Finite Group Databasebeta

← C3×#24147 #10642754

Subgroups — C3×#24147 #10642754

Order 768.

1006 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3509C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic, complemented
3C3×C4×#137C211yesmaximal, characteristic
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2038C212yesmaximal
6C3×#2005C212yesmaximal
7C3×#1986C211yesmaximal, characteristic, complemented
8C3×#1966C211yesmaximal, characteristic, complemented
9C3×#1886C212yesmaximal, complemented
10C3×#1857C212yesmaximal, complemented
11C3×#1806C211yesmaximal, characteristic
12#24147C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2211yescharacteristic
14C3×#557—21no—
15C3×#557—21no—
16C3×#556—21no—
17C3×#556—21no—
18C3×#552—22no—
19C3×#552—22no—
20C3×#541—22no—
21C3×#536C2212yes—
22C3×#533—21no—
23C3×#533—21no—
24C3×#533C2211yescharacteristic
25C2×C3×C4⋊Q8C2212yes—
26C2×C3×C4⋊Q8C2211yescharacteristic
27C3×#519C2212yes—
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×#101C2212yes—
45C2×C3×#101C2212yes—
46C3×C4×M16C2211yescharacteristic
47C3×#398—22no—
48C3×#391C2212yes—
49C3×#389—22no—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×#339C2212yes—
52C3×#339C2212yes—
53C3×#339C2212yes—
54#3509C2×C311yescharacteristic
55#3078C2×C311yescharacteristic, complemented
56C4×#137C2×C311yescharacteristic
57#2546C2×C311yescharacteristic, complemented
58#2038C2×C312yes—
59#2005C2×C312yes—
60#1986C2×C311yescharacteristic, complemented
61#1966C2×C311yescharacteristic, complemented
62#1886C2×C312yescomplemented
63#1857C2×C312yescomplemented
64#1806C2×C311yescharacteristic
65C2×C3×#41C2311yescharacteristic
66C22×C3×Q8C2311yescharacteristic
67C22×C3×Q8C2312yes—
68C3×#137—22no—
69C3×#137—22no—
70C2×C3×Q16—21no—
71C2×C3×SD16—21no—
72C2×C3×M16C2311yescharacteristic
73C2×C3×M16C2312yes—
74C2×C3×M16C2311yescharacteristic
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8—24no—
77C3×C4⋊Q8—21no—
78C3×C4⋊Q8—21no—
79C3×C4⋊Q8—21no—
80C3×C4⋊Q8—21no—
81C3×C4⋊Q8—42no—
82C3×C4⋊Q8D411yescharacteristic
83C3×C4⋊Q8D411yescharacteristic
84C3×#126—24no—
85C3×#126—24no—
86C3×#125—22no—
87C3×#125—22no—
88C3×#125—24no—
89C3×#125—24no—
90C3×#125D412yes—
91C3×#124—22no—
92C3×#123—24no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8—21no—
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—22no—
99C3×C4×Q8—22no—
100C3×C4×Q8—22no—
101C3×C4×Q8—22no—
102C3×C4×Q8—22no—
103C3×C4×Q8—22no—
104C3×C4×Q8—22no—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8—21no—
109C3×C4×Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8D411yescharacteristic
112C3×C4×Q8D411yescharacteristic
113C3×C4×D4—21no—
114C3×C4×D4—21no—
115C3×C4×D4D411yescharacteristic
116C3×C4×D4D411yescharacteristic
117C3×#117—22no—
118C3×#117—22no—
119C3×#117—24no—
120C3×#117—21no—
121C3×#117C2311yescharacteristic
122C3×#117C2312yes—
123C2×C3×C4⋊C4—21no—
124C2×C3×C4⋊C4—24no—
125C2×C3×C4⋊C4—21no—
126C2×C3×C4⋊C4D412yes—
127C2×C3×C4⋊C4D412yes—
128C2×C3×C4⋊C4C2311yescharacteristic
129C2×C3×C42—21no—
130C2×C3×C42—21no—
131C2×C3×C42C2312yes—
132C2×C3×C42C2311yescharacteristic
133C3×#107—21no—
134C3×#107—22no—
135C3×#106—21no—
136C3×#106—22no—
137C3×C4⋊C8—21no—
138C3×C4⋊C8—21no—
139C3×#104—42no—
140C3×#103—21no—
141C3×#103—21no—
142C3×#103—21no—
143C3×#103—22no—
144C3×#103—22no—
145C3×#102—21no—
146C3×#101—24no—
147C3×#101—42no—
148C3×#97—21no—
149C3×C4×C8—21no—
150C3×#95—21no—
151C3×#95—24no—
152C2×#137C22×C311yescharacteristic
153#557—21no—
154#557—21no—
155#556—21no—
156#556—21no—
157#552—22no—
158#552—22no—
159#541—22no—
160#536C22×C312yes—
161#533—21no—
162#533—21no—
163#533C22×C311yescharacteristic
164C2×C4⋊Q8C22×C312yes—
165C2×C4⋊Q8C22×C311yescharacteristic
166#519C22×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×#101C22×C312yes—
184C2×#101C22×C312yes—
185C4×M16C22×C311yescharacteristic
186#398—22no—
187#391C22×C312yes—
188#389—22no—
189C4×C4⋊C4C22×C311yescharacteristic
190#339C22×C312yes—
191#339C22×C312yes—
192#339C22×C312yes—
193C3×#41—21no—
194C3×#41—21no—
195C3×#41—21no—
196C3×#41—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—22no—
200C2×C3×Q8—22no—
201C2×C3×Q8—22no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—21no—
205C2×C3×Q8—22no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8C2×D411yescharacteristic
211C2×C3×Q8#4112yes—
212C2×C3×D4—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4C2×D414yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C3×Q16—24no—
225C3×SD16—24no—
226C3×M16—21no—
227C3×M16—22no—
228C3×M16—21no—
229C3×M16—22no—
230C3×M16—21no—
231C3×M16#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—22no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—48no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D412yes—
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D412yes—
259C3×C22⋊C4—24no—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—21no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—21no—
265C3×C42—24no—
266C3×C42—24no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—22no—
271C3×C42—22no—
272C3×C42—22no—
273C3×C42—41no—
274C3×C42—41no—
275C3×C42C2×D411yescharacteristic
276C3×C42C2×D411yescharacteristic
277C3×C42C2×D411yescharacteristic
278C3×C42C2×D411yescharacteristic
279C2×#41C23×C311yescharacteristic
280C22×Q8C23×C311yescharacteristic
281C22×Q8C23×C312yes—
282#137—22no—
283#137—22no—
284C2×Q16—21no—
285C2×SD16—21no—
286C2×M16C23×C311yescharacteristic
287C2×M16C23×C312yes—
288C2×M16C23×C311yescharacteristic
289C4⋊Q8—22no—
290C4⋊Q8—24no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—42no—
296C4⋊Q8C3×D411yescharacteristic
297C4⋊Q8C3×D411yescharacteristic
298#126—24no—
299#126—24no—
300#125—22no—
301#125—22no—
302#125—24no—
303#125—24no—
304#125C3×D412yes—
305#124—22no—
306#123—24no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8C3×D411yescharacteristic
326C4×Q8C3×D411yescharacteristic
327C4×D4—21no—
328C4×D4—21no—
329C4×D4C3×D411yescharacteristic
330C4×D4C3×D411yescharacteristic
331#117—22no—
332#117—22no—
333#117—24no—
334#117—21no—
335#117C23×C311yescharacteristic
336#117C23×C312yes—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4C3×D412yes—
341C2×C4⋊C4C3×D412yes—
342C2×C4⋊C4C23×C311yescharacteristic
343C2×C42—21no—
344C2×C42—21no—
345C2×C42C23×C312yes—
346C2×C42C23×C311yescharacteristic
347#107—21no—
348#107—22no—
349#106—21no—
350#106—22no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353#104—42no—
354#103—21no—
355#103—21no—
356#103—21no—
357#103—22no—
358#103—22no—
359#102—21no—
360#101—24no—
361#101—42no—
362#97—21no—
363C4×C8—21no—
364#95—21no—
365#95—24no—
366C23×C3—21no—
367C23×C3C22×D411yescharacteristic
368C3×Q8—22no—
369C3×Q8—21no—
370C3×Q8—21no—
371C3×Q8—22no—
372C3×Q8—42no—
373C3×Q8—42no—
374C3×Q8—42no—
375C3×Q8—41no—
376C3×Q8—42no—
377C3×Q8—42no—
378C3×Q8—41no—
379C3×Q8—42no—
380C3×D4—21no—
381C3×D4—21no—
382C3×D4—41no—
383C3×D4—41no—
384C2×C3×C4—22no—
385C2×C3×C4—24no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—24no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—24no—
394C2×C3×C4—24no—
395C2×C3×C4—41no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4—41no—
399C2×C3×C4—42no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—42no—
404C2×C3×C4—42no—
405C2×C3×C4—41no—
406C2×C3×C4#12011yescharacteristic
407C2×C3×C4#12011yescharacteristic
408C2×C3×C4#12011yescharacteristic
409C2×C3×C4#12011yescharacteristic
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4H(5,𝔽2)11yescharacteristic
415C2×C3×C42−1+411yescharacteristic
416C3×C8—22no—
417C3×C8—41no—
418C3×C8—42no—
419#41—21no—
420#41—21no—
421#41—21no—
422#41—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C3×D411yescharacteristic
437C2×Q8C3×#4112yes—
438C2×D4—21no—
439C2×D4C2×C3×D411yescharacteristic
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4C2×C3×D414yes—
447C22×C4C2×C3×D411yescharacteristic
448C22×C4C2×C3×D411yescharacteristic
449C22×C4C24×C311yescharacteristic
450Q16—24no—
451SD16—24no—
452M16—21no—
453M16—22no—
454M16—21no—
455M16—22no—
456M16—21no—
457M16C3×#4112yes—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—48no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4C2×C3×D411yescharacteristic
482C4⋊C4C2×C3×D412yes—
483C4⋊C4C2×C3×D411yescharacteristic
484C4⋊C4C2×C3×D412yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—21no—
488C22⋊C4—22no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C42—24no—
492C42—24no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—41no—
500C42—41no—
501C42C2×C3×D411yescharacteristic
502C42C2×C3×D411yescharacteristic
503C42C2×C3×D411yescharacteristic
504C42C2×C3×D411yescharacteristic
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3C2×#12011yescharacteristic
511C22×C3#54611yescharacteristic
512C22×C3#54711yescharacteristic
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—44no—
517C3×C4—44no—
518C3×C4—41no—
519C3×C4—42no—
520C3×C4—41no—
521C3×C4—42no—
522C3×C4—41no—
523C3×C4—42no—
524C3×C4—81no—
525C3×C4#45612yes—
526C3×C4#52511yescharacteristic
527C3×C4#53711yescharacteristic
528C3×C4#54811yescharacteristic
529C3×C4#55011yescharacteristic
530C23—21no—
531C23C22×C3×D411yescharacteristic
532Q8—22no—
533Q8—21no—
534Q8—21no—
535Q8—22no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—41no—
541Q8—42no—
542Q8—42no—
543Q8—41no—
544D4—21no—
545D4—21no—
546D4—41no—
547D4—41no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4C3×#12011yescharacteristic
571C2×C4C3×#12011yescharacteristic
572C2×C4C3×#12011yescharacteristic
573C2×C4C3×#12011yescharacteristic
574C2×C4C22×C3×D411yescharacteristic
575C2×C4C22×C3×D411yescharacteristic
576C2×C4C2×C3×#4111yescharacteristic
577C2×C4C2×C3×#4111yescharacteristic
578C2×C4C3×H(5,𝔽2)11yescharacteristic
579C2×C4C3×2−1+411yescharacteristic
580C8—22no—
581C8—41no—
582C8—42no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—81no—
586C2×C3#240711yescharacteristic
587C2×C3C2×#45611yescharacteristic
588C2×C3#302011yescharacteristic
589C22—21no—
590C22—21no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22C2×C3×#12011yescharacteristic
595C22C3×#54611yescharacteristic
596C22C3×#54711yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—44no—
601C4—44no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—42no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4C3×#45612yes—
610C4C3×#52511yescharacteristic
611C4C3×#53711yescharacteristic
612C4C3×#54811yescharacteristic
613C4C3×#55011yescharacteristic
614C3#2414711yescharacteristic, direct product factor, Sylow
615C2—21no—
616C2—21no—
617C2—81no—
618C2C3×#240711yescharacteristic
619C2C2×C3×#45611yescharacteristic
620C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#41C23
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
4. C3×#2546C2semidirect
7. C3×#1986C2semidirect
8. C3×#1966C2semidirect
9. C3×#1886C2semidirect
10. C3×#1857C2semidirect
12. #24147C3direct (N × H)
55. #3078C2×C3semidirect
57. #2546C2×C3semidirect
60. #1986C2×C3semidirect
61. #1966C2×C3semidirect
62. #1886C2×C3semidirect
63. #1857C2×C3semidirect
614. C3#24147direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35092
C3×#30782
C3×C4×#1372
C3×#25462
C3×#20382
C3×#20052
C3×#19862
C3×#19662
C3×#18862
C3×#18572
C3×#18062
#241473