Finite Group Databasebeta

← C3×#29683 #10648290

Subgroups — C3×#29683 #10648290

Order 768.

1056 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic, complemented
2C3×#3040C212yesmaximal, complemented
3C3×#2545C211yesmaximal, characteristic, complemented
4C3×#2028C211yesmaximal, characteristic, complemented
5C3×#2027C211yesmaximal, characteristic, complemented
6C3×#2007C211yesmaximal, characteristic, complemented
7C3×#2006C211yesmaximal, characteristic
8C3×#2005C212yesmaximal, complemented
9C3×#2003C212yesmaximal, complemented
10C3×#1806C212yesmaximal, complemented
11C3×#1801C211yesmaximal, characteristic, complemented
12#29683C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C2×C3×AGL(1,ℤ/8)C2212yes—
15C3×#544C2211yescharacteristic
16C3×#542—22no—
17C3×#541—22no—
18C3×#539—22no—
19C3×#537C2211yescharacteristic
20C3×#536C2212yes—
21C3×#533—22no—
22C2×C3×C4⋊Q8C2211yescharacteristic
23C3×#529—22no—
24C3×#529—22no—
25C2×C3×#124C2211yescharacteristic
26C2×C3×#124C2211yescharacteristic
27C3×C4×#41C2212yes—
28C2×C3×C4×Q8C2211yescharacteristic
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×#101C2212yes—
39C3×#396—22no—
40C3×#391C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#388—22no—
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3461C2×C311yescharacteristic, complemented
49#3040C2×C312yescomplemented
50#2545C2×C311yescharacteristic, complemented
51#2028C2×C311yescharacteristic, complemented
52#2027C2×C311yescharacteristic, complemented
53#2007C2×C311yescharacteristic, complemented
54#2006C2×C311yescharacteristic
55#2005C2×C312yescomplemented
56#2003C2×C312yescomplemented
57#1806C2×C312yescomplemented
58#1801C2×C311yescharacteristic, complemented
59C2×C3×#41C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C3×#137—42no—
64C3×#137—42no—
65C3×AGL(1,ℤ/8)—42no—
66C3×AGL(1,ℤ/8)—42no—
67C2×C3×Q16—22no—
68C2×C3×SD16—22no—
69C2×C3×SD16—22no—
70C2×C3×D8—22no—
71C2×C3×M16C2312yes—
72C2×C3×M16C2312yes—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8—42no—
76C3×#126—24no—
77C3×#126—24no—
78C3×#125—22no—
79C3×#125—24no—
80C3×#125—22no—
81C3×#124—22no—
82C3×#124—22no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124—42no—
86C3×#123—24no—
87C3×#123—24no—
88C3×C22⋊Q8—24no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—22no—
93C3×#121—22no—
94C3×#121—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×Q8D412yes—
99C3×C4×Q8D412yes—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4D412yes—
104C3×C4×D4D412yes—
105C3×#117—24no—
106C3×#117—22no—
107C3×#117C2312yes—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—22no—
110C2×C3×C4⋊C4D412yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C42—22no—
117C2×C3×C42D412yes—
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×#101—42no—
130C3×#95—21no—
131C3×#95—22no—
132C2×#137C22×C312yes—
133C2×AGL(1,ℤ/8)C22×C312yes—
134#544C22×C311yescharacteristic
135#542—22no—
136#541—22no—
137#539—22no—
138#537C22×C311yescharacteristic
139#536C22×C312yes—
140#533—22no—
141C2×C4⋊Q8C22×C311yescharacteristic
142#529—22no—
143#529—22no—
144C2×#124C22×C311yescharacteristic
145C2×#124C22×C311yescharacteristic
146C4×#41C22×C312yes—
147C2×C4×Q8C22×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×#101C22×C312yes—
158#396—22no—
159#391C22×C311yescharacteristic
160#390C22×C311yescharacteristic
161#388—22no—
162C4×C4⋊C4C22×C311yescharacteristic
163#343C22×C311yescharacteristic
164#343C22×C311yescharacteristic
165#339C22×C312yes—
166#339C22×C312yes—
167C24×C3—21no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—21no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8C2×D412yes—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4C2×D412yes—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C3×Q16—44no—
201C3×SD16—44no—
202C3×SD16—44no—
203C3×D8—44no—
204C3×M16—22no—
205C3×M16—22no—
206C3×M16—22no—
207C3×M16—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4—44no—
221C3×C4⋊C4C2×D412yes—
222C3×C4⋊C4C2×D412yes—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—44no—
230C3×C22⋊C4—44no—
231C3×C22⋊C4—41no—
232C3×C22⋊C4—41no—
233C3×C22⋊C4—42no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42—22no—
240C3×C42—22no—
241C3×C42C2×D412yes—
242C3×C42C2×D412yes—
243C2×#41C23×C312yes—
244C22×Q8C23×C311yescharacteristic
245C22×Q8C23×C311yescharacteristic
246C22×D4C23×C311yescharacteristic
247#137—42no—
248#137—42no—
249AGL(1,ℤ/8)—42no—
250AGL(1,ℤ/8)—42no—
251C2×Q16—22no—
252C2×SD16—22no—
253C2×SD16—22no—
254C2×D8—22no—
255C2×M16C23×C312yes—
256C2×M16C23×C312yes—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—42no—
260#126—24no—
261#126—24no—
262#125—22no—
263#125—24no—
264#125—22no—
265#124—22no—
266#124—22no—
267#124—22no—
268#124—22no—
269#124—42no—
270#123—24no—
271#123—24no—
272C22⋊Q8—24no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277#121—22no—
278#121—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8C3×D412yes—
283C4×Q8C3×D412yes—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4C3×D412yes—
288C4×D4C3×D412yes—
289#117—24no—
290#117—22no—
291#117C23×C312yes—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4C3×D412yes—
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C42—22no—
301C2×C42C3×D412yes—
302C2×C42C23×C311yescharacteristic
303C2×C42C23×C311yescharacteristic
304C2×C42C23×C311yescharacteristic
305C4⋊C8—22no—
306C4⋊C8—22no—
307#104—42no—
308#104—42no—
309#103—22no—
310#103—22no—
311#102—22no—
312#102—22no—
313#101—42no—
314#95—21no—
315#95—22no—
316C23×C3—22no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3C22×D411yescharacteristic
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×Q8—82no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—82no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—44no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—82no—
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4#12011yescharacteristic
362C2×C3×C4#12012yes—
363C2×C3×C4C22×D412yes—
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×C3×C4H(5,𝔽2)12yes—
366C2×C3×C42−1+411yescharacteristic
367C3×C8—42no—
368C3×C8—42no—
369C24—21no—
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8C2×C3×D412yes—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4C2×C3×D412yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4C2×C3×D412yes—
398C22×C4C2×C3×D412yes—
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C24×C311yescharacteristic
402Q16—44no—
403SD16—44no—
404SD16—44no—
405D8—44no—
406M16—22no—
407M16—22no—
408M16—22no—
409M16—22no—
410C2×C8—22no—
411C2×C8—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4C2×C3×D412yes—
424C4⋊C4C2×C3×D412yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—42no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42C2×C3×D412yes—
444C42C2×C3×D412yes—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3C2×#12011yescharacteristic
456C22×C3#53511yescharacteristic
457C22×C3#53711yescharacteristic
458C3×C4—22no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4#53512yes—
470C3×C4#53712yes—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23C22×C3×D411yescharacteristic
478Q8—22no—
479Q8—22no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—82no—
485D4—22no—
486D4—22no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—82no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—82no—
515C2×C4C3×#12011yescharacteristic
516C2×C4C3×#12011yescharacteristic
517C2×C4C3×#12012yes—
518C2×C4C22×C3×D412yes—
519C2×C4C3×H(5,𝔽2)11yescharacteristic
520C2×C4C3×H(5,𝔽2)12yes—
521C2×C4C3×2−1+411yescharacteristic
522C8—42no—
523C8—42no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—82no—
527C2×C3—161no—
528C2×C3#261511yescharacteristic
529C2×C3#301811yescharacteristic
530C2×C3#302011yescharacteristic
531C22—21no—
532C22—21no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—82no—
537C22—82no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C2×C3×#12011yescharacteristic
542C22C3×#53511yescharacteristic
543C22C3×#53711yescharacteristic
544C4—22no—
545C4—21no—
546C4—21no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—81no—
554C4—81no—
555C4C3×#53512yes—
556C4C3×#53712yes—
557C3#2968311yescharacteristic, direct product factor, Sylow
558C2—21no—
559C2—21no—
560C2—82no—
561C2—161no—
562C2C3×#261511yescharacteristic
563C2C3×#301811yescharacteristic
564C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2006C2
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×#3461C2semidirect
2. C3×#3040C2semidirect
3. C3×#2545C2semidirect
4. C3×#2028C2semidirect
5. C3×#2027C2semidirect
6. C3×#2007C2semidirect
8. C3×#2005C2semidirect
9. C3×#2003C2semidirect
10. C3×#1806C2semidirect
11. C3×#1801C2semidirect
12. #29683C3direct (N × H)
48. #3461C2×C3semidirect
49. #3040C2×C3semidirect
50. #2545C2×C3semidirect
51. #2028C2×C3semidirect
52. #2027C2×C3semidirect
53. #2007C2×C3semidirect
55. #2005C2×C3semidirect
56. #2003C2×C3semidirect
57. #1806C2×C3semidirect
58. #1801C2×C3semidirect
557. C3#29683direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34612
C3×#30402
C3×#25452
C3×#20282
C3×#20272
C3×#20072
C3×#20062
C3×#20052
C3×#20032
C3×#18062
C3×#18012
#296833