Finite Group Databasebeta

← C3×#24110 #10642717

Subgroups — C3×#24110 #10642717

Order 768.

1088 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#552C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic, complemented
3C2×C3×#428C211yesmaximal, characteristic, complemented
4C3×#2486C211yesmaximal, characteristic
5C3×#2059C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#1894C211yesmaximal, characteristic, complemented
8C3×#1870C211yesmaximal, characteristic
9C3×#1863C212yesmaximal, complemented
10C3×#1852C212yesmaximal, complemented
11C3×#1825C211yesmaximal, characteristic, complemented
12#24110C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2211yescharacteristic
14C3×#552—24no—
15C3×#552—24no—
16C2×C3×#126—21no—
17C2×C3×#125—21no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×#123—21no—
20C2×C3×#123—21no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117—21no—
26C2×C3×#117C2211yescharacteristic
27C22×C3×C4⋊C4C2211yescharacteristic
28C22×C3×C4⋊C4C2211yescharacteristic
29C3×#484—24no—
30C3×#428—24no—
31C2×C3×#107C2212yes—
32C2×C3×#107C2212yes—
33C2×C3×#106C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C2×C3×C22⋊C8C2211yescharacteristic
38C3×#398—21no—
39C3×#397—21no—
40C3×#389C2212yes—
41C3×#381—21no—
42C3×#380C2212yes—
43C3×#380C2211yescharacteristic
44C3×#340C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#552C2×C311yescharacteristic
49C2×#484C2×C311yescharacteristic, complemented
50C2×#428C2×C311yescharacteristic, complemented
51#2486C2×C311yescharacteristic
52#2059C2×C312yescomplemented
53#2035C2×C312yescomplemented
54#1894C2×C311yescharacteristic, complemented
55#1870C2×C311yescharacteristic
56#1863C2×C312yescomplemented
57#1852C2×C312yescomplemented
58#1825C2×C311yescharacteristic, complemented
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4D411yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×M16—22no—
66C2×C3×M16—24no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×#126—28no—
70C3×#125—28no—
71C3×#125—22no—
72C3×#125—22no—
73C3×#123—28no—
74C3×#123—44no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8D414yes—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×#117—28no—
84C3×#117—22no—
85C3×#117D414yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C22⋊C4D411yescharacteristic
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42C2311yescharacteristic
114C3×#107—24no—
115C3×#107—24no—
116C3×#106—24no—
117C3×#103—24no—
118C3×#103—24no—
119C3×C22⋊C8—22no—
120C3×C22⋊C8—24no—
121C3×#95—22no—
122C3×#95—21no—
123C3×#95—22no—
124C22×M16C22×C311yescharacteristic
125#552—24no—
126#552—24no—
127C2×#126—21no—
128C2×#125—21no—
129C2×#125C22×C311yescharacteristic
130C2×#123—21no—
131C2×#123—21no—
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×C22⋊Q8C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×D4C22×C311yescharacteristic
136C2×#117—21no—
137C2×#117C22×C311yescharacteristic
138C22×C4⋊C4C22×C311yescharacteristic
139C22×C4⋊C4C22×C311yescharacteristic
140#484—24no—
141#428—24no—
142C2×#107C22×C312yes—
143C2×#107C22×C312yes—
144C2×#106C22×C312yes—
145C2×#103C22×C312yes—
146C2×#103C22×C312yes—
147C2×C22⋊C8C22×C311yescharacteristic
148C2×C22⋊C8C22×C312yes—
149#398—21no—
150#397—21no—
151#389C22×C312yes—
152#381—21no—
153#380C22×C311yescharacteristic
154#380C22×C312yes—
155#340C22×C312yes—
156#340C22×C312yes—
157#340C22×C312yes—
158#336C22×C312yes—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8C2×D412yes—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—41no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4—42no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2411yescharacteristic
193C3×M16—28no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8#4114yes—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—44no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4C2×D412yes—
218C3×C4⋊C4C2×D412yes—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—44no—
226C3×C42—24no—
227C3×C42—24no—
228C3×C42—21no—
229C3×C42—21no—
230C22×Q8C23×C311yescharacteristic
231C22×D4C3×D411yescharacteristic
232C23×C4—21no—
233C23×C4—21no—
234C23×C4C23×C311yescharacteristic
235C23×C4C23×C311yescharacteristic
236C2×M16—22no—
237C2×M16—24no—
238C22×C8C23×C312yes—
239C22×C8C23×C312yes—
240#126—28no—
241#125—28no—
242#125—22no—
243#125—22no—
244#123—28no—
245#123—44no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8C3×D414yes—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×D4—22no—
253C4×D4—22no—
254#117—28no—
255#117—22no—
256#117C3×D414yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4C3×D411yescharacteristic
270C2×C4⋊C4C23×C312yes—
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4C3×D411yescharacteristic
281C2×C22⋊C4C3×D411yescharacteristic
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C23×C311yescharacteristic
285#107—24no—
286#107—24no—
287#106—24no—
288#103—24no—
289#103—24no—
290C22⋊C8—22no—
291C22⋊C8—24no—
292#95—22no—
293#95—21no—
294#95—22no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3#12012yes—
304C23×C3AGL(1,ℤ/8)12yes—
305C23×C3C22×D411yescharacteristic
306C3×Q8—24no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×D4—44no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—48no—
326C2×C3×C4—42no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—84no—
342C2×C3×C4#12012yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C2×#4112yes—
346C2×C3×C42−1+412yes—
347C3×C8—24no—
348C3×C8—42no—
349C24C2×C3×D411yescharacteristic
350C24C2×C3×D411yescharacteristic
351C2×Q8—21no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8C2×C3×D412yes—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4C2×C3×D411yescharacteristic
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D412yes—
381C22×C4C2×C3×D412yes—
382C22×C4C24×C311yescharacteristic
383M16—28no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8C3×#4114yes—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4C2×C3×D412yes—
407C4⋊C4C2×C3×D412yes—
408C4⋊C4C2×C3×D412yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C42—24no—
417C42—24no—
418C42—21no—
419C42—21no—
420C22×C3—24no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3C2×#12011yescharacteristic
431C22×C3#54712yes—
432C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
433C22×C3#57511yescharacteristic
434C22×C3#57812yes—
435C3×C4—22no—
436C3×C4—41no—
437C3×C4—42no—
438C3×C4—41no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4—42no—
443C3×C4—82no—
444C3×C4—82no—
445C3×C4#52511yescharacteristic
446C3×C4#53611yescharacteristic
447C3×C4#54712yes—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23C3×#12012yes—
457C23C3×AGL(1,ℤ/8)12yes—
458C23C22×C3×D411yescharacteristic
459Q8—24no—
460Q8—41no—
461Q8—41no—
462D4—44no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—48no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4C3×#12012yes—
496C2×C4C22×C3×D411yescharacteristic
497C2×C4C22×C3×D411yescharacteristic
498C2×C4C2×C3×#4112yes—
499C2×C4C3×2−1+412yes—
500C8—24no—
501C8—42no—
502C2×C3—42no—
503C2×C3—42no—
504C2×C3#240611yescharacteristic
505C2×C3#300611yescharacteristic
506C2×C3#301111yescharacteristic
507C2×C3#334312yes—
508C2×C3#335212yes—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—24no—
514C22—41no—
515C22—41no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22C2×C3×#12011yescharacteristic
520C22C3×#54712yes—
521C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
522C22C3×#57511yescharacteristic
523C22C3×#57812yes—
524C4—22no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—41no—
531C4—42no—
532C4—82no—
533C4—82no—
534C4C3×#52511yescharacteristic
535C4C3×#53611yescharacteristic
536C4C3×#54712yes—
537C3#2411011yescharacteristic, direct product factor, Sylow
538C2—42no—
539C2—42no—
540C2C3×#240611yescharacteristic
541C2C3×#300611yescharacteristic
542C2C3×#301111yescharacteristic
543C2C3×#334312yes—
544C2C3×#335212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#484C2semidirect
3. C2×C3×#428C2semidirect
5. C3×#2059C2semidirect
6. C3×#2035C2semidirect
7. C3×#1894C2semidirect
9. C3×#1863C2semidirect
10. C3×#1852C2semidirect
11. C3×#1825C2semidirect
12. #24110C3direct (N × H)
49. C2×#484C2×C3semidirect
50. C2×#428C2×C3semidirect
52. #2059C2×C3semidirect
53. #2035C2×C3semidirect
54. #1894C2×C3semidirect
56. #1863C2×C3semidirect
57. #1852C2×C3semidirect
58. #1825C2×C3semidirect
537. C3#24110direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5522
C2×C3×#4842
C2×C3×#4282
C3×#24862
C3×#20592
C3×#20352
C3×#18942
C3×#18702
C3×#18632
C3×#18522
C3×#18252
#241103