Finite Group Databasebeta

← C3×#24350 #10642957

Subgroups — C3×#24350 #10642957

Order 768.

1462 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3501C211yesmaximal, characteristic, complemented
2C3×#2848C212yesmaximal, complemented
3C3×C4×#120C211yesmaximal, characteristic
4C3×#2062C212yesmaximal, complemented
5C3×#2029C212yesmaximal, complemented
6C3×#1996C212yesmaximal, complemented
7C3×#1889C212yesmaximal
8C3×#1867C212yesmaximal, complemented
9C3×#1799C211yesmaximal, characteristic, complemented
10#24350C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C3×#554—22no—
13C3×#554—24no—
14C3×D4×Q8—22no—
15C3×#544—21no—
16C3×#543—21no—
17C3×#543C2212yes—
18C3×#533—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8—22no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×#120C2211yescharacteristic
23C3×#518C2212yes—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×#99C2212yes—
27C3×#386—21no—
28C3×#386C2211yescharacteristic
29C3×#385—21no—
30C3×#385—24no—
31C3×#385C2211yescharacteristic
32C3×#385C2214yes—
33C3×#380—22no—
34C3×#380C2212yes—
35C3×#379C2211yescharacteristic
36C3×#379C2214yes—
37C3×C4×C22⋊C4—21no—
38C3×C4×C22⋊C4C2211yescharacteristic
39C3×#342C2214yes—
40C3×#342C2214yes—
41C3×#342C2212yes—
42#3501C2×C311yescharacteristic, complemented
43#2848C2×C312yescomplemented
44C4×#120C2×C311yescharacteristic
45#2062C2×C312yescomplemented
46#2029C2×C312yescomplemented
47#1996C2×C312yescomplemented
48#1889C2×C312yes—
49#1867C2×C312yescomplemented
50#1799C2×C311yescharacteristic, complemented
51C25×C3C2311yescharacteristic
52C22×C3×Q8—21no—
53C22×C3×Q8D412yescomplemented
54C22×C3×D4—21no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4—22no—
58C23×C3×C4—22no—
59C23×C3×C4—21no—
60C23×C3×C4—44no—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—24no—
67C3×#125—22no—
68C3×#125—24no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—24no—
81C3×C22⋊Q8—48no—
82C3×#120—22no—
83C3×#120Q814yescomplemented
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—24no—
89C3×C4×D4—24no—
90C3×C4×D4—44no—
91C3×#117—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—24no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D412yescomplemented
99C2×C3×C4⋊C4D414yescomplemented
100C2×C3×C22⋊C4—22no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—24no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—24no—
107C2×C3×C22⋊C4D411yescharacteristic
108C2×C3×C22⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C22⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4C2314yes—
112C2×C3×C22⋊C4C2311yescharacteristic
113C2×C3×C42—21no—
114C2×C3×C42D411yescharacteristic
115C3×#99—24no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—24no—
120C3×#95—22no—
121C24×C4C22×C311yescharacteristic
122#554—22no—
123#554—24no—
124D4×Q8—22no—
125#544—21no—
126#543—21no—
127#543C22×C312yes—
128#533—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8C22×C312yes—
132C2×#120C22×C311yescharacteristic
133#518C22×C312yes—
134C2×C4×D4—21no—
135C2×C4×D4C22×C311yescharacteristic
136C2×#99C22×C312yes—
137#386—21no—
138#386C22×C311yescharacteristic
139#385—21no—
140#385—24no—
141#385C22×C311yescharacteristic
142#385C22×C314yes—
143#380—22no—
144#380C22×C312yes—
145#379C22×C314yes—
146#379C22×C311yescharacteristic
147C4×C22⋊C4—21no—
148C4×C22⋊C4C22×C311yescharacteristic
149#342C22×C312yes—
150#342C22×C314yes—
151#342C22×C314yes—
152C24×C3—22no—
153C24×C3—21no—
154C24×C3—21no—
155C24×C3—22no—
156C24×C3—42no—
157C24×C3—42no—
158C24×C3C2×D411yescharacteristic
159C24×C3C2×Q812yescomplemented
160C24×C3C2411yescharacteristic
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—44no—
166C2×C3×Q8—82no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—24no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4C2×Q812yes—
173C2×C3×D4#4112yes—
174C22×C3×C4—24no—
175C22×C3×C4—28no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—24no—
185C22×C3×C4—42no—
186C22×C3×C4—42no—
187C22×C3×C4—42no—
188C22×C3×C4—44no—
189C22×C3×C4—42no—
190C22×C3×C4—44no—
191C22×C3×C4—42no—
192C22×C3×C4—44no—
193C22×C3×C4—42no—
194C22×C3×C4—41no—
195C22×C3×C4—42no—
196C22×C3×C4—41no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D414yes—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D414yes—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D412yes—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—44no—
219C3×C4⋊C4—48no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—48no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—48no—
235C3×C22⋊C4C2×Q812yescomplemented
236C3×C42—24no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—42no—
240C25C23×C311yescharacteristic
241C22×Q8—21no—
242C22×Q8C3×D412yescomplemented
243C22×D4—21no—
244C22×D4C23×C311yescharacteristic
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—44no—
250C23×C4C3×D411yescharacteristic
251C23×C4C23×C311yescharacteristic
252C23×C4C23×C311yescharacteristic
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—24no—
256#125—22no—
257#125—24no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—48no—
271#120—22no—
272#120C3×Q814yescomplemented
273C4×Q8—22no—
274C4×Q8—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—44no—
280#117—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4C3×D411yescharacteristic
287C2×C4⋊C4C3×D412yescomplemented
288C2×C4⋊C4C3×D414yescomplemented
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4C3×D411yescharacteristic
297C2×C22⋊C4C23×C311yescharacteristic
298C2×C22⋊C4C23×C314yes—
299C2×C22⋊C4C23×C311yescharacteristic
300C2×C22⋊C4C23×C314yes—
301C2×C22⋊C4C23×C311yescharacteristic
302C2×C42—21no—
303C2×C42C3×D411yescharacteristic
304#99—24no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—24no—
309#95—22no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—24no—
317C23×C3—21no—
318C23×C3—41no—
319C23×C3—44no—
320C23×C3—41no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—42no—
324C23×C3—41no—
325C23×C3—44no—
326C23×C3—48no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—42no—
332C23×C3C22⋊Q812yescomplemented
333C23×C3C22×D411yescharacteristic
334C23×C3C22×D412yes—
335C23×C3C22×Q811yescharacteristic
336C23×C3C2×#4111yescharacteristic
337C23×C32−1+412yes—
338C3×Q8—44no—
339C3×Q8—82no—
340C3×Q8—82no—
341C3×Q8—82no—
342C3×D4—24no—
343C3×D4—24no—
344C3×D4—48no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—21no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—44no—
359C2×C3×C4—48no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—44no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—44no—
369C2×C3×C4—44no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—44no—
373C2×C3×C4—44no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—44no—
377C2×C3×C4—81no—
378C2×C3×C4—82no—
379C2×C3×C4—82no—
380C2×C3×C4#12011yescharacteristic
381C2×C3×C4#12012yes—
382C2×C3×C4#12011yescharacteristic
383C2×C3×C4C22⋊Q812yescomplemented
384C24—22no—
385C24—21no—
386C24—21no—
387C24—22no—
388C24—42no—
389C24—42no—
390C24C2×C3×D411yescharacteristic
391C24C2×C3×Q812yescomplemented
392C24C24×C311yescharacteristic
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—82no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4C2×C3×Q812yes—
405C2×D4C3×#4112yes—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—41no—
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C2×C3×D412yes—
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D414yes—
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D414yes—
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C2×C3×D412yes—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—21no—
457C22⋊C4—22no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—22no—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—42no—
467C22⋊C4C2×C3×Q812yescomplemented
468C42—24no—
469C42—21no—
470C42—21no—
471C42—42no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—21no—
476C22×C3—24no—
477C22×C3—28no—
478C22×C3—22no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—44no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—42no—
494C22×C3—81no—
495C22×C3C2×#12011yescharacteristic
496C22×C3C2×C22⋊Q811yescharacteristic
497C22×C3#53611yescharacteristic
498C22×C3#54812yes—
499C22×C3D4×Q812yes—
500C3×C4—22no—
501C3×C4—21no—
502C3×C4—22no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4—41no—
506C3×C4—44no—
507C3×C4—44no—
508C3×C4—41no—
509C3×C4—84no—
510C3×C4—81no—
511C3×C4B(4,𝔽2)12yes—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—21no—
517C23—22no—
518C23—24no—
519C23—21no—
520C23—41no—
521C23—44no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—44no—
528C23—48no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23C3×C22⋊Q812yescomplemented
535C23C22×C3×D411yescharacteristic
536C23C22×C3×D412yes—
537C23C22×C3×Q811yescharacteristic
538C23C2×C3×#4111yescharacteristic
539C23C3×2−1+412yes—
540Q8—44no—
541Q8—82no—
542Q8—82no—
543Q8—82no—
544D4—24no—
545D4—24no—
546D4—48no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4C3×#12011yescharacteristic
583C2×C4C3×#12012yes—
584C2×C4C3×#12011yescharacteristic
585C2×C4C3×C22⋊Q812yescomplemented
586C2×C3—21no—
587C2×C3—22no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—44no—
591C2×C3—42no—
592C2×C3#240911yescharacteristic
593C2×C3C2×B(4,𝔽2)11yescharacteristic
594C2×C3#302511yescharacteristic
595C22—22no—
596C22—21no—
597C22—22no—
598C22—21no—
599C22—24no—
600C22—28no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—81no—
618C22C2×C3×#12011yescharacteristic
619C22C2×C3×C22⋊Q811yescharacteristic
620C22C3×#53611yescharacteristic
621C22C3×#54812yes—
622C22C3×D4×Q812yes—
623C4—22no—
624C4—21no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—44no—
630C4—44no—
631C4—41no—
632C4—84no—
633C4—81no—
634C4C3×B(4,𝔽2)12yes—
635C3#2435011yescharacteristic, direct product factor, Sylow
636C2—21no—
637C2—22no—
638C2—21no—
639C2—22no—
640C2—44no—
641C2—42no—
642C2C3×#240911yescharacteristic
643C2C2×C3×B(4,𝔽2)11yescharacteristic
644C2C3×#302511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC2×C3×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3501C2semidirect
2. C3×#2848C2semidirect
4. C3×#2062C2semidirect
5. C3×#2029C2semidirect
6. C3×#1996C2semidirect
8. C3×#1867C2semidirect
9. C3×#1799C2semidirect
10. #24350C3direct (N × H)
42. #3501C2×C3semidirect
43. #2848C2×C3semidirect
45. #2062C2×C3semidirect
46. #2029C2×C3semidirect
47. #1996C2×C3semidirect
49. #1867C2×C3semidirect
50. #1799C2×C3semidirect
53. C22×C3×Q8D4semidirect
83. C3×#120Q8semidirect
98. C2×C3×C4⋊C4D4semidirect
99. C2×C3×C4⋊C4D4semidirect
159. C24×C3C2×Q8semidirect
235. C3×C22⋊C4C2×Q8semidirect
242. C22×Q8C3×D4semidirect
272. #120C3×Q8semidirect
287. C2×C4⋊C4C3×D4semidirect
288. C2×C4⋊C4C3×D4semidirect
332. C23×C3C22⋊Q8semidirect
383. C2×C3×C4C22⋊Q8semidirect
391. C24C2×C3×Q8semidirect
467. C22⋊C4C2×C3×Q8semidirect
534. C23C3×C22⋊Q8semidirect
585. C2×C4C3×C22⋊Q8semidirect
635. C3#24350direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35012
C3×#28482
C3×C4×#1202
C3×#20622
C3×#20292
C3×#19962
C3×#18892
C3×#18672
C3×#17992
#243503