C3×#127—21no—
86C3×#125D412yes—
87C3×#124—22no—
88C3×#124—22no—
89C3×#124—22no—
90C3×#124—24no—
91C3×#124—42no—
92C3×#123—24no—
93C3×#123—24no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—21no—
96C3×#121—22no—
97C3×#121—22no—
98C3×#121—24no—
99C3×#121—24no—
100C3×#121—22no—
101C3×#121—22no—
102C3×#120—24no—
103C3×C4×Q8—21no—
104C3×C4×Q8—21no—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8—21no—
109C3×C4×Q8D411yescharacteristic
110C3×C4×Q8D411yescharacteristic
111C3×C4×D4—22no—
112C3×C4×D4—22no—
113C3×C4×D4—21no—
114C3×C4×D4—21no—
115C3×C4×D4—22no—
116C3×C4×D4—22no—
117C3×C4×D4—22no—
118C3×C4×D4—22no—
119C3×C4×D4D411yescharacteristic
120C3×C4×D4D411yescharacteristic
121C3×#117—21no—
122C3×#117C2311yescharacteristic
123C3×#117C2312yes—
124C2×C3×C4⋊C4—21no—
125C2×C3×C4⋊C4—21no—
126C2×C3×C4⋊C4C2311yescharacteristic
127C2×C3×C22⋊C4—24no—
128C2×C3×C22⋊C4—24no—
129C2×C3×C22⋊C4—24no—
130C2×C3×C42—21no—
131C2×C3×C42—21no—
132C2×C3×C42C2312yes—
133C2×C3×C42C2311yescharacteristic
134C3×#107—21no—
135C3×#107—22no—
136C3×#106—21no—
137C3×#106—22no—
138C3×C4⋊C8—21no—
139C3×C4⋊C8—21no—
140C3×#104—42no—
141C3×#103—21no—
142C3×#103—21no—
143C3×#103—21no—
144C3×#103—22no—
145C3×#103—22no—
146C3×#102—21no—
147C3×#100—24no—
148C3×#100—42no—
149C3×#97—21no—
150C3×C4×C8—21no—
151C3×#95—21no—
152C2×#137C22×C311yescharacteristic
153#559—21no—
154#559—21no—
155#550—21no—
156#550—21no—
157#546—22no—
158#546—22no—
159#540—22no—
160#535C22×C312yes—
161#533C22×C311yescharacteristic
162#532—21no—
163#532—21no—
164C2×#127C22×C311yescharacteristic
165C2×#124C22×C312yes—
166#518C22×C312yes—
167C4×#41C22×C311yescharacteristic
168C2×C4×Q8C22×C311yescharacteristic
169#479—22no—
170#477—21no—
171#475—21no—
172#441—21no—
173#440—21no—
174C4×Q16—21no—
175C4×SD16—21no—
176#428C22×C312yes—
177#428C22×C311yescharacteristic
178#423C22×C311yescharacteristic
179C2×#104C22×C312yes—
180#419C22×C312yes—
181#419C22×C311yescharacteristic
182#417C22×C311yescharacteristic
183C2×#100C22×C312yes—
184C2×#100C22×C312yes—
185C4×M16C22×C311yescharacteristic
186#394—22no—
187#390—22no—
188#390C22×C312yes—
189C4×C4⋊C4C22×C311yescharacteristic
190#339C22×C312yes—
191#339C22×C312yes—
192#339C22×C312yes—
193C24×C3C2×D414yes—
194C3×#41—21no—
195C3×#41—21no—
196C3×#41—21no—
197C3×#41—21no—
198C3×#41—84no—
199C3×#41—84no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8C2×D411yescharacteristic
208C2×C3×Q8#4112yescomplemented
209C2×C3×D4—21no—
210C2×C3×D4—24no—
211C2×C3×D4—24no—
212C2×C3×D4—22no—
213C2×C3×D4—22no—
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4—22no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—44no—
223C2×C3×D4—82no—
224C2×C3×D4C2×D411yescharacteristic
225C22×C3×C4—21no—
226C22×C3×C4—24no—
227C22×C3×C4—24no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—22no—
232C22×C3×C4—21no—
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C3×Q16—24no—
237C3×SD16—24no—
238C3×M16—21no—
239C3×M16—22no—
240C3×M16—21no—
241C3×M16—22no—
242C3×M16—21no—
243C3×M16#4112yescomplemented
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—22no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—42no—
256C3×C4⋊C4—41no—
257C3×C4⋊C4—41no—
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D412yes—
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C4⋊C4C2×D412yes—
262C3×C22⋊C4—24no—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—48no—
271C3×C22⋊C4—84no—
272C3×C42—21no—
273C3×C42—21no—
274C3×C42—21no—
275C3×C42—22no—
276C3×C42—22no—
277C3×C42—22no—
278C3×C42—41no—
279C3×C42—41no—
280C3×C42C2×D411yescharacteristic
281C3×C42C2×D411yescharacteristic
282C3×C42C2×D411yescharacteristic
283C3×C42C2×D411yescharacteristic
284C2×#41—22no—
285C2×#41—22no—
286C2×#41C23×C311yescharacteristic
287C22×Q8C23×C311yescharacteristic
288C22×D4C3×D412yes—
289C22×D4C3×D412yes—
290C22×D4C23×C312yes—
291#137—22no—
292#137—22no—
293C2×Q16—21no—
294C2×SD16—21no—
295C2×M16C23×C311yescharacteristic
296C2×M16C23×C312yes—
297C2×M16C23×C311yescharacteristic
298C4⋊Q8C3×D411yescharacteristic
299C4⋊Q8C3×D411yescharacteristic
300#127—22no—
301#127—21no—
302#127—21no—
303#127—21no—
304#127—21no—
305#125C3×D412yes—
306#124—22no—
307#124—22no—
308#124—22no—
309#124—24no—
310#124—42no—
311#123—24no—
312#123—24no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315#121—22no—
316#121—22no—
317#121—24no—
318#121—24no—
319#121—22no—
320#121—22no—
321#120—24no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8C3×D411yescharacteristic
329C4×Q8C3×D411yescharacteristic
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4C3×D411yescharacteristic
339C4×D4C3×D411yescharacteristic
340#117—21no—
341#117C23×C311yescharacteristic
342#117C23×C312yes—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C23×C311yescharacteristic
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42C23×C312yes—
352C2×C42C23×C311yescharacteristic
353#107—21no—
354#107—22no—
355#106—21no—
356#106—22no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359#104—42no—
360#103—21no—
361#103—21no—
362#103—21no—
363#103—22no—
364#103—22no—
365#102—21no—
366#100—24no—
367#100—42no—
368#97—21no—
369C4×C8—21no—
370#95—21no—
371C23×C3—24no—
372C23×C3—21no—
373C23×C3—24no—
374C23×C3—24no—
375C23×C3—44no—
376C23×C3—44no—
377C23×C3C22×D411yescharacteristic
378C3×Q8—22no—
379C3×Q8—21no—
380C3×Q8—21no—
381C3×Q8—22no—
382C3×Q8—42no—
383C3×Q8—41no—
384C3×Q8—41no—
385C3×Q8—42no—
386C3×D4—21no—
387C3×D4—21no—
388C3×D4—44no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—42no—
392C3×D4—42no—
393C3×D4—42no—
394C3×D4—42no—
395C3×D4—44no—
396C3×D4—82no—
397C3×D4—82no—
398C3×D4—84no—
399C3×D4—84no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—41no—
408C2×C3×C4—42no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—42no—
414C2×C3×C4—42no—
415C2×C3×C4—42no—
416C2×C3×C4—42no—
417C2×C3×C4—41no—
418C2×C3×C4—42no—
419C2×C3×C4—84no—
420C2×C3×C4—84no—
421C2×C3×C4#12011yescharacteristic
422C2×C3×C4#12011yescharacteristic
423C2×C3×C4#12011yescharacteristic
424C2×C3×C4#12011yescharacteristic
425C2×C3×C4C22×D411yescharacteristic
426C2×C3×C4C22×D411yescharacteristic
427C2×C3×C4C2×#4111yescharacteristic
428C2×C3×C4C2×#4111yescharacteristic
429C2×C3×C4H(5,𝔽2)11yescharacteristic
430C2×C3×C42−1+411yescharacteristic
431C3×C8—22no—
432C3×C8—41no—
433C3×C8—42no—
434C24C2×C3×D414yes—
435#41—21no—
436#41—21no—
437#41—21no—
438#41—21no—
439#41—84no—
440#41—84no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×C3×D411yescharacteristic
449C2×Q8C3×#4112yescomplemented
450C2×D4—21no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—44no—
464C2×D4—82no—
465C2×D4C2×C3×D411yescharacteristic
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4C2×C3×D411yescharacteristic
475C22×C4C2×C3×D411yescharacteristic
476C22×C4C24×C311yescharacteristic
477Q16—24no—
478SD16—24no—
479M16—21no—
480M16—22no—
481M16—21no—
482M16—22no—
483M16—21no—
484M16C3×#4112yescomplemented
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4C2×C3×D411yescharacteristic
500C4⋊C4C2×C3×D412yes—
501C4⋊C4C2×C3×D411yescharacteristic
502C4⋊C4C2×C3×D412yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—21no—
508C22⋊C4—22no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—48no—
512C22⋊C4—84no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—41no—
520C42—41no—
521C42C2×C3×D411yescharacteristic
522C42C2×C3×D411yescharacteristic
523C42C2×C3×D411yescharacteristic
524C42C2×C3×D411yescharacteristic
525C22×C3—21no—
526C22×C3—21no—
527C22×C3—41no—
528C22×C3—44no—
529C22×C3—41no—
530C22×C3—44no—
531C22×C3—41no—
532C22×C3—44no—
533C22×C3—84no—
534C22×C3—84no—
535C22×C3C2×#12011yescharacteristic
536C22×C3#54611yescharacteristic
537C22×C3#54711yescharacteristic
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—41no—
542C3×C4—42no—
543C3×C4—41no—
544C3×C4—42no—
545C3×C4—41no—
546C3×C4—42no—
547C3×C4—82no—
548C3×C4—81no—
549C3×C4#45512yes—
550C3×C4#52511yescharacteristic
551C3×C4#53711yescharacteristic
552C3×C4#54811yescharacteristic
553C3×C4#55011yescharacteristic
554C23—24no—
555C23—21no—
556C23—24no—
557C23—24no—
558C23—44no—
559C23—44no—
560C23C22×C3×D411yescharacteristic
561Q8—22no—
562Q8—21no—
563Q8—21no—
564Q8—22no—
565Q8—42no—
566Q8—42no—
567Q8—41no—
568Q8—41no—
569D4—21no—
570D4—21no—
571D4—44no—
572D4—41no—
573D4—41no—
574D4—44no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—82no—
580D4—82no—
581D4—84no—
582D4—84no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4C3×#12011yescharacteristic
605C2×C4C3×#12011yescharacteristic
606C2×C4C3×#12011yescharacteristic
607C2×C4C3×#12011yescharacteristic
608C2×C4C22×C3×D411yescharacteristic
609C2×C4C22×C3×D411yescharacteristic
610C2×C4C2×C3×#4111yescharacteristic
611C2×C4C2×C3×#4111yescharacteristic
612C2×C4C3×H(5,𝔽2)11yescharacteristic
613C2×C4C3×2−1+411yescharacteristic
614C8—22no—
615C8—41no—
616C8—42no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—81no—
620C2×C3—84no—
621C2×C3#240711yescharacteristic
622C2×C3C2×#45511yescharacteristic
623C2×C3#301911yescharacteristic
624C22—21no—
625C22—21no—
626C22—41no—
627C22—44no—
628C22—41no—
629C22—44no—
630C22—41no—
631C22—44no—
632C22—84no—
633C22—84no—
634C22C2×C3×#12011yescharacteristic
635C22C3×#54611yescharacteristic
636C22C3×#54711yescharacteristic
637C4—21no—
638C4—21no—
639C4—21no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—82no—
647C4—81no—
648C4C3×#45512yes—
649C4C3×#52511yescharacteristic
650C4C3×#53711yescharacteristic
651C4C3×#54811yescharacteristic
652C4C3×#55011yescharacteristic
653C3#2414611yescharacteristic, direct product factor, Sylow
654C2—21no—
655C2—21no—
656C2—84no—
657C2—81no—
658C2C3×#240711yescharacteristic
659C2C2×C3×#45511yescharacteristic
660C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3508C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3078C2semidirect
3. C3×C4×#137C2semidirect
4. C3×#2548C2semidirect
5. C3×#2037C2semidirect
6. C3×#2004C2semidirect
7. C3×#1986C2semidirect
8. C3×#1965C2semidirect
9. C3×#1885C2semidirect
10. C3×#1856C2semidirect
11. C3×#1806C2semidirect
12. #24146C3direct (N × H)
55. #3078C2×C3semidirect
56. C4×#137C2×C3semidirect
57. #2548C2×C3semidirect
58. #2037C2×C3semidirect
59. #2004C2×C3semidirect
60. #1986C2×C3semidirect
61. #1965C2×C3semidirect
62. #1885C2×C3semidirect
63. #1856C2×C3semidirect
64. #1806C2×C3semidirect
208. C2×C3×Q8#41semidirect
243. C3×M16#41semidirect
449. C2×Q8C3×#41semidirect
484. M16C3×#41semidirect
653. C3#24146direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35082
C3×#30782
C3×C4×#1372
C3×#25482
C3×#20372
C3×#20042
C3×#19862
C3×#19652
C3×#18852
C3×#18562
C3×#18062
#241463