Finite Group Databasebeta

← C3×#24690 #10643297

Subgroups — C3×#24690 #10643297

Order 768.

808 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#478C211yesmaximal, characteristic
2C2×C3×#440C211yesmaximal, characteristic
3C3×#2565C211yesmaximal, characteristic, complemented
4C3×#2354C211yesmaximal, characteristic
5C3×#2073C211yesmaximal, characteristic
6C3×#2071C211yesmaximal, characteristic
7C3×#2055C211yesmaximal, characteristic
8C3×#2055C211yesmaximal, characteristic
9C3×#1992C211yesmaximal, characteristic, complemented
10C3×#1949C211yesmaximal, characteristic, complemented
11C3×#1947C211yesmaximal, characteristic, complemented
12C3×#1940C211yesmaximal, characteristic, complemented
13C3×#1928C211yesmaximal, characteristic, complemented
14C3×#1927C211yesmaximal, characteristic, complemented
15C3×#1803C211yesmaximal, characteristic, complemented
16#24690C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#117—21no—
22C3×#478—24no—
23C3×#440—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C2×C3×#97C2211yescharacteristic
37C3×#398—21no—
38C3×#398—21no—
39C3×#391C2211yescharacteristic
40C3×#391C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#390C2211yescharacteristic
43C3×#388—21no—
44C3×#388—21no—
45C3×#387—21no—
46C3×#387—21no—
47C3×#384C2211yescharacteristic
48C3×#384C2211yescharacteristic
49C3×#382C2211yescharacteristic
50C3×#380—21no—
51C3×C4×C4⋊C4C2211yescharacteristic
52C3×C4×C4⋊C4C2211yescharacteristic
53C3×C4×C4⋊C4C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62C2×#478C2×C311yescharacteristic
63C2×#440C2×C311yescharacteristic
64#2565C2×C311yescharacteristic, complemented
65#2354C2×C311yescharacteristic
66#2073C2×C311yescharacteristic
67#2071C2×C311yescharacteristic
68#2055C2×C311yescharacteristic
69#2055C2×C311yescharacteristic
70#1992C2×C311yescharacteristic, complemented
71#1949C2×C311yescharacteristic, complemented
72#1947C2×C311yescharacteristic, complemented
73#1940C2×C311yescharacteristic, complemented
74#1928C2×C311yescharacteristic, complemented
75#1927C2×C311yescharacteristic, complemented
76#1803C2×C311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C2×C3×SD16—22no—
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#125—22no—
88C3×#125—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×D4—22no—
92C3×C4×D4D414yes—
93C3×#117—44no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4D411yescharacteristic
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
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×C42—21no—
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×#107—22no—
121C3×#107—22no—
122C3×#106—22no—
123C3×C4⋊C8—22no—
124C3×C4⋊C8—22no—
125C3×C4⋊C8—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#103—22no—
129C3×#102—22no—
130C3×#102—22no—
131C3×#102—22no—
132C3×#97—22no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95D411yescharacteristic
142C3×#95D411yescharacteristic
143C22×SD16C22×C311yescharacteristic
144C2×#125C22×C311yescharacteristic
145C2×C4×Q8C22×C311yescharacteristic
146C2×C4×D4C22×C311yescharacteristic
147C2×#117—21no—
148#478—24no—
149#440—24no—
150C2×#107C22×C311yescharacteristic
151C2×#107C22×C311yescharacteristic
152C2×#106C22×C311yescharacteristic
153C2×C4⋊C8C22×C311yescharacteristic
154C2×C4⋊C8C22×C311yescharacteristic
155C2×C4⋊C8C22×C311yescharacteristic
156C2×#103C22×C311yescharacteristic
157C2×#103C22×C311yescharacteristic
158C2×#103C22×C311yescharacteristic
159C2×#102C22×C311yescharacteristic
160C2×#102C22×C311yescharacteristic
161C2×#102C22×C311yescharacteristic
162C2×#97C22×C311yescharacteristic
163#398—21no—
164#398—21no—
165#391C22×C311yescharacteristic
166#391C22×C311yescharacteristic
167#390C22×C311yescharacteristic
168#390C22×C311yescharacteristic
169#388—21no—
170#388—21no—
171#387—21no—
172#387—21no—
173#384C22×C311yescharacteristic
174#384C22×C311yescharacteristic
175#382C22×C311yescharacteristic
176#380—21no—
177C4×C4⋊C4C22×C311yescharacteristic
178C4×C4⋊C4C22×C311yescharacteristic
179C4×C4⋊C4C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#340C22×C311yescharacteristic
182#340C22×C311yescharacteristic
183#340C22×C311yescharacteristic
184#340C22×C311yescharacteristic
185#340C22×C311yescharacteristic
186#336C22×C311yescharacteristic
187#336C22×C311yescharacteristic
188C24×C3—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8#4112yes—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4C2×D412yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—24no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4C2×D411yescharacteristic
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—28no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8#4112yes—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—81no—
251C3×C4⋊C4C2×D412yes—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—42no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—81no—
256C3×C22⋊C4—81no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—42no—
262C3×C42—42no—
263C3×C42—42no—
264C3×C42C2×D412yes—
265C3×C42#4112yes—
266C22×Q8C23×C311yescharacteristic
267C22×D4C23×C311yescharacteristic
268C23×C4—21no—
269C2×SD16—22no—
270C2×SD16—22no—
271C2×SD16—22no—
272C22×C8C23×C311yescharacteristic
273C22×C8C23×C311yescharacteristic
274C22×C8C23×C311yescharacteristic
275C22×C8C23×C311yescharacteristic
276#125—22no—
277#125—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×D4—22no—
281C4×D4C3×D414yes—
282#117—44no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C3×D411yescharacteristic
291C2×C4⋊C4C3×D411yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42C23×C311yescharacteristic
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309#107—22no—
310#107—22no—
311#106—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315#103—22no—
316#103—22no—
317#103—22no—
318#102—22no—
319#102—22no—
320#102—22no—
321#97—22no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95C3×D411yescharacteristic
331#95C3×D411yescharacteristic
332C23×C3—22no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3C22×D411yescharacteristic
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—41no—
347C3×D4—41no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
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—48no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
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—41no—
384C2×C3×C4—41no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4AGL(1,ℤ/8)12yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394C3×C8—22no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8—41no—
398C24—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8C3×#4112yes—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4C2×C3×D412yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C24×C311yescharacteristic
428SD16—28no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8C3×#4112yes—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4C2×C3×D412yes—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42C2×C3×D412yes—
475C42C3×#4112yes—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3C2×#12111yescharacteristic
484C22×C3#54711yescharacteristic
485C22×C3#54811yescharacteristic
486C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
487C22×C3#57511yescharacteristic
488C22×C3#57711yescharacteristic
489C22×C3#57811yescharacteristic
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4#52511yescharacteristic
504C3×C4#53611yescharacteristic
505C3×C4#55211yescharacteristic
506C3×C4#55611yescharacteristic
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C22×C3×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—41no—
518Q8—41no—
519D4—22no—
520D4—22no—
521D4—41no—
522D4—41no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—48no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
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×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4C3×#12112yes—
561C2×C4C3×#12112yes—
562C2×C4C3×AGL(1,ℤ/8)12yes—
563C2×C4C22×C3×D411yescharacteristic
564C2×C4C2×C3×#4111yescharacteristic
565C2×C4C2×C3×#4111yescharacteristic
566C2×C4C2×C3×#4111yescharacteristic
567C2×C4C3×2−1+411yescharacteristic
568C2×C4C3×2−1+411yescharacteristic
569C8—22no—
570C8—41no—
571C8—41no—
572C8—41no—
573C2×C3—82no—
574C2×C3#241511yescharacteristic
575C2×C3#303611yescharacteristic
576C2×C3#304611yescharacteristic
577C2×C3#334311yescharacteristic
578C2×C3#334811yescharacteristic
579C2×C3#336811yescharacteristic
580C2×C3#337711yescharacteristic
581C22—42no—
582C22—42no—
583C22—42no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22C2×C3×#12111yescharacteristic
589C22C3×#54711yescharacteristic
590C22C3×#54811yescharacteristic
591C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
592C22C3×#57511yescharacteristic
593C22C3×#57711yescharacteristic
594C22C3×#57811yescharacteristic
595C4—22no—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4C3×#52511yescharacteristic
609C4C3×#53611yescharacteristic
610C4C3×#55211yescharacteristic
611C4C3×#55611yescharacteristic
612C3#2469011yescharacteristic, direct product factor, Sylow
613C2—82no—
614C2C3×#241511yescharacteristic
615C2C3×#303611yescharacteristic
616C2C3×#304611yescharacteristic
617C2C3×#334311yescharacteristic
618C2C3×#334811yescharacteristic
619C2C3×#336811yescharacteristic
620C2C3×#337711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2565C2semidirect
9. C3×#1992C2semidirect
10. C3×#1949C2semidirect
11. C3×#1947C2semidirect
12. C3×#1940C2semidirect
13. C3×#1928C2semidirect
14. C3×#1927C2semidirect
15. C3×#1803C2semidirect
16. #24690C3direct (N × H)
64. #2565C2×C3semidirect
70. #1992C2×C3semidirect
71. #1949C2×C3semidirect
72. #1947C2×C3semidirect
73. #1940C2×C3semidirect
74. #1928C2×C3semidirect
75. #1927C2×C3semidirect
76. #1803C2×C3semidirect
612. C3#24690direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4782
C2×C3×#4402
C3×#25652
C3×#23542
C3×#20732
C3×#20712
C3×#20552
C3×#19922
C3×#19492
C3×#19472
C3×#19402
C3×#19282
C3×#19272
C3×#18032
#246903