Finite Group Databasebeta

← C3×#24237 #10642844

Subgroups — C3×#24237 #10642844

Order 768.

1522 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic, complemented
3C2×C3×#449C212yesmaximal, complemented
4C2×C3×C4×D8C211yesmaximal, characteristic, complemented
5C3×#2537C211yesmaximal, characteristic, complemented
6C3×#2036C212yesmaximal, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1954C211yesmaximal, characteristic, complemented
10C3×#1802C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal, complemented
12#24237C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×SD16C2212yes—
15C22×C3×D8C2211yescharacteristic
16C3×#547—24no—
17C3×#547—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#123—22no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121—21no—
22C2×C3×#121—21no—
23C2×C3×#121C2212yes—
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C2×C3×C4×D4C2211yescharacteristic
27C22×C3×C4⋊C4—21no—
28C3×#461—24no—
29C3×#449—28no—
30C3×C4×D8—24no—
31C2×C3×#107C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#102C2211yescharacteristic
37C2×C3×#102C2211yescharacteristic
38C2×C3×#102C2212yes—
39C2×C3×C22⋊C8C2212yes—
40C2×C3×C22⋊C8C2212yes—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#394—22no—
43C3×#384C2211yescharacteristic
44C3×#380—21no—
45C3×#380C2212yes—
46C3×#340C2212yes—
47C2×#547C2×C311yescharacteristic, complemented
48C2×#461C2×C311yescharacteristic, complemented
49C2×#449C2×C312yescomplemented
50C2×C4×D8C2×C311yescharacteristic, complemented
51#2537C2×C311yescharacteristic, complemented
52#2036C2×C312yescomplemented
53#2000C2×C312yescomplemented
54#1956C2×C311yescharacteristic, complemented
55#1954C2×C311yescharacteristic, complemented
56#1802C2×C311yescharacteristic, complemented
57#1788C2×C312yescomplemented
58C2×C3×#41—24no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41—42no—
63C2×C3×#41—42no—
64C22×C3×Q8C2312yes—
65C22×C3×D4—22no—
66C22×C3×D4—22no—
67C22×C3×D4D411yescharacteristic
68C22×C3×D4C2311yescharacteristic
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—22no—
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4C2312yes—
74C2×C3×SD16—24no—
75C2×C3×SD16—24no—
76C2×C3×SD16—42no—
77C2×C3×D8—22no—
78C2×C3×D8—22no—
79C2×C3×D8D414yescomplemented
80C22×C3×C8C2312yes—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8D414yescomplemented
85C3×#123—216no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—24no—
88C3×#121—28no—
89C3×#121—24no—
90C3×#121—42no—
91C3×#121—44no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4D414yescomplemented
98C2×C3×C4⋊C4—28no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—22no—
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4D412yes—
114C2×C3×C42C2311yescharacteristic
115C3×#107—22no—
116C3×C4⋊C8—22no—
117C3×#103—24no—
118C3×#103—24no—
119C3×#103—42no—
120C3×#102—22no—
121C3×#102—24no—
122C3×#102—22no—
123C3×C22⋊C8—24no—
124C3×C22⋊C8—42no—
125C3×C4×C8—22no—
126C3×#95—22no—
127C22×#41C22×C312yes—
128C22×SD16C22×C312yes—
129C22×D8C22×C311yescharacteristic
130#547—24no—
131#547—24no—
132C2×C4⋊Q8C22×C311yescharacteristic
133C2×#123—22no—
134C2×C22⋊Q8C22×C312yes—
135C2×#121—21no—
136C2×#121—21no—
137C2×#121C22×C312yes—
138C2×C4×D4C22×C311yescharacteristic
139C2×C4×D4C22×C311yescharacteristic
140C2×C4×D4C22×C311yescharacteristic
141C22×C4⋊C4—21no—
142#461—24no—
143#449—28no—
144C4×D8—24no—
145C2×#107C22×C311yescharacteristic
146C2×C4⋊C8C22×C311yescharacteristic
147C2×#103C22×C312yes—
148C2×#103C22×C312yes—
149C2×#103C22×C312yes—
150C2×#102C22×C312yes—
151C2×#102C22×C311yescharacteristic
152C2×#102C22×C311yescharacteristic
153C2×C22⋊C8C22×C312yes—
154C2×C22⋊C8C22×C312yes—
155C2×C4×C8C22×C311yescharacteristic
156#394—22no—
157#384C22×C311yescharacteristic
158#380—21no—
159#380C22×C312yes—
160#340C22×C312yes—
161C24×C3—21no—
162C24×C3—21no—
163C24×C3C2×D412yes—
164C3×#41—48no—
165C3×#41—48no—
166C3×#41—48no—
167C3×#41—48no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8C2×D414yes—
174C2×C3×D4—28no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—44no—
190C2×C3×D4—82no—
191C2×C3×D4C2×D412yes—
192C2×C3×D4C2×D412yes—
193C2×C3×D4C2×D412yes—
194C2×C3×D4#4112yes—
195C22×C3×C4—28no—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—24no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—44no—
208C22×C3×C4—44no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C3×SD16—48no—
219C3×D8—28no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—21no—
226C2×C3×C8—22no—
227C2×C3×C8—21no—
228C2×C3×C8C2×D412yes—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4C2×D412yes—
241C3×C22⋊C4—28no—
242C3×C22⋊C4—24no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—44no—
245C3×C22⋊C4—44no—
246C3×C22⋊C4—42no—
247C3×C42C2×D412yes—
248C3×C42C2×D412yes—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—24no—
253C2×#41—42no—
254C2×#41—42no—
255C22×Q8C23×C312yes—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4C3×D411yescharacteristic
259C22×D4C23×C311yescharacteristic
260C22×D4C23×C311yescharacteristic
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4C23×C312yes—
265C2×SD16—24no—
266C2×SD16—24no—
267C2×SD16—42no—
268C2×D8—22no—
269C2×D8—22no—
270C2×D8C3×D414yescomplemented
271C22×C8C23×C312yes—
272C22×C8C23×C311yescharacteristic
273C22×C8C23×C311yescharacteristic
274C4⋊Q8—22no—
275C4⋊Q8C3×D414yescomplemented
276#123—216no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279#121—28no—
280#121—24no—
281#121—42no—
282#121—44no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4C3×D414yescomplemented
289C2×C4⋊C4—28no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C3×D411yescharacteristic
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C4⋊C4C23×C312yes—
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4C3×D412yes—
305C2×C42C23×C311yescharacteristic
306#107—22no—
307C4⋊C8—22no—
308#103—24no—
309#103—24no—
310#103—42no—
311#102—22no—
312#102—24no—
313#102—22no—
314C22⋊C8—24no—
315C22⋊C8—42no—
316C4×C8—22no—
317#95—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3C22×D411yescharacteristic
340C3×Q8—24no—
341C3×Q8—24no—
342C3×Q8—42no—
343C3×Q8—42no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—22no—
349C3×D4—416no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—44no—
353C3×D4—88no—
354C3×D4#12114yescomplemented
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—416no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—48no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—84no—
379C2×C3×C4#12012yes—
380C2×C3×C4#12012yes—
381C2×C3×C4#13512yes—
382C2×C3×C4AGL(1,ℤ/8)12yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4H(5,𝔽2)11yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C3×C8—22no—
390C3×C8—41no—
391C3×C8—42no—
392C24—21no—
393C24—21no—
394C24C2×C3×D412yes—
395#41—48no—
396#41—48no—
397#41—48no—
398#41—48no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8C2×C3×D414yes—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—82no—
422C2×D4C2×C3×D412yes—
423C2×D4C2×C3×D412yes—
424C2×D4C2×C3×D412yes—
425C2×D4C3×#4112yes—
426C22×C4—28no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4C2×C3×D412yes—
446C22×C4C2×C3×D411yescharacteristic
447C22×C4C2×C3×D411yescharacteristic
448C22×C4C24×C311yescharacteristic
449SD16—48no—
450D8—28no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—21no—
457C2×C8—22no—
458C2×C8—21no—
459C2×C8C2×C3×D412yes—
460C4⋊C4—28no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4C2×C3×D412yes—
472C22⋊C4—28no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—42no—
478C42C2×C3×D412yes—
479C42C2×C3×D412yes—
480C22×C3—24no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3C2×#12011yescharacteristic
502C22×C3D4211yescharacteristic
503C22×C3#54611yescharacteristic
504C22×C3C2×#13511yescharacteristic
505C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
506C22×C3#57711yescharacteristic
507C22×C3#57811yescharacteristic
508C3×C4—22no—
509C3×C4—42no—
510C3×C4—44no—
511C3×C4—42no—
512C3×C4—81no—
513C3×C4—82no—
514C3×C4#44914yescomplemented
515C3×C4C2×#12111yescharacteristic
516C3×C4#53511yescharacteristic
517C3×C4D4211yescharacteristic
518C3×C4#54611yescharacteristic
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×C3×D411yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—42no—
544Q8—42no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4—22no—
550D4—416no—
551D4—44no—
552D4—41no—
553D4—41no—
554D4—88no—
555D4C3×#12114yescomplemented
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—416no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—84no—
580C2×C4C3×#12012yes—
581C2×C4C3×#12012yes—
582C2×C4C3×#13512yes—
583C2×C4C3×AGL(1,ℤ/8)12yes—
584C2×C4C22×C3×D411yescharacteristic
585C2×C4C22×C3×D411yescharacteristic
586C2×C4C22×C3×D411yescharacteristic
587C2×C4C2×C3×#4111yescharacteristic
588C2×C4C3×H(5,𝔽2)11yescharacteristic
589C2×C4C3×H(5,𝔽2)11yescharacteristic
590C8—22no—
591C8—41no—
592C8—42no—
593C2×C3—44no—
594C2×C3—82no—
595C2×C3—82no—
596C2×C3#240811yescharacteristic
597C2×C3C2×#44911yescharacteristic
598C2×C3#301311yescharacteristic
599C2×C3#327211yescharacteristic
600C2×C3#328311yescharacteristic
601C2×C3#330311yescharacteristic
602C2×C3#333011yescharacteristic
603C22—24no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22C2×C3×#12011yescharacteristic
625C22C3×D4211yescharacteristic
626C22C3×#54611yescharacteristic
627C22C2×C3×#13511yescharacteristic
628C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
629C22C3×#57711yescharacteristic
630C22C3×#57811yescharacteristic
631C4—22no—
632C4—42no—
633C4—44no—
634C4—42no—
635C4—81no—
636C4—82no—
637C4C3×#44914yescomplemented
638C4C2×C3×#12111yescharacteristic
639C4C3×#53511yescharacteristic
640C4C3×D4211yescharacteristic
641C4C3×#54611yescharacteristic
642C3#2423711yescharacteristic, direct product factor, Sylow
643C2—44no—
644C2—82no—
645C2—82no—
646C2C3×#240811yescharacteristic
647C2C2×C3×#44911yescharacteristic
648C2C3×#301311yescharacteristic
649C2C3×#327211yescharacteristic
650C2C3×#328311yescharacteristic
651C2C3×#330311yescharacteristic
652C2C3×#333011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#547C2semidirect
2. C2×C3×#461C2semidirect
3. C2×C3×#449C2semidirect
4. C2×C3×C4×D8C2semidirect
5. C3×#2537C2semidirect
6. C3×#2036C2semidirect
7. C3×#2000C2semidirect
8. C3×#1956C2semidirect
9. C3×#1954C2semidirect
10. C3×#1802C2semidirect
11. C3×#1788C2semidirect
12. #24237C3direct (N × H)
47. C2×#547C2×C3semidirect
48. C2×#461C2×C3semidirect
49. C2×#449C2×C3semidirect
50. C2×C4×D8C2×C3semidirect
51. #2537C2×C3semidirect
52. #2036C2×C3semidirect
53. #2000C2×C3semidirect
54. #1956C2×C3semidirect
55. #1954C2×C3semidirect
56. #1802C2×C3semidirect
57. #1788C2×C3semidirect
79. C2×C3×D8D4semidirect
84. C3×C4⋊Q8D4semidirect
97. C3×C4×D4D4semidirect
270. C2×D8C3×D4semidirect
275. C4⋊Q8C3×D4semidirect
288. C4×D4C3×D4semidirect
354. C3×D4#121semidirect
514. C3×C4#449semidirect
555. D4C3×#121semidirect
637. C4C3×#449semidirect
642. C3#24237direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#4612
C2×C3×#4492
C2×C3×C4×D82
C3×#25372
C3×#20362
C3×#20002
C3×#19562
C3×#19542
C3×#18022
C3×#17882
#242373