Finite Group Databasebeta

← C3×#25577 #10644184

Subgroups — C3×#25577 #10644184

Order 768.

1422 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic, complemented
2C2×C3×#489C211yesmaximal, characteristic, complemented
3C2×C3×#463C211yesmaximal, characteristic, complemented
4C3×#2540C211yesmaximal, characteristic, complemented
5C3×#2000C212yesmaximal, complemented
6C3×#1999C212yesmaximal, complemented
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1955C211yesmaximal, characteristic, complemented
9C3×#1874C212yesmaximal, complemented
10C3×#1872C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25577C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×SD16C2211yescharacteristic
15C22×C3×D8C2211yescharacteristic
16C3×#547—24no—
17C3×#547—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×#123—22no—
21C2×C3×C22⋊Q8—21no—
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×#121C2212yes—
24C2×C3×#121C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C22×C3×C4⋊C4C2212yes—
28C3×#489—24no—
29C3×#463—24no—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2211yescharacteristic
36C2×C3×C22⋊C8C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C2×C3×#97C2211yescharacteristic
39C3×#399—21no—
40C3×#392—21no—
41C3×#390—22no—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#547C2×C311yescharacteristic, complemented
47C2×#489C2×C311yescharacteristic, complemented
48C2×#463C2×C311yescharacteristic, complemented
49#2540C2×C311yescharacteristic, complemented
50#2000C2×C312yescomplemented
51#1999C2×C312yescomplemented
52#1987C2×C311yescharacteristic, complemented
53#1955C2×C311yescharacteristic, complemented
54#1874C2×C312yescomplemented
55#1872C2×C312yescomplemented
56#1762C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4D411yescharacteristic
64C22×C3×D4C2311yescharacteristic
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—22no—
67C23×C3×C4—21no—
68C23×C3×C4C2312yes—
69C2×C3×SD16—22no—
70C2×C3×SD16—42no—
71C2×C3×D8—22no—
72C2×C3×D8—42no—
73C22×C3×C8C2311yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2312yes—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8D414yescomplemented
78C3×#124—22no—
79C3×#124—22no—
80C3×#123—216no—
81C3×C22⋊Q8—28no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—42no—
84C3×#121—24no—
85C3×#121—24no—
86C3×#121—24no—
87C3×#121—42no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4D414yescomplemented
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—24no—
95C2×C3×C4⋊C4—24no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4D412yes—
108C2×C3×C42—21no—
109C2×C3×C42C2311yescharacteristic
110C3×C4⋊C8—22no—
111C3×#103—22no—
112C3×#103—24no—
113C3×#102—22no—
114C3×#102—24no—
115C3×#102—24no—
116C3×C22⋊C8—42no—
117C3×C22⋊C8—42no—
118C3×#97—22no—
119C3×#95—21no—
120C22×#41—21no—
121C22×SD16C22×C311yescharacteristic
122C22×D8C22×C311yescharacteristic
123#547—24no—
124#547—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×#124C22×C311yescharacteristic
127C2×#123—22no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8C22×C312yes—
130C2×#121C22×C312yes—
131C2×#121C22×C312yes—
132C2×C4×D4C22×C311yescharacteristic
133C2×C4×D4C22×C311yescharacteristic
134C22×C4⋊C4C22×C312yes—
135#489—24no—
136#463—24no—
137C2×C4⋊C8C22×C311yescharacteristic
138C2×#103C22×C312yes—
139C2×#103C22×C311yescharacteristic
140C2×#102C22×C312yes—
141C2×#102C22×C311yescharacteristic
142C2×#102C22×C312yes—
143C2×C22⋊C8C22×C312yes—
144C2×C22⋊C8C22×C312yes—
145C2×#97C22×C311yescharacteristic
146#399—21no—
147#392—21no—
148#390—22no—
149C4×C4⋊C4C22×C311yescharacteristic
150#340C22×C312yes—
151#340C22×C312yes—
152#340C22×C312yes—
153C24×C3—21no—
154C24×C3—21no—
155C24×C3C2×D412yes—
156C3×#41—432no—
157C2×C3×Q8—24no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—44no—
163C2×C3×D4—24no—
164C2×C3×D4—28no—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—44no—
174C2×C3×D4—44no—
175C2×C3×D4—44no—
176C2×C3×D4—44no—
177C2×C3×D4—82no—
178C2×C3×D4C2×D412yes—
179C22×C3×C4—28no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—44no—
192C22×C3×C4—44no—
193C22×C3×C4—44no—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C3×SD16—44no—
199C3×D8—44no—
200C2×C3×C8—22no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C3×C4⋊C4—28no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—44no—
214C3×C4⋊C4C2×D412yes—
215C3×C4⋊C4C2×D414yes—
216C3×C4⋊C4#4114yescomplemented
217C3×C22⋊C4—28no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—44no—
221C3×C22⋊C4—44no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—42no—
224C3×C42—81no—
225C3×C42C2×D412yes—
226C3×C42C2×D412yes—
227C2×#41—28no—
228C2×#41—48no—
229C22×Q8—21no—
230C22×Q8C23×C311yescharacteristic
231C22×D4—22no—
232C22×D4—22no—
233C22×D4C3×D411yescharacteristic
234C22×D4C23×C311yescharacteristic
235C22×D4C23×C311yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C23×C4C23×C312yes—
239C2×SD16—22no—
240C2×SD16—42no—
241C2×D8—22no—
242C2×D8—42no—
243C22×C8C23×C311yescharacteristic
244C22×C8C23×C311yescharacteristic
245C22×C8C23×C312yes—
246C4⋊Q8—22no—
247C4⋊Q8C3×D414yescomplemented
248#124—22no—
249#124—22no—
250#123—216no—
251C22⋊Q8—28no—
252C22⋊Q8—24no—
253C22⋊Q8—42no—
254#121—24no—
255#121—24no—
256#121—24no—
257#121—42no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4C3×D414yescomplemented
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4C3×D411yescharacteristic
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C312yes—
270C2×C4⋊C4C23×C312yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4C3×D412yes—
278C2×C42—21no—
279C2×C42C23×C311yescharacteristic
280C4⋊C8—22no—
281#103—22no—
282#103—24no—
283#102—22no—
284#102—24no—
285#102—24no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#97—22no—
289#95—21no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3AGL(1,ℤ/8)14yes—
307C23×C3C22×D411yescharacteristic
308C3×Q8—48no—
309C3×Q8—41no—
310C3×Q8—41no—
311C3×Q8—42no—
312C3×D4—28no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—416no—
316C3×D4—41no—
317C3×D4—41no—
318C3×D4—42no—
319C3×D4—42no—
320C3×D4—88no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—48no—
331C2×C3×C4—416no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
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—41no—
342C2×C3×C4—42no—
343C2×C3×C4#12012yes—
344C2×C3×C4#12012yes—
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4112yes—
348C2×C3×C4H(5,𝔽2)12yes—
349C3×C8—41no—
350C3×C8—41no—
351C3×C8—42no—
352C24—21no—
353C24—21no—
354C24C2×C3×D412yes—
355#41—432no—
356C2×Q8—24no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—44no—
362C2×D4—24no—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—82no—
377C2×D4C2×C3×D412yes—
378C22×C4—28no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4C2×C3×D412yes—
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C24×C311yescharacteristic
397SD16—44no—
398D8—44no—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—44no—
413C4⋊C4C2×C3×D412yes—
414C4⋊C4C2×C3×D414yes—
415C4⋊C4C3×#4114yescomplemented
416C22⋊C4—28no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C42—81no—
424C42C2×C3×D412yes—
425C42C2×C3×D412yes—
426C22×C3—24no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3C2×#12011yescharacteristic
444C22×C3#54612yes—
445C22×C3C2×AGL(1,ℤ/8)12yes—
446C22×C3#57711yescharacteristic
447C22×C3#57811yescharacteristic
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—44no—
453C3×C4—44no—
454C3×C4—81no—
455C3×C4#53211yescharacteristic
456C3×C4#53511yescharacteristic
457C3×C4#55012yes—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23C3×AGL(1,ℤ/8)14yes—
475C23C22×C3×D411yescharacteristic
476Q8—48no—
477Q8—41no—
478Q8—41no—
479Q8—42no—
480D4—28no—
481D4—22no—
482D4—22no—
483D4—416no—
484D4—41no—
485D4—41no—
486D4—42no—
487D4—42no—
488D4—88no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—48no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4C3×#12012yes—
512C2×C4C3×#12012yes—
513C2×C4C22×C3×D411yescharacteristic
514C2×C4C22×C3×D411yescharacteristic
515C2×C4C2×C3×#4112yes—
516C2×C4C3×H(5,𝔽2)12yes—
517C8—41no—
518C8—41no—
519C8—42no—
520C2×C3—44no—
521C2×C3—82no—
522C2×C3—161no—
523C2×C3#243111yescharacteristic
524C2×C3#300311yescharacteristic
525C2×C3#301311yescharacteristic
526C2×C3#330612yes—
527C2×C3#330812yes—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—82no—
539C22—82no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—82no—
544C22—82no—
545C22C2×C3×#12011yescharacteristic
546C22C3×#54612yes—
547C22C2×C3×AGL(1,ℤ/8)12yes—
548C22C3×#57711yescharacteristic
549C22C3×#57811yescharacteristic
550C4—22no—
551C4—22no—
552C4—42no—
553C4—42no—
554C4—44no—
555C4—44no—
556C4—81no—
557C4C3×#53211yescharacteristic
558C4C3×#53511yescharacteristic
559C4C3×#55012yes—
560C3#2557711yescharacteristic, direct product factor, Sylow
561C2—44no—
562C2—82no—
563C2—161no—
564C2C3×#243111yescharacteristic
565C2C3×#300311yescharacteristic
566C2C3×#301311yescharacteristic
567C2C3×#330612yes—
568C2C3×#330812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C2×C3×#547C2semidirect
2. C2×C3×#489C2semidirect
3. C2×C3×#463C2semidirect
4. C3×#2540C2semidirect
5. C3×#2000C2semidirect
6. C3×#1999C2semidirect
7. C3×#1987C2semidirect
8. C3×#1955C2semidirect
9. C3×#1874C2semidirect
10. C3×#1872C2semidirect
11. C3×#1762C2semidirect
12. #25577C3direct (N × H)
46. C2×#547C2×C3semidirect
47. C2×#489C2×C3semidirect
48. C2×#463C2×C3semidirect
49. #2540C2×C3semidirect
50. #2000C2×C3semidirect
51. #1999C2×C3semidirect
52. #1987C2×C3semidirect
53. #1955C2×C3semidirect
54. #1874C2×C3semidirect
55. #1872C2×C3semidirect
56. #1762C2×C3semidirect
77. C3×C4⋊Q8D4semidirect
91. C3×C4×D4D4semidirect
216. C3×C4⋊C4#41semidirect
247. C4⋊Q8C3×D4semidirect
261. C4×D4C3×D4semidirect
415. C4⋊C4C3×#41semidirect
560. C3#25577direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#4892
C2×C3×#4632
C3×#25402
C3×#20002
C3×#19992
C3×#19872
C3×#19552
C3×#18742
C3×#18722
C3×#17622
#255773