Finite Group Databasebeta

← C3×#24181 #10642788

Subgroups — C3×#24181 #10642788

Order 768.

1050 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic, complemented
3C3×#3036C211yesmaximal, characteristic
4C3×#2326C211yesmaximal, characteristic, complemented
5C3×#2038C212yesmaximal, complemented
6C3×#2003C212yesmaximal, complemented
7C3×#1995C211yesmaximal, characteristic, complemented
8C3×#1963C211yesmaximal, characteristic, complemented
9C3×#1883C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C3×#1801C211yesmaximal, characteristic, complemented
12#24181C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C3×#559—21no—
15C3×#557—21no—
16C3×#556—21no—
17C3×#556—22no—
18C3×#554—21no—
19C3×#552—22no—
20C3×#539—22no—
21C3×#536C2212yes—
22C3×#533C2211yescharacteristic
23C3×#532C2211yescharacteristic
24C3×#529—22no—
25C2×C3×#124C2211yescharacteristic
26C3×#520C2212yes—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C3×#478—22no—
30C3×#476—21no—
31C3×#474—21no—
32C3×#463—22no—
33C3×#461—21no—
34C3×#459—21no—
35C3×#428C2212yes—
36C3×#423C2211yescharacteristic
37C3×#423C2211yescharacteristic
38C2×C3×#104C2212yes—
39C2×C3×#104C2212yes—
40C3×#418C2212yes—
41C3×#417C2212yes—
42C2×C3×#101C2212yes—
43C3×#391C2211yescharacteristic
44C3×#390C2211yescharacteristic
45C3×#388—22no—
46C3×#385—22no—
47C3×#384C2211yescharacteristic
48C3×#376C2211yescharacteristic
49C3×#343C2212yes—
50C3×#339C2212yes—
51C3×#339C2212yes—
52#3507C2×C311yescharacteristic, complemented
53#3077C2×C311yescharacteristic, complemented
54#3036C2×C311yescharacteristic
55#2326C2×C311yescharacteristic, complemented
56#2038C2×C312yescomplemented
57#2003C2×C312yescomplemented
58#1995C2×C311yescharacteristic, complemented
59#1963C2×C311yescharacteristic, complemented
60#1883C2×C312yescomplemented
61#1859C2×C312yescomplemented
62#1801C2×C311yescharacteristic, complemented
63C2×C3×#41C2312yes—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C3×AGL(1,ℤ/8)—24no—
68C3×AGL(1,ℤ/8)—42no—
69C2×C3×SD16—22no—
70C2×C3×D8—22no—
71C2×C3×M16C2312yes—
72C2×C3×M16C2312yes—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8D411yescharacteristic
75C3×C4⋊Q8D411yescharacteristic
76C3×C4⋊Q8D411yescharacteristic
77C3×#127D411yescharacteristic
78C3×#126—24no—
79C3×#126—24no—
80C3×#125—22no—
81C3×#125—24no—
82C3×#125D412yes—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124—22no—
86C3×#124—22no—
87C3×#124D412yes—
88C3×#123—24no—
89C3×#123—24no—
90C3×C22⋊Q8—24no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—22no—
95C3×#121—21no—
96C3×#121—21no—
97C3×C4×Q8—22no—
98C3×C4×Q8—22no—
99C3×C4×Q8—21no—
100C3×C4×Q8—21no—
101C3×C4×Q8—21no—
102C3×C4×Q8—21no—
103C3×C4×Q8—21no—
104C3×C4×Q8—21no—
105C3×C4×D4—22no—
106C3×C4×D4—22no—
107C3×C4×D4—24no—
108C3×C4×D4—21no—
109C3×C4×D4—21no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×#117—24no—
115C3×#117—22no—
116C3×#117C2312yes—
117C2×C3×C4⋊C4—22no—
118C2×C3×C4⋊C4—22no—
119C2×C3×C4⋊C4D412yes—
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C22⋊C4—22no—
123C2×C3×C22⋊C4—21no—
124C2×C3×C22⋊C4—22no—
125C2×C3×C42D412yes—
126C2×C3×C42C2311yescharacteristic
127C2×C3×C42C2311yescharacteristic
128C2×C3×C42C2311yescharacteristic
129C3×#107—22no—
130C3×#106—22no—
131C3×C4⋊C8—21no—
132C3×C4⋊C8—21no—
133C3×C4⋊C8—21no—
134C3×C4⋊C8—21no—
135C3×#104—24no—
136C3×#104—42no—
137C3×#103—22no—
138C3×#102—22no—
139C3×#102—22no—
140C3×#102—22no—
141C3×#101—42no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95—22no—
145C2×AGL(1,ℤ/8)C22×C312yes—
146#559—21no—
147#557—21no—
148#556—21no—
149#556—22no—
150#554—21no—
151#552—22no—
152#539—22no—
153#536C22×C312yes—
154#533C22×C311yescharacteristic
155#532C22×C311yescharacteristic
156#529—22no—
157C2×#124C22×C311yescharacteristic
158#520C22×C312yes—
159C2×C4×Q8C22×C311yescharacteristic
160C2×C4×D4C22×C311yescharacteristic
161#478—22no—
162#476—21no—
163#474—21no—
164#463—22no—
165#461—21no—
166#459—21no—
167#428C22×C312yes—
168#423C22×C311yescharacteristic
169#423C22×C311yescharacteristic
170C2×#104C22×C312yes—
171C2×#104C22×C312yes—
172#418C22×C312yes—
173#417C22×C312yes—
174C2×#101C22×C312yes—
175#391C22×C311yescharacteristic
176#390C22×C311yescharacteristic
177#388—22no—
178#385—22no—
179#384C22×C311yescharacteristic
180#376C22×C311yescharacteristic
181#343C22×C312yes—
182#339C22×C312yes—
183#339C22×C312yes—
184C24×C3—21no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—22no—
188C3×#41—22no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C2×D412yes—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4C2×D412yes—
207C2×C3×D4#4112yes—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—22no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—41no—
214C22×C3×C4—41no—
215C22×C3×C4—82no—
216C22×C3×C4C2×D412yes—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C3×SD16—44no—
222C3×D8—44no—
223C3×M16—22no—
224C3×M16—22no—
225C3×M16—22no—
226C3×M16—22no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—82no—
245C3×C4⋊C4C2×D412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C22⋊C4—28no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—41no—
255C3×C22⋊C4—41no—
256C3×C22⋊C4—82no—
257C3×C42—24no—
258C3×C42—22no—
259C3×C42—22no—
260C3×C42—22no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42C2×D411yescharacteristic
265C3×C42C2×D411yescharacteristic
266C3×C42C2×D411yescharacteristic
267C3×C42C2×D411yescharacteristic
268C3×C42#4112yes—
269C2×#41C23×C312yes—
270C22×Q8C23×C311yescharacteristic
271C22×D4C23×C311yescharacteristic
272C23×C4—21no—
273AGL(1,ℤ/8)—24no—
274AGL(1,ℤ/8)—42no—
275C2×SD16—22no—
276C2×D8—22no—
277C2×M16C23×C312yes—
278C2×M16C23×C312yes—
279C4⋊Q8—22no—
280C4⋊Q8C3×D411yescharacteristic
281C4⋊Q8C3×D411yescharacteristic
282C4⋊Q8C3×D411yescharacteristic
283#127C3×D411yescharacteristic
284#126—24no—
285#126—24no—
286#125—22no—
287#125—24no—
288#125C3×D412yes—
289#124—22no—
290#124—22no—
291#124—22no—
292#124—22no—
293#124C3×D412yes—
294#123—24no—
295#123—24no—
296C22⋊Q8—24no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301#121—21no—
302#121—21no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—24no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320#117—24no—
321#117—22no—
322#117C23×C312yes—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4C3×D412yes—
326C2×C4⋊C4C23×C311yescharacteristic
327C2×C4⋊C4C23×C311yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C42C3×D412yes—
332C2×C42C23×C311yescharacteristic
333C2×C42C23×C311yescharacteristic
334C2×C42C23×C311yescharacteristic
335#107—22no—
336#106—22no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341#104—24no—
342#104—42no—
343#103—22no—
344#102—22no—
345#102—22no—
346#102—22no—
347#101—42no—
348#95—21no—
349#95—21no—
350#95—22no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—42no—
354C23×C3—42no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—42no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—81no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—42no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—81no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—42no—
389C2×C3×C4—42no—
390C2×C3×C4—42no—
391C2×C3×C4—41no—
392C2×C3×C4—42no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—82no—
397C2×C3×C4—82no—
398C2×C3×C4—84no—
399C2×C3×C4#12011yescharacteristic
400C2×C3×C4#12011yescharacteristic
401C2×C3×C4#12011yescharacteristic
402C2×C3×C4#12011yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4H(5,𝔽2)11yescharacteristic
408C2×C3×C42−1+411yescharacteristic
409C3×C8—42no—
410C3×C8—42no—
411C24—21no—
412#41—22no—
413#41—22no—
414#41—22no—
415#41—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8C2×C3×D412yes—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4C2×C3×D412yes—
434C2×D4C3×#4112yes—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—82no—
443C22×C4C2×C3×D412yes—
444C22×C4C2×C3×D412yes—
445C22×C4C2×C3×D411yescharacteristic
446C22×C4C2×C3×D411yescharacteristic
447C22×C4C24×C311yescharacteristic
448SD16—44no—
449D8—44no—
450M16—22no—
451M16—22no—
452M16—22no—
453M16—22no—
454C2×C8—22no—
455C2×C8—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—82no—
472C4⋊C4C2×C3×D412yes—
473C4⋊C4C2×C3×D412yes—
474C22⋊C4—28no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—82no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42C2×C3×D411yescharacteristic
492C42C2×C3×D411yescharacteristic
493C42C2×C3×D411yescharacteristic
494C42C2×C3×D411yescharacteristic
495C42C3×#4112yes—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—44no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—82no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3C2×#12011yescharacteristic
507C22×C3#52511yescharacteristic
508C22×C3#53711yescharacteristic
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—42no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4#54611yescharacteristic
522C3×C4#54711yescharacteristic
523C3×C4#54811yescharacteristic
524C3×C4#55011yescharacteristic
525C23—22no—
526C23—22no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23C22×C3×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—42no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—81no—
540D4—22no—
541D4—22no—
542D4—22no—
543D4—22no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—41no—
548D4—41no—
549D4—81no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—84no—
573C2×C4C3×#12011yescharacteristic
574C2×C4C3×#12011yescharacteristic
575C2×C4C3×#12011yescharacteristic
576C2×C4C3×#12011yescharacteristic
577C2×C4C22×C3×D411yescharacteristic
578C2×C4C22×C3×D411yescharacteristic
579C2×C4C2×C3×#4111yescharacteristic
580C2×C4C2×C3×#4111yescharacteristic
581C2×C4C3×H(5,𝔽2)11yescharacteristic
582C2×C4C3×2−1+411yescharacteristic
583C8—42no—
584C8—42no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—82no—
588C2×C3—82no—
589C2×C3#240711yescharacteristic
590C2×C3#302011yescharacteristic
591C2×C3#302111yescharacteristic
592C22—21no—
593C22—21no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22C2×C3×#12011yescharacteristic
603C22C3×#52511yescharacteristic
604C22C3×#53711yescharacteristic
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4C3×#54611yescharacteristic
618C4C3×#54711yescharacteristic
619C4C3×#54811yescharacteristic
620C4C3×#55011yescharacteristic
621C3#2418111yescharacteristic, direct product factor, Sylow
622C2—21no—
623C2—21no—
624C2—82no—
625C2—82no—
626C2C3×#240711yescharacteristic
627C2C3×#302011yescharacteristic
628C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3036C2
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
1. C3×#3507C2semidirect
2. C3×#3077C2semidirect
4. C3×#2326C2semidirect
5. C3×#2038C2semidirect
6. C3×#2003C2semidirect
7. C3×#1995C2semidirect
8. C3×#1963C2semidirect
9. C3×#1883C2semidirect
10. C3×#1859C2semidirect
11. C3×#1801C2semidirect
12. #24181C3direct (N × H)
52. #3507C2×C3semidirect
53. #3077C2×C3semidirect
55. #2326C2×C3semidirect
56. #2038C2×C3semidirect
57. #2003C2×C3semidirect
58. #1995C2×C3semidirect
59. #1963C2×C3semidirect
60. #1883C2×C3semidirect
61. #1859C2×C3semidirect
62. #1801C2×C3semidirect
621. C3#24181direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35072
C3×#30772
C3×#30362
C3×#23262
C3×#20382
C3×#20032
C3×#19952
C3×#19632
C3×#18832
C3×#18592
C3×#18012
#241813