Finite Group Databasebeta

← C3×#139777 #72253574

Subgroups — C3×#139777 #72253574

Order 1536.

1780 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1780 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×#23907C211yesmaximal, characteristic
2C3×#14169C211yesmaximal, characteristic, complemented
3C3×#12694C211yesmaximal, characteristic, complemented
4C3×#8534C212yesmaximal
5C3×#7304C212yesmaximal, complemented
6#139777C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#3484C2211yescharacteristic
8C22×C3×#124C2211yescharacteristic
9C2×C3×#455C412yescomplemented
10C3×#3004—21no—
11C2×C3×#376—21no—
12C2×C3×#354—22no—
13C3×#2004—22no—
14C3×#1898—24no—
15C3×#1896—21no—
16C3×#1885C412yescomplemented
17C3×#1783C2211yescharacteristic
18C3×#1781C2212yes—
19C3×#1446—21no—
20C3×#1438C2212yes—
21C3×#1437—24no—
22#23907C2×C311yescharacteristic
23#14169C2×C311yescharacteristic, complemented
24#12694C2×C311yescharacteristic, complemented
25#8534C2×C312yes—
26#7304C2×C312yescomplemented
27C2×C3×H(5,𝔽2)C2×C411yescharacteristic
28C23×C3×Q8D411yescharacteristic
29C23×C3×D4C2311yescharacteristic
30C2×C3×#137—22no—
31C3×#546—42no—
32C3×D42—42no—
33C3×#535—21no—
34C3×#534—22no—
35C2×C3×#124—22no—
36C2×C3×#124—44no—
37C2×C3×#124C2×C412yes—
38C2×C3×#124D412yes—
39C3×#525—21no—
40C2×C3×#120—22no—
41C3×#518C2×C411yescharacteristic
42C22×C3×C22⋊C4—22no—
43C22×C3×C22⋊C4—22no—
44C22×C3×C22⋊C4D414yes—
45C22×C3×C42—22no—
46C22×C3×C42D411yescharacteristic
47C3×#455—44no—
48C3×#451—42no—
49C3×#448—42no—
50C2×C3×#104—22no—
51C3×#419—22no—
52C2×C3×#100—21no—
53C2×C3×#100C2×C412yes—
54C2×C3×#99—22no—
55C2×C3×#99—22no—
56C2×C3×#99—44no—
57C3×#407—21no—
58C3×#379—22no—
59C3×#376—44no—
60C2×C3×#95—22no—
61C2×C3×#95—22no—
62C2×C3×#95—22no—
63C2×C3×#95—22no—
64C3×#354—44no—
65C3×#342—44no—
66C3×#339—22no—
67#3484C22×C311yescharacteristic
68C22×#124C22×C311yescharacteristic
69C2×#455C3×C412yescomplemented
70#3004—21no—
71C2×#376—21no—
72C2×#354—22no—
73#2004—22no—
74#1898—24no—
75#1896—21no—
76#1885C3×C412yescomplemented
77#1783C22×C311yescharacteristic
78#1781C22×C312yes—
79#1446—21no—
80#1438C22×C312yes—
81#1437—24no—
82C25×C3C2×D412yes—
83C3×H(5,𝔽2)—42no—
84C3×H(5,𝔽2)—81no—
85C2×C3×#41—21no—
86C2×C3×#41—41no—
87C22×C3×Q8—21no—
88C22×C3×Q8—42no—
89C22×C3×Q8—42no—
90C22×C3×Q8—81no—
91C22×C3×Q8C22⋊C412yes—
92C22×C3×D4—24no—
93C22×C3×D4—22no—
94C22×C3×D4—42no—
95C22×C3×D4—42no—
96C22×C3×D4—41no—
97C22×C3×D4—81no—
98C22×C3×D4C22×C411yescharacteristic
99C22×C3×D4C2×D411yescharacteristic
100C22×C3×D4#4112yes—
101C23×C3×C4—21no—
102C23×C3×C4—21no—
103C23×C3×C4—24no—
104C23×C3×C4C2×D411yescharacteristic
105C3×#137—44no—
106C3×#137—82no—
107C2×C3×Q16—42no—
108C2×C3×SD16—42no—
109C2×C3×M16—22no—
110C3×#127—41no—
111C3×#124—24no—
112C3×#124—24no—
113C3×#124—42no—
114C3×#124—42no—
115C3×#124—48no—
116C3×#124—41no—
117C3×#123—42no—
118C3×#123—41no—
119C3×C22⋊Q8—22no—
120C3×#121—22no—
121C3×#121—42no—
122C3×#121—44no—
123C3×#120—44no—
124C3×#120—42no—
125C3×#120—41no—
126C3×#120—42no—
127C3×C4×D4—42no—
128C3×C4×D4—42no—
129C3×#117—21no—
130C2×C3×C22⋊C4—24no—
131C2×C3×C22⋊C4—24no—
132C2×C3×C22⋊C4—22no—
133C2×C3×C22⋊C4—24no—
134C2×C3×C22⋊C4—44no—
135C2×C3×C22⋊C4—44no—
136C2×C3×C22⋊C4—44no—
137C2×C3×C22⋊C4—48no—
138C2×C3×C22⋊C4—48no—
139C2×C3×C22⋊C4—44no—
140C2×C3×C22⋊C4—42no—
141C2×C3×C22⋊C4—82no—
142C2×C3×C42—21no—
143C2×C3×C42—24no—
144C2×C3×C42—44no—
145C2×C3×C42—48no—
146C2×C3×C42C22⋊C412yes—
147C3×#104—44no—
148C3×#103—44no—
149C3×#100—42no—
150C3×#100—81no—
151C3×#99—44no—
152C3×#99—88no—
153C3×#99—82no—
154C3×C22⋊C8—42no—
155C3×#95—44no—
156C3×#95—44no—
157C3×#95—82no—
158C3×#95—82no—
159C2×H(5,𝔽2)C2×C3×C411yescharacteristic
160C23×Q8C3×D411yescharacteristic
161C23×D4C23×C311yescharacteristic
162C2×#137—22no—
163#546—42no—
164D42—42no—
165#535—21no—
166#534—22no—
167C2×#124—22no—
168C2×#124—44no—
169C2×#124C2×C3×C412yes—
170C2×#124C3×D412yes—
171#525—21no—
172C2×#120—22no—
173#518C2×C3×C411yescharacteristic
174C22×C22⋊C4—22no—
175C22×C22⋊C4—22no—
176C22×C22⋊C4C3×D414yes—
177C22×C42—22no—
178C22×C42C3×D411yescharacteristic
179#455—44no—
180#451—42no—
181#448—42no—
182C2×#104—22no—
183#419—22no—
184C2×#100—21no—
185C2×#100C2×C3×C412yes—
186C2×#99—22no—
187C2×#99—22no—
188C2×#99—44no—
189#407—21no—
190#379—22no—
191#376—44no—
192C2×#95—22no—
193C2×#95—22no—
194C2×#95—22no—
195C2×#95—22no—
196#354—44no—
197#342—44no—
198#339—22no—
199C24×C3—24no—
200C24×C3—22no—
201C24×C3—42no—
202C24×C3—42no—
203C24×C3—44no—
204C24×C3—44no—
205C24×C3C4×D412yes—
206C24×C3#12011yescharacteristic
207C24×C3#12112yes—
208C3×#41—42no—
209C3×#41—42no—
210C3×#41—84no—
211C2×C3×Q8—21no—
212C2×C3×Q8—22no—
213C2×C3×Q8—22no—
214C2×C3×Q8—42no—
215C2×C3×Q8—42no—
216C2×C3×Q8—42no—
217C2×C3×Q8—81no—
218C2×C3×Q8—81no—
219C2×C3×Q8—88no—
220C2×C3×D4—21no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4—41no—
224C2×C3×D4—41no—
225C2×C3×D4—42no—
226C2×C3×D4—41no—
227C2×C3×D4—42no—
228C2×C3×D4—41no—
229C2×C3×D4—42no—
230C2×C3×D4—42no—
231C2×C3×D4—41no—
232C2×C3×D4—42no—
233C2×C3×D4—44no—
234C2×C3×D4—88no—
235C2×C3×D4—84no—
236C2×C3×D4—82no—
237C2×C3×D4—82no—
238C2×C3×D4—84no—
239C2×C3×D4—84no—
240C22×C3×C4—22no—
241C22×C3×C4—22no—
242C22×C3×C4—22no—
243C22×C3×C4—22no—
244C22×C3×C4—22no—
245C22×C3×C4—28no—
246C22×C3×C4—22no—
247C22×C3×C4—24no—
248C22×C3×C4—22no—
249C22×C3×C4—41no—
250C22×C3×C4—41no—
251C22×C3×C4—44no—
252C22×C3×C4—44no—
253C22×C3×C4—44no—
254C22×C3×C4—42no—
255C22×C3×C4—44no—
256C22×C3×C4—42no—
257C22×C3×C4—44no—
258C22×C3×C4—44no—
259C22×C3×C4—44no—
260C22×C3×C4—42no—
261C22×C3×C4—82no—
262C22×C3×C4C2×C22⋊C411yescharacteristic
263C22×C3×C4#12311yescharacteristic
264C3×Q16—84no—
265C3×SD16—84no—
266C3×M16—42no—
267C3×M16—42no—
268C2×C3×C8—42no—
269C3×C4⋊C4—22no—
270C3×C4⋊C4—42no—
271C3×C22⋊C4—48no—
272C3×C22⋊C4—48no—
273C3×C22⋊C4—42no—
274C3×C22⋊C4—42no—
275C3×C22⋊C4—44no—
276C3×C22⋊C4—41no—
277C3×C22⋊C4—42no—
278C3×C22⋊C4—48no—
279C3×C22⋊C4—84no—
280C3×C22⋊C4—88no—
281C3×C22⋊C4—82no—
282C3×C22⋊C4—82no—
283C3×C22⋊C4—84no—
284C3×C22⋊C4—84no—
285C3×C42—24no—
286C3×C42—22no—
287C3×C42—41no—
288C3×C42—416no—
289C3×C42#9914yescomplemented
290C25C2×C3×D412yes—
291H(5,𝔽2)—42no—
292H(5,𝔽2)—81no—
293C2×#41—21no—
294C2×#41—41no—
295C22×Q8—21no—
296C22×Q8—42no—
297C22×Q8—42no—
298C22×Q8—81no—
299C22×Q8C3×C22⋊C412yes—
300C22×D4—24no—
301C22×D4—22no—
302C22×D4—42no—
303C22×D4—42no—
304C22×D4—41no—
305C22×D4—81no—
306C22×D4C22×C3×C411yescharacteristic
307C22×D4C2×C3×D411yescharacteristic
308C22×D4C3×#4112yes—
309C23×C4—21no—
310C23×C4—21no—
311C23×C4—24no—
312C23×C4C2×C3×D411yescharacteristic
313#137—44no—
314#137—82no—
315C2×Q16—42no—
316C2×SD16—42no—
317C2×M16—22no—
318#127—41no—
319#124—24no—
320#124—24no—
321#124—42no—
322#124—42no—
323#124—48no—
324#124—41no—
325#123—42no—
326#123—41no—
327C22⋊Q8—22no—
328#121—22no—
329#121—42no—
330#121—44no—
331#120—44no—
332#120—42no—
333#120—41no—
334#120—42no—
335C4×D4—42no—
336C4×D4—42no—
337#117—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—82no—
350C2×C42—21no—
351C2×C42—24no—
352C2×C42—44no—
353C2×C42—48no—
354C2×C42C3×C22⋊C412yes—
355#104—44no—
356#103—44no—
357#100—42no—
358#100—81no—
359#99—44no—
360#99—88no—
361#99—82no—
362C22⋊C8—42no—
363#95—44no—
364#95—44no—
365#95—82no—
366#95—82no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—24no—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3—22no—
376C23×C3—44no—
377C23×C3—44no—
378C23×C3—44no—
379C23×C3—44no—
380C23×C3—41no—
381C23×C3—84no—
382C23×C3—84no—
383C23×C3—82no—
384C23×C3—84no—
385C23×C3—82no—
386C23×C3—82no—
387C23×C3—82no—
388C23×C3—84no—
389C23×C3—82no—
390C23×C3—84no—
391C23×C3#38611yescharacteristic
392C23×C3B(4,𝔽2)12yes—
393C3×Q8—44no—
394C3×Q8—42no—
395C3×Q8—82no—
396C3×Q8—84no—
397C3×D4—44no—
398C3×D4—42no—
399C3×D4—82no—
400C3×D4—84no—
401C3×D4—84no—
402C3×D4—81no—
403C3×D4—81no—
404C3×D4—84no—
405C3×D4—164no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—24no—
411C2×C3×C4—21no—
412C2×C3×C4—44no—
413C2×C3×C4—44no—
414C2×C3×C4—44no—
415C2×C3×C4—44no—
416C2×C3×C4—48no—
417C2×C3×C4—48no—
418C2×C3×C4—42no—
419C2×C3×C4—41no—
420C2×C3×C4—42no—
421C2×C3×C4—42no—
422C2×C3×C4—42no—
423C2×C3×C4—48no—
424C2×C3×C4—82no—
425C2×C3×C4—84no—
426C2×C3×C4—84no—
427C2×C3×C4—84no—
428C2×C3×C4—84no—
429C2×C3×C4—84no—
430C2×C3×C4—84no—
431C2×C3×C4—82no—
432C2×C3×C4—162no—
433C2×C3×C4—164no—
434C2×C3×C4C2×#9912yes—
435C3×C8—82no—
436C24—24no—
437C24—22no—
438C24—42no—
439C24—42no—
440C24—44no—
441C24—44no—
442C24C3×C4×D412yes—
443C24C3×#12011yescharacteristic
444C24C3×#12112yes—
445#41—42no—
446#41—42no—
447#41—84no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8—81no—
456C2×Q8—88no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—42no—
463C2×D4—41no—
464C2×D4—42no—
465C2×D4—41no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—41no—
469C2×D4—42no—
470C2×D4—44no—
471C2×D4—88no—
472C2×D4—84no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4—84no—
476C2×D4—84no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—28no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—82no—
499C22×C4C2×C3×C22⋊C411yescharacteristic
500C22×C4C3×#12311yescharacteristic
501Q16—84no—
502SD16—84no—
503M16—42no—
504M16—42no—
505C2×C8—42no—
506C4⋊C4—22no—
507C4⋊C4—42no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C22⋊C4—48no—
516C22⋊C4—84no—
517C22⋊C4—88no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C42—24no—
523C42—22no—
524C42—41no—
525C42—416no—
526C42C3×#9914yescomplemented
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—22no—
531C22×C3—22no—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—44no—
537C22×C3—44no—
538C22×C3—82no—
539C22×C3—82no—
540C22×C3—82no—
541C22×C3—82no—
542C22×C3—81no—
543C22×C3—82no—
544C22×C3—164no—
545C22×C3—164no—
546C22×C3—322no—
547C22×C3#188911yescharacteristic
548C22×C3#189811yescharacteristic
549C22×C3#191311yescharacteristic
550C22×C3#219714yes—
551C3×C4—22no—
552C3×C4—44no—
553C3×C4—41no—
554C3×C4—88no—
555C3×C4—82no—
556C3×C4—81no—
557C3×C4—82no—
558C3×C4—322no—
559C23—21no—
560C23—21no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—41no—
573C23—84no—
574C23—84no—
575C23—82no—
576C23—84no—
577C23—82no—
578C23—82no—
579C23—82no—
580C23—84no—
581C23—82no—
582C23—84no—
583C23C3×#38611yescharacteristic
584C23C3×B(4,𝔽2)12yes—
585Q8—44no—
586Q8—42no—
587Q8—82no—
588Q8—84no—
589D4—44no—
590D4—42no—
591D4—82no—
592D4—84no—
593D4—84no—
594D4—81no—
595D4—81no—
596D4—84no—
597D4—164no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—24no—
603C2×C4—21no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—48no—
616C2×C4—82no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—82no—
623C2×C4—84no—
624C2×C4—162no—
625C2×C4—164no—
626C2×C4C2×C3×#9912yes—
627C8—82no—
628C2×C3—21no—
629C2×C3—24no—
630C2×C3—21no—
631C2×C3—82no—
632C2×C3—82no—
633C2×C3—161no—
634C2×C3#949711yescharacteristic
635C2×C3#1304812yes—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—81no—
652C22—82no—
653C22—164no—
654C22—164no—
655C22—322no—
656C22C3×#188911yescharacteristic
657C22C3×#189811yescharacteristic
658C22C3×#191311yescharacteristic
659C22C3×#219714yes—
660C4—22no—
661C4—44no—
662C4—41no—
663C4—88no—
664C4—82no—
665C4—81no—
666C4—82no—
667C4—322no—
668C3#13977711yescharacteristic, direct product factor, Sylow
669C2—21no—
670C2—24no—
671C2—21no—
672C2—82no—
673C2—82no—
674C2—161no—
675C2C3×#949711yescharacteristic
676C2C3×#1304812yes—
subgroupG/H
center, socleC22×C3#1889
commutatorC22×D4C22×C3×C4
Frattini, cosocleC23×D4C23×C3
prime radicalC3×#3484C22
LCSG ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#14169C2semidirect
3. C3×#12694C2semidirect
5. C3×#7304C2semidirect
6. #139777C3direct (N × H)
9. C2×C3×#455C4semidirect
16. C3×#1885C4semidirect
23. #14169C2×C3semidirect
24. #12694C2×C3semidirect
26. #7304C2×C3semidirect
69. C2×#455C3×C4semidirect
76. #1885C3×C4semidirect
289. C3×C42#99semidirect
526. C42C3×#99semidirect
668. C3#139777direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#239072
C3×#141692
C3×#126942
C3×#85342
C3×#73042
#1397773