Finite Group Databasebeta

← C3×#29682 #10648289

Subgroups — C3×#29682 #10648289

Order 768.

1152 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic, complemented
2C3×#3040C212yesmaximal, complemented
3C3×#2544C211yesmaximal, characteristic, complemented
4C3×#2028C211yesmaximal, characteristic, complemented
5C3×#2027C211yesmaximal, characteristic, complemented
6C3×#2007C211yesmaximal, characteristic, complemented
7C3×#2006C211yesmaximal, characteristic, complemented
8C3×#2004C212yesmaximal, complemented
9C3×#2002C212yesmaximal, complemented
10C3×#1806C212yesmaximal, complemented
11C3×#1800C211yesmaximal, characteristic, complemented
12#29682C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C2×C3×AGL(1,ℤ/8)C2212yes—
15C3×#544C2211yescharacteristic
16C3×#542—22no—
17C3×#540—22no—
18C3×#538—22no—
19C3×#537C2211yescharacteristic
20C3×#535C2212yes—
21C3×#532—22no—
22C2×C3×#127C2211yescharacteristic
23C3×#529—22no—
24C2×C3×#124C2211yescharacteristic
25C2×C3×#124C2211yescharacteristic
26C3×#525—22no—
27C3×C4×#41C2212yes—
28C2×C3×C4×D4C2211yescharacteristic
29C3×#464—22no—
30C3×#464—22no—
31C3×#463—22no—
32C3×#463—22no—
33C3×#423C2212yes—
34C2×C3×#104C2212yes—
35C2×C3×#104C2212yes—
36C3×#419C2212yes—
37C3×#418C2212yes—
38C2×C3×#100C2212yes—
39C3×#394—22no—
40C3×#391C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#386—22no—
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3460C2×C311yescharacteristic, complemented
49#3040C2×C312yescomplemented
50#2544C2×C311yescharacteristic, complemented
51#2028C2×C311yescharacteristic, complemented
52#2027C2×C311yescharacteristic, complemented
53#2007C2×C311yescharacteristic, complemented
54#2006C2×C311yescharacteristic, complemented
55#2004C2×C312yescomplemented
56#2002C2×C312yescomplemented
57#1806C2×C312yescomplemented
58#1800C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—22no—
63C22×C3×D4C2311yescharacteristic
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4D412yes—
66C3×#137—42no—
67C3×#137—42no—
68C3×AGL(1,ℤ/8)—42no—
69C3×AGL(1,ℤ/8)—42no—
70C2×C3×Q16—22no—
71C2×C3×SD16—22no—
72C2×C3×SD16—22no—
73C2×C3×D8—22no—
74C2×C3×M16C2312yes—
75C2×C3×M16C2312yes—
76C3×C4⋊Q8—22no—
77C3×#127—22no—
78C3×#127—42no—
79C3×#126—24no—
80C3×#125—22no—
81C3×#124—24no—
82C3×#124—22no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124—42no—
86C3×#123—24no—
87C3×#123—24no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×#121—22no—
92C3×#121—24no—
93C3×#121—22no—
94C3×#121—22no—
95C3×#121—22no—
96C3×#121—22no—
97C3×#120—24no—
98C3×C4×Q8D412yescomplemented
99C3×C4×Q8D412yescomplemented
100C3×C4×D4—24no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—22no—
105C3×C4×D4D412yescomplemented
106C3×C4×D4D412yescomplemented
107C3×#117—22no—
108C3×#117—22no—
109C3×#117C2312yes—
110C2×C3×C4⋊C4—21no—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4D412yes—
117C2×C3×C42—22no—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C3×C4⋊C8—22no—
122C3×C4⋊C8—22no—
123C3×#104—42no—
124C3×#104—42no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#102—22no—
128C3×#102—22no—
129C3×#100—42no—
130C3×#95—21no—
131C3×#95—22no—
132C2×#137C22×C312yes—
133C2×AGL(1,ℤ/8)C22×C312yes—
134#544C22×C311yescharacteristic
135#542—22no—
136#540—22no—
137#538—22no—
138#537C22×C311yescharacteristic
139#535C22×C312yes—
140#532—22no—
141C2×#127C22×C311yescharacteristic
142#529—22no—
143C2×#124C22×C311yescharacteristic
144C2×#124C22×C311yescharacteristic
145#525—22no—
146C4×#41C22×C312yes—
147C2×C4×D4C22×C311yescharacteristic
148#464—22no—
149#464—22no—
150#463—22no—
151#463—22no—
152#423C22×C312yes—
153C2×#104C22×C312yes—
154C2×#104C22×C312yes—
155#419C22×C312yes—
156#418C22×C312yes—
157C2×#100C22×C312yes—
158#394—22no—
159#391C22×C311yescharacteristic
160#390C22×C311yescharacteristic
161#386—22no—
162C4×C4⋊C4C22×C311yescharacteristic
163#343C22×C311yescharacteristic
164#343C22×C311yescharacteristic
165#339C22×C312yes—
166#339C22×C312yes—
167C24×C3—21no—
168C24×C3C2×D412yes—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—22no—
172C3×#41—22no—
173C3×#41—84no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8C2×D412yes—
180C2×C3×D4—22no—
181C2×C3×D4—24no—
182C2×C3×D4—24no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—44no—
193C2×C3×D4—44no—
194C2×C3×D4—82no—
195C2×C3×D4C2×D412yes—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—82no—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C3×Q16—44no—
210C3×SD16—44no—
211C3×SD16—44no—
212C3×D8—44no—
213C3×M16—22no—
214C3×M16—22no—
215C3×M16—22no—
216C3×M16—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4C2×D412yes—
227C3×C4⋊C4C2×D412yes—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—44no—
236C3×C22⋊C4—41no—
237C3×C22⋊C4—42no—
238C3×C22⋊C4—41no—
239C3×C22⋊C4—82no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42—22no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42C2×D412yes—
246C3×C42C2×D412yes—
247C2×#41—22no—
248C2×#41C23×C312yes—
249C22×Q8C23×C311yescharacteristic
250C22×D4—22no—
251C22×D4C23×C311yescharacteristic
252C22×D4C23×C311yescharacteristic
253C23×C4C3×D412yes—
254#137—42no—
255#137—42no—
256AGL(1,ℤ/8)—42no—
257AGL(1,ℤ/8)—42no—
258C2×Q16—22no—
259C2×SD16—22no—
260C2×SD16—22no—
261C2×D8—22no—
262C2×M16C23×C312yes—
263C2×M16C23×C312yes—
264C4⋊Q8—22no—
265#127—22no—
266#127—42no—
267#126—24no—
268#125—22no—
269#124—24no—
270#124—22no—
271#124—22no—
272#124—22no—
273#124—42no—
274#123—24no—
275#123—24no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279#121—22no—
280#121—24no—
281#121—22no—
282#121—22no—
283#121—22no—
284#121—22no—
285#120—24no—
286C4×Q8C3×D412yescomplemented
287C4×Q8C3×D412yescomplemented
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4C3×D412yescomplemented
294C4×D4C3×D412yescomplemented
295#117—22no—
296#117—22no—
297#117C23×C312yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4C23×C311yescharacteristic
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4C3×D412yes—
305C2×C42—22no—
306C2×C42C23×C311yescharacteristic
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309C4⋊C8—22no—
310C4⋊C8—22no—
311#104—42no—
312#104—42no—
313#103—22no—
314#103—22no—
315#102—22no—
316#102—22no—
317#100—42no—
318#95—21no—
319#95—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3C22×D411yescharacteristic
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—42no—
335C3×Q8—42no—
336C3×Q8—82no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—42no—
344C3×D4—42no—
345C3×D4—84no—
346C3×D4—82no—
347C3×D4—84no—
348C3×D4—88no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—44no—
357C2×C3×C4—44no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—82no—
370C2×C3×C4—82no—
371C2×C3×C4#12011yescharacteristic
372C2×C3×C4#12012yes—
373C2×C3×C4#12011yescharacteristic
374C2×C3×C4C22×D412yes—
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C4H(5,𝔽2)12yes—
377C2×C3×C42−1+411yescharacteristic
378C3×C8—42no—
379C3×C8—42no—
380C24—21no—
381C24C2×C3×D412yes—
382#41—22no—
383#41—22no—
384#41—22no—
385#41—22no—
386#41—84no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8C2×C3×D412yes—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4C2×C3×D412yes—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—82no—
418C22×C4C2×C3×D412yes—
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C24×C311yescharacteristic
422Q16—44no—
423SD16—44no—
424SD16—44no—
425D8—44no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16—22no—
430C2×C8—22no—
431C2×C8—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4C2×C3×D412yes—
440C4⋊C4C2×C3×D412yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—24no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—44no—
449C22⋊C4—41no—
450C22⋊C4—42no—
451C22⋊C4—41no—
452C22⋊C4—82no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42C2×C3×D412yes—
459C42C2×C3×D412yes—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—44no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—82no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3C2×#12011yescharacteristic
476C22×C3#53511yescharacteristic
477C22×C3#53711yescharacteristic
478C3×C4—22no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4#53512yes—
489C3×C4#53712yes—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C22×C3×D411yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—42no—
505Q8—42no—
506Q8—82no—
507D4—22no—
508D4—22no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—82no—
516D4—84no—
517D4—84no—
518D4—88no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4C3×#12011yescharacteristic
542C2×C4C3×#12012yes—
543C2×C4C3×#12011yescharacteristic
544C2×C4C22×C3×D412yes—
545C2×C4C3×H(5,𝔽2)11yescharacteristic
546C2×C4C3×H(5,𝔽2)12yes—
547C2×C4C3×2−1+411yescharacteristic
548C8—42no—
549C8—42no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—82no—
553C2×C3—82no—
554C2×C3—161no—
555C2×C3#261511yescharacteristic
556C2×C3#301811yescharacteristic
557C2×C3#301911yescharacteristic
558C22—21no—
559C22—21no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×C3×#12011yescharacteristic
574C22C3×#53511yescharacteristic
575C22C3×#53711yescharacteristic
576C4—22no—
577C4—21no—
578C4—21no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—81no—
585C4—81no—
586C4C3×#53512yes—
587C4C3×#53712yes—
588C3#2968211yescharacteristic, direct product factor, Sylow
589C2—21no—
590C2—21no—
591C2—82no—
592C2—82no—
593C2—161no—
594C2C3×#261511yescharacteristic
595C2C3×#301811yescharacteristic
596C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
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×#3460C2semidirect
2. C3×#3040C2semidirect
3. C3×#2544C2semidirect
4. C3×#2028C2semidirect
5. C3×#2027C2semidirect
6. C3×#2007C2semidirect
7. C3×#2006C2semidirect
8. C3×#2004C2semidirect
9. C3×#2002C2semidirect
10. C3×#1806C2semidirect
11. C3×#1800C2semidirect
12. #29682C3direct (N × H)
48. #3460C2×C3semidirect
49. #3040C2×C3semidirect
50. #2544C2×C3semidirect
51. #2028C2×C3semidirect
52. #2027C2×C3semidirect
53. #2007C2×C3semidirect
54. #2006C2×C3semidirect
55. #2004C2×C3semidirect
56. #2002C2×C3semidirect
57. #1806C2×C3semidirect
58. #1800C2×C3semidirect
98. C3×C4×Q8D4semidirect
99. C3×C4×Q8D4semidirect
105. C3×C4×D4D4semidirect
106. C3×C4×D4D4semidirect
286. C4×Q8C3×D4semidirect
287. C4×Q8C3×D4semidirect
293. C4×D4C3×D4semidirect
294. C4×D4C3×D4semidirect
588. C3#29682direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34602
C3×#30402
C3×#25442
C3×#20282
C3×#20272
C3×#20072
C3×#20062
C3×#20042
C3×#20022
C3×#18062
C3×#18002
#296823