Finite Group Databasebeta

← C3×#23950 #10642557

Subgroups — C3×#23950 #10642557

Order 768.

1242 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
1C3×#3493C211yesmaximal, characteristic, complemented
2C3×#3005C212yesmaximal, complemented
3C3×#2297C211yesmaximal, characteristic, complemented
4C3×#2028C211yesmaximal, characteristic, complemented
5C3×#2027C211yesmaximal, characteristic, complemented
6C3×#2007C211yesmaximal, characteristic, complemented
7C3×#2006C211yesmaximal, characteristic, complemented
8C3×#1896C212yesmaximal, complemented
9C3×#1883C212yesmaximal, complemented
10C3×#1882C212yesmaximal, complemented
11C3×#1822C211yesmaximal, characteristic, complemented
12#23950C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#137C2212yes—
15C2×C3×AGL(1,ℤ/8)C2212yes—
16C3×#556—22no—
17C3×#555—22no—
18C3×#553—22no—
19C3×#552—22no—
20C3×#544C2211yescharacteristic
21C3×#538—22no—
22C3×#537C2211yescharacteristic
23C3×#529—22no—
24C3×#525C2212yes—
25C3×#520C2212yes—
26C2×C3×#117C2211yescharacteristic
27C22×C3×C42C2211yescharacteristic
28C3×#452—22no—
29C3×#452—22no—
30C3×#449—22no—
31C3×#449—22no—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#419C2212yes—
35C3×#418C2212yes—
36C3×#407C2212yes—
37C3×#391C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#390C2211yescharacteristic
41C3×#386—22no—
42C3×C4×C22⋊C4—22no—
43C3×#343C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2212yes—
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3493C2×C311yescharacteristic, complemented
49#3005C2×C312yescomplemented
50#2297C2×C311yescharacteristic, complemented
51#2028C2×C311yescharacteristic, complemented
52#2027C2×C311yescharacteristic, complemented
53#2007C2×C311yescharacteristic, complemented
54#2006C2×C311yescharacteristic, complemented
55#1896C2×C312yescomplemented
56#1883C2×C312yescomplemented
57#1882C2×C312yescomplemented
58#1822C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—42no—
64C2×C3×#41C2312yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C3×#137—24no—
72C3×#137—42no—
73C3×AGL(1,ℤ/8)—24no—
74C3×AGL(1,ℤ/8)—42no—
75C2×C3×Q16—22no—
76C2×C3×SD16—22no—
77C2×C3×SD16—22no—
78C2×C3×D8—22no—
79C2×C3×M16C2312yes—
80C2×C3×M16C2312yes—
81C3×C4⋊Q8—22no—
82C3×#126—24no—
83C3×#126—24no—
84C3×#125—22no—
85C3×#125—22no—
86C3×#124—22no—
87C3×#124—24no—
88C3×#123—24no—
89C3×#123—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8D412yes—
93C3×C22⋊Q8D412yes—
94C3×#121—24no—
95C3×#121—22no—
96C3×#121—22no—
97C3×#121D412yes—
98C3×#121D412yes—
99C3×#120—22no—
100C3×C4×Q8—22no—
101C3×C4×Q8—22no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—24no—
105C3×C4×D4—22no—
106C3×C4×D4—22no—
107C3×#117—24no—
108C3×#117—22no—
109C3×#117—22no—
110C3×#117—22no—
111C3×#117C2312yes—
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C22⋊C4D412yes—
118C2×C3×C42—21no—
119C2×C3×C42—42no—
120C2×C3×C42D412yes—
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C3×#104—42no—
124C3×#104—42no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#102—22no—
128C3×#102—22no—
129C3×C22⋊C8—22no—
130C3×C22⋊C8—22no—
131C3×#95—22no—
132C3×#95—22no—
133C22×#41C22×C311yescharacteristic
134C2×#137C22×C312yes—
135C2×AGL(1,ℤ/8)C22×C312yes—
136#556—22no—
137#555—22no—
138#553—22no—
139#552—22no—
140#544C22×C311yescharacteristic
141#538—22no—
142#537C22×C311yescharacteristic
143#529—22no—
144#525C22×C312yes—
145#520C22×C312yes—
146C2×#117C22×C311yescharacteristic
147C22×C42C22×C311yescharacteristic
148#452—22no—
149#452—22no—
150#449—22no—
151#449—22no—
152C2×#104C22×C312yes—
153C2×#104C22×C312yes—
154#419C22×C312yes—
155#418C22×C312yes—
156#407C22×C312yes—
157#391C22×C311yescharacteristic
158#391C22×C311yescharacteristic
159#390C22×C311yescharacteristic
160#390C22×C311yescharacteristic
161#386—22no—
162C4×C22⋊C4—22no—
163#343C22×C312yes—
164#343C22×C311yescharacteristic
165#343C22×C311yescharacteristic
166#339C22×C312yes—
167#339C22×C312yes—
168C24×C3—21no—
169C24×C3C2×D411yescharacteristic
170C3×#41—22no—
171C3×#41—22no—
172C3×#41—22no—
173C3×#41—22no—
174C3×#41—44no—
175C3×#41—44no—
176C3×#41—44no—
177C3×#41—44no—
178C2×C3×Q8—22no—
179C2×C3×Q8—21no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8C2×D412yes—
183C2×C3×Q8#4112yes—
184C2×C3×D4—24no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—21no—
191C2×C3×D4—22no—
192C2×C3×D4—42no—
193C2×C3×D4—42no—
194C2×C3×D4C2×D412yes—
195C2×C3×D4#4112yes—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—22no—
199C22×C3×C4—24no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—24no—
204C22×C3×C4—22no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—42no—
209C22×C3×C4—41no—
210C22×C3×C4—41no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2411yescharacteristic
217C3×Q16—44no—
218C3×SD16—44no—
219C3×SD16—44no—
220C3×D8—44no—
221C3×M16—22no—
222C3×M16—22no—
223C3×M16—22no—
224C3×M16—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4C2×D412yes—
236C3×C4⋊C4C2×D412yes—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—44no—
246C3×C22⋊C4—44no—
247C3×C42—24no—
248C3×C42—22no—
249C3×C42—22no—
250C3×C42—22no—
251C3×C42—22no—
252C3×C42—22no—
253C3×C42—42no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—42no—
259C2×#41C23×C312yes—
260C22×Q8C23×C311yescharacteristic
261C22×D4—21no—
262C22×D4C23×C311yescharacteristic
263C23×C4—21no—
264C23×C4—21no—
265C23×C4C23×C311yescharacteristic
266#137—24no—
267#137—42no—
268AGL(1,ℤ/8)—24no—
269AGL(1,ℤ/8)—42no—
270C2×Q16—22no—
271C2×SD16—22no—
272C2×SD16—22no—
273C2×D8—22no—
274C2×M16C23×C312yes—
275C2×M16C23×C312yes—
276C4⋊Q8—22no—
277#126—24no—
278#126—24no—
279#125—22no—
280#125—22no—
281#124—22no—
282#124—24no—
283#123—24no—
284#123—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8C3×D412yes—
288C22⋊Q8C3×D412yes—
289#121—24no—
290#121—22no—
291#121—22no—
292#121C3×D412yes—
293#121C3×D412yes—
294#120—22no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—24no—
300C4×D4—22no—
301C4×D4—22no—
302#117—24no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117C23×C312yes—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C23×C311yescharacteristic
309C2×C4⋊C4C23×C311yescharacteristic
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4C3×D412yes—
313C2×C42—21no—
314C2×C42—42no—
315C2×C42C3×D412yes—
316C2×C42C23×C311yescharacteristic
317C2×C42C23×C311yescharacteristic
318#104—42no—
319#104—42no—
320#103—22no—
321#103—22no—
322#102—22no—
323#102—22no—
324C22⋊C8—22no—
325C22⋊C8—22no—
326#95—22no—
327#95—22no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3#12012yes—
339C23×C3C22×D411yescharacteristic
340C3×Q8—22no—
341C3×Q8—24no—
342C3×Q8—22no—
343C3×Q8—42no—
344C3×Q8—82no—
345C3×D4—22no—
346C3×D4—24no—
347C3×D4—22no—
348C3×D4—48no—
349C3×D4—42no—
350C3×D4—42no—
351C3×D4—42no—
352C3×D4—82no—
353C3×D4—82no—
354C3×D4—82no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—44no—
370C2×C3×C4—44no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—48no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—84no—
383C2×C3×C4#12011yescharacteristic
384C2×C3×C4#12011yescharacteristic
385C2×C3×C4C22×D412yes—
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)12yes—
389C3×C8—42no—
390C3×C8—42no—
391C24—21no—
392C24C2×C3×D411yescharacteristic
393#41—22no—
394#41—22no—
395#41—22no—
396#41—22no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C2×C3×D412yes—
406C2×Q8C3×#4112yes—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C2×C3×D412yes—
418C2×D4C3×#4112yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D412yes—
436C22×C4C2×C3×D412yes—
437C22×C4C2×C3×D412yes—
438C22×C4C2×C3×D412yes—
439C22×C4C24×C311yescharacteristic
440Q16—44no—
441SD16—44no—
442SD16—44no—
443D8—44no—
444M16—22no—
445M16—22no—
446M16—22no—
447M16—22no—
448C2×C8—22no—
449C2×C8—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4C2×C3×D412yes—
459C4⋊C4C2×C3×D412yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—42no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—44no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—82no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3C2×#12011yescharacteristic
493C22×C3#52511yescharacteristic
494C22×C3#53511yescharacteristic
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4#54612yes—
506C3×C4#54612yes—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23C3×#12012yes—
518C23C22×C3×D411yescharacteristic
519Q8—22no—
520Q8—24no—
521Q8—22no—
522Q8—42no—
523Q8—82no—
524D4—22no—
525D4—24no—
526D4—22no—
527D4—48no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—82no—
532D4—82no—
533D4—82no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—84no—
562C2×C4C3×#12011yescharacteristic
563C2×C4C3×#12011yescharacteristic
564C2×C4C22×C3×D412yes—
565C2×C4C2×C3×#4111yescharacteristic
566C2×C4C2×C3×#4111yescharacteristic
567C2×C4C3×H(5,𝔽2)12yes—
568C8—42no—
569C8—42no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—42no—
573C2×C3—82no—
574C2×C3—82no—
575C2×C3#240411yescharacteristic
576C2×C3#301811yescharacteristic
577C2×C3#301811yescharacteristic
578C22—21no—
579C22—21no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—82no—
591C22—81no—
592C22—81no—
593C22C2×C3×#12011yescharacteristic
594C22C3×#52511yescharacteristic
595C22C3×#53511yescharacteristic
596C4—21no—
597C4—21no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4C3×#54612yes—
607C4C3×#54612yes—
608C3#2395011yescharacteristic, direct product factor, Sylow
609C2—21no—
610C2—21no—
611C2—42no—
612C2—82no—
613C2—82no—
614C2C3×#240411yescharacteristic
615C2C3×#301811yescharacteristic
616C2C3×#301811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C3×#3493C2semidirect
2. C3×#3005C2semidirect
3. C3×#2297C2semidirect
4. C3×#2028C2semidirect
5. C3×#2027C2semidirect
6. C3×#2007C2semidirect
7. C3×#2006C2semidirect
8. C3×#1896C2semidirect
9. C3×#1883C2semidirect
10. C3×#1882C2semidirect
11. C3×#1822C2semidirect
12. #23950C3direct (N × H)
48. #3493C2×C3semidirect
49. #3005C2×C3semidirect
50. #2297C2×C3semidirect
51. #2028C2×C3semidirect
52. #2027C2×C3semidirect
53. #2007C2×C3semidirect
54. #2006C2×C3semidirect
55. #1896C2×C3semidirect
56. #1883C2×C3semidirect
57. #1882C2×C3semidirect
58. #1822C2×C3semidirect
608. C3#23950direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34932
C3×#30052
C3×#22972
C3×#20282
C3×#20272
C3×#20072
C3×#20062
C3×#18962
C3×#18832
C3×#18822
C3×#18222
#239503