Finite Group Databasebeta

← C3×#18316 #10636923

Subgroups — C3×#18316 #10636923

Order 768.

1516 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#534C211yesmaximal, characteristic, complemented
2C3×#3192C212yesmaximal, complemented
3C3×#2539C211yesmaximal, characteristic, complemented
4C3×#1613C214yesmaximal, complemented
5C3×#1595C214yesmaximal, complemented
6C3×#1507C212yesmaximal, complemented
7C3×#1462C211yesmaximal, characteristic, complemented
8#18316C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4—21no—
10C3×#534—24no—
11C3×#534—22no—
12C3×#534C2214yes—
13C2×C3×#123—21no—
14C2×C3×#123—22no—
15C2×C3×#121—21no—
16C2×C3×#121C2212yes—
17C2×C3×#121C2211yescharacteristic, complemented
18C2×C3×#120—22no—
19C2×C3×#120—21no—
20C22×C3×C22⋊C4C2211yescharacteristic
21C3×#481—22no—
22C3×#481C2214yescomplemented
23C3×#469—24no—
24C3×#447—22no—
25C3×#447C2214yescomplemented
26C3×#407C2212yescomplemented
27C3×#399C2212yescomplemented
28C3×#392C2212yescomplemented
29C3×#390—22no—
30C3×C4×C22⋊C4C2211yescharacteristic, complemented
31C3×#327C2218yescomplemented
32C3×#323C2214yescomplemented
33C2×#534C2×C311yescharacteristic, complemented
34#3192C2×C312yescomplemented
35#2539C2×C311yescharacteristic, complemented
36#1613C2×C314yescomplemented
37#1595C2×C314yescomplemented
38#1507C2×C312yescomplemented
39#1462C2×C311yescharacteristic, complemented
40C25×C3—21no—
41C25×C3D411yescharacteristic
42C22×C3×D4—22no—
43C22×C3×D4—24no—
44C22×C3×D4—22no—
45C22×C3×D4—24no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—22no—
49C22×C3×D4—48no—
50C22×C3×D4D412yescomplemented
51C22×C3×D4C2311yescharacteristic, complemented
52C23×C3×C4—21no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C2×C3×D8—24no—
56C2×C3×M16—22no—
57C3×#123—28no—
58C3×#123—28no—
59C3×#123—44no—
60C3×#121—24no—
61C3×#121—24no—
62C3×#121—22no—
63C3×#121—22no—
64C3×#121—22no—
65C3×#121—22no—
66C3×#121D414yescomplemented
67C3×#121C2314yes—
68C3×#120—28no—
69C3×#120—28no—
70C3×#120—44no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4D412yescomplemented
74C2×C3×C4⋊C4C2311yescharacteristic, complemented
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4D412yes—
84C2×C3×C22⋊C4C2312yes—
85C2×C3×C42—21no—
86C3×#106—24no—
87C3×#102—24no—
88C3×#102—24no—
89C3×C22⋊C8—22no—
90C3×C22⋊C8C2314yescomplemented
91C3×#95C2312yescomplemented
92C23×D4—21no—
93#534—24no—
94#534—22no—
95#534C22×C314yes—
96C2×#123—21no—
97C2×#123—22no—
98C2×#121—21no—
99C2×#121C22×C312yes—
100C2×#121C22×C311yescharacteristic, complemented
101C2×#120—22no—
102C2×#120—21no—
103C22×C22⋊C4C22×C311yescharacteristic
104#481—22no—
105#481C22×C314yescomplemented
106#469—24no—
107#447—22no—
108#447C22×C314yescomplemented
109#407C22×C312yescomplemented
110#399C22×C312yescomplemented
111#392C22×C312yescomplemented
112#390—22no—
113C4×C22⋊C4C22×C311yescharacteristic, complemented
114#327C22×C318yescomplemented
115#323C22×C314yescomplemented
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—24no—
119C24×C3—24no—
120C24×C3—22no—
121C24×C3—21no—
122C24×C3—44no—
123C24×C3—42no—
124C24×C3—81no—
125C24×C3—82no—
126C24×C3—82no—
127C24×C3C2×D411yescharacteristic
128C24×C3C2×D412yes—
129C2×C3×D4—28no—
130C2×C3×D4—24no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—44no—
136C2×C3×D4—48no—
137C2×C3×D4—416no—
138C2×C3×D4—48no—
139C2×C3×D4—48no—
140C2×C3×D4—48no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—44no—
146C2×C3×D4—81no—
147C2×C3×D4—82no—
148C2×C3×D4—82no—
149C2×C3×D4—82no—
150C2×C3×D4C2×D414yes—
151C22×C3×C4—24no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—44no—
158C22×C3×C4—42no—
159C22×C3×C4—81no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2411yescharacteristic
165C3×D8—84no—
166C3×M16—82no—
167C2×C3×C8—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4C2×D414yes—
173C3×C22⋊C4—28no—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—44no—
180C3×C22⋊C4—44no—
181C3×C22⋊C4—42no—
182C3×C42—81no—
183C25—21no—
184C25C3×D411yescharacteristic
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—48no—
193C22×D4C3×D412yescomplemented
194C22×D4C23×C311yescharacteristic, complemented
195C23×C4—21no—
196C23×C4C3×D411yescharacteristic
197C23×C4C23×C311yescharacteristic
198C2×D8—24no—
199C2×M16—22no—
200#123—28no—
201#123—28no—
202#123—44no—
203#121—24no—
204#121—24no—
205#121—22no—
206#121—22no—
207#121—22no—
208#121—22no—
209#121C3×D414yescomplemented
210#121C23×C314yes—
211#120—28no—
212#120—28no—
213#120—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4C3×D412yescomplemented
217C2×C4⋊C4C23×C311yescharacteristic, complemented
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4C3×D412yes—
227C2×C22⋊C4C23×C312yes—
228C2×C42—21no—
229#106—24no—
230#102—24no—
231#102—24no—
232C22⋊C8—22no—
233C22⋊C8C23×C314yescomplemented
234#95C23×C312yescomplemented
235C23×C3—24no—
236C23×C3—24no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—24no—
242C23×C3—41no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—44no—
247C23×C3—48no—
248C23×C3—44no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—44no—
254C23×C3—41no—
255C23×C3—42no—
256C23×C3—81no—
257C23×C3—81no—
258C23×C3—88no—
259C23×C3—84no—
260C23×C3—84no—
261C23×C3—816no—
262C23×C3#12011yescharacteristic
263C23×C3#12011yescharacteristic
264C23×C3AGL(1,ℤ/8)14yes—
265C23×C3C22×D411yescharacteristic
266C3×D4—44no—
267C3×D4—44no—
268C3×D4—416no—
269C3×D4—48no—
270C3×D4—48no—
271C3×D4—82no—
272C3×D4—84no—
273C3×D4—88no—
274C3×D4—82no—
275C3×D4—82no—
276C2×C3×C4—24no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—44no—
285C2×C3×C4—48no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—81no—
293C2×C3×C4#12012yes—
294C2×C3×C4C22×D412yes—
295C3×C8—44no—
296C24—21no—
297C24—21no—
298C24—24no—
299C24—24no—
300C24—22no—
301C24—21no—
302C24—44no—
303C24—42no—
304C24—81no—
305C24—82no—
306C24—82no—
307C24C2×C3×D411yescharacteristic
308C24C2×C3×D412yes—
309C2×D4—28no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—44no—
316C2×D4—48no—
317C2×D4—416no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—44no—
326C2×D4—81no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4C2×C3×D414yes—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—81no—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D412yes—
343C22×C4C2×C3×D412yes—
344C22×C4C24×C311yescharacteristic
345D8—84no—
346M16—82no—
347C2×C8—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—44no—
352C4⋊C4C2×C3×D414yes—
353C22⋊C4—28no—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C42—81no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—24no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—44no—
372C22×C3—44no—
373C22×C3—48no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—82no—
380C22×C3—88no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3—84no—
384C22×C3—816no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3#45312yes—
389C22×C3C2×#12011yescharacteristic
390C22×C3C2×AGL(1,ℤ/8)12yes—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—44no—
394C3×C4—44no—
395C3×C4—42no—
396C3×C4—81no—
397C23—24no—
398C23—24no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—24no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—48no—
410C23—44no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—42no—
417C23—41no—
418C23—81no—
419C23—81no—
420C23—88no—
421C23—84no—
422C23—84no—
423C23—816no—
424C23C3×#12011yescharacteristic
425C23C3×#12011yescharacteristic
426C23C3×AGL(1,ℤ/8)14yes—
427C23C22×C3×D411yescharacteristic
428D4—44no—
429D4—44no—
430D4—416no—
431D4—48no—
432D4—48no—
433D4—82no—
434D4—84no—
435D4—88no—
436D4—82no—
437D4—82no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4C3×#12012yes—
456C2×C4C22×C3×D412yes—
457C8—44no—
458C2×C3—22no—
459C2×C3—22no—
460C2×C3—41no—
461C2×C3—44no—
462C2×C3—84no—
463C2×C3—161no—
464C2×C3#300311yescharacteristic
465C2×C3C2×#45311yescharacteristic
466C2×C3#302611yescharacteristic
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—24no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—44no—
476C22—44no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—82no—
484C22—88no—
485C22—82no—
486C22—82no—
487C22—84no—
488C22—816no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22C3×#45312yes—
493C22C2×C3×#12011yescharacteristic
494C22C2×C3×AGL(1,ℤ/8)12yes—
495C4—22no—
496C4—22no—
497C4—44no—
498C4—44no—
499C4—42no—
500C4—81no—
501C3#1831611yescharacteristic, direct product factor, Sylow
502C2—22no—
503C2—22no—
504C2—41no—
505C2—44no—
506C2—84no—
507C2—161no—
508C2C3×#300311yescharacteristic
509C2C2×C3×#45311yescharacteristic
510C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C22 ▷ 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×#534C2semidirect
2. C3×#3192C2semidirect
3. C3×#2539C2semidirect
4. C3×#1613C2semidirect
5. C3×#1595C2semidirect
6. C3×#1507C2semidirect
7. C3×#1462C2semidirect
8. #18316C3direct (N × H)
17. C2×C3×#121C22semidirect
22. C3×#481C22semidirect
25. C3×#447C22semidirect
26. C3×#407C22semidirect
27. C3×#399C22semidirect
28. C3×#392C22semidirect
30. C3×C4×C22⋊C4C22semidirect
31. C3×#327C22semidirect
32. C3×#323C22semidirect
33. C2×#534C2×C3semidirect
34. #3192C2×C3semidirect
35. #2539C2×C3semidirect
36. #1613C2×C3semidirect
37. #1595C2×C3semidirect
38. #1507C2×C3semidirect
39. #1462C2×C3semidirect
50. C22×C3×D4D4semidirect
51. C22×C3×D4C23semidirect
66. C3×#121D4semidirect
73. C2×C3×C4⋊C4D4semidirect
74. C2×C3×C4⋊C4C23semidirect
90. C3×C22⋊C8C23semidirect
91. C3×#95C23semidirect
100. C2×#121C22×C3semidirect
105. #481C22×C3semidirect
108. #447C22×C3semidirect
109. #407C22×C3semidirect
110. #399C22×C3semidirect
111. #392C22×C3semidirect
113. C4×C22⋊C4C22×C3semidirect
114. #327C22×C3semidirect
115. #323C22×C3semidirect
193. C22×D4C3×D4semidirect
194. C22×D4C23×C3semidirect
209. #121C3×D4semidirect
216. C2×C4⋊C4C3×D4semidirect
217. C2×C4⋊C4C23×C3semidirect
233. C22⋊C8C23×C3semidirect
234. #95C23×C3semidirect
501. C3#18316direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5342
C3×#31922
C3×#25392
C3×#16132
C3×#15952
C3×#15072
C3×#14622
#183163