Finite Group Databasebeta

← C3×#24105 #10642712

Subgroups — C3×#24105 #10642712

Order 768.

1126 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#555C211yesmaximal, characteristic
2C2×C3×#480C211yesmaximal, characteristic, complemented
3C2×C3×#428C211yesmaximal, characteristic, complemented
4C3×#2487C211yesmaximal, characteristic, complemented
5C3×#2057C212yesmaximal, complemented
6C3×#2033C212yesmaximal, complemented
7C3×#1895C211yesmaximal, characteristic, complemented
8C3×#1870C211yesmaximal, characteristic, complemented
9C3×#1864C212yesmaximal, complemented
10C3×#1854C212yesmaximal, complemented
11C3×#1825C211yesmaximal, characteristic, complemented
12#24105C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2211yescharacteristic
14C3×#555—24no—
15C3×#555—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#124—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C2×C3×#117C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C22×C3×C4⋊C4C2211yescharacteristic
28C3×#480—24no—
29C3×#428—24no—
30C2×C3×#107C2212yes—
31C2×C3×#107C2212yes—
32C2×C3×#106C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×C22⋊C8C2212yes—
36C2×C3×C22⋊C8C2211yescharacteristic
37C3×#400—21no—
38C3×#399—21no—
39C3×#389C2212yes—
40C3×#386—21no—
41C3×#385—21no—
42C3×#380C2212yes—
43C3×#380C2211yescharacteristic
44C3×#340C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#555C2×C311yescharacteristic
49C2×#480C2×C311yescharacteristic, complemented
50C2×#428C2×C311yescharacteristic, complemented
51#2487C2×C311yescharacteristic, complemented
52#2057C2×C312yescomplemented
53#2033C2×C312yescomplemented
54#1895C2×C311yescharacteristic, complemented
55#1870C2×C311yescharacteristic, complemented
56#1864C2×C312yescomplemented
57#1854C2×C312yescomplemented
58#1825C2×C311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4D411yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C2×C3×M16—22no—
69C2×C3×M16—24no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8—22no—
74C3×#124—28no—
75C3×#124—28no—
76C3×#121—28no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121D414yes—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×#117—28no—
86C3×#117—22no—
87C3×#117—22no—
88C3×#117—22no—
89C3×#117D414yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4D411yescharacteristic
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C3×#107—24no—
114C3×#107—24no—
115C3×#106—24no—
116C3×#102—24no—
117C3×#102—24no—
118C3×C22⋊C8—22no—
119C3×C22⋊C8—24no—
120C3×#95—22no—
121C3×#95—21no—
122C3×#95—22no—
123C22×M16C22×C311yescharacteristic
124#555—24no—
125#555—24no—
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#124—21no—
128C2×#124—21no—
129C2×#121—21no—
130C2×#121C22×C311yescharacteristic
131C2×#121C22×C311yescharacteristic
132C2×C4×D4C22×C311yescharacteristic
133C2×C4×D4C22×C311yescharacteristic
134C2×#117—21no—
135C2×#117C22×C311yescharacteristic
136C2×#117C22×C311yescharacteristic
137C22×C4⋊C4C22×C311yescharacteristic
138#480—24no—
139#428—24no—
140C2×#107C22×C312yes—
141C2×#107C22×C312yes—
142C2×#106C22×C312yes—
143C2×#102C22×C312yes—
144C2×#102C22×C312yes—
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C312yes—
147#400—21no—
148#399—21no—
149#389C22×C312yes—
150#386—21no—
151#385—21no—
152#380C22×C311yescharacteristic
153#380C22×C312yes—
154#340C22×C312yes—
155#340C22×C312yes—
156#340C22×C312yes—
157#336C22×C312yes—
158C24×C3—21no—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—24no—
162C2×C3×Q8—82no—
163C2×C3×D4—22no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—41no—
171C2×C3×D4—82no—
172C2×C3×D4—82no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—42no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4—82no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C3×M16—28no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8#4114yes—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—82no—
215C3×C4⋊C4C2×D412yes—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—82no—
227C3×C42—24no—
228C3×C42—24no—
229C3×C42—21no—
230C3×C42—21no—
231C22×Q8—21no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C3×D411yescharacteristic
235C22×D4C23×C311yescharacteristic
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C23×C311yescharacteristic
239C23×C4C23×C311yescharacteristic
240C2×M16—22no—
241C2×M16—24no—
242C22×C8C23×C312yes—
243C22×C8C23×C312yes—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246#124—28no—
247#124—28no—
248#121—28no—
249#121—22no—
250#121—22no—
251#121—22no—
252#121C3×D414yes—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257#117—28no—
258#117—22no—
259#117—22no—
260#117—22no—
261#117C3×D414yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4C3×D411yescharacteristic
268C2×C4⋊C4C23×C312yes—
269C2×C4⋊C4C23×C312yes—
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
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#102—24no—
289#102—24no—
290C22⋊C8—22no—
291C22⋊C8—24no—
292#95—22no—
293#95—21no—
294#95—22no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—21no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—22no—
305C23×C3—42no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—42no—
309C23×C3#12012yes—
310C23×C3C22×D411yescharacteristic
311C3×Q8—84no—
312C3×D4—24no—
313C3×D4—44no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—84no—
317C3×D4—84no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—48no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—84no—
346C2×C3×C4#12012yes—
347C2×C3×C4#13712yes—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4112yes—
351C2×C3×C42−1+412yes—
352C3×C8—24no—
353C3×C8—42no—
354C24—21no—
355C24C2×C3×D411yescharacteristic
356C24C2×C3×D411yescharacteristic
357C2×Q8—24no—
358C2×Q8—82no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—41no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4C2×C3×D412yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—82no—
386C22×C4C2×C3×D411yescharacteristic
387C22×C4C2×C3×D411yescharacteristic
388C22×C4C2×C3×D411yescharacteristic
389C22×C4C2×C3×D412yes—
390C22×C4C2×C3×D412yes—
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C24×C311yescharacteristic
393M16—28no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8C3×#4114yes—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—82no—
411C4⋊C4C2×C3×D412yes—
412C4⋊C4C2×C3×D412yes—
413C4⋊C4C2×C3×D412yes—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—24no—
420C22⋊C4—22no—
421C22⋊C4—42no—
422C22⋊C4—82no—
423C42—24no—
424C42—24no—
425C42—21no—
426C42—21no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—41no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—42no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3C2×#12011yescharacteristic
443C22×C3#54712yes—
444C22×C3C2×#13711yescharacteristic
445C22×C3#57511yescharacteristic
446C22×C3#57612yes—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—82no—
455C3×C4—82no—
456C3×C4#52511yescharacteristic
457C3×C4#53611yescharacteristic
458C3×C4#54712yes—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—21no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23C3×#12012yes—
474C23C22×C3×D411yescharacteristic
475Q8—84no—
476D4—24no—
477D4—44no—
478D4—41no—
479D4—41no—
480D4—84no—
481D4—84no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—48no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—84no—
510C2×C4C3×#12012yes—
511C2×C4C3×#13712yes—
512C2×C4C22×C3×D411yescharacteristic
513C2×C4C22×C3×D411yescharacteristic
514C2×C4C2×C3×#4112yes—
515C2×C4C3×2−1+412yes—
516C8—24no—
517C8—42no—
518C2×C3—42no—
519C2×C3—81no—
520C2×C3—82no—
521C2×C3#240611yescharacteristic
522C2×C3#300711yescharacteristic
523C2×C3#300811yescharacteristic
524C2×C3#334712yes—
525C2×C3#335712yes—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—82no—
540C22—82no—
541C22C2×C3×#12011yescharacteristic
542C22C3×#54712yes—
543C22C2×C3×#13711yescharacteristic
544C22C3×#57511yescharacteristic
545C22C3×#57612yes—
546C4—22no—
547C4—22no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—42no—
552C4—42no—
553C4—82no—
554C4—82no—
555C4C3×#52511yescharacteristic
556C4C3×#53611yescharacteristic
557C4C3×#54712yes—
558C3#2410511yescharacteristic, direct product factor, Sylow
559C2—42no—
560C2—81no—
561C2—82no—
562C2C3×#240611yescharacteristic
563C2C3×#300711yescharacteristic
564C2C3×#300811yescharacteristic
565C2C3×#334712yes—
566C2C3×#335712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#555C2
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×#480C2semidirect
3. C2×C3×#428C2semidirect
4. C3×#2487C2semidirect
5. C3×#2057C2semidirect
6. C3×#2033C2semidirect
7. C3×#1895C2semidirect
8. C3×#1870C2semidirect
9. C3×#1864C2semidirect
10. C3×#1854C2semidirect
11. C3×#1825C2semidirect
12. #24105C3direct (N × H)
49. C2×#480C2×C3semidirect
50. C2×#428C2×C3semidirect
51. #2487C2×C3semidirect
52. #2057C2×C3semidirect
53. #2033C2×C3semidirect
54. #1895C2×C3semidirect
55. #1870C2×C3semidirect
56. #1864C2×C3semidirect
57. #1854C2×C3semidirect
58. #1825C2×C3semidirect
558. C3#24105direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5552
C2×C3×#4802
C2×C3×#4282
C3×#24872
C3×#20572
C3×#20332
C3×#18952
C3×#18702
C3×#18642
C3×#18542
C3×#18252
#241053