Finite Group Databasebeta

← C3×#24108 #10642715

Subgroups — C3×#24108 #10642715

Order 768.

1062 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic, complemented
3C2×C3×#428C211yesmaximal, characteristic, complemented
4C3×#2487C211yesmaximal, characteristic
5C3×#2058C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#1894C211yesmaximal, characteristic, complemented
8C3×#1870C211yesmaximal, characteristic
9C3×#1863C212yesmaximal, complemented
10C3×#1854C212yesmaximal, complemented
11C3×#1825C211yesmaximal, characteristic, complemented
12#24108C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2211yescharacteristic
14C3×#556—24no—
15C3×#556—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#126—21no—
18C2×C3×#126—21no—
19C2×C3×#124—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C4×Q8—21no—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C22×C3×C4⋊C4C2211yescharacteristic
28C3×#483—24no—
29C3×#428—24no—
30C2×C3×#107C2212yes—
31C2×C3×#106C2212yes—
32C2×C3×#106C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×C22⋊C8C2211yescharacteristic
37C3×#400—21no—
38C3×#399—21no—
39C3×#389C2212yes—
40C3×#386—21no—
41C3×#385—21no—
42C3×#380C2212yes—
43C3×#380C2211yescharacteristic
44C3×#340C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#556C2×C311yescharacteristic
49C2×#483C2×C311yescharacteristic, complemented
50C2×#428C2×C311yescharacteristic, complemented
51#2487C2×C311yescharacteristic
52#2058C2×C312yescomplemented
53#2035C2×C312yescomplemented
54#1894C2×C311yescharacteristic, complemented
55#1870C2×C311yescharacteristic
56#1863C2×C312yescomplemented
57#1854C2×C312yescomplemented
58#1825C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4D411yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×M16—22no—
66C2×C3×M16—24no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#126—28no—
72C3×#126—28no—
73C3×#124—28no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8D414yes—
78C3×C4×Q8—28no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×#117—22no—
84C3×#117—22no—
85C3×#117—22no—
86C3×#117D414yes—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C22⋊C4D411yescharacteristic
109C2×C3×C42—21no—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C3×#107—24no—
114C3×#106—24no—
115C3×#106—24no—
116C3×#103—24no—
117C3×#103—24no—
118C3×C22⋊C8—22no—
119C3×C22⋊C8—24no—
120C3×#95—22no—
121C3×#95—21no—
122C3×#95—22no—
123C22×M16C22×C311yescharacteristic
124#556—24no—
125#556—24no—
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#126—21no—
128C2×#126—21no—
129C2×#124—21no—
130C2×C22⋊Q8C22×C311yescharacteristic
131C2×C22⋊Q8C22×C311yescharacteristic
132C2×C4×Q8—21no—
133C2×C4×Q8C22×C311yescharacteristic
134C2×C4×D4C22×C311yescharacteristic
135C2×#117C22×C311yescharacteristic
136C2×#117C22×C311yescharacteristic
137C22×C4⋊C4C22×C311yescharacteristic
138#483—24no—
139#428—24no—
140C2×#107C22×C312yes—
141C2×#106C22×C312yes—
142C2×#106C22×C312yes—
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C312yes—
147#400—21no—
148#399—21no—
149#389C22×C312yes—
150#386—21no—
151#385—21no—
152#380C22×C311yescharacteristic
153#380C22×C312yes—
154#340C22×C312yes—
155#340C22×C312yes—
156#340C22×C312yes—
157#336C22×C312yes—
158C24×C3C2×D411yescharacteristic
159C24×C3C2×D411yescharacteristic
160C2×C3×Q8—24no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—42no—
165C2×C3×Q8—42no—
166C2×C3×Q8C2×D412yes—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—41no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—42no—
183C22×C3×C4—82no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C3×M16—28no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8#4114yes—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4—82no—
212C3×C4⋊C4C2×D412yes—
213C3×C4⋊C4C2×D412yes—
214C3×C4⋊C4C2×D412yes—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—82no—
223C3×C42—24no—
224C3×C42—24no—
225C3×C42—24no—
226C3×C42—21no—
227C3×C42—21no—
228C22×Q8—21no—
229C22×Q8C23×C311yescharacteristic
230C22×D4C3×D411yescharacteristic
231C23×C4—21no—
232C23×C4C23×C311yescharacteristic
233C23×C4C23×C311yescharacteristic
234C2×M16—22no—
235C2×M16—24no—
236C22×C8C23×C312yes—
237C22×C8C23×C312yes—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240#126—28no—
241#126—28no—
242#124—28no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8C3×D414yes—
247C4×Q8—28no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×D4—22no—
251C4×D4—22no—
252#117—22no—
253#117—22no—
254#117—22no—
255#117C3×D414yes—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4C3×D411yescharacteristic
265C2×C4⋊C4C23×C312yes—
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4C3×D411yescharacteristic
277C2×C22⋊C4C3×D411yescharacteristic
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42C23×C311yescharacteristic
282#107—24no—
283#106—24no—
284#106—24no—
285#103—24no—
286#103—24no—
287C22⋊C8—22no—
288C22⋊C8—24no—
289#95—22no—
290#95—21no—
291#95—22no—
292C23×C3—22no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—22no—
301C23×C3#12012yes—
302C23×C3C22×D411yescharacteristic
303C3×Q8—24no—
304C3×Q8—48no—
305C3×Q8—41no—
306C3×Q8—41no—
307C3×D4—44no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—84no—
338C2×C3×C4#12012yes—
339C2×C3×C4#13712yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4112yes—
343C2×C3×C42−1+412yes—
344C3×C8—24no—
345C3×C8—42no—
346C24C2×C3×D411yescharacteristic
347C24C2×C3×D411yescharacteristic
348C2×Q8—24no—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8C2×C3×D412yes—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—82no—
372C22×C4C2×C3×D411yescharacteristic
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C2×C3×D412yes—
376C22×C4C2×C3×D412yes—
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C24×C311yescharacteristic
379M16—28no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8C3×#4114yes—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—42no—
399C4⋊C4—82no—
400C4⋊C4C2×C3×D412yes—
401C4⋊C4C2×C3×D412yes—
402C4⋊C4C2×C3×D412yes—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—42no—
410C22⋊C4—82no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—21no—
415C42—21no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3C2×#12011yescharacteristic
427C22×C3#54712yes—
428C22×C3C2×#13711yescharacteristic
429C22×C3#57511yescharacteristic
430C22×C3#57712yes—
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—41no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—41no—
438C3×C4—42no—
439C3×C4—82no—
440C3×C4—82no—
441C3×C4#52511yescharacteristic
442C3×C4#53611yescharacteristic
443C3×C4#54712yes—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—22no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—21no—
452C23—22no—
453C23C3×#12012yes—
454C23C22×C3×D411yescharacteristic
455Q8—24no—
456Q8—48no—
457Q8—41no—
458Q8—41no—
459D4—44no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—84no—
490C2×C4C3×#12012yes—
491C2×C4C3×#13712yes—
492C2×C4C22×C3×D411yescharacteristic
493C2×C4C22×C3×D411yescharacteristic
494C2×C4C2×C3×#4112yes—
495C2×C4C3×2−1+412yes—
496C8—24no—
497C8—42no—
498C2×C3—42no—
499C2×C3—81no—
500C2×C3#240611yescharacteristic
501C2×C3#300711yescharacteristic
502C2×C3#301011yescharacteristic
503C2×C3#335012yes—
504C2×C3#335612yes—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—42no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—42no—
515C22C2×C3×#12011yescharacteristic
516C22C3×#54712yes—
517C22C2×C3×#13711yescharacteristic
518C22C3×#57511yescharacteristic
519C22C3×#57712yes—
520C4—22no—
521C4—22no—
522C4—41no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—41no—
527C4—42no—
528C4—82no—
529C4—82no—
530C4C3×#52511yescharacteristic
531C4C3×#53611yescharacteristic
532C4C3×#54712yes—
533C3#2410811yescharacteristic, direct product factor, Sylow
534C2—42no—
535C2—81no—
536C2C3×#240611yescharacteristic
537C2C3×#300711yescharacteristic
538C2C3×#301011yescharacteristic
539C2C3×#335012yes—
540C2C3×#335612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#483C2semidirect
3. C2×C3×#428C2semidirect
5. C3×#2058C2semidirect
6. C3×#2035C2semidirect
7. C3×#1894C2semidirect
9. C3×#1863C2semidirect
10. C3×#1854C2semidirect
11. C3×#1825C2semidirect
12. #24108C3direct (N × H)
49. C2×#483C2×C3semidirect
50. C2×#428C2×C3semidirect
52. #2058C2×C3semidirect
53. #2035C2×C3semidirect
54. #1894C2×C3semidirect
56. #1863C2×C3semidirect
57. #1854C2×C3semidirect
58. #1825C2×C3semidirect
533. C3#24108direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4832
C2×C3×#4282
C3×#24872
C3×#20582
C3×#20352
C3×#18942
C3×#18702
C3×#18632
C3×#18542
C3×#18252
#241083