Finite Group Databasebeta

← C3×C4×#480 #10640179

Subgroups — C3×C4×#480 #10640179

Order 768.

1060 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#480C211yesmaximal, characteristic
2C3×C4×#121C211yesmaximal, characteristic
3C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#1925C212yesmaximal, complemented
5C3×#1918C212yesmaximal, complemented
6C3×#1874C211yesmaximal, characteristic
7C3×#1852C211yesmaximal, characteristic, complemented
8C3×#1823C211yesmaximal, characteristic, complemented
9C3×C4×#107C212yesmaximal, complemented
10C3×C4×#102C212yesmaximal, complemented
11C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
12C4×#480C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#121C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42—21no—
19C22×C3×C42C2211yescharacteristic
20C3×#480C418yesdirect product factor
21C2×C3×#107C2212yes—
22C2×C3×#107C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2212yes—
28C3×#390C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#388—21no—
31C3×#386—21no—
32C3×#380C2211yescharacteristic
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×#480C2×C311yescharacteristic
46C4×#121C2×C311yescharacteristic
47C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
48#1925C2×C312yescomplemented
49#1918C2×C312yescomplemented
50#1874C2×C311yescharacteristic
51#1852C2×C311yescharacteristic, complemented
52#1823C2×C311yescharacteristic, complemented
53C4×#107C2×C312yescomplemented
54C4×#102C2×C312yescomplemented
55C4×C22⋊C8C2×C311yescharacteristic, complemented
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×#121—22no—
67C3×#121C2×C414yes—
68C3×C4×D4—24no—
69C3×C4×D4—44no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2×C414yescomplemented
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42—24no—
87C2×C3×C42—22no—
88C2×C3×C42—21no—
89C2×C3×C42—22no—
90C2×C3×C42—44no—
91C2×C3×C42D411yescharacteristic
92C2×C3×C42D411yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C3×#107—24no—
96C3×#107C2×C418yescomplemented
97C3×#102—24no—
98C3×#102C2×C418yescomplemented
99C3×C22⋊C8—22no—
100C3×C22⋊C8C2×C414yescomplemented
101C3×C4×C8—24no—
102C3×#95—22no—
103C3×#95—22no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107C2×#121C22×C311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4C22×C311yescharacteristic
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C42—21no—
113C22×C42C22×C311yescharacteristic
114#480C3×C418yesdirect product factor
115C2×#107C22×C312yes—
116C2×#107C22×C312yes—
117C2×#102C22×C312yes—
118C2×#102C22×C312yes—
119C2×C22⋊C8C22×C311yescharacteristic
120C2×C22⋊C8C22×C311yescharacteristic
121C2×C4×C8C22×C312yes—
122#390C22×C311yescharacteristic
123#389C22×C311yescharacteristic
124#388—21no—
125#386—21no—
126#380C22×C311yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C22×C312yes—
130C4×C4⋊C4C22×C312yes—
131C4×C4⋊C4C22×C311yescharacteristic
132C4×C22⋊C4—21no—
133C2×#95—21no—
134#340C22×C312yes—
135#340C22×C312yes—
136#340C22×C312yes—
137#340C22×C312yes—
138#336C22×C312yes—
139C24×C3—21no—
140C24×C3C2×D411yescharacteristic
141C2×C3×D4—24no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—21no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4C22×C412yes—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—41no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4D814yes—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4#4112yes—
177C22×C3×C4C2411yescharacteristic
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8C22×C414yes—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4C22×C414yes—
193C3×C4⋊C4C22×C412yes—
194C3×C4⋊C4#4114yes—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—42no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—42no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—44no—
204C3×C42—44no—
205C3×C42—42no—
206C3×C42#4114yes—
207C22×D4—21no—
208C22×D4C23×C311yescharacteristic
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4C3×D411yescharacteristic
213C23×C4C3×D411yescharacteristic
214C23×C4C23×C311yescharacteristic
215C22×C8C23×C312yes—
216C22×C8C23×C312yes—
217#121—22no—
218#121C2×C3×C414yes—
219C4×D4—24no—
220C4×D4—44no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C2×C3×C414yescomplemented
227C2×C4⋊C4C23×C311yescharacteristic
228C2×C4⋊C4C23×C311yescharacteristic
229C2×C4⋊C4C23×C312yes—
230C2×C4⋊C4C23×C311yescharacteristic
231C2×C4⋊C4C23×C312yes—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—24no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—44no—
242C2×C42C3×D411yescharacteristic
243C2×C42C3×D411yescharacteristic
244C2×C42C23×C311yescharacteristic
245C2×C42C23×C311yescharacteristic
246#107—24no—
247#107C2×C3×C418yescomplemented
248#102—24no—
249#102C2×C3×C418yescomplemented
250C22⋊C8—22no—
251C22⋊C8C2×C3×C414yescomplemented
252C4×C8—24no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—21no—
257#95—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3C4×D412yes—
270C23×C3C2×D812yes—
271C23×C3#13512yes—
272C23×C3C22×D411yescharacteristic
273C3×D4—24no—
274C3×D4—48no—
275C3×D4—42no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—28no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—21no—
288C2×C3×C4—24no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—44no—
292C2×C3×C4—48no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—48no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—84no—
307C2×C3×C4C4×D412yes—
308C2×C3×C4#12312yes—
309C2×C3×C4#12312yes—
310C2×C3×C4C2×D812yes—
311C2×C3×C4C2×D812yes—
312C2×C3×C4#13712yes—
313C2×C3×C4C23×C411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4112yes—
316C2×C3×C4C2×#4111yescharacteristic
317C3×C8—24no—
318C3×C8—42no—
319C24—21no—
320C24C2×C3×D411yescharacteristic
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4C22×C3×C412yes—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4C3×D814yes—
350C22×C4C22×C3×C412yes—
351C22×C4C2×C3×D411yescharacteristic
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D411yescharacteristic
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C2×C3×D411yescharacteristic
356C22×C4C3×#4112yes—
357C22×C4C24×C311yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8C22×C3×C414yes—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4C22×C3×C414yes—
373C4⋊C4C22×C3×C412yes—
374C4⋊C4C3×#4114yes—
375C22⋊C4—24no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—44no—
384C42—44no—
385C42—42no—
386C42C3×#4114yes—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3C4×D814yes—
401C22×C3C2×C4×D411yescharacteristic
402C22×C3C2×#12311yescharacteristic
403C22×C3#52511yescharacteristic
404C22×C3C22×D811yescharacteristic
405C22×C3C2×#13511yescharacteristic
406C22×C3C2×#13711yescharacteristic
407C22×C3#57511yescharacteristic
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—41no—
412C3×C4—44no—
413C3×C4—44no—
414C3×C4—42no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4#48014yesdirect product factor
418C3×C4C4×#4112yes—
419C3×C4#52912yes—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23C3×C4×D412yes—
432C23C2×C3×D812yes—
433C23C3×#13512yes—
434C23C22×C3×D411yescharacteristic
435D4—24no—
436D4—48no—
437D4—42no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—48no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—84no—
469C2×C4C3×C4×D412yes—
470C2×C4C3×#12312yes—
471C2×C4C3×#12312yes—
472C2×C4C2×C3×D812yes—
473C2×C4C2×C3×D812yes—
474C2×C4C3×#13712yes—
475C2×C4C23×C3×C411yescharacteristic
476C2×C4C2×C3×#4112yes—
477C2×C4C2×C3×#4112yes—
478C2×C4C2×C3×#4111yescharacteristic
479C8—24no—
480C8—42no—
481C2×C3—24no—
482C2×C3—82no—
483C2×C3C4×#12311yescharacteristic
484C2×C3C2×C4×D811yescharacteristic
485C2×C3C4×#13711yescharacteristic
486C2×C3C2×#48011yescharacteristic
487C2×C3#309111yescharacteristic
488C2×C3#310011yescharacteristic
489C2×C3#310811yescharacteristic
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22C3×C4×D814yes—
504C22C2×C3×C4×D411yescharacteristic
505C22C2×C3×#12311yescharacteristic
506C22C3×#52511yescharacteristic
507C22C22×C3×D811yescharacteristic
508C22C2×C3×#13511yescharacteristic
509C22C2×C3×#13711yescharacteristic
510C22C3×#57511yescharacteristic
511C4—22no—
512C4—22no—
513C4—22no—
514C4—44no—
515C4—42no—
516C4—41no—
517C4—44no—
518C4—81no—
519C4—81no—
520C4C3×#48014yesdirect product factor
521C4C3×C4×#4112yes—
522C4C3×#52912yes—
523C3C4×#48011yescharacteristic, direct product factor, Sylow
524C2—24no—
525C2—82no—
526C2C3×C4×#12311yescharacteristic
527C2C2×C3×C4×D811yescharacteristic
528C2C3×C4×#13711yescharacteristic
529C2C2×C3×#48011yescharacteristic
530C2C3×#309111yescharacteristic
531C2C3×#310011yescharacteristic
532C2C3×#310811yescharacteristic
subgroupG/H
centerC22×C3×C4C2×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
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
3. C2×C3×C4×C4⋊C4C2semidirect
4. C3×#1925C2semidirect
5. C3×#1918C2semidirect
7. C3×#1852C2semidirect
8. C3×#1823C2semidirect
9. C3×C4×#107C2semidirect
10. C3×C4×#102C2semidirect
11. C3×C4×C22⋊C8C2semidirect
12. C4×#480C3direct (N × H)
20. C3×#480C4direct (N × H)
47. C2×C4×C4⋊C4C2×C3semidirect
48. #1925C2×C3semidirect
49. #1918C2×C3semidirect
51. #1852C2×C3semidirect
52. #1823C2×C3semidirect
53. C4×#107C2×C3semidirect
54. C4×#102C2×C3semidirect
55. C4×C22⋊C8C2×C3semidirect
75. C2×C3×C4⋊C4C2×C4semidirect
96. C3×#107C2×C4semidirect
98. C3×#102C2×C4semidirect
100. C3×C22⋊C8C2×C4semidirect
114. #480C3×C4direct (N × H)
226. C2×C4⋊C4C2×C3×C4semidirect
247. #107C2×C3×C4semidirect
249. #102C2×C3×C4semidirect
251. C22⋊C8C2×C3×C4semidirect
417. C3×C4#480direct (N × H)
520. C4C3×#480direct (N × H)
523. C3C4×#480direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4802
C3×C4×#1212
C2×C3×C4×C4⋊C42
C3×#19252
C3×#19182
C3×#18742
C3×#18522
C3×#18232
C3×C4×#1072
C3×C4×#1022
C3×C4×C22⋊C82
C4×#4803