Finite Group Databasebeta

← C3×#14840 #10633447

Subgroups — C3×#14840 #10633447

Order 768.

1622 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#398C211yesmaximal, characteristic, complemented
2C2×C3×#385C211yesmaximal, characteristic
3C2×C3×#380C211yesmaximal, characteristic
4C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
5C3×#1445C211yesmaximal, characteristic, complemented
6C3×#1444C211yesmaximal, characteristic, complemented
7C3×#1441C211yesmaximal, characteristic, complemented
8C3×#1439C212yesmaximal
9C3×#1439C212yesmaximal
10C3×#1436C212yesmaximal, complemented
11C3×C4⋊#95C212yesmaximal, complemented
12#14840C311yesmaximal, characteristic, direct product factor, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C22⋊C4C2211yescharacteristic
24C22×C3×C42C2212yes—
25C3×#398—24no—
26C3×#385—24no—
27C3×#380—24no—
28C3×C4×C22⋊C4—24no—
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2211yescharacteristic
40C2×#398C2×C311yescharacteristic, complemented
41C2×#385C2×C311yescharacteristic
42C2×#380C2×C311yescharacteristic
43C2×C4×C22⋊C4C2×C311yescharacteristic
44#1445C2×C311yescharacteristic, complemented
45#1444C2×C311yescharacteristic, complemented
46#1441C2×C311yescharacteristic, complemented
47#1439C2×C312yes—
48#1439C2×C312yes—
49#1436C2×C312yescomplemented
50C4⋊#95C2×C312yescomplemented
51C25×C3C2311yescharacteristic
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—24no—
55C23×C3×C4—42no—
56C23×C3×C4D414yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—42no—
80C2×C3×C4⋊C4—42no—
81C2×C3×C4⋊C4—42no—
82C2×C3×C4⋊C4—42no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—42no—
91C2×C3×C22⋊C4—42no—
92C2×C3×C22⋊C4—42no—
93C2×C3×C22⋊C4Q818yes—
94C2×C3×C42—24no—
95C2×C3×C42—24no—
96C2×C3×C42—42no—
97C3×#95—22no—
98C3×#95—24no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—24no—
102C3×#95—24no—
103C3×#95—24no—
104C3×#95—42no—
105C3×#95—42no—
106C3×#95—42no—
107C3×#95D414yes—
108C24×C4C22×C311yescharacteristic
109C22×C4⋊C4C22×C311yescharacteristic
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C4⋊C4C22×C311yescharacteristic
112C22×C4⋊C4C22×C312yes—
113C22×C4⋊C4C22×C312yes—
114C22×C4⋊C4C22×C311yescharacteristic
115C22×C22⋊C4C22×C311yescharacteristic
116C22×C22⋊C4C22×C312yes—
117C22×C22⋊C4C22×C312yes—
118C22×C22⋊C4C22×C311yescharacteristic
119C22×C42C22×C312yes—
120#398—24no—
121#385—24no—
122#380—24no—
123C4×C22⋊C4—24no—
124C2×#95C22×C312yes—
125C2×#95C22×C311yescharacteristic
126C2×#95C22×C312yes—
127C2×#95C22×C312yes—
128C2×#95C22×C312yes—
129C2×#95C22×C311yescharacteristic
130C2×#95C22×C311yescharacteristic
131C2×#95C22×C312yes—
132C2×#95C22×C312yes—
133C2×#95C22×C312yes—
134C2×#95C22×C311yescharacteristic
135C24×C3—21no—
136C24×C3—22no—
137C24×C3—21no—
138C24×C3—21no—
139C24×C3—21no—
140C24×C3—22no—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3—22no—
144C24×C3C2×D412yes—
145C24×C3C2×Q814yes—
146C24×C3C2411yescharacteristic
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—28no—
173C22×C3×C4—28no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—44no—
199C22×C3×C4—48no—
200C22×C3×C4—48no—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D414yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×Q814yes—
206C22×C3×C4C2×Q814yes—
207C22×C3×C4#4112yes—
208C22×C3×C4#4112yes—
209C22×C3×C4#4112yes—
210C22×C3×C4#4114yes—
211C22×C3×C4#4114yes—
212C3×C4⋊C4—48no—
213C3×C4⋊C4—44no—
214C3×C4⋊C4—48no—
215C3×C4⋊C4—44no—
216C3×C4⋊C4—44no—
217C3×C4⋊C4—44no—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—216no—
220C3×C22⋊C4—44no—
221C3×C22⋊C4—48no—
222C3×C42—48no—
223C25C23×C311yescharacteristic
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—42no—
228C23×C4C3×D414yes—
229C23×C4C23×C311yescharacteristic
230C23×C4C23×C311yescharacteristic
231C23×C4C23×C312yes—
232C23×C4C23×C312yes—
233C23×C4C23×C312yes—
234C23×C4C23×C311yescharacteristic
235C23×C4C23×C311yescharacteristic
236C23×C4C23×C311yescharacteristic
237C23×C4C23×C312yes—
238C23×C4C23×C311yescharacteristic
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4C3×Q818yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—42no—
269#95—22no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95C3×D414yes—
280C23×C3—22no—
281C23×C3—24no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—24no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—24no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3C4⋊Q814yes—
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D411yescharacteristic
305C23×C3C22×Q812yes—
306C23×C3C2×#4111yescharacteristic
307C23×C3C2×#4112yes—
308C23×C3C2×#4112yes—
309C23×C3C2×#4111yescharacteristic
310C23×C3C2×#4111yescharacteristic
311C23×C3H(5,𝔽2)12yes—
312C23×C3H(5,𝔽2)11yescharacteristic
313C23×C3H(5,𝔽2)11yescharacteristic
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—416no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4#12114yes—
373C2×C3×C4C22⋊Q818yes—
374C2×C3×C4#12314yes—
375C24—21no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—21no—
382C24—21no—
383C24—22no—
384C24C2×C3×D412yes—
385C24C2×C3×Q814yes—
386C24C24×C311yescharacteristic
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4C2×C3×D412yes—
442C22×C4C2×C3×D412yes—
443C22×C4C2×C3×D414yes—
444C22×C4C2×C3×D412yes—
445C22×C4C2×C3×Q814yes—
446C22×C4C2×C3×Q814yes—
447C22×C4C3×#4112yes—
448C22×C4C3×#4112yes—
449C22×C4C3×#4112yes—
450C22×C4C3×#4114yes—
451C22×C4C3×#4114yes—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C22⋊C4—28no—
459C22⋊C4—216no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C42—48no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—44no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—44no—
472C22×C3—44no—
473C22×C3—44no—
474C22×C3C2×#12111yescharacteristic
475C22×C3C2×C22⋊Q812yes—
476C22×C3C2×#12311yescharacteristic
477C22×C3#52511yescharacteristic
478C22×C3#52912yes—
479C22×C3#52912yes—
480C22×C3C2×C4⋊Q811yescharacteristic
481C22×C3#53211yescharacteristic
482C22×C3#53312yes—
483C22×C3#53411yescharacteristic
484C22×C3#53511yescharacteristic
485C22×C3#53812yes—
486C22×C3#54012yes—
487C22×C3#54612yes—
488C22×C3#54611yescharacteristic
489C22×C3#55011yescharacteristic
490C22×C3#55112yes—
491C22×C3#55311yescharacteristic
492C22×C3#55412yes—
493C22×C3#55412yes—
494C22×C3#55412yes—
495C22×C3#55412yes—
496C22×C3#55511yescharacteristic
497C3×C4—24no—
498C3×C4—24no—
499C3×C4—28no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—44no—
503C3×C4—81no—
504C3×C4—82no—
505C3×C4—81no—
506C3×C4—82no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23C3×C4⋊Q814yes—
530C23C22×C3×D411yescharacteristic
531C23C22×C3×D411yescharacteristic
532C23C22×C3×Q812yes—
533C23C2×C3×#4111yescharacteristic
534C23C2×C3×#4112yes—
535C23C2×C3×#4112yes—
536C23C2×C3×#4111yescharacteristic
537C23C2×C3×#4111yescharacteristic
538C23C3×H(5,𝔽2)12yes—
539C23C3×H(5,𝔽2)11yescharacteristic
540C23C3×H(5,𝔽2)11yescharacteristic
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—416no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4C3×#12114yes—
600C2×C4C3×C22⋊Q818yes—
601C2×C4C3×#12314yes—
602C2×C3—44no—
603C2×C3#239511yescharacteristic
604C2×C3#241211yescharacteristic
605C2×C3#241411yescharacteristic
606C2×C3#245012yes—
607C2×C3#245412yes—
608C2×C3#250211yescharacteristic
609C2×C3#252212yes—
610C2×C3#257412yes—
611C2×C3#258311yescharacteristic
612C2×C3#266111yescharacteristic
613C2×C3#285711yescharacteristic
614C22—24no—
615C22—24no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22C2×C3×#12111yescharacteristic
626C22C2×C3×C22⋊Q812yes—
627C22C2×C3×#12311yescharacteristic
628C22C3×#52511yescharacteristic
629C22C3×#52912yes—
630C22C3×#52912yes—
631C22C2×C3×C4⋊Q811yescharacteristic
632C22C3×#53211yescharacteristic
633C22C3×#53312yes—
634C22C3×#53411yescharacteristic
635C22C3×#53511yescharacteristic
636C22C3×#53812yes—
637C22C3×#54012yes—
638C22C3×#54612yes—
639C22C3×#54611yescharacteristic
640C22C3×#55011yescharacteristic
641C22C3×#55112yes—
642C22C3×#55311yescharacteristic
643C22C3×#55412yes—
644C22C3×#55412yes—
645C22C3×#55412yes—
646C22C3×#55412yes—
647C22C3×#55511yescharacteristic
648C4—24no—
649C4—24no—
650C4—28no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4—82no—
658C3#1484011yescharacteristic, direct product factor, Sylow
659C2—44no—
660C2C3×#239511yescharacteristic
661C2C3×#241211yescharacteristic
662C2C3×#241411yescharacteristic
663C2C3×#245012yes—
664C2C3×#245412yes—
665C2C3×#250211yescharacteristic
666C2C3×#252212yes—
667C2C3×#257412yes—
668C2C3×#258311yescharacteristic
669C2C3×#266111yescharacteristic
670C2C3×#285711yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#398C2semidirect
5. C3×#1445C2semidirect
6. C3×#1444C2semidirect
7. C3×#1441C2semidirect
10. C3×#1436C2semidirect
11. C3×C4⋊#95C2semidirect
12. #14840C3direct (N × H)
40. C2×#398C2×C3semidirect
44. #1445C2×C3semidirect
45. #1444C2×C3semidirect
46. #1441C2×C3semidirect
49. #1436C2×C3semidirect
50. C4⋊#95C2×C3semidirect
658. C3#14840direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3982
C2×C3×#3852
C2×C3×#3802
C2×C3×C4×C22⋊C42
C3×#14452
C3×#14442
C3×#14412
C3×#14392
C3×#14362
C3×C4⋊#952
#148403