Finite Group Databasebeta

← C3×#139787 #72271633

Subgroups — C3×#139787 #72271633

Order 1536.

1584 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1584 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#23910C211yesmaximal, characteristic
2C3×#14662C211yesmaximal, characteristic, complemented
3C3×#12699C211yesmaximal, characteristic, complemented
4C3×#8455C212yesmaximal, complemented
5C3×#7308C212yesmaximal, complemented
6#139787C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#3486C2211yescharacteristic
8C22×C3×#124C2211yescharacteristic
9C2×C3×#454C412yescomplemented
10C3×#3003—21no—
11C2×C3×#382—21no—
12C2×C3×#355—22no—
13C3×#2003—22no—
14C3×#1896—21no—
15C3×#1888C412yescomplemented
16C3×#1784C2212yes—
17C3×#1784C2211yescharacteristic
18C3×#1504—24no—
19C3×#1446C2212yes—
20C3×#1439—24no—
21C3×#1438—21no—
22#23910C2×C311yescharacteristic
23#14662C2×C311yescharacteristic, complemented
24#12699C2×C311yescharacteristic, complemented
25#8455C2×C312yescomplemented
26#7308C2×C312yescomplemented
27C2×C3×2−1+4C2×C411yescharacteristic
28C23×C3×Q8C2311yescharacteristic
29C23×C3×D4D411yescharacteristic
30C2×C3×AGL(1,ℤ/8)—22no—
31C3×D4×Q8—42no—
32C3×#548—42no—
33C3×#536—21no—
34C3×#536—22no—
35C2×C3×#124—22no—
36C2×C3×#124—44no—
37C2×C3×#124C2×C412yes—
38C2×C3×#124D412yes—
39C3×#525—22no—
40C3×#525—21no—
41C3×#519C2×C411yescharacteristic
42C22×C3×C4⋊C4—22no—
43C22×C3×C22⋊C4—22no—
44C22×C3×C22⋊C4—22no—
45C22×C3×C22⋊C4—22no—
46C22×C3×C42D411yescharacteristic
47C3×#454—44no—
48C3×#450—42no—
49C3×#447—42no—
50C2×C3×#104—22no—
51C3×#418—22no—
52C2×C3×#101—22no—
53C2×C3×#101—21no—
54C2×C3×#101—22no—
55C2×C3×#101C2×C412yes—
56C3×#407—21no—
57C3×#407—22no—
58C3×#382—44no—
59C2×C3×#95—22no—
60C2×C3×#95—22no—
61C2×C3×#95—22no—
62C2×C3×#95D414yes—
63C3×#355—44no—
64C3×#339—22no—
65C3×#328—44no—
66C3×#327—44no—
67#3486C22×C311yescharacteristic
68C22×#124C22×C311yescharacteristic
69C2×#454C3×C412yescomplemented
70#3003—21no—
71C2×#382—21no—
72C2×#355—22no—
73#2003—22no—
74#1896—21no—
75#1888C3×C412yescomplemented
76#1784C22×C312yes—
77#1784C22×C311yescharacteristic
78#1504—24no—
79#1446C22×C312yes—
80#1439—24no—
81#1438—21no—
82C25×C3—21no—
83C3×2−1+4—42no—
84C3×2−1+4—81no—
85C2×C3×#41—21no—
86C2×C3×#41—22no—
87C2×C3×#41—41no—
88C22×C3×Q8—42no—
89C22×C3×Q8—42no—
90C22×C3×Q8—41no—
91C22×C3×Q8—81no—
92C22×C3×Q8C22×C411yescharacteristic
93C22×C3×Q8C2×D411yescharacteristic
94C22×C3×Q8#4112yes—
95C22×C3×D4—21no—
96C22×C3×D4—42no—
97C22×C3×D4—42no—
98C22×C3×D4—81no—
99C22×C3×D4C22⋊C412yes—
100C23×C3×C4—21no—
101C23×C3×C4—24no—
102C23×C3×C4C2×D411yescharacteristic
103C23×C3×C4C2×D412yes—
104C3×AGL(1,ℤ/8)—44no—
105C3×AGL(1,ℤ/8)—82no—
106C2×C3×SD16—42no—
107C2×C3×D8—42no—
108C2×C3×M16—22no—
109C2×C3×M16—42no—
110C3×C4⋊Q8—41no—
111C3×C4⋊Q8—42no—
112C3×#124—24no—
113C3×#124—24no—
114C3×#124—42no—
115C3×#124—42no—
116C3×#124—48no—
117C3×#124—41no—
118C3×#124—42no—
119C3×#123—42no—
120C3×#123—41no—
121C3×C22⋊Q8—22no—
122C3×C22⋊Q8—24no—
123C3×C22⋊Q8—42no—
124C3×C22⋊Q8—44no—
125C3×#121—22no—
126C3×#121—24no—
127C3×#120—41no—
128C3×C4×Q8—42no—
129C3×C4×D4—42no—
130C3×C4×D4—44no—
131C3×#117—21no—
132C3×#117—22no—
133C2×C3×C4⋊C4—44no—
134C2×C3×C4⋊C4—82no—
135C2×C3×C4⋊C4—82no—
136C2×C3×C22⋊C4—24no—
137C2×C3×C22⋊C4—24no—
138C2×C3×C22⋊C4—44no—
139C2×C3×C22⋊C4—44no—
140C2×C3×C22⋊C4—82no—
141C2×C3×C22⋊C4—84no—
142C2×C3×C22⋊C4—84no—
143C2×C3×C22⋊C4—82no—
144C2×C3×C42—21no—
145C2×C3×C42—24no—
146C2×C3×C42C22⋊C412yes—
147C3×#104—44no—
148C3×#102—44no—
149C3×#101—42no—
150C3×#101—44no—
151C3×#101—81no—
152C3×#101—82no—
153C3×C22⋊C8—42no—
154C3×C22⋊C8—44no—
155C3×#95—44no—
156C3×#95—44no—
157C3×#95—44no—
158C3×#95—82no—
159C2×2−1+4C2×C3×C411yescharacteristic
160C23×Q8C23×C311yescharacteristic
161C23×D4C3×D411yescharacteristic
162C2×AGL(1,ℤ/8)—22no—
163D4×Q8—42no—
164#548—42no—
165#536—21no—
166#536—22no—
167C2×#124—22no—
168C2×#124—44no—
169C2×#124C2×C3×C412yes—
170C2×#124C3×D412yes—
171#525—21no—
172#525—22no—
173#519C2×C3×C411yescharacteristic
174C22×C4⋊C4—22no—
175C22×C22⋊C4—22no—
176C22×C22⋊C4—22no—
177C22×C22⋊C4—22no—
178C22×C42C3×D411yescharacteristic
179#454—44no—
180#450—42no—
181#447—42no—
182C2×#104—22no—
183#418—22no—
184C2×#101—22no—
185C2×#101—21no—
186C2×#101—22no—
187C2×#101C2×C3×C412yes—
188#407—21no—
189#407—22no—
190#382—44no—
191C2×#95—22no—
192C2×#95—22no—
193C2×#95—22no—
194C2×#95C3×D414yes—
195#355—44no—
196#339—22no—
197#328—44no—
198#327—44no—
199C24×C3—22no—
200C24×C3—44no—
201C24×C3—42no—
202C24×C3—44no—
203C24×C3—42no—
204C24×C3—42no—
205C24×C3#12011yescharacteristic
206C3×#41—42no—
207C3×#41—42no—
208C3×#41—82no—
209C3×#41—82no—
210C3×#41—84no—
211C2×C3×Q8—22no—
212C2×C3×Q8—22no—
213C2×C3×Q8—21no—
214C2×C3×Q8—22no—
215C2×C3×Q8—41no—
216C2×C3×Q8—41no—
217C2×C3×Q8—41no—
218C2×C3×Q8—42no—
219C2×C3×Q8—41no—
220C2×C3×Q8—42no—
221C2×C3×Q8—42no—
222C2×C3×Q8—88no—
223C2×C3×Q8—84no—
224C2×C3×D4—21no—
225C2×C3×D4—22no—
226C2×C3×D4—22no—
227C2×C3×D4—22no—
228C2×C3×D4—42no—
229C2×C3×D4—42no—
230C2×C3×D4—42no—
231C2×C3×D4—41no—
232C2×C3×D4—42no—
233C2×C3×D4—81no—
234C2×C3×D4—81no—
235C2×C3×D4—88no—
236C22×C3×C4—22no—
237C22×C3×C4—22no—
238C22×C3×C4—22no—
239C22×C3×C4—22no—
240C22×C3×C4—24no—
241C22×C3×C4—22no—
242C22×C3×C4—41no—
243C22×C3×C4—42no—
244C22×C3×C4—42no—
245C22×C3×C4—41no—
246C22×C3×C4—42no—
247C22×C3×C4—42no—
248C22×C3×C4—42no—
249C22×C3×C4—44no—
250C22×C3×C4—44no—
251C22×C3×C4—44no—
252C22×C3×C4—44no—
253C22×C3×C4—44no—
254C22×C3×C4—44no—
255C22×C3×C4—44no—
256C22×C3×C4C2×C22⋊C411yescharacteristic
257C22×C3×C4C4×D412yes—
258C22×C3×C4#12112yes—
259C22×C3×C4#12311yescharacteristic
260C3×SD16—84no—
261C3×D8—84no—
262C3×M16—42no—
263C3×M16—42no—
264C3×M16—82no—
265C3×M16—82no—
266C2×C3×C8—42no—
267C2×C3×C8—82no—
268C3×C4⋊C4—22no—
269C3×C4⋊C4—24no—
270C3×C4⋊C4—44no—
271C3×C4⋊C4—42no—
272C3×C4⋊C4—88no—
273C3×C22⋊C4—48no—
274C3×C22⋊C4—48no—
275C3×C22⋊C4—44no—
276C3×C22⋊C4—42no—
277C3×C22⋊C4—41no—
278C3×C22⋊C4—42no—
279C3×C22⋊C4—88no—
280C3×C22⋊C4—88no—
281C3×C42—24no—
282C3×C42—22no—
283C3×C42—41no—
284C3×C42—42no—
285C3×C42#9914yes—
286C25—21no—
2872−1+4—42no—
2882−1+4—81no—
289C2×#41—21no—
290C2×#41—22no—
291C2×#41—41no—
292C22×Q8—42no—
293C22×Q8—42no—
294C22×Q8—41no—
295C22×Q8—81no—
296C22×Q8C22×C3×C411yescharacteristic
297C22×Q8C2×C3×D411yescharacteristic
298C22×Q8C3×#4112yes—
299C22×D4—21no—
300C22×D4—42no—
301C22×D4—42no—
302C22×D4—81no—
303C22×D4C3×C22⋊C412yes—
304C23×C4—21no—
305C23×C4—24no—
306C23×C4C2×C3×D411yescharacteristic
307C23×C4C2×C3×D412yes—
308AGL(1,ℤ/8)—44no—
309AGL(1,ℤ/8)—82no—
310C2×SD16—42no—
311C2×D8—42no—
312C2×M16—22no—
313C2×M16—42no—
314C4⋊Q8—41no—
315C4⋊Q8—42no—
316#124—24no—
317#124—24no—
318#124—42no—
319#124—42no—
320#124—48no—
321#124—41no—
322#124—42no—
323#123—42no—
324#123—41no—
325C22⋊Q8—24no—
326C22⋊Q8—22no—
327C22⋊Q8—42no—
328C22⋊Q8—44no—
329#121—24no—
330#121—22no—
331#120—41no—
332C4×Q8—42no—
333C4×D4—42no—
334C4×D4—44no—
335#117—21no—
336#117—22no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—82no—
339C2×C4⋊C4—82no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—82no—
345C2×C22⋊C4—84no—
346C2×C22⋊C4—84no—
347C2×C22⋊C4—82no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42C3×C22⋊C412yes—
351#104—44no—
352#102—44no—
353#101—42no—
354#101—44no—
355#101—81no—
356#101—82no—
357C22⋊C8—42no—
358C22⋊C8—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—82no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—22no—
366C23×C3—22no—
367C23×C3—48no—
368C23×C3—41no—
369C23×C3—42no—
370C23×C3—42no—
371C23×C3—42no—
372C23×C3—82no—
373C23×C3—84no—
374C23×C3—82no—
375C23×C3—84no—
376C23×C3—82no—
377C23×C3—84no—
378C23×C3—82no—
379C23×C3—84no—
380C23×C3—82no—
381C23×C3—82no—
382C23×C3#38611yescharacteristic
383C23×C3B(4,𝔽2)12yes—
384C3×Q8—44no—
385C3×Q8—42no—
386C3×Q8—82no—
387C3×Q8—84no—
388C3×Q8—82no—
389C3×D4—42no—
390C3×D4—44no—
391C3×D4—82no—
392C3×D4—82no—
393C3×D4—84no—
394C3×D4—81no—
395C3×D4—81no—
396C3×D4—82no—
397C3×D4—82no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—21no—
410C2×C3×C4—44no—
411C2×C3×C4—42no—
412C2×C3×C4—44no—
413C2×C3×C4—42no—
414C2×C3×C4—44no—
415C2×C3×C4—42no—
416C2×C3×C4—41no—
417C2×C3×C4—82no—
418C2×C3×C4—84no—
419C2×C3×C4—84no—
420C2×C3×C4—84no—
421C2×C3×C4—84no—
422C2×C3×C4—84no—
423C2×C3×C4—84no—
424C2×C3×C4—84no—
425C2×C3×C4—84no—
426C2×C3×C4—84no—
427C2×C3×C4C2×#9912yes—
428C3×C8—82no—
429C3×C8—162no—
430C24—22no—
431C24—42no—
432C24—44no—
433C24—42no—
434C24—44no—
435C24—42no—
436C24C3×#12011yescharacteristic
437#41—42no—
438#41—42no—
439#41—82no—
440#41—82no—
441#41—84no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8—41no—
449C2×Q8—42no—
450C2×Q8—41no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—88no—
454C2×Q8—84no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—41no—
463C2×D4—42no—
464C2×D4—81no—
465C2×D4—81no—
466C2×D4—88no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4C2×C3×C22⋊C411yescharacteristic
488C22×C4C3×C4×D412yes—
489C22×C4C3×#12112yes—
490C22×C4C3×#12311yescharacteristic
491SD16—84no—
492D8—84no—
493M16—42no—
494M16—42no—
495M16—82no—
496M16—82no—
497C2×C8—42no—
498C2×C8—82no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—88no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—41no—
509C22⋊C4—42no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C42—24no—
513C42—22no—
514C42—41no—
515C42—42no—
516C42C3×#9914yes—
517C22×C3—21no—
518C22×C3—22no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—22no—
522C22×C3—21no—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—84no—
527C22×C3—84no—
528C22×C3—81no—
529C22×C3—82no—
530C22×C3—82no—
531C22×C3—84no—
532C22×C3—164no—
533C22×C3—164no—
534C22×C3#188911yescharacteristic
535C22×C3#189811yescharacteristic
536C22×C3#191311yescharacteristic
537C22×C3#220212yes—
538C22×C3#220312yes—
539C3×C4—22no—
540C3×C4—44no—
541C3×C4—42no—
542C3×C4—41no—
543C3×C4—42no—
544C3×C4—81no—
545C3×C4—82no—
546C3×C4—164no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—48no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—82no—
557C23—84no—
558C23—82no—
559C23—84no—
560C23—82no—
561C23—84no—
562C23—82no—
563C23—84no—
564C23—82no—
565C23—82no—
566C23C3×#38611yescharacteristic
567C23C3×B(4,𝔽2)12yes—
568Q8—44no—
569Q8—42no—
570Q8—82no—
571Q8—84no—
572Q8—82no—
573D4—42no—
574D4—44no—
575D4—82no—
576D4—82no—
577D4—84no—
578D4—81no—
579D4—81no—
580D4—82no—
581D4—82no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—82no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4C2×C3×#9912yes—
612C8—82no—
613C8—162no—
614C2×C3—21no—
615C2×C3—22no—
616C2×C3—21no—
617C2×C3—22no—
618C2×C3—161no—
619C2×C3—162no—
620C2×C3#949711yescharacteristic
621C2×C3#1305512yes—
622C22—21no—
623C22—22no—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—21no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—84no—
632C22—84no—
633C22—81no—
634C22—82no—
635C22—82no—
636C22—84no—
637C22—164no—
638C22—164no—
639C22C3×#188911yescharacteristic
640C22C3×#189811yescharacteristic
641C22C3×#191311yescharacteristic
642C22C3×#220212yes—
643C22C3×#220312yes—
644C4—22no—
645C4—44no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—81no—
650C4—82no—
651C4—164no—
652C3#13978711yescharacteristic, direct product factor, Sylow
653C2—21no—
654C2—22no—
655C2—21no—
656C2—22no—
657C2—162no—
658C2—161no—
659C2C3×#949711yescharacteristic
660C2C3×#1305512yes—
subgroupG/H
center, socleC22×C3#1889
commutatorC22×Q8C22×C3×C4
Frattini, cosocleC23×Q8C23×C3
prime radicalC3×#23910C2
LCSG ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#14662C2semidirect
3. C3×#12699C2semidirect
4. C3×#8455C2semidirect
5. C3×#7308C2semidirect
6. #139787C3direct (N × H)
9. C2×C3×#454C4semidirect
15. C3×#1888C4semidirect
23. #14662C2×C3semidirect
24. #12699C2×C3semidirect
25. #8455C2×C3semidirect
26. #7308C2×C3semidirect
69. C2×#454C3×C4semidirect
75. #1888C3×C4semidirect
652. C3#139787direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#239102
C3×#146622
C3×#126992
C3×#84552
C3×#73082
#1397873