Finite Group Databasebeta

← C3×#24658 #10643265

Subgroups — C3×#24658 #10643265

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic, complemented
2C2×C3×#432C211yesmaximal, characteristic, complemented
3C3×#2856C211yesmaximal, characteristic, complemented
4C3×C4×#121C211yesmaximal, characteristic, complemented
5C3×#2034C212yesmaximal, complemented
6C3×#1999C212yesmaximal, complemented
7C3×#1982C211yesmaximal, characteristic, complemented
8C3×#1972C212yesmaximal, complemented
9C3×#1948C211yesmaximal, characteristic, complemented
10C3×#1936C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24658C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×#121—21no—
16C2×C3×#121—21no—
17C2×C3×#121C2211yescharacteristic
18C2×C3×#121C2212yes—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C3×#459—24no—
23C3×#432—24no—
24C2×C3×#106C2212yes—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×C22⋊C8C2212yes—
32C2×C3×#97C2212yes—
33C3×#390—21no—
34C3×#390—21no—
35C3×#390C2211yescharacteristic
36C3×#390C2212yes—
37C3×#389C2211yescharacteristic
38C3×#388—21no—
39C3×#386—21no—
40C3×#384C2211yescharacteristic
41C3×#380C2211yescharacteristic
42C3×C4×C4⋊C4C2211yescharacteristic
43C3×C4×C22⋊C4—21no—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#459C2×C311yescharacteristic, complemented
47C2×#432C2×C311yescharacteristic, complemented
48#2856C2×C311yescharacteristic, complemented
49C4×#121C2×C311yescharacteristic, complemented
50#2034C2×C312yescomplemented
51#1999C2×C312yescomplemented
52#1982C2×C311yescharacteristic, complemented
53#1972C2×C312yescomplemented
54#1948C2×C311yescharacteristic, complemented
55#1936C2×C312yescomplemented
56#1802C2×C311yescharacteristic, complemented
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C2×C3×D8—24no—
68C2×C3×D8—24no—
69C2×C3×D8—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×#127—22no—
73C3×#127—22no—
74C3×#121—22no—
75C3×#121—22no—
76C3×#121—42no—
77C3×#121—42no—
78C3×#121—44no—
79C3×#121—44no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—44no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C42—22no—
99C2×C3×C42—22no—
100C2×C3×C42—21no—
101C2×C3×C42D411yescharacteristic
102C2×C3×C42D411yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C3×#106—42no—
106C3×C4⋊C8—22no—
107C3×C4⋊C8D414yescomplemented
108C3×#102—24no—
109C3×#102—24no—
110C3×#102—42no—
111C3×#102—42no—
112C3×C22⋊C8—24no—
113C3×#97—24no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C22×D8C22×C312yes—
118C2×#127C22×C311yescharacteristic
119C2×#121—21no—
120C2×#121—21no—
121C2×#121C22×C311yescharacteristic
122C2×#121C22×C312yes—
123C2×C4×D4—21no—
124C2×C4×D4C22×C311yescharacteristic
125C22×C42C22×C311yescharacteristic
126#459—24no—
127#432—24no—
128C2×#106C22×C312yes—
129C2×C4⋊C8C22×C311yescharacteristic
130C2×C4⋊C8C22×C311yescharacteristic
131C2×#102C22×C312yes—
132C2×#102C22×C312yes—
133C2×#102C22×C312yes—
134C2×#102C22×C312yes—
135C2×C22⋊C8C22×C312yes—
136C2×#97C22×C312yes—
137#390—21no—
138#390—21no—
139#390C22×C312yes—
140#390C22×C311yescharacteristic
141#389C22×C311yescharacteristic
142#388—21no—
143#386—21no—
144#384C22×C311yescharacteristic
145#380C22×C311yescharacteristic
146C4×C4⋊C4C22×C311yescharacteristic
147C4×C22⋊C4—21no—
148#340C22×C312yes—
149#340C22×C312yes—
150C24×C3—21no—
151C24×C3—22no—
152C24×C3C2×D411yescharacteristic
153C2×C3×D4—24no—
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C2×C3×D4—44no—
170C2×C3×D4—44no—
171C2×C3×D4—44no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C2×C3×D4#4112yes—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—41no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C3×D8—48no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8C2×D414yes—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4—42no—
219C3×C4⋊C4#4112yes—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—44no—
225C3×C22⋊C4—44no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—42no—
233C3×C42C2×D412yes—
234C3×C42#4112yes—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C23×C312yes—
240C22×D4C23×C311yescharacteristic
241C23×C4—21no—
242C23×C4C3×D411yescharacteristic
243C23×C4C3×D411yescharacteristic
244C23×C4C23×C311yescharacteristic
245C2×D8—24no—
246C2×D8—24no—
247C2×D8—42no—
248C22×C8C23×C312yes—
249C22×C8C23×C312yes—
250#127—22no—
251#127—22no—
252#121—22no—
253#121—22no—
254#121—42no—
255#121—42no—
256#121—44no—
257#121—44no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—44no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C4⋊C4C23×C312yes—
266C2×C4⋊C4C23×C311yescharacteristic
267C2×C4⋊C4C23×C311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42C3×D411yescharacteristic
280C2×C42C3×D411yescharacteristic
281C2×C42C23×C311yescharacteristic
282C2×C42C23×C311yescharacteristic
283#106—42no—
284C4⋊C8—22no—
285C4⋊C8C3×D414yescomplemented
286#102—24no—
287#102—24no—
288#102—42no—
289#102—42no—
290C22⋊C8—24no—
291#97—24no—
292#95—21no—
293#95—21no—
294#95—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3AGL(1,ℤ/8)12yes—
316C23×C3C22×D411yescharacteristic
317C3×D4—22no—
318C3×D4—22no—
319C3×D4—48no—
320C3×D4—42no—
321C3×D4—41no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—41no—
325C3×D4—42no—
326C3×D4—84no—
327C3×D4—88no—
328C3×D4—84no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—44no—
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—44no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—84no—
359C2×C3×C4#12112yes—
360C2×C3×C4#12112yes—
361C2×C3×C4AGL(1,ℤ/8)12yes—
362C2×C3×C4AGL(1,ℤ/8)12yes—
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C3×C8—24no—
370C3×C8—42no—
371C24—21no—
372C24—22no—
373C24C2×C3×D411yescharacteristic
374C2×D4—24no—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4C3×#4112yes—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C24×C311yescharacteristic
423D8—48no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8C2×C3×D414yes—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4C3×#4112yes—
441C22⋊C4—24no—
442C22⋊C4—42no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C42—22no—
450C42—22no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42C2×C3×D412yes—
455C42C3×#4112yes—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3—82no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3C2×#12111yescharacteristic
478C22×C3#52511yescharacteristic
479C22×C3#53511yescharacteristic
480C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
481C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
482C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
483C22×C3#57511yescharacteristic
484C3×C4—22no—
485C3×C4—42no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—81no—
490C3×C4—82no—
491C3×C4—81no—
492C3×C4#46914yes—
493C3×C4#54611yescharacteristic
494C3×C4#55011yescharacteristic
495C3×C4#55311yescharacteristic
496C3×C4#55511yescharacteristic
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23C3×AGL(1,ℤ/8)12yes—
518C23C22×C3×D411yescharacteristic
519D4—22no—
520D4—22no—
521D4—48no—
522D4—42no—
523D4—41no—
524D4—42no—
525D4—42no—
526D4—41no—
527D4—42no—
528D4—84no—
529D4—88no—
530D4—84no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4C3×#12112yes—
562C2×C4C3×#12112yes—
563C2×C4C3×AGL(1,ℤ/8)12yes—
564C2×C4C3×AGL(1,ℤ/8)12yes—
565C2×C4C22×C3×D411yescharacteristic
566C2×C4C2×C3×#4111yescharacteristic
567C2×C4C2×C3×#4111yescharacteristic
568C2×C4C2×C3×#4111yescharacteristic
569C2×C4C3×H(5,𝔽2)11yescharacteristic
570C2×C4C3×H(5,𝔽2)11yescharacteristic
571C8—24no—
572C8—42no—
573C2×C3—42no—
574C2×C3—82no—
575C2×C3—162no—
576C2×C3#241411yescharacteristic
577C2×C3C2×#46911yescharacteristic
578C2×C3#305511yescharacteristic
579C2×C3#310911yescharacteristic
580C2×C3#311511yescharacteristic
581C2×C3#321411yescharacteristic
582C2×C3#321711yescharacteristic
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22C2×C3×#12111yescharacteristic
605C22C3×#52511yescharacteristic
606C22C3×#53511yescharacteristic
607C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
608C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
609C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
610C22C3×#57511yescharacteristic
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—81no—
617C4—82no—
618C4—81no—
619C4C3×#46914yes—
620C4C3×#54611yescharacteristic
621C4C3×#55011yescharacteristic
622C4C3×#55311yescharacteristic
623C4C3×#55511yescharacteristic
624C3#2465811yescharacteristic, direct product factor, Sylow
625C2—42no—
626C2—82no—
627C2—162no—
628C2C3×#241411yescharacteristic
629C2C2×C3×#46911yescharacteristic
630C2C3×#305511yescharacteristic
631C2C3×#310911yescharacteristic
632C2C3×#311511yescharacteristic
633C2C3×#321411yescharacteristic
634C2C3×#321711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C2×C3×#459C2semidirect
2. C2×C3×#432C2semidirect
3. C3×#2856C2semidirect
4. C3×C4×#121C2semidirect
5. C3×#2034C2semidirect
6. C3×#1999C2semidirect
7. C3×#1982C2semidirect
8. C3×#1972C2semidirect
9. C3×#1948C2semidirect
10. C3×#1936C2semidirect
11. C3×#1802C2semidirect
12. #24658C3direct (N × H)
46. C2×#459C2×C3semidirect
47. C2×#432C2×C3semidirect
48. #2856C2×C3semidirect
49. C4×#121C2×C3semidirect
50. #2034C2×C3semidirect
51. #1999C2×C3semidirect
52. #1982C2×C3semidirect
53. #1972C2×C3semidirect
54. #1948C2×C3semidirect
55. #1936C2×C3semidirect
56. #1802C2×C3semidirect
107. C3×C4⋊C8D4semidirect
285. C4⋊C8C3×D4semidirect
624. C3#24658direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4592
C2×C3×#4322
C3×#28562
C3×C4×#1212
C3×#20342
C3×#19992
C3×#19822
C3×#19722
C3×#19482
C3×#19362
C3×#18022
#246583