Finite Group Databasebeta

← C3×#24705 #10643312

Subgroups — C3×#24705 #10643312

Order 768.

804 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×#479C211yesmaximal, characteristic, complemented
2C2×C3×#431C211yesmaximal, characteristic
3C3×#2851C211yesmaximal, characteristic
4C3×C4×C22⋊Q8C211yesmaximal, characteristic
5C3×#2077C211yesmaximal, characteristic
6C3×#2075C211yesmaximal, characteristic
7C3×#2036C211yesmaximal, characteristic
8C3×#2035C211yesmaximal, characteristic
9C3×#1989C211yesmaximal, characteristic, complemented
10C3×#1949C211yesmaximal, characteristic, complemented
11C3×#1942C211yesmaximal, characteristic, complemented
12C3×#1941C211yesmaximal, characteristic, complemented
13C3×#1933C211yesmaximal, characteristic, complemented
14C3×#1933C211yesmaximal, characteristic, complemented
15C3×#1805C211yesmaximal, characteristic, complemented
16#24705C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C3×#479—24no—
22C3×#431—24no—
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C4×C8C2211yescharacteristic
36C2×C3×C4×C8C2211yescharacteristic
37C3×#391C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#389C2211yescharacteristic
41C3×#387—21no—
42C3×#387—21no—
43C3×#387—21no—
44C3×#385—21no—
45C3×#385—21no—
46C3×#385—21no—
47C3×#382C2211yescharacteristic
48C3×#380C2211yescharacteristic
49C3×#380C2211yescharacteristic
50C3×#380C2211yescharacteristic
51C3×C4×C4⋊C4—21no—
52C3×C4×C4⋊C4C2211yescharacteristic
53C3×C4×C22⋊C4—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60C3×#340C2211yescharacteristic
61C3×#340C2211yescharacteristic
62C2×#479C2×C311yescharacteristic, complemented
63C2×#431C2×C311yescharacteristic
64#2851C2×C311yescharacteristic
65C4×C22⋊Q8C2×C311yescharacteristic
66#2077C2×C311yescharacteristic
67#2075C2×C311yescharacteristic
68#2036C2×C311yescharacteristic
69#2035C2×C311yescharacteristic
70#1989C2×C311yescharacteristic, complemented
71#1949C2×C311yescharacteristic, complemented
72#1942C2×C311yescharacteristic, complemented
73#1941C2×C311yescharacteristic, complemented
74#1933C2×C311yescharacteristic, complemented
75#1933C2×C311yescharacteristic, complemented
76#1805C2×C311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4D411yescharacteristic
79C23×C3×C4D411yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#125—22no—
86C3×#125—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C42—22no—
110C2×C3×C42—22no—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42D411yescharacteristic
114C2×C3×C42D411yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#107—22no—
119C3×#106—22no—
120C3×#106—22no—
121C3×C4⋊C8—22no—
122C3×C4⋊C8D414yes—
123C3×#103—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#103—22no—
127C3×C22⋊C8—22no—
128C3×C22⋊C8—22no—
129C3×C4×C8—22no—
130C3×C4×C8—22no—
131C3×#95—21no—
132C3×#95—21no—
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×#95—21no—
142C2×#125C22×C311yescharacteristic
143C2×C22⋊Q8C22×C311yescharacteristic
144C2×C4×Q8C22×C311yescharacteristic
145C22×C42C22×C311yescharacteristic
146#479—24no—
147#431—24no—
148C2×#107C22×C311yescharacteristic
149C2×#107C22×C311yescharacteristic
150C2×#106C22×C311yescharacteristic
151C2×#106C22×C311yescharacteristic
152C2×C4⋊C8C22×C311yescharacteristic
153C2×C4⋊C8C22×C311yescharacteristic
154C2×#103C22×C311yescharacteristic
155C2×#103C22×C311yescharacteristic
156C2×#103C22×C311yescharacteristic
157C2×#103C22×C311yescharacteristic
158C2×C22⋊C8C22×C311yescharacteristic
159C2×C22⋊C8C22×C311yescharacteristic
160C2×C4×C8C22×C311yescharacteristic
161C2×C4×C8C22×C311yescharacteristic
162#391C22×C311yescharacteristic
163#389C22×C311yescharacteristic
164#389C22×C311yescharacteristic
165#389C22×C311yescharacteristic
166#387—21no—
167#387—21no—
168#387—21no—
169#385—21no—
170#385—21no—
171#385—21no—
172#382C22×C311yescharacteristic
173#380C22×C311yescharacteristic
174#380C22×C311yescharacteristic
175#380C22×C311yescharacteristic
176C4×C4⋊C4—21no—
177C4×C4⋊C4C22×C311yescharacteristic
178C4×C22⋊C4—21no—
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#340C22×C311yescharacteristic
182#340C22×C311yescharacteristic
183#340C22×C311yescharacteristic
184#340C22×C311yescharacteristic
185#340C22×C311yescharacteristic
186#340C22×C311yescharacteristic
187C24×C3C2×D411yescharacteristic
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8#4112yes—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—41no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
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×C8C2×D412yes—
231C2×C3×C8C2×D412yes—
232C3×C4⋊C4—24no—
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—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4—42no—
252C3×C4⋊C4#4112yes—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—42no—
257C3×C42—22no—
258C3×C42—22no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—42no—
262C3×C42—42no—
263C3×C42C2×D412yes—
264C3×C42#4112yes—
265C22×Q8C23×C311yescharacteristic
266C23×C4C3×D411yescharacteristic
267C23×C4C3×D411yescharacteristic
268C23×C4C23×C311yescharacteristic
269C22×C8C23×C311yescharacteristic
270C22×C8C23×C311yescharacteristic
271C22×C8C23×C311yescharacteristic
272C22×C8C23×C311yescharacteristic
273#125—22no—
274#125—22no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42C3×D411yescharacteristic
302C2×C42C3×D411yescharacteristic
303C2×C42C23×C311yescharacteristic
304C2×C42C23×C311yescharacteristic
305#107—22no—
306#107—22no—
307#106—22no—
308#106—22no—
309C4⋊C8—22no—
310C4⋊C8C3×D414yes—
311#103—22no—
312#103—22no—
313#103—22no—
314#103—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C4×C8—22no—
318C4×C8—22no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3#13512yes—
337C23×C3C22×D411yescharacteristic
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C2×C3×C4—28no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—44no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
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#12112yes—
384C2×C3×C4#12112yes—
385C2×C3×C4#13512yes—
386C2×C3×C4#13712yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C42−1+411yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C3×C8—22no—
394C3×C8—22no—
395C3×C8—41no—
396C3×C8—41no—
397C24C2×C3×D411yescharacteristic
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8C3×#4112yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—41no—
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D411yescharacteristic
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C24×C311yescharacteristic
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×C8C2×C3×D412yes—
441C2×C8C2×C3×D412yes—
442C4⋊C4—24no—
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—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4C3×#4112yes—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—42no—
473C42C2×C3×D412yes—
474C42C3×#4112yes—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3C2×#12111yescharacteristic
483C22×C3#52511yescharacteristic
484C22×C3#53611yescharacteristic
485C22×C3C2×#13511yescharacteristic
486C22×C3C2×#13511yescharacteristic
487C22×C3C2×#13711yescharacteristic
488C22×C3#57511yescharacteristic
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—42no—
494C3×C4—41no—
495C3×C4—42no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4—81no—
502C3×C4#54711yescharacteristic
503C3×C4#54811yescharacteristic
504C3×C4#55211yescharacteristic
505C3×C4#55611yescharacteristic
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23C3×#13512yes—
513C23C22×C3×D411yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—41no—
517Q8—41no—
518C2×C4—28no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
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×C4C3×#12112yes—
560C2×C4C3×#12112yes—
561C2×C4C3×#13512yes—
562C2×C4C3×#13712yes—
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—22no—
571C8—41no—
572C8—41no—
573C2×C3—42no—
574C2×C3#241511yescharacteristic
575C2×C3#305011yescharacteristic
576C2×C3#305611yescharacteristic
577C2×C3#310211yescharacteristic
578C2×C3#310811yescharacteristic
579C2×C3#323111yescharacteristic
580C2×C3#323211yescharacteristic
581C22—22no—
582C22—22no—
583C22—22no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22C2×C3×#12111yescharacteristic
589C22C3×#52511yescharacteristic
590C22C3×#53611yescharacteristic
591C22C2×C3×#13511yescharacteristic
592C22C2×C3×#13511yescharacteristic
593C22C2×C3×#13711yescharacteristic
594C22C3×#57511yescharacteristic
595C4—22no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4C3×#54711yescharacteristic
609C4C3×#54811yescharacteristic
610C4C3×#55211yescharacteristic
611C4C3×#55611yescharacteristic
612C3#2470511yescharacteristic, direct product factor, Sylow
613C2—42no—
614C2C3×#241511yescharacteristic
615C2C3×#305011yescharacteristic
616C2C3×#305611yescharacteristic
617C2C3×#310211yescharacteristic
618C2C3×#310811yescharacteristic
619C2C3×#323111yescharacteristic
620C2C3×#323211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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×#479C2semidirect
9. C3×#1989C2semidirect
10. C3×#1949C2semidirect
11. C3×#1942C2semidirect
12. C3×#1941C2semidirect
13. C3×#1933C2semidirect
14. C3×#1933C2semidirect
15. C3×#1805C2semidirect
16. #24705C3direct (N × H)
62. C2×#479C2×C3semidirect
70. #1989C2×C3semidirect
71. #1949C2×C3semidirect
72. #1942C2×C3semidirect
73. #1941C2×C3semidirect
74. #1933C2×C3semidirect
75. #1933C2×C3semidirect
76. #1805C2×C3semidirect
612. C3#24705direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4792
C2×C3×#4312
C3×#28512
C3×C4×C22⋊Q82
C3×#20772
C3×#20752
C3×#20362
C3×#20352
C3×#19892
C3×#19492
C3×#19422
C3×#19412
C3×#19332
C3×#18052
#247053