Finite Group Databasebeta

← C3×#32996 #10651603

Subgroups — C3×#32996 #10651603

Order 768.

1382 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×#520C211yesmaximal, characteristic, complemented
2C3×#3094C211yesmaximal, characteristic, complemented
3C3×#3006C211yesmaximal, characteristic
4C3×#2501C211yesmaximal, characteristic, complemented
5C3×#1883C214yesmaximal, complemented
6C3×#1859C214yesmaximal, complemented
7C3×#1826C211yesmaximal, characteristic, complemented
8C3×#1821C211yesmaximal, characteristic, complemented
9C3×#1789C211yesmaximal, characteristic, complemented
10#32996C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C2×C3×AGL(1,ℤ/8)C2212yes—
13C2×C3×#123—22no—
14C2×C3×#121C2211yescharacteristic
15C3×#520—22no—
16C3×#520—22no—
17C3×#520—22no—
18C3×#520C2214yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4—22no—
22C2×C3×#117—21no—
23C2×C3×#117C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C22×C3×C4⋊C4C2211yescharacteristic
26C3×#482—22no—
27C3×#481—21no—
28C3×#480—21no—
29C3×#449—22no—
30C3×#448—21no—
31C3×#447—21no—
32C3×#428C2212yescomplemented
33C2×C3×#104C2214yes—
34C3×#418C2212yescomplemented
35C3×#417C2212yes—
36C3×#407C2211yescharacteristic
37C3×#407C2211yescharacteristic
38C3×#399—22no—
39C3×#390C2212yescomplemented
40C3×#384C2212yescomplemented
41C3×C4×C22⋊C4C2211yescharacteristic
42C3×#343C2214yescomplemented
43C3×#339C2214yescomplemented
44C2×#520C2×C311yescharacteristic, complemented
45#3094C2×C311yescharacteristic, complemented
46#3006C2×C311yescharacteristic
47#2501C2×C311yescharacteristic, complemented
48#1883C2×C314yescomplemented
49#1859C2×C314yescomplemented
50#1826C2×C311yescharacteristic, complemented
51#1821C2×C311yescharacteristic, complemented
52#1789C2×C311yescharacteristic, complemented
53C2×C3×#41—28no—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41D412yes—
57C2×C3×#41C2312yes—
58C22×C3×Q8D411yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4—22no—
65C23×C3×C4D412yes—
66C23×C3×C4C2311yescharacteristic
67C3×AGL(1,ℤ/8)—28no—
68C2×C3×SD16—22no—
69C2×C3×D8—22no—
70C2×C3×M16C2312yes—
71C2×C3×M16C2312yes—
72C3×#123—48no—
73C3×#121—21no—
74C3×#121—21no—
75C3×#121—21no—
76C3×#121—21no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×C4×D4—28no—
80C3×C4×D4—24no—
81C3×C4×D4—24no—
82C3×C4×D4—28no—
83C3×#117—24no—
84C3×#117—21no—
85C3×#117—21no—
86C3×#117—24no—
87C3×#117D412yes—
88C3×#117C2312yes—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—24no—
96C2×C3×C4⋊C4—24no—
97C2×C3×C4⋊C4D412yes—
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—24no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C42—22no—
108C2×C3×C42—21no—
109C2×C3×C42C2312yes—
110C3×#107—22no—
111C3×#106—22no—
112C3×#104—28no—
113C3×#103—22no—
114C3×#102—22no—
115C3×#102—22no—
116C3×#102—22no—
117C3×C22⋊C8—21no—
118C3×C22⋊C8—21no—
119C3×C22⋊C8—21no—
120C3×C22⋊C8—21no—
121C3×#95—21no—
122C22×#41C22×C311yescharacteristic
123C2×AGL(1,ℤ/8)C22×C312yes—
124C2×#123—22no—
125C2×#121C22×C311yescharacteristic
126#520—22no—
127#520—22no—
128#520—22no—
129#520C22×C314yes—
130C2×C4×Q8—21no—
131C2×C4×D4—21no—
132C2×C4×D4—22no—
133C2×#117—21no—
134C2×#117C22×C311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4C22×C311yescharacteristic
137#482—22no—
138#481—21no—
139#480—21no—
140#449—22no—
141#448—21no—
142#447—21no—
143#428C22×C312yescomplemented
144C2×#104C22×C314yes—
145#418C22×C312yescomplemented
146#417C22×C312yes—
147#407C22×C311yescharacteristic
148#407C22×C311yescharacteristic
149#399—22no—
150#390C22×C312yescomplemented
151#384C22×C312yescomplemented
152C4×C22⋊C4C22×C311yescharacteristic
153#343C22×C314yescomplemented
154#339C22×C314yescomplemented
155C24×C3—21no—
156C24×C3—21no—
157C24×C3C2×D411yescharacteristic
158C3×#41—28no—
159C3×#41—28no—
160C3×#41—24no—
161C3×#41—24no—
162C3×#41—24no—
163C3×#41#4118yes—
164C2×C3×Q8—24no—
165C2×C3×Q8—21no—
166C2×C3×Q8C2×D412yes—
167C2×C3×D4—28no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—24no—
175C2×C3×D4—44no—
176C2×C3×D4—41no—
177C2×C3×D4—41no—
178C2×C3×D4C2×D412yes—
179C22×C3×C4—28no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—22no—
188C22×C3×C4—28no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—44no—
192C22×C3×C4—84no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D414yes—
199C22×C3×C4C2411yescharacteristic
200C3×SD16—44no—
201C3×D8—44no—
202C3×M16—24no—
203C3×M16—24no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C3×C4⋊C4—28no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—48no—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—48no—
223C3×C22⋊C4—41no—
224C3×C22⋊C4—41no—
225C3×C42—28no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—24no—
229C2×#41—28no—
230C2×#41—21no—
231C2×#41—21no—
232C2×#41C3×D412yes—
233C2×#41C23×C312yes—
234C22×Q8C3×D411yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C3×D411yescharacteristic
238C22×D4C23×C311yescharacteristic
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C3×D412yes—
242C23×C4C23×C311yescharacteristic
243AGL(1,ℤ/8)—28no—
244C2×SD16—22no—
245C2×D8—22no—
246C2×M16C23×C312yes—
247C2×M16C23×C312yes—
248#123—48no—
249#121—21no—
250#121—21no—
251#121—21no—
252#121—21no—
253C4×Q8—24no—
254C4×Q8—24no—
255C4×D4—28no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—28no—
259#117—24no—
260#117—21no—
261#117—21no—
262#117—24no—
263#117C3×D412yes—
264#117C23×C312yes—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4C3×D412yes—
274C2×C4⋊C4C3×D411yescharacteristic
275C2×C4⋊C4C3×D411yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C312yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—24no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42C23×C312yes—
286#107—22no—
287#106—22no—
288#104—28no—
289#103—22no—
290#102—22no—
291#102—22no—
292#102—22no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297#95—21no—
298C23×C3—24no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—41no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—42no—
309C23×C3#12011yescharacteristic
310C23×C3#12011yescharacteristic
311C23×C3C22×D411yescharacteristic
312C3×Q8—24no—
313C3×Q8—22no—
314C3×Q8#12314yescomplemented
315C3×D4—28no—
316C3×D4—24no—
317C3×D4—28no—
318C3×D4—22no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—82no—
322C3×D4—82no—
323C3×D4#12314yescomplemented
324C2×C3×C4—24no—
325C2×C3×C4—28no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—28no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—44no—
341C2×C3×C4—42no—
342C2×C3×C4—44no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—82no—
346C2×C3×C4—88no—
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4114yes—
352C3×C8—42no—
353C3×C8—42no—
354C24—21no—
355C24—21no—
356C24C2×C3×D411yescharacteristic
357#41—28no—
358#41—28no—
359#41—24no—
360#41—24no—
361#41—24no—
362#41C3×#4118yes—
363C2×Q8—24no—
364C2×Q8—21no—
365C2×Q8C2×C3×D412yes—
366C2×D4—28no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—24no—
374C2×D4—44no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4C2×C3×D412yes—
378C22×C4—28no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—44no—
391C22×C4—84no—
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D414yes—
398C22×C4C24×C311yescharacteristic
399SD16—44no—
400D8—44no—
401M16—24no—
402M16—24no—
403C2×C8—22no—
404C2×C8—22no—
405C4⋊C4—28no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—48no—
415C4⋊C4C2×C3×D412yes—
416C4⋊C4C2×C3×D412yes—
417C22⋊C4—28no—
418C22⋊C4—24no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—48no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C42—28no—
425C42—22no—
426C42—22no—
427C42—24no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—24no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—44no—
437C22×C3—48no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—84no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3—81no—
444C22×C3#45312yes—
445C22×C3C2×#12011yescharacteristic
446C22×C3#52512yes—
447C3×C4—21no—
448C3×C4—24no—
449C3×C4—21no—
450C3×C4—44no—
451C3×C4—44no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—81no—
455C3×C4C2×#12312yes—
456C3×C4#52512yes—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—42no—
468C23C3×#12011yescharacteristic
469C23C3×#12011yescharacteristic
470C23C22×C3×D411yescharacteristic
471Q8—24no—
472Q8—22no—
473Q8C3×#12314yescomplemented
474D4—28no—
475D4—24no—
476D4—28no—
477D4—22no—
478D4—42no—
479D4—42no—
480D4—82no—
481D4—82no—
482D4C3×#12314yescomplemented
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—82no—
505C2×C4—88no—
506C2×C4C3×#12011yescharacteristic
507C2×C4C3×#12011yescharacteristic
508C2×C4C22×C3×D411yescharacteristic
509C2×C4C22×C3×D411yescharacteristic
510C2×C4C2×C3×#4114yes—
511C8—42no—
512C8—42no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—41no—
516C2×C3—44no—
517C2×C3—161no—
518C2×C3#284911yescharacteristic
519C2×C3C2×#45311yescharacteristic
520C2×C3#302011yescharacteristic
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—24no—
529C22—44no—
530C22—48no—
531C22—42no—
532C22—42no—
533C22—84no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22C3×#45312yes—
538C22C2×C3×#12011yescharacteristic
539C22C3×#52512yes—
540C4—21no—
541C4—24no—
542C4—21no—
543C4—44no—
544C4—44no—
545C4—42no—
546C4—42no—
547C4—81no—
548C4C2×C3×#12312yes—
549C4C3×#52512yes—
550C3#3299611yescharacteristic, direct product factor, Sylow
551C2—22no—
552C2—22no—
553C2—41no—
554C2—44no—
555C2—161no—
556C2C3×#284911yescharacteristic
557C2C2×C3×#45311yescharacteristic
558C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#520C2semidirect
2. C3×#3094C2semidirect
4. C3×#2501C2semidirect
5. C3×#1883C2semidirect
6. C3×#1859C2semidirect
7. C3×#1826C2semidirect
8. C3×#1821C2semidirect
9. C3×#1789C2semidirect
10. #32996C3direct (N × H)
32. C3×#428C22semidirect
34. C3×#418C22semidirect
39. C3×#390C22semidirect
40. C3×#384C22semidirect
42. C3×#343C22semidirect
43. C3×#339C22semidirect
44. C2×#520C2×C3semidirect
45. #3094C2×C3semidirect
47. #2501C2×C3semidirect
48. #1883C2×C3semidirect
49. #1859C2×C3semidirect
50. #1826C2×C3semidirect
51. #1821C2×C3semidirect
52. #1789C2×C3semidirect
143. #428C22×C3semidirect
145. #418C22×C3semidirect
150. #390C22×C3semidirect
151. #384C22×C3semidirect
153. #343C22×C3semidirect
154. #339C22×C3semidirect
314. C3×Q8#123semidirect
323. C3×D4#123semidirect
473. Q8C3×#123semidirect
482. D4C3×#123semidirect
550. C3#32996direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C3×#30942
C3×#30062
C3×#25012
C3×#18832
C3×#18592
C3×#18262
C3×#18212
C3×#17892
#329963