Finite Group Databasebeta

← C3×#25637 #10644244

Subgroups — C3×#25637 #10644244

Order 768.

1446 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3C2×C3×#463C211yesmaximal, characteristic, complemented
4C3×#2592C211yesmaximal, characteristic, complemented
5C3×#2014C212yesmaximal, complemented
6C3×#2011C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1982C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1877C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25637C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×SD16C2212yes—
15C22×C3×SD16C2211yescharacteristic
16C22×C3×D8C2212yes—
17C22×C3×D8C2211yescharacteristic
18C3×#550—24no—
19C3×#550—24no—
20C2×C3×#127C2212yes—
21C2×C3×#127C2211yescharacteristic
22C2×C3×#124C2211yescharacteristic
23C2×C3×#121—22no—
24C2×C3×#121—21no—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2212yes—
27C2×C3×C4×D4C2211yescharacteristic
28C3×#496—24no—
29C3×#463—24no—
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#102C2212yes—
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#97C2211yescharacteristic
37C3×#397—21no—
38C3×#393—21no—
39C3×#391C2212yes—
40C3×#390C2212yes—
41C3×#388—22no—
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#550C2×C311yescharacteristic, complemented
47C2×#496C2×C311yescharacteristic, complemented
48C2×#463C2×C311yescharacteristic, complemented
49#2592C2×C311yescharacteristic, complemented
50#2014C2×C312yescomplemented
51#2011C2×C312yescomplemented
52#1985C2×C311yescharacteristic, complemented
53#1982C2×C311yescharacteristic, complemented
54#1879C2×C312yescomplemented
55#1877C2×C312yescomplemented
56#1762C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4C2311yescharacteristic
66C22×C3×D4C2311yescharacteristic
67C22×C3×D4C2312yes—
68C23×C3×C4—22no—
69C23×C3×C4—21no—
70C2×C3×SD16—22no—
71C2×C3×SD16—24no—
72C2×C3×SD16—24no—
73C2×C3×SD16—42no—
74C2×C3×SD16—42no—
75C2×C3×D8—22no—
76C2×C3×D8—24no—
77C2×C3×D8—24no—
78C2×C3×D8—42no—
79C2×C3×D8—42no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2312yes—
83C3×#127—24no—
84C3×#127—24no—
85C3×#127—22no—
86C3×#127D414yes—
87C3×#124—22no—
88C3×#124—22no—
89C3×#121—28no—
90C3×#121—216no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×D4—24no—
94C3×C4×D4—24no—
95C3×C4×D4—22no—
96C3×C4×D4D414yes—
97C2×C3×C4⋊C4D412yes—
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C42—21no—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C3×C4⋊C8—22no—
111C3×#103—22no—
112C3×#102—24no—
113C3×#102—22no—
114C3×#102—42no—
115C3×#102—42no—
116C3×#97—22no—
117C3×#95—21no—
118C3×#95—22no—
119C3×#95—22no—
120C22×#41—21no—
121C22×SD16C22×C312yes—
122C22×SD16C22×C311yescharacteristic
123C22×D8C22×C312yes—
124C22×D8C22×C311yescharacteristic
125#550—24no—
126#550—24no—
127C2×#127C22×C312yes—
128C2×#127C22×C311yescharacteristic
129C2×#124C22×C311yescharacteristic
130C2×#121—22no—
131C2×#121—21no—
132C2×C4×Q8C22×C311yescharacteristic
133C2×C4×D4C22×C312yes—
134C2×C4×D4C22×C311yescharacteristic
135#496—24no—
136#463—24no—
137C2×C4⋊C8C22×C311yescharacteristic
138C2×#103C22×C311yescharacteristic
139C2×#102C22×C311yescharacteristic
140C2×#102C22×C312yes—
141C2×#102C22×C312yes—
142C2×#102C22×C312yes—
143C2×#97C22×C311yescharacteristic
144#397—21no—
145#393—21no—
146#391C22×C312yes—
147#390C22×C312yes—
148#388—22no—
149C4×C4⋊C4C22×C311yescharacteristic
150#340C22×C312yes—
151#336C22×C312yes—
152#336C22×C312yes—
153C24×C3—22no—
154C24×C3—21no—
155C24×C3—21no—
156C3×#41—432no—
157C2×C3×Q8—22no—
158C2×C3×Q8—24no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×D4—28no—
164C2×C3×D4—28no—
165C2×C3×D4—24no—
166C2×C3×D4—24no—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—44no—
180C2×C3×D4—44no—
181C2×C3×D4—44no—
182C2×C3×D4—81no—
183C2×C3×D4—81no—
184C2×C3×D4—82no—
185C2×C3×D4—82no—
186C2×C3×D4C2×D414yes—
187C2×C3×D4C2×D412yes—
188C22×C3×C4—28no—
189C22×C3×C4—21no—
190C22×C3×C4—24no—
191C22×C3×C4—21no—
192C22×C3×C4—22no—
193C22×C3×C4—44no—
194C22×C3×C4—44no—
195C22×C3×C4—44no—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2411yescharacteristic
201C3×SD16—48no—
202C3×SD16—44no—
203C3×D8—48no—
204C3×D8—44no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8#4114yescomplemented
212C3×C4⋊C4—24no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4C2×D412yes—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—44no—
223C3×C22⋊C4—82no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42—81no—
227C3×C42C2×D412yes—
228C3×C42C2×D412yes—
229C2×#41—28no—
230C2×#41—48no—
231C22×Q8C3×D411yescharacteristic
232C22×Q8C23×C311yescharacteristic
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4C23×C311yescharacteristic
238C22×D4C23×C311yescharacteristic
239C22×D4C23×C312yes—
240C23×C4—22no—
241C23×C4—21no—
242C2×SD16—22no—
243C2×SD16—24no—
244C2×SD16—24no—
245C2×SD16—42no—
246C2×SD16—42no—
247C2×D8—22no—
248C2×D8—24no—
249C2×D8—24no—
250C2×D8—42no—
251C2×D8—42no—
252C22×C8C23×C311yescharacteristic
253C22×C8C23×C311yescharacteristic
254C22×C8C23×C312yes—
255#127—24no—
256#127—24no—
257#127—22no—
258#127C3×D414yes—
259#124—22no—
260#124—22no—
261#121—28no—
262#121—216no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—22no—
268C4×D4C3×D414yes—
269C2×C4⋊C4C3×D412yes—
270C2×C4⋊C4C3×D411yescharacteristic
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C42—21no—
280C2×C42C23×C312yes—
281C2×C42C23×C311yescharacteristic
282C4⋊C8—22no—
283#103—22no—
284#102—24no—
285#102—22no—
286#102—42no—
287#102—42no—
288#97—22no—
289#95—21no—
290#95—22no—
291#95—22no—
292C23×C3—24no—
293C23×C3—22no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3C22×D411yescharacteristic
310C3×Q8—28no—
311C3×Q8—41no—
312C3×Q8—41no—
313C3×Q8—42no—
314C3×D4—22no—
315C3×D4—22no—
316C3×D4—24no—
317C3×D4—24no—
318C3×D4—48no—
319C3×D4—416no—
320C3×D4—42no—
321C3×D4—41no—
322C3×D4—42no—
323C3×D4—41no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—84no—
327C3×D4—88no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—48no—
338C2×C3×C4—416no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4#12012yes—
346C2×C3×C4#12012yes—
347C2×C3×C4AGL(1,ℤ/8)14yes—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4112yes—
351C2×C3×C4H(5,𝔽2)12yes—
352C3×C8—24no—
353C3×C8—41no—
354C3×C8—41no—
355C24—22no—
356C24—21no—
357C24—21no—
358#41—432no—
359C2×Q8—22no—
360C2×Q8—24no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4C2×C3×D414yes—
389C2×D4C2×C3×D412yes—
390C22×C4—28no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4C2×C3×D412yes—
399C22×C4C2×C3×D411yescharacteristic
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D412yes—
402C22×C4C24×C311yescharacteristic
403SD16—48no—
404SD16—44no—
405D8—48no—
406D8—44no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8C3×#4114yescomplemented
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4C2×C3×D412yes—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—82no—
426C42—22no—
427C42—22no—
428C42—81no—
429C42C2×C3×D412yes—
430C42C2×C3×D412yes—
431C22×C3—42no—
432C22×C3—44no—
433C22×C3—42no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—42no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3C2×#12011yescharacteristic
449C22×C3#55012yes—
450C22×C3C2×AGL(1,ℤ/8)12yes—
451C22×C3#57611yescharacteristic
452C22×C3#57711yescharacteristic
453C3×C4—22no—
454C3×C4—24no—
455C3×C4—22no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4—82no—
459C3×C4—81no—
460C3×C4#53211yescharacteristic
461C3×C4#53511yescharacteristic
462C3×C4#54612yes—
463C23—24no—
464C23—22no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23C22×C3×D411yescharacteristic
481Q8—28no—
482Q8—41no—
483Q8—41no—
484Q8—42no—
485D4—22no—
486D4—22no—
487D4—24no—
488D4—24no—
489D4—48no—
490D4—416no—
491D4—42no—
492D4—41no—
493D4—42no—
494D4—41no—
495D4—42no—
496D4—42no—
497D4—84no—
498D4—88no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—48no—
509C2×C4—416no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4C3×#12012yes—
517C2×C4C3×#12012yes—
518C2×C4C3×AGL(1,ℤ/8)14yes—
519C2×C4C22×C3×D411yescharacteristic
520C2×C4C22×C3×D411yescharacteristic
521C2×C4C2×C3×#4112yes—
522C2×C4C3×H(5,𝔽2)12yes—
523C8—24no—
524C8—41no—
525C8—41no—
526C2×C3—82no—
527C2×C3—84no—
528C2×C3—161no—
529C2×C3#243111yescharacteristic
530C2×C3#300311yescharacteristic
531C2×C3#301211yescharacteristic
532C2×C3#339712yes—
533C2×C3#339912yes—
534C22—42no—
535C22—44no—
536C22—42no—
537C22—44no—
538C22—44no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22C2×C3×#12011yescharacteristic
552C22C3×#55012yes—
553C22C2×C3×AGL(1,ℤ/8)12yes—
554C22C3×#57611yescharacteristic
555C22C3×#57711yescharacteristic
556C4—22no—
557C4—24no—
558C4—22no—
559C4—42no—
560C4—42no—
561C4—82no—
562C4—81no—
563C4C3×#53211yescharacteristic
564C4C3×#53511yescharacteristic
565C4C3×#54612yes—
566C3#2563711yescharacteristic, direct product factor, Sylow
567C2—82no—
568C2—84no—
569C2—161no—
570C2C3×#243111yescharacteristic
571C2C3×#300311yescharacteristic
572C2C3×#301211yescharacteristic
573C2C3×#339712yes—
574C2C3×#339912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#550C2semidirect
2. C2×C3×#496C2semidirect
3. C2×C3×#463C2semidirect
4. C3×#2592C2semidirect
5. C3×#2014C2semidirect
6. C3×#2011C2semidirect
7. C3×#1985C2semidirect
8. C3×#1982C2semidirect
9. C3×#1879C2semidirect
10. C3×#1877C2semidirect
11. C3×#1762C2semidirect
12. #25637C3direct (N × H)
46. C2×#550C2×C3semidirect
47. C2×#496C2×C3semidirect
48. C2×#463C2×C3semidirect
49. #2592C2×C3semidirect
50. #2014C2×C3semidirect
51. #2011C2×C3semidirect
52. #1985C2×C3semidirect
53. #1982C2×C3semidirect
54. #1879C2×C3semidirect
55. #1877C2×C3semidirect
56. #1762C2×C3semidirect
211. C2×C3×C8#41semidirect
413. C2×C8C3×#41semidirect
566. C3#25637direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4962
C2×C3×#4632
C3×#25922
C3×#20142
C3×#20112
C3×#19852
C3×#19822
C3×#18792
C3×#18772
C3×#17622
#256373