Finite Group Databasebeta

← C3×#23573 #10642180

Subgroups — C3×#23573 #10642180

Order 768.

1508 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×#532C211yesmaximal, characteristic, complemented
2C2×C3×#442C212yesmaximal, complemented
3C3×C4×#121C211yesmaximal, characteristic, complemented
4C3×#2058C211yesmaximal, characteristic, complemented
5C3×#2041C211yesmaximal, characteristic, complemented
6C3×#2034C212yesmaximal, complemented
7C3×#2023C211yesmaximal, characteristic, complemented
8C3×#2011C211yesmaximal, characteristic, complemented
9C3×#1999C212yesmaximal, complemented
10C3×#1762C212yesmaximal, complemented
11C3×#1749C211yesmaximal, characteristic, complemented
12#23573C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×D8C2212yes—
15C3×#532—28no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#121C2212yes—
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2212yes—
26C22×C3×C42C2211yescharacteristic
27C3×#442—28no—
28C2×C3×#106C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×C22⋊C8C2212yes—
34C2×C3×#97C2212yes—
35C3×#390C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#388—21no—
38C3×#386—21no—
39C3×#380C2211yescharacteristic
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4—21no—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2211yescharacteristic
45C3×#336C2211yescharacteristic
46C2×#532C2×C311yescharacteristic, complemented
47C2×#442C2×C312yescomplemented
48C4×#121C2×C311yescharacteristic, complemented
49#2058C2×C311yescharacteristic, complemented
50#2041C2×C311yescharacteristic, complemented
51#2034C2×C312yescomplemented
52#2023C2×C311yescharacteristic, complemented
53#2011C2×C311yescharacteristic, complemented
54#1999C2×C312yescomplemented
55#1762C2×C312yescomplemented
56#1749C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×D4—22no—
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4—22no—
68C23×C3×C4—21no—
69C23×C3×C4D411yescharacteristic
70C23×C3×C4D411yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C2×C3×D8—24no—
73C2×C3×D8—24no—
74C2×C3×D8—24no—
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C3×C4⋊Q8—24no—
78C3×#127—24no—
79C3×#124—28no—
80C3×#121—28no—
81C3×#121—24no—
82C3×#121—22no—
83C3×#121—24no—
84C3×#121—22no—
85C3×C4×D4—28no—
86C3×C4×D4—24no—
87C3×C4×D4—24no—
88C3×C4×D4—44no—
89C3×C4×D4D418yescomplemented
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C42—24no—
104C2×C3×C42—21no—
105C2×C3×C42D411yescharacteristic
106C2×C3×C42D411yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C3×#106—24no—
110C3×#102—24no—
111C3×#102—24no—
112C3×#102—24no—
113C3×#102—24no—
114C3×C22⋊C8—24no—
115C3×#97—24no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C22×#41—21no—
120C22×D8C22×C312yes—
121#532—28no—
122C2×C4⋊Q8C22×C311yescharacteristic
123C2×#127C22×C311yescharacteristic
124C2×#124—21no—
125C2×#121—21no—
126C2×#121C22×C312yes—
127C2×#121C22×C311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C22×C311yescharacteristic
131C2×C4×D4C22×C312yes—
132C22×C42C22×C311yescharacteristic
133#442—28no—
134C2×#106C22×C312yes—
135C2×#102C22×C312yes—
136C2×#102C22×C312yes—
137C2×#102C22×C312yes—
138C2×#102C22×C312yes—
139C2×C22⋊C8C22×C312yes—
140C2×#97C22×C312yes—
141#390C22×C311yescharacteristic
142#389C22×C311yescharacteristic
143#388—21no—
144#386—21no—
145#380C22×C311yescharacteristic
146C4×C4⋊C4C22×C311yescharacteristic
147C4×C22⋊C4—21no—
148#340C22×C312yes—
149#340C22×C312yes—
150#336C22×C311yescharacteristic
151#336C22×C311yescharacteristic
152C24×C3—21no—
153C24×C3—22no—
154C24×C3C2×D411yescharacteristic
155C3×#41—432no—
156C2×C3×Q8—24no—
157C2×C3×Q8—42no—
158C2×C3×Q8—42no—
159C2×C3×D4—24no—
160C2×C3×D4—28no—
161C2×C3×D4—24no—
162C2×C3×D4—24no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—21no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4—44no—
175C2×C3×D4—42no—
176C2×C3×D4—44no—
177C2×C3×D4—42no—
178C2×C3×D4—42no—
179C2×C3×D4C2×D414yes—
180C2×C3×D4#4112yescomplemented
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—28no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—41no—
196C22×C3×C4—44no—
197C22×C3×C4—44no—
198C22×C3×C4—42no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×D8—216no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8#4114yescomplemented
214C3×C4⋊C4—24no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4C2×D414yes—
221C3×C4⋊C4#4112yescomplemented
222C3×C22⋊C4—28no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—28no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—22no—
232C3×C42—42no—
233C3×C42C2×D414yes—
234C2×#41—28no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C23×C312yes—
243C22×D4C23×C311yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4C3×D411yescharacteristic
247C23×C4C3×D411yescharacteristic
248C23×C4C23×C311yescharacteristic
249C2×D8—24no—
250C2×D8—24no—
251C2×D8—24no—
252C22×C8C23×C312yes—
253C22×C8C23×C312yes—
254C4⋊Q8—24no—
255#127—24no—
256#124—28no—
257#121—28no—
258#121—24no—
259#121—22no—
260#121—24no—
261#121—22no—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—44no—
266C4×D4C3×D418yescomplemented
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42—24no—
281C2×C42—21no—
282C2×C42C3×D411yescharacteristic
283C2×C42C3×D411yescharacteristic
284C2×C42C23×C311yescharacteristic
285C2×C42C23×C311yescharacteristic
286#106—24no—
287#102—24no—
288#102—24no—
289#102—24no—
290#102—24no—
291C22⋊C8—24no—
292#97—24no—
293#95—21no—
294#95—21no—
295#95—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—24no—
303C23×C3—22no—
304C23×C3—41no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—42no—
316C23×C3AGL(1,ℤ/8)12yes—
317C23×C3C22×D411yescharacteristic
318C3×Q8—48no—
319C3×D4—24no—
320C3×D4—24no—
321C3×D4—24no—
322C3×D4—48no—
323C3×D4—48no—
324C3×D4—48no—
325C3×D4—416no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—42no—
329C2×C3×C4—28no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—24no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—44no—
344C2×C3×C4—416no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—84no—
358C2×C3×C4#12012yes—
359C2×C3×C4#12012yes—
360C2×C3×C4AGL(1,ℤ/8)12yes—
361C2×C3×C4AGL(1,ℤ/8)12yes—
362C2×C3×C4C22×D412yes—
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4112yes—
365C2×C3×C4C2×#4111yescharacteristic
366C3×C8—24no—
367C3×C8—42no—
368C24—21no—
369C24—22no—
370C24C2×C3×D411yescharacteristic
371#41—432no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4C2×C3×D414yes—
396C2×D4C3×#4112yescomplemented
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—41no—
412C22×C4—44no—
413C22×C4—44no—
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—216no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8C3×#4114yescomplemented
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4C2×C3×D414yes—
437C4⋊C4C3×#4112yescomplemented
438C22⋊C4—28no—
439C22⋊C4—24no—
440C22⋊C4—28no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—42no—
449C42C2×C3×D414yes—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—44no—
460C22×C3—42no—
461C22×C3—44no—
462C22×C3—44no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—82no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3C2×#12011yescharacteristic
472C22×C3#52511yescharacteristic
473C22×C3#53211yescharacteristic
474C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
475C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
476C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
477C22×C3#57511yescharacteristic
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—41no—
484C3×C4—44no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4#52512yes—
488C3×C4#53212yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—24no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23C3×AGL(1,ℤ/8)12yes—
510C23C22×C3×D411yescharacteristic
511Q8—48no—
512D4—24no—
513D4—24no—
514D4—24no—
515D4—48no—
516D4—48no—
517D4—416no—
518D4—48no—
519D4—42no—
520D4—42no—
521D4—42no—
522C2×C4—28no—
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—24no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—44no—
537C2×C4—416no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4C3×#12012yes—
552C2×C4C3×#12012yes—
553C2×C4C3×AGL(1,ℤ/8)12yes—
554C2×C4C3×AGL(1,ℤ/8)12yes—
555C2×C4C22×C3×D412yes—
556C2×C4C2×C3×#4111yescharacteristic
557C2×C4C2×C3×#4112yes—
558C2×C4C2×C3×#4111yescharacteristic
559C8—24no—
560C8—42no—
561C2×C3—42no—
562C2×C3—82no—
563C2×C3—84no—
564C2×C3#238811yescharacteristic
565C2×C3#300311yescharacteristic
566C2×C3#300611yescharacteristic
567C2×C3#310911yescharacteristic
568C2×C3#311511yescharacteristic
569C2×C3#318011yescharacteristic
570C2×C3#318411yescharacteristic
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—44no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—82no—
585C22—81no—
586C22—81no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22—82no—
592C22C2×C3×#12011yescharacteristic
593C22C3×#52511yescharacteristic
594C22C3×#53211yescharacteristic
595C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
596C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
597C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
598C22C3×#57511yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—41no—
604C4—44no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4C3×#52512yes—
609C4C3×#53212yes—
610C3#2357311yescharacteristic, direct product factor, Sylow
611C2—42no—
612C2—84no—
613C2—82no—
614C2C3×#238811yescharacteristic
615C2C3×#300311yescharacteristic
616C2C3×#300611yescharacteristic
617C2C3×#310911yescharacteristic
618C2C3×#311511yescharacteristic
619C2C3×#318011yescharacteristic
620C2C3×#318411yescharacteristic
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×#532C2semidirect
2. C2×C3×#442C2semidirect
3. C3×C4×#121C2semidirect
4. C3×#2058C2semidirect
5. C3×#2041C2semidirect
6. C3×#2034C2semidirect
7. C3×#2023C2semidirect
8. C3×#2011C2semidirect
9. C3×#1999C2semidirect
10. C3×#1762C2semidirect
11. C3×#1749C2semidirect
12. #23573C3direct (N × H)
46. C2×#532C2×C3semidirect
47. C2×#442C2×C3semidirect
48. C4×#121C2×C3semidirect
49. #2058C2×C3semidirect
50. #2041C2×C3semidirect
51. #2034C2×C3semidirect
52. #2023C2×C3semidirect
53. #2011C2×C3semidirect
54. #1999C2×C3semidirect
55. #1762C2×C3semidirect
56. #1749C2×C3semidirect
89. C3×C4×D4D4semidirect
180. C2×C3×D4#41semidirect
213. C2×C3×C8#41semidirect
221. C3×C4⋊C4#41semidirect
266. C4×D4C3×D4semidirect
396. C2×D4C3×#41semidirect
429. C2×C8C3×#41semidirect
437. C4⋊C4C3×#41semidirect
610. C3#23573direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4422
C3×C4×#1212
C3×#20582
C3×#20412
C3×#20342
C3×#20232
C3×#20112
C3×#19992
C3×#17622
C3×#17492
#235733