Finite Group Databasebeta

← C3×C4×#483 #10640180

Subgroups — C3×C4×#483 #10640180

Order 768.

976 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#483C211yesmaximal, characteristic
2C3×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
4C3×#1927C212yesmaximal, complemented
5C3×#1920C212yesmaximal, complemented
6C3×#1872C211yesmaximal, characteristic
7C3×#1853C211yesmaximal, characteristic
8C3×#1823C211yesmaximal, characteristic
9C3×C4×#106C212yesmaximal, complemented
10C3×C4×#103C212yesmaximal, complemented
11C3×C4×C22⋊C8C211yesmaximal, characteristic
12C4×#483C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42—21no—
18C22×C3×C42C2211yescharacteristic
19C3×#483C418yesdirect product factor
20C2×C3×#106C2212yes—
21C2×C3×#106C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C4×C8C2212yes—
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#387—21no—
30C3×#385—21no—
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4—21no—
33C3×C4×C4⋊C4—21no—
34C3×C4×C4⋊C4—21no—
35C3×C4×C4⋊C4C2212yes—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C4⋊C4C2212yes—
38C3×C4×C22⋊C4—21no—
39C2×C3×#95—21no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C2×#483C2×C311yescharacteristic
46C4×C22⋊Q8C2×C311yescharacteristic
47C2×C4×C4⋊C4C2×C311yescharacteristic
48#1927C2×C312yescomplemented
49#1920C2×C312yescomplemented
50#1872C2×C311yescharacteristic
51#1853C2×C311yescharacteristic
52#1823C2×C311yescharacteristic
53C4×#106C2×C312yescomplemented
54C4×#103C2×C312yescomplemented
55C4×C22⋊C8C2×C311yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C23×C3×C4—21no—
58C23×C3×C4—21no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×C3×C8C2312yes—
63C22×C3×C8C2312yes—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8C2×C414yes—
66C3×C4×Q8—24no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2×C414yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—24no—
83C2×C3×C42—22no—
84C2×C3×C42—21no—
85C2×C3×C42—21no—
86C2×C3×C42—22no—
87C2×C3×C42—44no—
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#106—24no—
93C3×#106C2×C418yescomplemented
94C3×#103—24no—
95C3×#103C2×C418yescomplemented
96C3×C22⋊C8—22no—
97C3×C22⋊C8C2×C414yes—
98C3×C4×C8—24no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—21no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C2×C22⋊Q8C22×C311yescharacteristic
107C2×C4×Q8C22×C311yescharacteristic
108C22×C4⋊C4C22×C311yescharacteristic
109C22×C4⋊C4C22×C311yescharacteristic
110C22×C42—21no—
111C22×C42C22×C311yescharacteristic
112#483C3×C418yesdirect product factor
113C2×#106C22×C312yes—
114C2×#106C22×C312yes—
115C2×#103C22×C312yes—
116C2×#103C22×C312yes—
117C2×C22⋊C8C22×C311yescharacteristic
118C2×C22⋊C8C22×C311yescharacteristic
119C2×C4×C8C22×C312yes—
120#391C22×C311yescharacteristic
121#389C22×C311yescharacteristic
122#387—21no—
123#385—21no—
124#380C22×C311yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C22×C312yes—
129C4×C4⋊C4C22×C312yes—
130C4×C4⋊C4C22×C311yescharacteristic
131C4×C22⋊C4—21no—
132C2×#95—21no—
133#340C22×C312yes—
134#340C22×C312yes—
135#340C22×C312yes—
136#340C22×C312yes—
137#336C22×C312yes—
138C24×C3C2×D411yescharacteristic
139C2×C3×Q8—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8C22×C412yes—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—41no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4SD1614yes—
163C22×C3×C4C22×C412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4#4112yes—
170C22×C3×C4C2411yescharacteristic
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8C22×C414yes—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4C22×C414yes—
188C3×C4⋊C4C22×C412yes—
189C3×C4⋊C4#4114yes—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—42no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—42no—
196C3×C42—44no—
197C3×C42—44no—
198C3×C42—42no—
199C3×C42#4114yes—
200C22×Q8C23×C311yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4C3×D411yescharacteristic
204C23×C4C3×D411yescharacteristic
205C23×C4C23×C311yescharacteristic
206C22×C8C23×C312yes—
207C22×C8C23×C312yes—
208C22⋊Q8—22no—
209C22⋊Q8C2×C3×C414yes—
210C4×Q8—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C2×C3×C414yes—
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C311yescharacteristic
223C2×C4⋊C4C23×C312yes—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—24no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—44no—
232C2×C42C3×D411yescharacteristic
233C2×C42C3×D411yescharacteristic
234C2×C42C23×C311yescharacteristic
235C2×C42C23×C311yescharacteristic
236#106—24no—
237#106C2×C3×C418yescomplemented
238#103—24no—
239#103C2×C3×C418yescomplemented
240C22⋊C8—22no—
241C22⋊C8C2×C3×C414yes—
242C4×C8—24no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3C4×D412yes—
255C23×C3C2×SD1612yes—
256C23×C3#13512yes—
257C23×C3C22×D411yescharacteristic
258C3×Q8—24no—
259C3×Q8—42no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—28no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—21no—
273C2×C3×C4—24no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—44no—
277C2×C3×C4—48no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—48no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4C4×D412yes—
294C2×C3×C4#12312yes—
295C2×C3×C4#12312yes—
296C2×C3×C4C2×SD1612yes—
297C2×C3×C4C2×SD1612yes—
298C2×C3×C4#13712yes—
299C2×C3×C4C23×C411yescharacteristic
300C2×C3×C4C2×#4112yes—
301C2×C3×C4C2×#4112yes—
302C2×C3×C4C2×#4111yescharacteristic
303C3×C8—24no—
304C3×C8—42no—
305C24C2×C3×D411yescharacteristic
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8C22×C3×C412yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4C3×SD1614yes—
330C22×C4C22×C3×C412yes—
331C22×C4C2×C3×D411yescharacteristic
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D411yescharacteristic
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C3×#4112yes—
337C22×C4C24×C311yescharacteristic
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8C22×C3×C414yes—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4C22×C3×C414yes—
355C4⋊C4C22×C3×C412yes—
356C4⋊C4C3×#4114yes—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—42no—
363C42—44no—
364C42—44no—
365C42—42no—
366C42C3×#4114yes—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3C4×SD1614yes—
374C22×C3C2×C4×D411yescharacteristic
375C22×C3C2×#12311yescharacteristic
376C22×C3#52511yescharacteristic
377C22×C3C22×SD1611yescharacteristic
378C22×C3C2×#13511yescharacteristic
379C22×C3C2×#13711yescharacteristic
380C22×C3#57511yescharacteristic
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—41no—
385C3×C4—44no—
386C3×C4—44no—
387C3×C4—42no—
388C3×C4—42no—
389C3×C4—81no—
390C3×C4—81no—
391C3×C4#48314yesdirect product factor
392C3×C4C4×#4112yes—
393C3×C4#52912yes—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23C3×C4×D412yes—
399C23C2×C3×SD1612yes—
400C23C3×#13512yes—
401C23C22×C3×D411yescharacteristic
402Q8—24no—
403Q8—42no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—48no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4C3×C4×D412yes—
438C2×C4C3×#12312yes—
439C2×C4C3×#12312yes—
440C2×C4C2×C3×SD1612yes—
441C2×C4C2×C3×SD1612yes—
442C2×C4C3×#13712yes—
443C2×C4C23×C3×C411yescharacteristic
444C2×C4C2×C3×#4112yes—
445C2×C4C2×C3×#4112yes—
446C2×C4C2×C3×#4111yescharacteristic
447C8—24no—
448C8—42no—
449C2×C3—24no—
450C2×C3C4×#12311yescharacteristic
451C2×C3C2×C4×SD1611yescharacteristic
452C2×C3C4×#13711yescharacteristic
453C2×C3C2×#48311yescharacteristic
454C2×C3#309111yescharacteristic
455C2×C3#310111yescharacteristic
456C2×C3#310811yescharacteristic
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22C3×C4×SD1614yes—
464C22C2×C3×C4×D411yescharacteristic
465C22C2×C3×#12311yescharacteristic
466C22C3×#52511yescharacteristic
467C22C22×C3×SD1611yescharacteristic
468C22C2×C3×#13511yescharacteristic
469C22C2×C3×#13711yescharacteristic
470C22C3×#57511yescharacteristic
471C4—22no—
472C4—22no—
473C4—22no—
474C4—44no—
475C4—42no—
476C4—42no—
477C4—41no—
478C4—44no—
479C4—81no—
480C4—81no—
481C4C3×#48314yesdirect product factor
482C4C3×C4×#4112yes—
483C4C3×#52912yes—
484C3C4×#48311yescharacteristic, direct product factor, Sylow
485C2—24no—
486C2C3×C4×#12311yescharacteristic
487C2C2×C3×C4×SD1611yescharacteristic
488C2C3×C4×#13711yescharacteristic
489C2C2×C3×#48311yescharacteristic
490C2C3×#309111yescharacteristic
491C2C3×#310111yescharacteristic
492C2C3×#310811yescharacteristic
subgroupG/H
centerC22×C3×C4C2×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#1927C2semidirect
5. C3×#1920C2semidirect
9. C3×C4×#106C2semidirect
10. C3×C4×#103C2semidirect
12. C4×#483C3direct (N × H)
19. C3×#483C4direct (N × H)
48. #1927C2×C3semidirect
49. #1920C2×C3semidirect
53. C4×#106C2×C3semidirect
54. C4×#103C2×C3semidirect
93. C3×#106C2×C4semidirect
95. C3×#103C2×C4semidirect
112. #483C3×C4direct (N × H)
237. #106C2×C3×C4semidirect
239. #103C2×C3×C4semidirect
391. C3×C4#483direct (N × H)
481. C4C3×#483direct (N × H)
484. C3C4×#483direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4832
C3×C4×C22⋊Q82
C2×C3×C4×C4⋊C42
C3×#19272
C3×#19202
C3×#18722
C3×#18532
C3×#18232
C3×C4×#1062
C3×C4×#1032
C3×C4×C22⋊C82
C4×#4833