Finite Group Databasebeta

← C3×#32917 #10651524

Subgroups — C3×#32917 #10651524

Order 768.

1588 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic, complemented
2C2×C3×#463C212yesmaximal, complemented
3C2×C3×#450C211yesmaximal, characteristic, complemented
4C2×C3×#447C211yesmaximal, characteristic
5C3×#2620C211yesmaximal, characteristic, complemented
6C3×#1879C212yesmaximal, complemented
7C3×#1877C212yesmaximal, complemented
8C3×#1875C211yesmaximal, characteristic, complemented
9C3×#1873C211yesmaximal, characteristic, complemented
10C3×#1802C212yesmaximal, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#32917C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C22×C3×D8C2212yes—
16C2×C3×#124C2211yescharacteristic, complemented
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C3×#518—28no—
20C2×C3×C4×D4—22no—
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2212yes—
23C2×C3×#117C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C22×C3×C22⋊C4—22no—
26C3×#463—28no—
27C3×#450—24no—
28C3×#447—24no—
29C2×C3×C4⋊C8C2212yescomplemented
30C2×C3×#103C2212yescomplemented
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yescomplemented
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#396—22no—
37C3×#394—22no—
38C3×#391C2211yescharacteristic, complemented
39C3×#390C2211yescharacteristic, complemented
40C3×#384C2211yescharacteristic, complemented
41C3×#340C2212yescomplemented
42C3×#340C2212yescomplemented
43C3×#336C2212yescomplemented
44C2×#518C2×C311yescharacteristic, complemented
45C2×#463C2×C312yescomplemented
46C2×#450C2×C311yescharacteristic, complemented
47C2×#447C2×C311yescharacteristic
48#2620C2×C311yescharacteristic, complemented
49#1879C2×C312yescomplemented
50#1877C2×C312yescomplemented
51#1875C2×C311yescharacteristic, complemented
52#1873C2×C311yescharacteristic, complemented
53#1802C2×C312yescomplemented
54#1798C2×C311yescharacteristic, complemented
55C25×C3—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—24no—
58C22×C3×D4—24no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4D414yes—
63C22×C3×D4C2311yescharacteristic
64C22×C3×D4C2312yes—
65C23×C3×C4—22no—
66C23×C3×C4—22no—
67C23×C3×C4C2311yescharacteristic
68C2×C3×SD16—24no—
69C2×C3×SD16—24no—
70C2×C3×SD16—24no—
71C2×C3×D8—24no—
72C2×C3×D8—24no—
73C2×C3×D8—24no—
74C22×C3×C8C2312yes—
75C22×C3×C8C2312yes—
76C3×#124—24no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×C4×D4—216no—
82C3×C4×D4—24no—
83C3×C4×D4—24no—
84C3×C4×D4—24no—
85C3×C4×D4D418yescomplemented
86C3×#117—24no—
87C3×#117—24no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4D412yes—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4—216no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4D412yes—
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C3×C4⋊C8—24no—
107C3×#103—24no—
108C3×#102—24no—
109C3×#102—24no—
110C3×#102—24no—
111C3×C22⋊C8—22no—
112C3×C22⋊C8—22no—
113C3×#95—22no—
114C23×D4C22×C311yescharacteristic
115C22×SD16C22×C312yes—
116C22×D8C22×C312yes—
117C2×#124C22×C311yescharacteristic, complemented
118C2×C22⋊Q8C22×C311yescharacteristic
119C2×#121C22×C311yescharacteristic
120#518—28no—
121C2×C4×D4—22no—
122C2×C4×D4C22×C312yes—
123C2×C4×D4C22×C312yes—
124C2×#117C22×C311yescharacteristic
125C2×#117C22×C311yescharacteristic
126C22×C22⋊C4—22no—
127#463—28no—
128#450—24no—
129#447—24no—
130C2×C4⋊C8C22×C312yescomplemented
131C2×#103C22×C312yescomplemented
132C2×#102C22×C312yes—
133C2×#102C22×C312yes—
134C2×#102C22×C312yescomplemented
135C2×C22⋊C8C22×C311yescharacteristic
136C2×C22⋊C8C22×C311yescharacteristic
137#396—22no—
138#394—22no—
139#391C22×C311yescharacteristic, complemented
140#390C22×C311yescharacteristic, complemented
141#384C22×C311yescharacteristic, complemented
142#340C22×C312yescomplemented
143#340C22×C312yescomplemented
144#336C22×C312yescomplemented
145C24×C3—24no—
146C24×C3—24no—
147C24×C3—22no—
148C24×C3—24no—
149C24×C3—21no—
150C24×C3—82no—
151C24×C3—82no—
152C24×C3C2×D411yescharacteristic
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×D4—24no—
158C2×C3×D4—24no—
159C2×C3×D4—216no—
160C2×C3×D4—216no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4—82no—
171C2×C3×D4C2×D414yes—
172C2×C3×D4C2×D414yes—
173C2×C3×D4#4114yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—28no—
178C22×C3×C4—28no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—82no—
183C22×C3×C4—82no—
184C22×C3×C4—82no—
185C22×C3×C4—82no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4#4112yes—
191C22×C3×C4C2411yescharacteristic
192C3×SD16—48no—
193C3×D8—48no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—82no—
208C3×C4⋊C4C2×D414yes—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—28no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—44no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—82no—
217C3×C42—24no—
218C3×C42—22no—
219C3×C42C2×D414yes—
220C25—21no—
221C22×Q8C23×C311yescharacteristic
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4C3×D414yes—
228C22×D4C23×C311yescharacteristic
229C22×D4C23×C312yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4C23×C311yescharacteristic
233C2×SD16—24no—
234C2×SD16—24no—
235C2×SD16—24no—
236C2×D8—24no—
237C2×D8—24no—
238C2×D8—24no—
239C22×C8C23×C312yes—
240C22×C8C23×C312yes—
241#124—24no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244#121—22no—
245#121—22no—
246C4×D4—216no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4C3×D418yescomplemented
251#117—24no—
252#117—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4C3×D412yes—
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C4⋊C4C23×C312yes—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4C3×D412yes—
268C2×C42—21no—
269C2×C42C23×C311yescharacteristic
270C2×C42C23×C311yescharacteristic
271C4⋊C8—24no—
272#103—24no—
273#102—24no—
274#102—24no—
275#102—24no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278#95—22no—
279C23×C3—216no—
280C23×C3—21no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—21no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—84no—
297C23×C3—84no—
298C23×C3—88no—
299C23×C3—88no—
300C23×C3#12112yes—
301C23×C3C22×D411yescharacteristic
302C3×Q8—42no—
303C3×Q8—42no—
304C3×D4—216no—
305C3×D4—24no—
306C3×D4—24no—
307C3×D4—24no—
308C3×D4—42no—
309C3×D4—42no—
310C3×D4—84no—
311C3×D4#12118yes—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—88no—
332C2×C3×C4—88no—
333C2×C3×C4#12014yes—
334C2×C3×C4#12112yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D412yes—
337C2×C3×C4C2×#4112yes—
338C2×C3×C4C2×#4111yescharacteristic
339C3×C8—42no—
340C3×C8—42no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—21no—
346C24—82no—
347C24—82no—
348C24C2×C3×D411yescharacteristic
349C2×Q8—22no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—22no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—42no—
366C2×D4—82no—
367C2×D4C2×C3×D414yes—
368C2×D4C2×C3×D414yes—
369C2×D4C3×#4114yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4C2×C3×D412yes—
383C22×C4C2×C3×D412yes—
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D412yes—
386C22×C4C3×#4112yes—
387C22×C4C24×C311yescharacteristic
388SD16—48no—
389D8—48no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—42no—
403C4⋊C4—82no—
404C4⋊C4C2×C3×D414yes—
405C22⋊C4—24no—
406C22⋊C4—28no—
407C22⋊C4—24no—
408C22⋊C4—28no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—82no—
413C42—24no—
414C42—22no—
415C42C2×C3×D414yes—
416C22×C3—22no—
417C22×C3—28no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—816no—
429C22×C3—81no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3C2×#12011yescharacteristic
435C22×C3C2×#12111yescharacteristic
436C22×C3#52511yescharacteristic
437C22×C3#57612yes—
438C22×C3#57712yes—
439C3×C4—22no—
440C3×C4—24no—
441C3×C4—42no—
442C3×C4—44no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4—81no—
446C3×C4C2×#12112yes—
447C3×C4#52512yes—
448C23—216no—
449C23—22no—
450C23—22no—
451C23—21no—
452C23—24no—
453C23—24no—
454C23—21no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—84no—
466C23—84no—
467C23—88no—
468C23—88no—
469C23C3×#12112yes—
470C23C22×C3×D411yescharacteristic
471Q8—42no—
472Q8—42no—
473D4—216no—
474D4—24no—
475D4—24no—
476D4—24no—
477D4—42no—
478D4—42no—
479D4—84no—
480D4C3×#12118yes—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—88no—
501C2×C4—88no—
502C2×C4C3×#12014yes—
503C2×C4C3×#12112yes—
504C2×C4C22×C3×D411yescharacteristic
505C2×C4C22×C3×D412yes—
506C2×C4C2×C3×#4112yes—
507C2×C4C2×C3×#4111yescharacteristic
508C8—42no—
509C8—42no—
510C2×C3—42no—
511C2×C3—48no—
512C2×C3—161no—
513C2×C3#284711yescharacteristic
514C2×C3#301212yes—
515C2×C3#304111yescharacteristic
516C2×C3#304211yescharacteristic
517C2×C3#311812yes—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—28no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—44no—
526C22—44no—
527C22—42no—
528C22—44no—
529C22—44no—
530C22—816no—
531C22—82no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—82no—
536C22C2×C3×#12011yescharacteristic
537C22C2×C3×#12111yescharacteristic
538C22C3×#52511yescharacteristic
539C22C3×#57612yes—
540C22C3×#57712yes—
541C4—24no—
542C4—22no—
543C4—42no—
544C4—44no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4C2×C3×#12112yes—
549C4C3×#52512yes—
550C3#3291711yescharacteristic, direct product factor, Sylow
551C2—42no—
552C2—48no—
553C2—161no—
554C2C3×#284711yescharacteristic
555C2C3×#301212yes—
556C2C3×#304111yescharacteristic
557C2C3×#304211yescharacteristic
558C2C3×#311812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#518C2semidirect
2. C2×C3×#463C2semidirect
3. C2×C3×#450C2semidirect
5. C3×#2620C2semidirect
6. C3×#1879C2semidirect
7. C3×#1877C2semidirect
8. C3×#1875C2semidirect
9. C3×#1873C2semidirect
10. C3×#1802C2semidirect
11. C3×#1798C2semidirect
12. #32917C3direct (N × H)
16. C2×C3×#124C22semidirect
29. C2×C3×C4⋊C8C22semidirect
30. C2×C3×#103C22semidirect
33. C2×C3×#102C22semidirect
38. C3×#391C22semidirect
39. C3×#390C22semidirect
40. C3×#384C22semidirect
41. C3×#340C22semidirect
42. C3×#340C22semidirect
43. C3×#336C22semidirect
44. C2×#518C2×C3semidirect
45. C2×#463C2×C3semidirect
46. C2×#450C2×C3semidirect
48. #2620C2×C3semidirect
49. #1879C2×C3semidirect
50. #1877C2×C3semidirect
51. #1875C2×C3semidirect
52. #1873C2×C3semidirect
53. #1802C2×C3semidirect
54. #1798C2×C3semidirect
85. C3×C4×D4D4semidirect
117. C2×#124C22×C3semidirect
130. C2×C4⋊C8C22×C3semidirect
131. C2×#103C22×C3semidirect
134. C2×#102C22×C3semidirect
139. #391C22×C3semidirect
140. #390C22×C3semidirect
141. #384C22×C3semidirect
142. #340C22×C3semidirect
143. #340C22×C3semidirect
144. #336C22×C3semidirect
250. C4×D4C3×D4semidirect
550. C3#32917direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C2×C3×#4632
C2×C3×#4502
C2×C3×#4472
C3×#26202
C3×#18792
C3×#18772
C3×#18752
C3×#18732
C3×#18022
C3×#17982
#329173