Finite Group Databasebeta

← C3×#32985 #10651592

Subgroups — C3×#32985 #10651592

Order 768.

1430 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#117C211yesmaximal, characteristic
2C2×C3×#485C212yesmaximal, complemented
3C3×#2848C211yesmaximal, characteristic
4C3×#1873C212yesmaximal, complemented
5C3×#1872C211yesmaximal, characteristic, complemented
6C3×#1872C211yesmaximal, characteristic, complemented
7C3×#1854C212yesmaximal, complemented
8C3×#1853C211yesmaximal, characteristic, complemented
9C3×#1852C211yesmaximal, characteristic, complemented
10C3×#1825C212yesmaximal, complemented
11C3×#1779C211yesmaximal, characteristic
12#32985C311yesmaximal, characteristic, direct product factor, Sylow
13C24×C3×C4C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#117—21no—
18C2×C3×#117—21no—
19C2×C3×#117—28no—
20C2×C3×#117C2212yes—
21C2×C3×#117C2212yes—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C22⋊C4—21no—
25C22×C3×C42—21no—
26C3×#485—28no—
27C2×C3×#107C2212yescomplemented
28C2×C3×#106C2212yescomplemented
29C2×C3×#103C2212yescomplemented
30C2×C3×#103C2212yescomplemented
31C2×C3×C22⋊C8C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C2×C3×C22⋊C8C2212yes—
35C3×#385—21no—
36C3×#385—21no—
37C3×#385—21no—
38C3×#380C2211yescharacteristic
39C3×#380C2211yescharacteristic
40C3×#380C2211yescharacteristic
41C3×#379—21no—
42C3×#340C2212yescomplemented
43C3×#340C2212yescomplemented
44C3×#340C2212yescomplemented
45C3×#340C2212yescomplemented
46C22×#117C2×C311yescharacteristic
47C2×#485C2×C312yescomplemented
48#2848C2×C311yescharacteristic
49#1873C2×C312yescomplemented
50#1872C2×C311yescharacteristic, complemented
51#1872C2×C311yescharacteristic, complemented
52#1854C2×C312yescomplemented
53#1853C2×C311yescharacteristic, complemented
54#1852C2×C311yescharacteristic, complemented
55#1825C2×C312yescomplemented
56#1779C2×C311yescharacteristic
57C25×C3D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4—24no—
60C23×C3×C4—24no—
61C23×C3×C4—22no—
62C23×C3×C4—22no—
63C23×C3×C4—21no—
64C23×C3×C4—21no—
65C23×C3×C4D411yescharacteristic
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C22×C3×C8C2312yes—
72C22×C3×C8C2312yes—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—24no—
77C3×#117—28no—
78C3×#117—28no—
79C3×#117—24no—
80C3×#117—24no—
81C3×#117—24no—
82C3×#117D418yes—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—28no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C42—28no—
105C2×C3×C42—22no—
106C2×C3×C42—22no—
107C3×#107—24no—
108C3×#106—24no—
109C3×#103—24no—
110C3×#103—24no—
111C3×C22⋊C8—24no—
112C3×C22⋊C8—22no—
113C3×C22⋊C8—22no—
114C3×C22⋊C8—24no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C24×C4C22×C311yescharacteristic
119C2×C22⋊Q8C22×C311yescharacteristic
120C2×C22⋊Q8C22×C311yescharacteristic
121C2×C22⋊Q8C22×C311yescharacteristic
122C2×#117—21no—
123C2×#117—21no—
124C2×#117—28no—
125C2×#117C22×C312yes—
126C2×#117C22×C312yes—
127C22×C4⋊C4C22×C311yescharacteristic
128C22×C4⋊C4C22×C311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C42—21no—
131#485—28no—
132C2×#107C22×C312yescomplemented
133C2×#106C22×C312yescomplemented
134C2×#103C22×C312yescomplemented
135C2×#103C22×C312yescomplemented
136C2×C22⋊C8C22×C312yes—
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C311yescharacteristic
139C2×C22⋊C8C22×C312yes—
140#385—21no—
141#385—21no—
142#385—21no—
143#380C22×C311yescharacteristic
144#380C22×C311yescharacteristic
145#380C22×C311yescharacteristic
146#379—21no—
147#340C22×C312yescomplemented
148#340C22×C312yescomplemented
149#340C22×C312yescomplemented
150#340C22×C312yescomplemented
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—41no—
157C24×C3—41no—
158C24×C3C2×D411yescharacteristic
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C22×C3×C4—216no—
166C22×C3×C4—216no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4#4114yes—
201C22×C3×C4#4112yes—
202C22×C3×C4#4112yes—
203C22×C3×C4C2411yescharacteristic
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—42no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4C2×D414yes—
222C3×C4⋊C4C2×D414yes—
223C3×C22⋊C4—28no—
224C3×C22⋊C4—28no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C42—28no—
232C3×C42—28no—
233C25C3×D411yescharacteristic
234C22×Q8C23×C311yescharacteristic
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C3×D411yescharacteristic
242C23×C4C3×D411yescharacteristic
243C23×C4C3×D411yescharacteristic
244C23×C4C23×C311yescharacteristic
245C23×C4C23×C311yescharacteristic
246C23×C4C23×C311yescharacteristic
247C22×C8C23×C312yes—
248C22×C8C23×C312yes—
249C22⋊Q8—24no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—24no—
253#117—28no—
254#117—28no—
255#117—24no—
256#117—24no—
257#117—24no—
258#117C3×D418yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C23×C311yescharacteristic
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C312yes—
270C2×C4⋊C4C23×C312yes—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42—28no—
281C2×C42—22no—
282C2×C42—22no—
283#107—24no—
284#106—24no—
285#103—24no—
286#103—24no—
287C22⋊C8—24no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—24no—
291#95—21no—
292#95—21no—
293#95—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—28no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—44no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3#12012yes—
318C23×C3#12312yes—
319C23×C3#12312yes—
320C23×C3#13512yes—
321C23×C3#13712yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—42no—
324C3×Q8—42no—
325C2×C3×C4—28no—
326C2×C3×C4—28no—
327C2×C3×C4—28no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—48no—
343C2×C3×C4—48no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4#12012yes—
361C2×C3×C4#12312yes—
362C2×C3×C4#12312yes—
363C2×C3×C4C22×D412yes—
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4112yes—
367C3×C8—42no—
368C3×C8—42no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—41no—
375C24—41no—
376C24C2×C3×D411yescharacteristic
377C24C2×C3×D411yescharacteristic
378C24C2×C3×D411yescharacteristic
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—22no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D414yes—
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C3×#4114yes—
419C22×C4C3×#4112yes—
420C22×C4C3×#4112yes—
421C22×C4C24×C311yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4C2×C3×D414yes—
440C4⋊C4C2×C3×D414yes—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C42—28no—
450C42—28no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—416no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3#48514yescomplemented
473C22×C3C2×#12011yescharacteristic
474C22×C3C2×#12311yescharacteristic
475C22×C3C2×#12311yescharacteristic
476C22×C3C2×#13511yescharacteristic
477C22×C3C2×#13711yescharacteristic
478C22×C3#57511yescharacteristic
479C22×C3#57511yescharacteristic
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—44no—
484C3×C4—44no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4#52512yes—
490C3×C4#52512yes—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—28no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C3×#12012yes—
515C23C3×#12312yes—
516C23C3×#12312yes—
517C23C3×#13512yes—
518C23C3×#13712yes—
519C23C22×C3×D411yescharacteristic
520Q8—42no—
521Q8—42no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4C3×#12012yes—
558C2×C4C3×#12312yes—
559C2×C4C3×#12312yes—
560C2×C4C22×C3×D412yes—
561C2×C4C2×C3×#4111yescharacteristic
562C2×C4C2×C3×#4111yescharacteristic
563C2×C4C2×C3×#4112yes—
564C8—42no—
565C8—42no—
566C2×C3—24no—
567C2×C3—42no—
568C2×C3—42no—
569C2×C3#284911yescharacteristic
570C2×C3#300211yescharacteristic
571C2×C3#300711yescharacteristic
572C2×C3C2×#48511yescharacteristic
573C2×C3#309111yescharacteristic
574C2×C3#309311yescharacteristic
575C2×C3#309511yescharacteristic
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—416no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C3×#48514yescomplemented
598C22C2×C3×#12011yescharacteristic
599C22C2×C3×#12311yescharacteristic
600C22C2×C3×#12311yescharacteristic
601C22C2×C3×#13511yescharacteristic
602C22C2×C3×#13711yescharacteristic
603C22C3×#57511yescharacteristic
604C22C3×#57511yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—44no—
609C4—44no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4C3×#52512yes—
615C4C3×#52512yes—
616C3#3298511yescharacteristic, direct product factor, Sylow
617C2—24no—
618C2—42no—
619C2—42no—
620C2C3×#284911yescharacteristic
621C2C3×#300211yescharacteristic
622C2C3×#300711yescharacteristic
623C2C2×C3×#48511yescharacteristic
624C2C3×#309111yescharacteristic
625C2C3×#309311yescharacteristic
626C2C3×#309511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#485C2semidirect
4. C3×#1873C2semidirect
5. C3×#1872C2semidirect
6. C3×#1872C2semidirect
7. C3×#1854C2semidirect
8. C3×#1853C2semidirect
9. C3×#1852C2semidirect
10. C3×#1825C2semidirect
12. #32985C3direct (N × H)
27. C2×C3×#107C22semidirect
28. C2×C3×#106C22semidirect
29. C2×C3×#103C22semidirect
30. C2×C3×#103C22semidirect
42. C3×#340C22semidirect
43. C3×#340C22semidirect
44. C3×#340C22semidirect
45. C3×#340C22semidirect
47. C2×#485C2×C3semidirect
49. #1873C2×C3semidirect
50. #1872C2×C3semidirect
51. #1872C2×C3semidirect
52. #1854C2×C3semidirect
53. #1853C2×C3semidirect
54. #1852C2×C3semidirect
55. #1825C2×C3semidirect
132. C2×#107C22×C3semidirect
133. C2×#106C22×C3semidirect
134. C2×#103C22×C3semidirect
135. C2×#103C22×C3semidirect
147. #340C22×C3semidirect
148. #340C22×C3semidirect
149. #340C22×C3semidirect
150. #340C22×C3semidirect
472. C22×C3#485semidirect
597. C22C3×#485semidirect
616. C3#32985direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1172
C2×C3×#4852
C3×#28482
C3×#18732
C3×#18722
C3×#18542
C3×#18532
C3×#18522
C3×#18252
C3×#17792
#329853