Finite Group Databasebeta

← C3×#139785 #72271635

Subgroups — C3×#139785 #72271635

Order 1536.

1664 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1664 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×#14169C211yesmaximal, characteristic, complemented
3C3×#12699C211yesmaximal, characteristic, complemented
4C3×#8455C212yesmaximal, complemented
5C3×#7308C212yesmaximal, complemented
6#139785C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#3486C2211yescharacteristic
8C22×C3×#124C2211yescharacteristic
9C2×C3×#454C412yescomplemented
10C3×#3003—21no—
11C2×C3×#376—21no—
12C2×C3×#355—22no—
13C3×#2003—22no—
14C3×#1896—21no—
15C3×#1888C412yescomplemented
16C3×#1784C2212yes—
17C3×#1784C2211yescharacteristic
18C3×#1504—24no—
19C3×#1446C2212yes—
20C3×#1438—21no—
21C3×#1437—24no—
22#23910C2×C311yescharacteristic
23#14169C2×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×C22⋊C4—22no—
43C22×C3×C22⋊C4—22no—
44C22×C3×C22⋊C4—22no—
45C22×C3×C42—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×#376—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×#376—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#1438—21no—
81#1437—24no—
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×C22⋊C4—24no—
134C2×C3×C22⋊C4—24no—
135C2×C3×C22⋊C4—44no—
136C2×C3×C22⋊C4—44no—
137C2×C3×C22⋊C4—48no—
138C2×C3×C22⋊C4—48no—
139C2×C3×C22⋊C4—82no—
140C2×C3×C22⋊C4—82no—
141C2×C3×C42—21no—
142C2×C3×C42—24no—
143C2×C3×C42—44no—
144C2×C3×C42—48no—
145C2×C3×C42C22⋊C412yes—
146C3×#104—44no—
147C3×#102—44no—
148C3×#101—42no—
149C3×#101—44no—
150C3×#101—81no—
151C3×#101—82no—
152C3×C22⋊C8—42no—
153C3×C22⋊C8—44no—
154C3×#95—44no—
155C3×#95—44no—
156C3×#95—44no—
157C3×#95—82no—
158C2×2−1+4C2×C3×C411yescharacteristic
159C23×Q8C23×C311yescharacteristic
160C23×D4C3×D411yescharacteristic
161C2×AGL(1,ℤ/8)—22no—
162D4×Q8—42no—
163#548—42no—
164#536—21no—
165#536—22no—
166C2×#124—22no—
167C2×#124—44no—
168C2×#124C2×C3×C412yes—
169C2×#124C3×D412yes—
170#525—21no—
171#525—22no—
172#519C2×C3×C411yescharacteristic
173C22×C22⋊C4—22no—
174C22×C22⋊C4—22no—
175C22×C22⋊C4—22no—
176C22×C42—22no—
177C22×C42C3×D411yescharacteristic
178#454—44no—
179#450—42no—
180#447—42no—
181C2×#104—22no—
182#418—22no—
183C2×#101—22no—
184C2×#101—21no—
185C2×#101—22no—
186C2×#101C2×C3×C412yes—
187#407—21no—
188#407—22no—
189#376—44no—
190C2×#95—22no—
191C2×#95—22no—
192C2×#95—22no—
193C2×#95C3×D414yes—
194#355—44no—
195#339—22no—
196#328—44no—
197#327—44no—
198C24×C3—22no—
199C24×C3—44no—
200C24×C3—44no—
201C24×C3—42no—
202C24×C3—42no—
203C24×C3—42no—
204C24×C3#12011yescharacteristic
205C3×#41—42no—
206C3×#41—42no—
207C3×#41—82no—
208C3×#41—82no—
209C3×#41—84no—
210C2×C3×Q8—22no—
211C2×C3×Q8—22no—
212C2×C3×Q8—21no—
213C2×C3×Q8—22no—
214C2×C3×Q8—41no—
215C2×C3×Q8—41no—
216C2×C3×Q8—41no—
217C2×C3×Q8—42no—
218C2×C3×Q8—41no—
219C2×C3×Q8—42no—
220C2×C3×Q8—42no—
221C2×C3×Q8—88no—
222C2×C3×Q8—84no—
223C2×C3×D4—21no—
224C2×C3×D4—22no—
225C2×C3×D4—22no—
226C2×C3×D4—22no—
227C2×C3×D4—42no—
228C2×C3×D4—42no—
229C2×C3×D4—42no—
230C2×C3×D4—41no—
231C2×C3×D4—42no—
232C2×C3×D4—81no—
233C2×C3×D4—81no—
234C2×C3×D4—88no—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—22no—
238C22×C3×C4—22no—
239C22×C3×C4—24no—
240C22×C3×C4—22no—
241C22×C3×C4—28no—
242C22×C3×C4—41no—
243C22×C3×C4—42no—
244C22×C3×C4—42no—
245C22×C3×C4—41no—
246C22×C3×C4—42no—
247C22×C3×C4—44no—
248C22×C3×C4—44no—
249C22×C3×C4—44no—
250C22×C3×C4—44no—
251C22×C3×C4—44no—
252C22×C3×C4—44no—
253C22×C3×C4—44no—
254C22×C3×C4C2×C22⋊C411yescharacteristic
255C22×C3×C4C4×D412yes—
256C22×C3×C4#12112yes—
257C22×C3×C4#12311yescharacteristic
258C3×SD16—84no—
259C3×D8—84no—
260C3×M16—42no—
261C3×M16—42no—
262C3×M16—82no—
263C3×M16—82no—
264C2×C3×C8—42no—
265C2×C3×C8—82no—
266C3×C4⋊C4—22no—
267C3×C4⋊C4—24no—
268C3×C4⋊C4—44no—
269C3×C4⋊C4—42no—
270C3×C22⋊C4—48no—
271C3×C22⋊C4—48no—
272C3×C22⋊C4—44no—
273C3×C22⋊C4—42no—
274C3×C22⋊C4—41no—
275C3×C22⋊C4—42no—
276C3×C22⋊C4—88no—
277C3×C22⋊C4—88no—
278C3×C42—24no—
279C3×C42—22no—
280C3×C42—41no—
281C3×C42—42no—
282C3×C42—416no—
283C3×C42#9914yes—
284C25—21no—
2852−1+4—42no—
2862−1+4—81no—
287C2×#41—21no—
288C2×#41—22no—
289C2×#41—41no—
290C22×Q8—42no—
291C22×Q8—42no—
292C22×Q8—41no—
293C22×Q8—81no—
294C22×Q8C22×C3×C411yescharacteristic
295C22×Q8C2×C3×D411yescharacteristic
296C22×Q8C3×#4112yes—
297C22×D4—21no—
298C22×D4—42no—
299C22×D4—42no—
300C22×D4—81no—
301C22×D4C3×C22⋊C412yes—
302C23×C4—21no—
303C23×C4—24no—
304C23×C4C2×C3×D411yescharacteristic
305C23×C4C2×C3×D412yes—
306AGL(1,ℤ/8)—44no—
307AGL(1,ℤ/8)—82no—
308C2×SD16—42no—
309C2×D8—42no—
310C2×M16—22no—
311C2×M16—42no—
312C4⋊Q8—41no—
313C4⋊Q8—42no—
314#124—24no—
315#124—24no—
316#124—42no—
317#124—42no—
318#124—48no—
319#124—41no—
320#124—42no—
321#123—42no—
322#123—41no—
323C22⋊Q8—24no—
324C22⋊Q8—22no—
325C22⋊Q8—42no—
326C22⋊Q8—44no—
327#121—24no—
328#121—22no—
329#120—41no—
330C4×Q8—42no—
331C4×D4—42no—
332C4×D4—44no—
333#117—21no—
334#117—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—48no—
340C2×C22⋊C4—48no—
341C2×C22⋊C4—82no—
342C2×C22⋊C4—82no—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42—44no—
346C2×C42—48no—
347C2×C42C3×C22⋊C412yes—
348#104—44no—
349#102—44no—
350#101—42no—
351#101—44no—
352#101—81no—
353#101—82no—
354C22⋊C8—42no—
355C22⋊C8—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—82no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—24no—
363C23×C3—48no—
364C23×C3—41no—
365C23×C3—42no—
366C23×C3—42no—
367C23×C3—42no—
368C23×C3—82no—
369C23×C3—82no—
370C23×C3—84no—
371C23×C3—84no—
372C23×C3—82no—
373C23×C3—82no—
374C23×C3—82no—
375C23×C3—84no—
376C23×C3—82no—
377C23×C3—84no—
378C23×C3#38611yescharacteristic
379C23×C3B(4,𝔽2)12yes—
380C3×Q8—44no—
381C3×Q8—42no—
382C3×Q8—82no—
383C3×Q8—84no—
384C3×Q8—82no—
385C3×D4—42no—
386C3×D4—44no—
387C3×D4—82no—
388C3×D4—82no—
389C3×D4—84no—
390C3×D4—81no—
391C3×D4—81no—
392C3×D4—84no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—44no—
406C2×C3×C4—44no—
407C2×C3×C4—44no—
408C2×C3×C4—44no—
409C2×C3×C4—48no—
410C2×C3×C4—48no—
411C2×C3×C4—48no—
412C2×C3×C4—42no—
413C2×C3×C4—41no—
414C2×C3×C4—82no—
415C2×C3×C4—84no—
416C2×C3×C4—84no—
417C2×C3×C4—84no—
418C2×C3×C4—84no—
419C2×C3×C4—84no—
420C2×C3×C4—84no—
421C2×C3×C4C2×#9912yes—
422C3×C8—82no—
423C3×C8—162no—
424C24—22no—
425C24—42no—
426C24—44no—
427C24—44no—
428C24—42no—
429C24—42no—
430C24C3×#12011yescharacteristic
431#41—42no—
432#41—42no—
433#41—82no—
434#41—82no—
435#41—84no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—41no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—88no—
448C2×Q8—84no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—41no—
457C2×D4—42no—
458C2×D4—81no—
459C2×D4—81no—
460C2×D4—88no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—28no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4C2×C3×C22⋊C411yescharacteristic
481C22×C4C3×C4×D412yes—
482C22×C4C3×#12112yes—
483C22×C4C3×#12311yescharacteristic
484SD16—84no—
485D8—84no—
486M16—42no—
487M16—42no—
488M16—82no—
489M16—82no—
490C2×C8—42no—
491C2×C8—82no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—24no—
505C42—22no—
506C42—41no—
507C42—42no—
508C42—416no—
509C42C3×#9914yes—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—84no—
518C22×C3—84no—
519C22×C3—82no—
520C22×C3—84no—
521C22×C3—81no—
522C22×C3—82no—
523C22×C3—164no—
524C22×C3—164no—
525C22×C3#188911yescharacteristic
526C22×C3#189811yescharacteristic
527C22×C3#191311yescharacteristic
528C22×C3#220414yes—
529C3×C4—22no—
530C3×C4—44no—
531C3×C4—42no—
532C3×C4—41no—
533C3×C4—42no—
534C3×C4—88no—
535C3×C4—82no—
536C3×C4—81no—
537C23—21no—
538C23—21no—
539C23—24no—
540C23—48no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—82no—
546C23—82no—
547C23—84no—
548C23—84no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23—84no—
553C23—82no—
554C23—84no—
555C23C3×#38611yescharacteristic
556C23C3×B(4,𝔽2)12yes—
557Q8—44no—
558Q8—42no—
559Q8—82no—
560Q8—84no—
561Q8—82no—
562D4—42no—
563D4—44no—
564D4—82no—
565D4—82no—
566D4—84no—
567D4—81no—
568D4—81no—
569D4—84no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—82no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4C2×C3×#9912yes—
599C8—82no—
600C8—162no—
601C2×C3—21no—
602C2×C3—24no—
603C2×C3—21no—
604C2×C3—161no—
605C2×C3—162no—
606C2×C3#949711yescharacteristic
607C2×C3#1305712yes—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—84no—
616C22—84no—
617C22—81no—
618C22—82no—
619C22—82no—
620C22—84no—
621C22—164no—
622C22—164no—
623C22C3×#188911yescharacteristic
624C22C3×#189811yescharacteristic
625C22C3×#191311yescharacteristic
626C22C3×#220414yes—
627C4—22no—
628C4—44no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—88no—
633C4—81no—
634C4—82no—
635C3#13978511yescharacteristic, direct product factor, Sylow
636C2—21no—
637C2—24no—
638C2—21no—
639C2—162no—
640C2—161no—
641C2C3×#949711yescharacteristic
642C2C3×#1305712yes—
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×#14169C2semidirect
3. C3×#12699C2semidirect
4. C3×#8455C2semidirect
5. C3×#7308C2semidirect
6. #139785C3direct (N × H)
9. C2×C3×#454C4semidirect
15. C3×#1888C4semidirect
23. #14169C2×C3semidirect
24. #12699C2×C3semidirect
25. #8455C2×C3semidirect
26. #7308C2×C3semidirect
69. C2×#454C3×C4semidirect
75. #1888C3×C4semidirect
635. C3#139785direct (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×#141692
C3×#126992
C3×#84552
C3×#73082
#1397853