Finite Group Databasebeta

← C3×#25605 #10644212

Subgroups — C3×#25605 #10644212

Order 768.

1318 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3C2×C3×#490C211yesmaximal, characteristic, complemented
4C3×#2856C211yesmaximal, characteristic, complemented
5C3×#2000C212yesmaximal, complemented
6C3×#1999C212yesmaximal, complemented
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1959C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1879C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic, complemented
12#25605C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C3×#529—24no—
17C3×#529—24no—
18C2×C3×#126—21no—
19C2×C3×#125C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×#123—21no—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×#121—21no—
24C2×C3×#121—21no—
25C2×C3×#121C2211yescharacteristic
26C2×C3×#121C2212yes—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4—21no—
29C2×C3×C4×D4C2211yescharacteristic
30C2×C3×#117—21no—
31C22×C3×C42C2211yescharacteristic
32C3×#496—24no—
33C3×#490—24no—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C2×C3×#97C2211yescharacteristic
40C2×C3×#97C2211yescharacteristic
41C3×#390—21no—
42C3×#390—21no—
43C3×#390C2212yes—
44C3×#384C2211yescharacteristic
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#529C2×C311yescharacteristic, complemented
49C2×#496C2×C311yescharacteristic, complemented
50C2×#490C2×C311yescharacteristic, complemented
51#2856C2×C311yescharacteristic, complemented
52#2000C2×C312yescomplemented
53#1999C2×C312yescomplemented
54#1992C2×C311yescharacteristic, complemented
55#1959C2×C311yescharacteristic, complemented
56#1880C2×C312yescomplemented
57#1879C2×C312yescomplemented
58#1749C2×C311yescharacteristic, complemented
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2312yes—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4D411yescharacteristic
68C23×C3×C4D411yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C2×C3×D8—24no—
74C2×C3×D8—24no—
75C2×C3×D8—42no—
76C22×C3×C8C2312yes—
77C22×C3×C8C2312yes—
78C3×#127—22no—
79C3×#127—22no—
80C3×#126—28no—
81C3×#125—22no—
82C3×#125D414yescomplemented
83C3×#124—22no—
84C3×#124D414yescomplemented
85C3×#123—28no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#121—42no—
91C3×#121—42no—
92C3×#121—44no—
93C3×#121—44no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×D4—28no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×#117—28no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—22no—
117C2×C3×C42—21no—
118C2×C3×C42—22no—
119C2×C3×C42—22no—
120C2×C3×C42D411yescharacteristic
121C2×C3×C42D411yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C2×C3×C42C2311yescharacteristic
124C3×#103—42no—
125C3×#102—24no—
126C3×#102—24no—
127C3×#102—42no—
128C3×C22⋊C8—24no—
129C3×#97—22no—
130C3×#97—22no—
131C22×SD16C22×C312yes—
132C22×D8C22×C312yes—
133C2×#127C22×C311yescharacteristic
134#529—24no—
135#529—24no—
136C2×#126—21no—
137C2×#125C22×C311yescharacteristic
138C2×#124C22×C311yescharacteristic
139C2×#123—21no—
140C2×C22⋊Q8C22×C311yescharacteristic
141C2×#121—21no—
142C2×#121—21no—
143C2×#121C22×C311yescharacteristic
144C2×#121C22×C312yes—
145C2×C4×Q8C22×C311yescharacteristic
146C2×C4×D4—21no—
147C2×C4×D4C22×C311yescharacteristic
148C2×#117—21no—
149C22×C42C22×C311yescharacteristic
150#496—24no—
151#490—24no—
152C2×#103C22×C312yes—
153C2×#102C22×C312yes—
154C2×#102C22×C312yes—
155C2×#102C22×C312yes—
156C2×C22⋊C8C22×C312yes—
157C2×#97C22×C311yescharacteristic
158C2×#97C22×C311yescharacteristic
159#390—21no—
160#390—21no—
161#390C22×C312yes—
162#384C22×C311yescharacteristic
163#340C22×C312yes—
164#340C22×C312yes—
165#336C22×C312yes—
166C24×C3—21no—
167C24×C3—22no—
168C24×C3C2×D411yescharacteristic
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8C2×D412yes—
173C2×C3×D4—24no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—42no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—44no—
192C2×C3×D4—42no—
193C2×C3×D4—42no—
194C2×C3×D4C2×D412yes—
195C22×C3×C4—24no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—41no—
209C22×C3×C4—41no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C3×SD16—48no—
219C3×D8—48no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8#4114yes—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4C2×D412yes—
237C3×C4⋊C4C2×D412yes—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—24no—
243C3×C22⋊C4—44no—
244C3×C22⋊C4—44no—
245C3×C22⋊C4—44no—
246C3×C42—24no—
247C3×C42—22no—
248C3×C42—22no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42C2×D412yes—
252C3×C42C2×D412yes—
253C22×Q8C23×C311yescharacteristic
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4C23×C312yes—
259C22×D4C23×C311yescharacteristic
260C23×C4—21no—
261C23×C4C3×D411yescharacteristic
262C23×C4C3×D411yescharacteristic
263C23×C4C23×C311yescharacteristic
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C2×D8—24no—
268C2×D8—24no—
269C2×D8—42no—
270C22×C8C23×C312yes—
271C22×C8C23×C312yes—
272#127—22no—
273#127—22no—
274#126—28no—
275#125—22no—
276#125C3×D414yescomplemented
277#124—22no—
278#124C3×D414yescomplemented
279#123—28no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282#121—22no—
283#121—22no—
284#121—42no—
285#121—42no—
286#121—44no—
287#121—44no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×D4—28no—
291C4×D4—22no—
292C4×D4—22no—
293#117—28no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C23×C312yes—
301C2×C4⋊C4C23×C311yescharacteristic
302C2×C4⋊C4C23×C311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42C3×D411yescharacteristic
315C2×C42C3×D411yescharacteristic
316C2×C42C23×C311yescharacteristic
317C2×C42C23×C311yescharacteristic
318#103—42no—
319#102—24no—
320#102—24no—
321#102—42no—
322C22⋊C8—24no—
323#97—22no—
324#97—22no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3AGL(1,ℤ/8)12yes—
346C23×C3C22×D411yescharacteristic
347C3×Q8—24no—
348C3×Q8—42no—
349C3×D4—24no—
350C3×D4—48no—
351C3×D4—42no—
352C3×D4—42no—
353C3×D4—42no—
354C3×D4—42no—
355C3×D4—42no—
356C3×D4—84no—
357C3×D4—84no—
358C3×D4—88no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—44no—
375C2×C3×C4—44no—
376C2×C3×C4—48no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—42no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—42no—
388C2×C3×C4#12012yes—
389C2×C3×C4#12012yes—
390C2×C3×C4AGL(1,ℤ/8)12yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4112yes—
394C2×C3×C4H(5,𝔽2)12yes—
395C3×C8—24no—
396C3×C8—42no—
397C24—21no—
398C24—22no—
399C24C2×C3×D411yescharacteristic
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C2×C3×D412yes—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4C2×C3×D412yes—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4C2×C3×D411yescharacteristic
444C22×C4C2×C3×D411yescharacteristic
445C22×C4C2×C3×D411yescharacteristic
446C22×C4C2×C3×D411yescharacteristic
447C22×C4C2×C3×D411yescharacteristic
448C22×C4C24×C311yescharacteristic
449SD16—48no—
450D8—48no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8C3×#4114yes—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4C2×C3×D412yes—
468C4⋊C4C2×C3×D412yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42C2×C3×D412yes—
483C42C2×C3×D412yes—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—82no—
499C22×C3—82no—
500C22×C3—82no—
501C22×C3—82no—
502C22×C3—82no—
503C22×C3—82no—
504C22×C3—82no—
505C22×C3C2×#12011yescharacteristic
506C22×C3#53211yescharacteristic
507C22×C3#53511yescharacteristic
508C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
509C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57511yescharacteristic
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—41no—
516C3×C4—42no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—82no—
521C3×C4#54612yes—
522C3×C4#55012yes—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C3×AGL(1,ℤ/8)12yes—
544C23C22×C3×D411yescharacteristic
545Q8—24no—
546Q8—42no—
547D4—24no—
548D4—48no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—84no—
555D4—84no—
556D4—88no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4C3×#12012yes—
587C2×C4C3×#12012yes—
588C2×C4C3×AGL(1,ℤ/8)12yes—
589C2×C4C22×C3×D411yescharacteristic
590C2×C4C22×C3×D411yescharacteristic
591C2×C4C2×C3×#4112yes—
592C2×C4C3×H(5,𝔽2)12yes—
593C8—24no—
594C8—42no—
595C2×C3—42no—
596C2×C3—82no—
597C2×C3—162no—
598C2×C3#243111yescharacteristic
599C2×C3#300611yescharacteristic
600C2×C3#300611yescharacteristic
601C2×C3#317111yescharacteristic
602C2×C3#318011yescharacteristic
603C2×C3#321711yescharacteristic
604C2×C3#321911yescharacteristic
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C2×C3×#12011yescharacteristic
627C22C3×#53211yescharacteristic
628C22C3×#53511yescharacteristic
629C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
630C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
631C22C3×#57511yescharacteristic
632C22C3×#57511yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4C3×#54612yes—
643C4C3×#55012yes—
644C3#2560511yescharacteristic, direct product factor, Sylow
645C2—42no—
646C2—82no—
647C2—162no—
648C2C3×#243111yescharacteristic
649C2C3×#300611yescharacteristic
650C2C3×#300611yescharacteristic
651C2C3×#317111yescharacteristic
652C2C3×#318011yescharacteristic
653C2C3×#321711yescharacteristic
654C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#529C2semidirect
2. C2×C3×#496C2semidirect
3. C2×C3×#490C2semidirect
4. C3×#2856C2semidirect
5. C3×#2000C2semidirect
6. C3×#1999C2semidirect
7. C3×#1992C2semidirect
8. C3×#1959C2semidirect
9. C3×#1880C2semidirect
10. C3×#1879C2semidirect
11. C3×#1749C2semidirect
12. #25605C3direct (N × H)
48. C2×#529C2×C3semidirect
49. C2×#496C2×C3semidirect
50. C2×#490C2×C3semidirect
51. #2856C2×C3semidirect
52. #2000C2×C3semidirect
53. #1999C2×C3semidirect
54. #1992C2×C3semidirect
55. #1959C2×C3semidirect
56. #1880C2×C3semidirect
57. #1879C2×C3semidirect
58. #1749C2×C3semidirect
82. C3×#125D4semidirect
84. C3×#124D4semidirect
276. #125C3×D4semidirect
278. #124C3×D4semidirect
644. C3#25605direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×C3×#4962
C2×C3×#4902
C3×#28562
C3×#20002
C3×#19992
C3×#19922
C3×#19592
C3×#18802
C3×#18792
C3×#17492
#256053