Finite Group Databasebeta

← C3×#32959 #10651566

Subgroups — C3×#32959 #10651566

Order 768.

1720 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×#478C212yesmaximal, complemented
3C3×#2851C211yesmaximal, characteristic, complemented
4C3×#1864C212yesmaximal, complemented
5C3×#1862C212yesmaximal, complemented
6C3×#1853C211yesmaximal, characteristic, complemented
7C3×#1852C211yesmaximal, characteristic, complemented
8C3×#1804C212yesmaximal, complemented
9C3×#1797C211yesmaximal, characteristic, complemented
10C3×#1786C211yesmaximal, characteristic
11C3×#1786C211yesmaximal, characteristic
12#32959C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×D4C2211yescharacteristic
14C24×C3×C4—21no—
15C2×C3×#125C2211yescharacteristic, complemented
16C2×C3×C4×D4—28no—
17C2×C3×C4×D4—21no—
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4C2212yes—
20C2×C3×C4×D4C2212yes—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4—21no—
23C22×C3×C42C2211yescharacteristic
24C3×#478—28no—
25C2×C3×#107C2212yescomplemented
26C2×C3×#106C2212yescomplemented
27C2×C3×C4⋊C8C2212yescomplemented
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#389C2211yescharacteristic, complemented
35C3×#389C2211yescharacteristic, complemented
36C3×#387—21no—
37C3×#387—21no—
38C3×#385—21no—
39C3×#385—21no—
40C3×#382C2211yescharacteristic, complemented
41C3×#380C2211yescharacteristic
42C3×#380C2211yescharacteristic
43C3×#340C2212yescomplemented
44C3×#340C2212yescomplemented
45C3×#336C2212yescomplemented
46C22×C4×D4C2×C311yescharacteristic
47C2×#478C2×C312yescomplemented
48#2851C2×C311yescharacteristic, complemented
49#1864C2×C312yescomplemented
50#1862C2×C312yescomplemented
51#1853C2×C311yescharacteristic, complemented
52#1852C2×C311yescharacteristic, complemented
53#1804C2×C312yescomplemented
54#1797C2×C311yescharacteristic, complemented
55#1786C2×C311yescharacteristic
56#1786C2×C311yescharacteristic
57C25×C3—21no—
58C22×C3×D4—24no—
59C22×C3×D4—24no—
60C22×C3×D4—21no—
61C22×C3×D4—22no—
62C22×C3×D4Q814yes—
63C22×C3×D4C2312yes—
64C23×C3×C4—28no—
65C23×C3×C4—21no—
66C23×C3×C4—22no—
67C23×C3×C4—22no—
68C23×C3×C4—44no—
69C23×C3×C4—44no—
70C23×C3×C4D411yescharacteristic
71C23×C3×C4D411yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C22×C3×C8C2312yes—
74C22×C3×C8C2312yes—
75C3×#125—24no—
76C3×C4×D4—28no—
77C3×C4×D4—28no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×C4×D4—24no—
81C3×C4×D4D418yes—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C22⋊C4—28no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—24no—
99C2×C3×C42D411yescharacteristic
100C2×C3×C42D411yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C3×#107—24no—
104C3×#106—24no—
105C3×C4⋊C8—24no—
106C3×#102—24no—
107C3×#102—24no—
108C3×#102—24no—
109C3×#102—24no—
110C3×C22⋊C8—22no—
111C3×C22⋊C8—22no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C23×D4C22×C311yescharacteristic
119C24×C4—21no—
120C2×#125C22×C311yescharacteristic, complemented
121C2×C4×D4—28no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C22×C312yes—
125C2×C4×D4C22×C312yes—
126C22×C4⋊C4C22×C311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C42C22×C311yescharacteristic
129#478—28no—
130C2×#107C22×C312yescomplemented
131C2×#106C22×C312yescomplemented
132C2×C4⋊C8C22×C312yescomplemented
133C2×#102C22×C312yes—
134C2×#102C22×C312yes—
135C2×#102C22×C312yes—
136C2×#102C22×C312yes—
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C311yescharacteristic
139#389C22×C311yescharacteristic, complemented
140#389C22×C311yescharacteristic, complemented
141#387—21no—
142#387—21no—
143#385—21no—
144#385—21no—
145#382C22×C311yescharacteristic, complemented
146#380C22×C311yescharacteristic
147#380C22×C311yescharacteristic
148#340C22×C312yescomplemented
149#340C22×C312yescomplemented
150#336C22×C312yescomplemented
151C24×C3—24no—
152C24×C3—24no—
153C24×C3—24no—
154C24×C3—22no—
155C24×C3—42no—
156C24×C3—42no—
157C24×C3—42no—
158C24×C3—42no—
159C24×C3C2×D411yescharacteristic
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—216no—
163C2×C3×D4—216no—
164C2×C3×D4—24no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4C2×D414yes—
169C2×C3×D4C2×Q814yes—
170C2×C3×D4#4114yes—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—28no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—28no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—41no—
187C22×C3×C4—432no—
188C22×C3×C4—44no—
189C22×C3×C4—44no—
190C22×C3×C4—44no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4—44no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×Q812yes—
200C22×C3×C4#4112yes—
201C22×C3×C4C2411yescharacteristic
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4C2×D414yes—
219C3×C22⋊C4—28no—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—42no—
223C3×C42—22no—
224C3×C42—22no—
225C3×C42—22no—
226C3×C42C2×D414yes—
227C25—21no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4C3×Q814yes—
233C22×D4C23×C312yes—
234C23×C4—28no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4C3×D411yescharacteristic
241C23×C4C3×D411yescharacteristic
242C23×C4C23×C311yescharacteristic
243C22×C8C23×C312yes—
244C22×C8C23×C312yes—
245#125—24no—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4C3×D418yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C4⋊C4C23×C311yescharacteristic
261C2×C4⋊C4C23×C311yescharacteristic
262C2×C4⋊C4C23×C312yes—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42—24no—
269C2×C42C3×D411yescharacteristic
270C2×C42C3×D411yescharacteristic
271C2×C42C23×C311yescharacteristic
272C2×C42C23×C311yescharacteristic
273#107—24no—
274#106—24no—
275C4⋊C8—24no—
276#102—24no—
277#102—24no—
278#102—24no—
279#102—24no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—216no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—48no—
297C23×C3—48no—
298C23×C3—48no—
299C23×C3—48no—
300C23×C3—48no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—48no—
306C23×C3C22⋊Q812yes—
307C23×C3#13512yes—
308C23×C3AGL(1,ℤ/8)12yes—
309C23×C3C22×D411yescharacteristic
310C3×D4—216no—
311C3×D4—24no—
312C3×D4—24no—
313C3×D4—24no—
314C3×D4C22⋊Q818yes—
315C2×C3×C4—28no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—21no—
327C2×C3×C4—24no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—44no—
331C2×C3×C4—416no—
332C2×C3×C4—416no—
333C2×C3×C4—48no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4#12012yes—
349C2×C3×C4#12012yes—
350C2×C3×C4C22⋊Q812yes—
351C2×C3×C4AGL(1,ℤ/8)12yes—
352C2×C3×C4C22×D412yes—
353C2×C3×C4C22×Q811yescharacteristic
354C2×C3×C4C2×#4112yes—
355C2×C3×C4C2×#4111yescharacteristic
356C3×C8—42no—
357C3×C8—42no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24C2×C3×D411yescharacteristic
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4C2×C3×D414yes—
376C2×D4C2×C3×Q814yes—
377C2×D4C3×#4114yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—41no—
394C22×C4—432no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×Q812yes—
407C22×C4C3×#4112yes—
408C22×C4C24×C311yescharacteristic
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4C2×C3×D414yes—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42C2×C3×D414yes—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—28no—
441C22×C3—432no—
442C22×C3—44no—
443C22×C3—44no—
444C22×C3—44no—
445C22×C3—44no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3#47814yescomplemented
449C22×C3C2×#12011yescharacteristic
450C22×C3C2×C22⋊Q811yescharacteristic
451C22×C3#52511yescharacteristic
452C22×C3C2×#13511yescharacteristic
453C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
454C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
455C22×C3#57511yescharacteristic
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—41no—
461C3×C4—44no—
462C3×C4—81no—
463C3×C4—81no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4C2×C22⋊Q812yes—
467C3×C4#52512yes—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—216no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—48no—
477C23—48no—
478C23—48no—
479C23—48no—
480C23—48no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—48no—
486C23C3×C22⋊Q812yes—
487C23C3×#13512yes—
488C23C3×AGL(1,ℤ/8)12yes—
489C23C22×C3×D411yescharacteristic
490D4—216no—
491D4—24no—
492D4—24no—
493D4—24no—
494D4C3×C22⋊Q818yes—
495C2×C4—28no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—44no—
511C2×C4—416no—
512C2×C4—416no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4C3×#12012yes—
529C2×C4C3×#12012yes—
530C2×C4C3×C22⋊Q812yes—
531C2×C4C3×AGL(1,ℤ/8)12yes—
532C2×C4C22×C3×D412yes—
533C2×C4C22×C3×Q811yescharacteristic
534C2×C4C2×C3×#4112yes—
535C2×C4C2×C3×#4111yescharacteristic
536C8—42no—
537C8—42no—
538C2×C3—24no—
539C2×C3—48no—
540C2×C3#284811yescharacteristic
541C2×C3#300211yescharacteristic
542C2×C3#300311yescharacteristic
543C2×C3C2×#47811yescharacteristic
544C2×C3#307711yescharacteristic
545C2×C3#310511yescharacteristic
546C2×C3#311411yescharacteristic
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—28no—
554C22—432no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22C3×#47814yescomplemented
562C22C2×C3×#12011yescharacteristic
563C22C2×C3×C22⋊Q811yescharacteristic
564C22C3×#52511yescharacteristic
565C22C2×C3×#13511yescharacteristic
566C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
567C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
568C22C3×#57511yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—44no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4C2×C3×C22⋊Q812yes—
580C4C3×#52512yes—
581C3#3295911yescharacteristic, direct product factor, Sylow
582C2—24no—
583C2—48no—
584C2C3×#284811yescharacteristic
585C2C3×#300211yescharacteristic
586C2C3×#300311yescharacteristic
587C2C2×C3×#47811yescharacteristic
588C2C3×#307711yescharacteristic
589C2C3×#310511yescharacteristic
590C2C3×#311411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
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
2. C2×C3×#478C2semidirect
3. C3×#2851C2semidirect
4. C3×#1864C2semidirect
5. C3×#1862C2semidirect
6. C3×#1853C2semidirect
7. C3×#1852C2semidirect
8. C3×#1804C2semidirect
9. C3×#1797C2semidirect
12. #32959C3direct (N × H)
15. C2×C3×#125C22semidirect
25. C2×C3×#107C22semidirect
26. C2×C3×#106C22semidirect
27. C2×C3×C4⋊C8C22semidirect
34. C3×#389C22semidirect
35. C3×#389C22semidirect
40. C3×#382C22semidirect
43. C3×#340C22semidirect
44. C3×#340C22semidirect
45. C3×#336C22semidirect
47. C2×#478C2×C3semidirect
48. #2851C2×C3semidirect
49. #1864C2×C3semidirect
50. #1862C2×C3semidirect
51. #1853C2×C3semidirect
52. #1852C2×C3semidirect
53. #1804C2×C3semidirect
54. #1797C2×C3semidirect
120. C2×#125C22×C3semidirect
130. C2×#107C22×C3semidirect
131. C2×#106C22×C3semidirect
132. C2×C4⋊C8C22×C3semidirect
139. #389C22×C3semidirect
140. #389C22×C3semidirect
145. #382C22×C3semidirect
148. #340C22×C3semidirect
149. #340C22×C3semidirect
150. #336C22×C3semidirect
448. C22×C3#478semidirect
561. C22C3×#478semidirect
581. C3#32959direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×C3×#4782
C3×#28512
C3×#18642
C3×#18622
C3×#18532
C3×#18522
C3×#18042
C3×#17972
C3×#17862
#329593