Finite Group Databasebeta

← C3×#23923 #10642530

Subgroups — C3×#23923 #10642530

Order 768.

1948 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#525C212yesmaximal, complemented
2C2×C3×#452C212yesmaximal, complemented
3C2×C3×#449C212yesmaximal, complemented
4C3×#2001C211yesmaximal, characteristic, complemented
5C3×#2000C212yesmaximal, complemented
6C3×#1999C211yesmaximal, characteristic, complemented
7C3×#1875C212yesmaximal, complemented
8C3×#1873C212yesmaximal, complemented
9C3×#1779C211yesmaximal, characteristic, complemented
10#23923C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C24×C3×C4C2211yescharacteristic
13C22×C3×Q16C2211yescharacteristic, complemented
14C22×C3×SD16C2212yescomplemented
15C22×C3×D8C2211yescharacteristic, complemented
16C3×#525—28no—
17C3×#525—28no—
18C2×C3×#123—22no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121C2212yes—
22C2×C3×#121C2212yes—
23C2×C3×#120—22no—
24C2×C3×C4×D4—22no—
25C2×C3×C4×D4—22no—
26C2×C3×#117C2212yes—
27C3×#452—28no—
28C3×#449—28no—
29C2×C3×#103C2212yescomplemented
30C2×C3×#103C2212yescomplemented
31C2×C3×#102C2212yescomplemented
32C2×C3×#102C2212yescomplemented
33C2×C3×C22⋊C8C2212yes—
34C2×C3×C22⋊C8C2212yes—
35C2×C3×C22⋊C8C2211yescharacteristic
36C2×C3×C22⋊C8C2211yescharacteristic
37C3×#340C2211yescharacteristic, complemented
38C3×#340C2212yescomplemented
39C3×#340C2211yescharacteristic, complemented
40C2×#525C2×C312yescomplemented
41C2×#452C2×C312yescomplemented
42C2×#449C2×C312yescomplemented
43#2001C2×C311yescharacteristic, complemented
44#2000C2×C312yescomplemented
45#1999C2×C311yescharacteristic, complemented
46#1875C2×C312yescomplemented
47#1873C2×C312yescomplemented
48#1779C2×C311yescharacteristic, complemented
49C25×C3D411yescharacteristic
50C2×C3×#41—24no—
51C2×C3×#41—24no—
52C2×C3×#41—24no—
53C2×C3×#41—24no—
54C2×C3×#41—42no—
55C2×C3×#41—42no—
56C22×C3×Q8C2312yes—
57C22×C3×D4—22no—
58C22×C3×D4—22no—
59C22×C3×D4—22no—
60C22×C3×D4C2312yes—
61C23×C3×C4—24no—
62C23×C3×C4—24no—
63C23×C3×C4—22no—
64C23×C3×C4—41no—
65C23×C3×C4—41no—
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D412yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2312yes—
70C2×C3×Q16—24no—
71C2×C3×Q16—41no—
72C2×C3×SD16—24no—
73C2×C3×SD16—24no—
74C2×C3×SD16—42no—
75C2×C3×D8—24no—
76C2×C3×D8—41no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2312yes—
79C22×C3×C8C2311yescharacteristic
80C3×#123—216no—
81C3×C22⋊Q8—24no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—24no—
84C3×C22⋊Q8D418yescomplemented
85C3×#121—24no—
86C3×#121—24no—
87C3×#121—24no—
88C3×#121D418yescomplemented
89C3×#120—216no—
90C3×C4×D4—216no—
91C3×C4×D4—216no—
92C3×#117—24no—
93C3×#117—24no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C42—22no—
104C3×#103—24no—
105C3×#103—24no—
106C3×#102—24no—
107C3×#102—24no—
108C3×C22⋊C8—24no—
109C3×C22⋊C8—24no—
110C3×C22⋊C8—41no—
111C3×C22⋊C8—41no—
112C22×#41C22×C312yes—
113C24×C4C22×C311yescharacteristic
114C22×Q16C22×C311yescharacteristic, complemented
115C22×SD16C22×C312yescomplemented
116C22×D8C22×C311yescharacteristic, complemented
117#525—28no—
118#525—28no—
119C2×#123—22no—
120C2×C22⋊Q8C22×C312yes—
121C2×C22⋊Q8C22×C312yes—
122C2×#121C22×C312yes—
123C2×#121C22×C312yes—
124C2×#120—22no—
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×#117C22×C312yes—
128#452—28no—
129#449—28no—
130C2×#103C22×C312yescomplemented
131C2×#103C22×C312yescomplemented
132C2×#102C22×C312yescomplemented
133C2×#102C22×C312yescomplemented
134C2×C22⋊C8C22×C312yes—
135C2×C22⋊C8C22×C312yes—
136C2×C22⋊C8C22×C311yescharacteristic
137C2×C22⋊C8C22×C311yescharacteristic
138#340C22×C311yescharacteristic, complemented
139#340C22×C312yescomplemented
140#340C22×C311yescharacteristic, complemented
141C24×C3—24no—
142C24×C3—22no—
143C24×C3—22no—
144C24×C3—42no—
145C24×C3—41no—
146C24×C3—41no—
147C24×C3C2×D412yes—
148C24×C3C2×D411yescharacteristic
149C3×#41—48no—
150C3×#41—48no—
151C3×#41—48no—
152C3×#41—48no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8C2×D414yes—
159C2×C3×D4—28no—
160C2×C3×D4—28no—
161C2×C3×D4—28no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—44no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4—44no—
171C2×C3×D4—44no—
172C2×C3×D4—44no—
173C2×C3×D4C2×D414yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—28no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—24no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—41no—
190C22×C3×C4—42no—
191C22×C3×C4—44no—
192C22×C3×C4—48no—
193C22×C3×C4—44no—
194C22×C3×C4—44no—
195C22×C3×C4—44no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D414yes—
199C22×C3×C4C2×D414yes—
200C22×C3×C4C2411yescharacteristic
201C3×Q16—44no—
202C3×SD16—48no—
203C3×D8—44no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—21no—
209C2×C3×C8—22no—
210C2×C3×C8—21no—
211C3×C4⋊C4—28no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4C2×D414yes—
216C3×C4⋊C4C2×D414yes—
217C3×C22⋊C4—28no—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—28no—
222C3×C42—28no—
223C25C3×D411yescharacteristic
224C2×#41—24no—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—42no—
229C2×#41—42no—
230C22×Q8C23×C312yes—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4C23×C312yes—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—41no—
239C23×C4—41no—
240C23×C4C3×D411yescharacteristic
241C23×C4C3×D412yes—
242C23×C4C23×C311yescharacteristic
243C23×C4C23×C312yes—
244C2×Q16—24no—
245C2×Q16—41no—
246C2×SD16—24no—
247C2×SD16—24no—
248C2×SD16—42no—
249C2×D8—24no—
250C2×D8—41no—
251C22×C8C23×C311yescharacteristic
252C22×C8C23×C312yes—
253C22×C8C23×C311yescharacteristic
254#123—216no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8C3×D418yescomplemented
259#121—24no—
260#121—24no—
261#121—24no—
262#121C3×D418yescomplemented
263#120—216no—
264C4×D4—216no—
265C4×D4—216no—
266#117—24no—
267#117—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C23×C312yes—
271C2×C4⋊C4C23×C312yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C42—22no—
278#103—24no—
279#103—24no—
280#102—24no—
281#102—24no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—44no—
298C23×C3—48no—
299C23×C3—44no—
300C23×C3—48no—
301C23×C3—44no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3#12014yes—
309C23×C3#13512yes—
310C23×C3C22×D411yescharacteristic
311C3×Q8—24no—
312C3×Q8—24no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×D4—24no—
316C3×D4—24no—
317C3×D4—416no—
318C3×D4—416no—
319C3×D4—416no—
320C3×D4—42no—
321C3×D4—42no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—21no—
333C2×C3×C4—44no—
334C2×C3×C4—48no—
335C2×C3×C4—416no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4#12014yes—
346C2×C3×C4C22×D412yes—
347C2×C3×C4C22×D412yes—
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349C2×C3×C4H(5,𝔽2)11yescharacteristic
350C3×C8—41no—
351C3×C8—42no—
352C3×C8—41no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—41no—
357C24—42no—
358C24—41no—
359C24C2×C3×D412yes—
360C24C2×C3×D411yescharacteristic
361#41—48no—
362#41—48no—
363#41—48no—
364#41—48no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8C2×C3×D414yes—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4C2×C3×D414yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D412yes—
410C22×C4C2×C3×D414yes—
411C22×C4C2×C3×D414yes—
412C22×C4C24×C311yescharacteristic
413Q16—44no—
414SD16—48no—
415D8—44no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—21no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4C2×C3×D414yes—
428C4⋊C4C2×C3×D414yes—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—28no—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C42—28no—
435C22×C3—24no—
436C22×C3—22no—
437C22×C3—24no—
438C22×C3—24no—
439C22×C3—24no—
440C22×C3—416no—
441C22×C3—41no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—41no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3—82no—
454C22×C3—82no—
455C22×C3C2×#12012yes—
456C22×C3#53411yescharacteristic
457C22×C3C2×#13511yescharacteristic
458C22×C3#57511yescharacteristic
459C22×C3#57512yes—
460C3×C4—24no—
461C3×C4—41no—
462C3×C4—44no—
463C3×C4—44no—
464C3×C4—44no—
465C3×C4D4211yescharacteristic
466C3×C4D4212yes—
467C3×C4D4211yescharacteristic
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—21no—
478C23—24no—
479C23—44no—
480C23—48no—
481C23—44no—
482C23—48no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23C3×#12014yes—
491C23C3×#13512yes—
492C23C22×C3×D411yescharacteristic
493Q8—24no—
494Q8—24no—
495Q8—42no—
496Q8—42no—
497D4—24no—
498D4—24no—
499D4—416no—
500D4—416no—
501D4—416no—
502D4—42no—
503D4—42no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—416no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4C3×#12014yes—
528C2×C4C22×C3×D412yes—
529C2×C4C22×C3×D412yes—
530C2×C4C3×H(5,𝔽2)11yescharacteristic
531C2×C4C3×H(5,𝔽2)11yescharacteristic
532C8—41no—
533C8—42no—
534C8—41no—
535C2×C3—44no—
536C2×C3—42no—
537C2×C3—84no—
538C2×C3#240311yescharacteristic
539C2×C3#300212yes—
540C2×C3#300512yes—
541C2×C3#319111yescharacteristic
542C2×C3#319711yescharacteristic
543C22—24no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—416no—
549C22—41no—
550C22—42no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22C2×C3×#12012yes—
564C22C3×#53411yescharacteristic
565C22C2×C3×#13511yescharacteristic
566C22C3×#57511yescharacteristic
567C22C3×#57512yes—
568C4—24no—
569C4—41no—
570C4—44no—
571C4—44no—
572C4—44no—
573C4C3×D4211yescharacteristic
574C4C3×D4212yes—
575C4C3×D4211yescharacteristic
576C3#2392311yescharacteristic, direct product factor, Sylow
577C2—44no—
578C2—42no—
579C2—84no—
580C2C3×#240311yescharacteristic
581C2C3×#300212yes—
582C2C3×#300512yes—
583C2C3×#319111yescharacteristic
584C2C3×#319711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#525C2semidirect
2. C2×C3×#452C2semidirect
3. C2×C3×#449C2semidirect
4. C3×#2001C2semidirect
5. C3×#2000C2semidirect
6. C3×#1999C2semidirect
7. C3×#1875C2semidirect
8. C3×#1873C2semidirect
9. C3×#1779C2semidirect
10. #23923C3direct (N × H)
13. C22×C3×Q16C22semidirect
14. C22×C3×SD16C22semidirect
15. C22×C3×D8C22semidirect
29. C2×C3×#103C22semidirect
30. C2×C3×#103C22semidirect
31. C2×C3×#102C22semidirect
32. C2×C3×#102C22semidirect
37. C3×#340C22semidirect
38. C3×#340C22semidirect
39. C3×#340C22semidirect
40. C2×#525C2×C3semidirect
41. C2×#452C2×C3semidirect
42. C2×#449C2×C3semidirect
43. #2001C2×C3semidirect
44. #2000C2×C3semidirect
45. #1999C2×C3semidirect
46. #1875C2×C3semidirect
47. #1873C2×C3semidirect
48. #1779C2×C3semidirect
84. C3×C22⋊Q8D4semidirect
88. C3×#121D4semidirect
114. C22×Q16C22×C3semidirect
115. C22×SD16C22×C3semidirect
116. C22×D8C22×C3semidirect
130. C2×#103C22×C3semidirect
131. C2×#103C22×C3semidirect
132. C2×#102C22×C3semidirect
133. C2×#102C22×C3semidirect
138. #340C22×C3semidirect
139. #340C22×C3semidirect
140. #340C22×C3semidirect
258. C22⋊Q8C3×D4semidirect
262. #121C3×D4semidirect
576. C3#23923direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5252
C2×C3×#4522
C2×C3×#4492
C3×#20012
C3×#20002
C3×#19992
C3×#18752
C3×#18732
C3×#17792
#239233