Finite Group Databasebeta

← C3×#32997 #10651604

Subgroups — C3×#32997 #10651604

Order 768.

1326 conjugacy classes of subgroups.

The subgroup lattice has 556 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×#2504C211yesmaximal, characteristic, complemented
5C3×#1883C214yesmaximal, complemented
6C3×#1860C214yesmaximal, complemented
7C3×#1826C211yesmaximal, characteristic, complemented
8C3×#1821C211yesmaximal, characteristic, complemented
9C3×#1789C211yesmaximal, characteristic, complemented
10#32997C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C2×C3×AGL(1,ℤ/8)C2212yes—
13C2×C3×#121C2211yescharacteristic
14C3×#520—22no—
15C3×#520—22no—
16C3×#520—22no—
17C3×#520C2214yes—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4—22no—
21C2×C3×#117—21no—
22C2×C3×#117C2211yescharacteristic
23C22×C3×C4⋊C4—21no—
24C22×C3×C4⋊C4C2211yescharacteristic
25C3×#482—22no—
26C3×#481—21no—
27C3×#480—21no—
28C3×#449—22no—
29C3×#448—21no—
30C3×#447—21no—
31C3×#428C2212yescomplemented
32C2×C3×#104C2214yes—
33C3×#418C2212yescomplemented
34C3×#417C2212yes—
35C3×#407C2211yescharacteristic
36C3×#407C2211yescharacteristic
37C3×#397—22no—
38C3×#390C2212yescomplemented
39C3×#386—22no—
40C3×#382C2212yescomplemented
41C3×C4×C22⋊C4C2211yescharacteristic
42C3×#343C2214yescomplemented
43C3×#339C2214yescomplemented
44C2×#520C2×C311yescharacteristic, complemented
45#3094C2×C311yescharacteristic, complemented
46#3006C2×C311yescharacteristic
47#2504C2×C311yescharacteristic, complemented
48#1883C2×C314yescomplemented
49#1860C2×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×#121—21no—
73C3×#121—21no—
74C3×#121—21no—
75C3×#121—21no—
76C3×C4×Q8—24no—
77C3×C4×Q8—24no—
78C3×C4×D4—28no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×C4×D4—28no—
82C3×#117—24no—
83C3×#117—21no—
84C3×#117—21no—
85C3×#117—24no—
86C3×#117D412yes—
87C3×#117C2312yes—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—24no—
95C2×C3×C4⋊C4D412yes—
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—24no—
105C2×C3×C42—22no—
106C2×C3×C42—21no—
107C2×C3×C42C2312yes—
108C3×#107—22no—
109C3×#106—22no—
110C3×#104—28no—
111C3×#103—22no—
112C3×#102—22no—
113C3×#102—22no—
114C3×#102—22no—
115C3×C22⋊C8—21no—
116C3×C22⋊C8—21no—
117C3×C22⋊C8—21no—
118C3×C22⋊C8—21no—
119C3×#95—21no—
120C3×#95—24no—
121C22×#41C22×C311yescharacteristic
122C2×AGL(1,ℤ/8)C22×C312yes—
123C2×#121C22×C311yescharacteristic
124#520—22no—
125#520—22no—
126#520—22no—
127#520C22×C314yes—
128C2×C4×Q8—21no—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×#117—21no—
132C2×#117C22×C311yescharacteristic
133C22×C4⋊C4—21no—
134C22×C4⋊C4C22×C311yescharacteristic
135#482—22no—
136#481—21no—
137#480—21no—
138#449—22no—
139#448—21no—
140#447—21no—
141#428C22×C312yescomplemented
142C2×#104C22×C314yes—
143#418C22×C312yescomplemented
144#417C22×C312yes—
145#407C22×C311yescharacteristic
146#407C22×C311yescharacteristic
147#397—22no—
148#390C22×C312yescomplemented
149#386—22no—
150#382C22×C312yescomplemented
151C4×C22⋊C4C22×C311yescharacteristic
152#343C22×C314yescomplemented
153#339C22×C314yescomplemented
154C24×C3—21no—
155C24×C3—21no—
156C24×C3C2×D411yescharacteristic
157C3×#41—28no—
158C3×#41—28no—
159C3×#41—24no—
160C3×#41—24no—
161C3×#41—24no—
162C3×#41#4118yes—
163C2×C3×Q8—24no—
164C2×C3×Q8—21no—
165C2×C3×Q8C2×D412yes—
166C2×C3×D4—28no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—24no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—24no—
174C2×C3×D4—44no—
175C2×C3×D4—41no—
176C2×C3×D4—41no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—28no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—28no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—82no—
191C22×C3×C4—84no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2411yescharacteristic
199C3×SD16—44no—
200C3×D8—44no—
201C3×M16—24no—
202C3×M16—24no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C3×C4⋊C4—28no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4C2×D412yes—
215C3×C4⋊C4C2×D412yes—
216C3×C22⋊C4—28no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—41no—
221C3×C22⋊C4—41no—
222C3×C22⋊C4—84no—
223C3×C42—28no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—24no—
227C2×#41—28no—
228C2×#41—21no—
229C2×#41—21no—
230C2×#41C3×D412yes—
231C2×#41C23×C312yes—
232C22×Q8C3×D411yescharacteristic
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C3×D411yescharacteristic
236C22×D4C23×C311yescharacteristic
237C23×C4—21no—
238C23×C4—22no—
239C23×C4C3×D412yes—
240C23×C4C23×C311yescharacteristic
241AGL(1,ℤ/8)—28no—
242C2×SD16—22no—
243C2×D8—22no—
244C2×M16C23×C312yes—
245C2×M16C23×C312yes—
246#121—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×D4—28no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—28no—
256#117—24no—
257#117—21no—
258#117—21no—
259#117—24no—
260#117C3×D412yes—
261#117C23×C312yes—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4C3×D412yes—
270C2×C4⋊C4C3×D411yescharacteristic
271C2×C4⋊C4C3×D411yescharacteristic
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C312yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—24no—
279C2×C42—22no—
280C2×C42—21no—
281C2×C42C23×C312yes—
282#107—22no—
283#106—22no—
284#104—28no—
285#103—22no—
286#102—22no—
287#102—22no—
288#102—22no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293#95—21no—
294#95—24no—
295C23×C3—24no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—41no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3—42no—
306C23×C3#12011yescharacteristic
307C23×C3#12011yescharacteristic
308C23×C3C22×D411yescharacteristic
309C3×Q8—24no—
310C3×Q8—22no—
311C3×Q8#12314yes—
312C3×D4—28no—
313C3×D4—24no—
314C3×D4—28no—
315C3×D4—22no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—82no—
319C3×D4—82no—
320C3×D4#12314yes—
321C2×C3×C4—24no—
322C2×C3×C4—28no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—28no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—44no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—82no—
343C2×C3×C4—88no—
344C2×C3×C4#12011yescharacteristic
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4114yes—
349C3×C8—42no—
350C3×C8—42no—
351C24—21no—
352C24—21no—
353C24C2×C3×D411yescharacteristic
354#41—28no—
355#41—28no—
356#41—24no—
357#41—24no—
358#41—24no—
359#41C3×#4118yes—
360C2×Q8—24no—
361C2×Q8—21no—
362C2×Q8C2×C3×D412yes—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—24no—
371C2×D4—44no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4C2×C3×D412yes—
375C22×C4—28no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—82no—
388C22×C4—84no—
389C22×C4C2×C3×D411yescharacteristic
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D414yes—
395C22×C4C24×C311yescharacteristic
396SD16—44no—
397D8—44no—
398M16—24no—
399M16—24no—
400C2×C8—22no—
401C2×C8—22no—
402C4⋊C4—28no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4C2×C3×D412yes—
412C4⋊C4C2×C3×D412yes—
413C22⋊C4—28no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—42no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—84no—
420C42—28no—
421C42—22no—
422C42—22no—
423C42—24no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—24no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—44no—
433C22×C3—48no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—84no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3#45412yes—
441C22×C3C2×#12011yescharacteristic
442C22×C3#52512yes—
443C3×C4—21no—
444C3×C4—24no—
445C3×C4—21no—
446C3×C4—42no—
447C3×C4—44no—
448C3×C4—42no—
449C3×C4—82no—
450C3×C4—81no—
451C3×C4C2×#12312yes—
452C3×C4#52512yes—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—42no—
464C23C3×#12011yescharacteristic
465C23C3×#12011yescharacteristic
466C23C22×C3×D411yescharacteristic
467Q8—24no—
468Q8—22no—
469Q8C3×#12314yes—
470D4—28no—
471D4—24no—
472D4—28no—
473D4—22no—
474D4—42no—
475D4—42no—
476D4—82no—
477D4—82no—
478D4C3×#12314yes—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—82no—
501C2×C4—88no—
502C2×C4C3×#12011yescharacteristic
503C2×C4C3×#12011yescharacteristic
504C2×C4C22×C3×D411yescharacteristic
505C2×C4C22×C3×D411yescharacteristic
506C2×C4C2×C3×#4114yes—
507C8—42no—
508C8—42no—
509C2×C3—22no—
510C2×C3—22no—
511C2×C3—41no—
512C2×C3—44no—
513C2×C3—161no—
514C2×C3#284911yescharacteristic
515C2×C3C2×#45411yescharacteristic
516C2×C3#301911yescharacteristic
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—24no—
525C22—44no—
526C22—48no—
527C22—42no—
528C22—42no—
529C22—84no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22C3×#45412yes—
534C22C2×C3×#12011yescharacteristic
535C22C3×#52512yes—
536C4—21no—
537C4—24no—
538C4—21no—
539C4—42no—
540C4—44no—
541C4—42no—
542C4—82no—
543C4—81no—
544C4C2×C3×#12312yes—
545C4C3×#52512yes—
546C3#3299711yescharacteristic, direct product factor, Sylow
547C2—22no—
548C2—22no—
549C2—41no—
550C2—44no—
551C2—161no—
552C2C3×#284911yescharacteristic
553C2C2×C3×#45411yescharacteristic
554C2C3×#301911yescharacteristic
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×#2504C2semidirect
5. C3×#1883C2semidirect
6. C3×#1860C2semidirect
7. C3×#1826C2semidirect
8. C3×#1821C2semidirect
9. C3×#1789C2semidirect
10. #32997C3direct (N × H)
31. C3×#428C22semidirect
33. C3×#418C22semidirect
38. C3×#390C22semidirect
40. C3×#382C22semidirect
42. C3×#343C22semidirect
43. C3×#339C22semidirect
44. C2×#520C2×C3semidirect
45. #3094C2×C3semidirect
47. #2504C2×C3semidirect
48. #1883C2×C3semidirect
49. #1860C2×C3semidirect
50. #1826C2×C3semidirect
51. #1821C2×C3semidirect
52. #1789C2×C3semidirect
141. #428C22×C3semidirect
143. #418C22×C3semidirect
148. #390C22×C3semidirect
150. #382C22×C3semidirect
152. #343C22×C3semidirect
153. #339C22×C3semidirect
546. C3#32997direct (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×#25042
C3×#18832
C3×#18602
C3×#18262
C3×#18212
C3×#17892
#329973