Finite Group Databasebeta

← C3×#25724 #10644331

Subgroups — C3×#25724 #10644331

Order 768.

1700 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#124C211yesmaximal, characteristic, complemented
2C2×C3×#495C212yesmaximal, complemented
3C2×C3×#451C211yesmaximal, characteristic, complemented
4C2×C3×#448C211yesmaximal, characteristic, complemented
5C2×C3×#450C211yesmaximal, characteristic, complemented
6C2×C3×#447C211yesmaximal, characteristic
7C3×#2850C211yesmaximal, characteristic, complemented
8C3×#2015C212yesmaximal, complemented
9C3×#2012C212yesmaximal, complemented
10C3×#1958C212yesmaximal, complemented
11C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
12#25724C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×Q16C2212yescomplemented
16C22×C3×SD16C2212yes—
17C22×C3×SD16C2212yescomplemented
18C22×C3×D8C2212yes—
19C2×C3×#124—28no—
20C2×C3×#124—22no—
21C2×C3×#124C2212yes—
22C2×C3×#124C2212yes—
23C2×C3×#124C2211yescharacteristic, complemented
24C2×C3×C22⋊Q8C2211yescharacteristic
25C2×C3×#121C2211yescharacteristic
26C22×C3×C22⋊C4—22no—
27C22×C3×C42C2211yescharacteristic
28C3×#495—28no—
29C3×#451—24no—
30C3×#450—24no—
31C3×#448—24no—
32C3×#447—24no—
33C2×C3×#103C2212yescomplemented
34C2×C3×#103C2212yescomplemented
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C22⋊C8C2211yescharacteristic
38C2×C3×C22⋊C8C2211yescharacteristic
39C2×C3×C4×C8C2212yescomplemented
40C3×#391C2211yescharacteristic, complemented
41C3×#390C2211yescharacteristic, complemented
42C3×#388—22no—
43C3×#386—22no—
44C3×#382C2211yescharacteristic, complemented
45C3×#336C2212yescomplemented
46C22×#124C2×C311yescharacteristic, complemented
47C2×#495C2×C312yescomplemented
48C2×#451C2×C311yescharacteristic, complemented
49C2×#448C2×C311yescharacteristic, complemented
50C2×#450C2×C311yescharacteristic, complemented
51C2×#447C2×C311yescharacteristic
52#2850C2×C311yescharacteristic, complemented
53#2015C2×C312yescomplemented
54#2012C2×C312yescomplemented
55#1958C2×C312yescomplemented
56C4×C22⋊C8C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C22×C3×Q8—24no—
59C22×C3×Q8—21no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—42no—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×Q8C2312yes—
65C22×C3×D4—24no—
66C22×C3×D4—21no—
67C22×C3×D4—21no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4—42no—
71C22×C3×D4C2311yescharacteristic
72C22×C3×D4C2312yes—
73C23×C3×C4—21no—
74C23×C3×C4D412yes—
75C23×C3×C4C2311yescharacteristic
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×Q16—42no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×SD16—42no—
84C2×C3×SD16—42no—
85C2×C3×D8—24no—
86C2×C3×D8—24no—
87C2×C3×D8—42no—
88C22×C3×C8C2312yes—
89C22×C3×C8C2312yes—
90C3×#124—216no—
91C3×#124—24no—
92C3×#124—24no—
93C3×#124—24no—
94C3×#124—24no—
95C3×#124D418yescomplemented
96C3×C22⋊Q8—22no—
97C3×C22⋊Q8—22no—
98C3×#121—22no—
99C3×#121—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—216no—
104C2×C3×C22⋊C4—24no—
105C2×C3×C22⋊C4—24no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C42—24no—
111C2×C3×C42D412yes—
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C3×#103—24no—
115C3×#103—24no—
116C3×#102—24no—
117C3×#102—24no—
118C3×C22⋊C8D414yes—
119C3×C22⋊C8D414yes—
120C3×C4×C8—24no—
121C3×#95—22no—
122C3×#95—22no—
123C23×Q8C22×C311yescharacteristic
124C23×D4C22×C311yescharacteristic
125C22×Q16C22×C312yescomplemented
126C22×SD16C22×C312yes—
127C22×SD16C22×C312yescomplemented
128C22×D8C22×C312yes—
129C2×#124—28no—
130C2×#124—22no—
131C2×#124C22×C312yes—
132C2×#124C22×C312yes—
133C2×#124C22×C311yescharacteristic, complemented
134C2×C22⋊Q8C22×C311yescharacteristic
135C2×#121C22×C311yescharacteristic
136C22×C22⋊C4—22no—
137C22×C42C22×C311yescharacteristic
138#495—28no—
139#451—24no—
140#450—24no—
141#448—24no—
142#447—24no—
143C2×#103C22×C312yescomplemented
144C2×#103C22×C312yescomplemented
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×C22⋊C8C22×C311yescharacteristic
148C2×C22⋊C8C22×C311yescharacteristic
149C2×C4×C8C22×C312yescomplemented
150#391C22×C311yescharacteristic, complemented
151#390C22×C311yescharacteristic, complemented
152#388—22no—
153#386—22no—
154#382C22×C311yescharacteristic, complemented
155#336C22×C312yescomplemented
156C24×C3—24no—
157C24×C3—22no—
158C24×C3—21no—
159C24×C3—44no—
160C24×C3—42no—
161C24×C3—42no—
162C24×C3—44no—
163C24×C3C2×D411yescharacteristic
164C2×C3×Q8—24no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—48no—
175C2×C3×Q8—48no—
176C2×C3×Q8—42no—
177C2×C3×Q8—42no—
178C2×C3×Q8C2×D414yes—
179C2×C3×D4—24no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C2×C3×D4—22no—
189C2×C3×D4—48no—
190C2×C3×D4—48no—
191C2×C3×D4—42no—
192C2×C3×D4—42no—
193C2×C3×D4—44no—
194C2×C3×D4—42no—
195C2×C3×D4C2×D414yes—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—22no—
201C22×C3×C4—24no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2411yescharacteristic
214C3×Q16—48no—
215C3×SD16—48no—
216C3×SD16—48no—
217C3×D8—48no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8C2×D414yes—
223C2×C3×C8C2×D414yes—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C22⋊C4—216no—
230C3×C22⋊C4—216no—
231C3×C22⋊C4—44no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—44no—
234C3×C22⋊C4—42no—
235C3×C42—24no—
236C3×C42—22no—
237C3×C42C2×D414yes—
238C25—21no—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—42no—
244C22×Q8C23×C311yescharacteristic
245C22×Q8C23×C312yes—
246C22×D4—24no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—42no—
252C22×D4C23×C311yescharacteristic
253C22×D4C23×C312yes—
254C23×C4—21no—
255C23×C4C3×D412yes—
256C23×C4C23×C311yescharacteristic
257C2×Q16—24no—
258C2×Q16—24no—
259C2×Q16—42no—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—42no—
265C2×SD16—42no—
266C2×D8—24no—
267C2×D8—24no—
268C2×D8—42no—
269C22×C8C23×C312yes—
270C22×C8C23×C312yes—
271#124—216no—
272#124—24no—
273#124—24no—
274#124—24no—
275#124—24no—
276#124C3×D418yescomplemented
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279#121—22no—
280#121—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C22⋊C4—216no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C42—24no—
292C2×C42C3×D412yes—
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295#103—24no—
296#103—24no—
297#102—24no—
298#102—24no—
299C22⋊C8C3×D414yes—
300C22⋊C8C3×D414yes—
301C4×C8—24no—
302#95—22no—
303#95—22no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—22no—
307C23×C3—24no—
308C23×C3—48no—
309C23×C3—416no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—84no—
320C23×C3—88no—
321C23×C3—84no—
322C23×C3#12712yes—
323C23×C3#13514yes—
324C23×C3C22×D411yescharacteristic
325C3×Q8—24no—
326C3×Q8—24no—
327C3×Q8—48no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—48no—
335C3×D4—42no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—84no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—48no—
351C2×C3×C4—48no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4#12014yes—
362C2×C3×C4#12712yes—
363C2×C3×C4C22×D412yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4H(5,𝔽2)12yes—
367C3×C8—24no—
368C3×C8—24no—
369C24—24no—
370C24—22no—
371C24—21no—
372C24—42no—
373C24—44no—
374C24—42no—
375C24—44no—
376C24C2×C3×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8C2×C3×D414yes—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4C2×C3×D414yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C2×C3×D412yes—
422C22×C4C2×C3×D412yes—
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D412yes—
425C22×C4C2×C3×D412yes—
426C22×C4C24×C311yescharacteristic
427Q16—48no—
428SD16—48no—
429SD16—48no—
430D8—48no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8C2×C3×D414yes—
436C2×C8C2×C3×D414yes—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—42no—
442C22⋊C4—216no—
443C22⋊C4—216no—
444C22⋊C4—44no—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—42no—
448C42—24no—
449C42—22no—
450C42C2×C3×D414yes—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—24no—
454C22×C3—24no—
455C22×C3—44no—
456C22×C3—48no—
457C22×C3—816no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3—84no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3#49514yescomplemented
467C22×C3C2×#12011yescharacteristic
468C22×C3C2×#12711yescharacteristic
469C22×C3#53511yescharacteristic
470C22×C3C2×#13512yes—
471C22×C3#57512yes—
472C3×C4—22no—
473C3×C4—24no—
474C3×C4—44no—
475C3×C4—42no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4D4212yes—
480C3×C4D4212yes—
481C23—22no—
482C23—21no—
483C23—21no—
484C23—24no—
485C23—48no—
486C23—416no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—84no—
497C23—88no—
498C23—84no—
499C23C3×#12712yes—
500C23C3×#13514yes—
501C23C22×C3×D411yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—48no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509D4—24no—
510D4—24no—
511D4—48no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—84no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—48no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4C3×#12014yes—
539C2×C4C3×#12712yes—
540C2×C4C22×C3×D412yes—
541C2×C4C22×C3×D411yescharacteristic
542C2×C4C22×C3×D411yescharacteristic
543C2×C4C3×H(5,𝔽2)12yes—
544C8—24no—
545C8—24no—
546C2×C3—24no—
547C2×C3—84no—
548C2×C3—161no—
549C2×C3#243311yescharacteristic
550C2×C3#300212yes—
551C2×C3C2×#49511yescharacteristic
552C2×C3#315311yescharacteristic
553C2×C3#321112yes—
554C22—22no—
555C22—22no—
556C22—24no—
557C22—24no—
558C22—44no—
559C22—48no—
560C22—816no—
561C22—82no—
562C22—82no—
563C22—84no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22C3×#49514yescomplemented
570C22C2×C3×#12011yescharacteristic
571C22C2×C3×#12711yescharacteristic
572C22C3×#53511yescharacteristic
573C22C2×C3×#13512yes—
574C22C3×#57512yes—
575C4—22no—
576C4—24no—
577C4—44no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4C3×D4212yes—
583C4C3×D4212yes—
584C3#2572411yescharacteristic, direct product factor, Sylow
585C2—24no—
586C2—84no—
587C2—161no—
588C2C3×#243311yescharacteristic
589C2C3×#300212yes—
590C2C2×C3×#49511yescharacteristic
591C2C3×#315311yescharacteristic
592C2C3×#321112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#447C2
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. C22×C3×#124C2semidirect
2. C2×C3×#495C2semidirect
3. C2×C3×#451C2semidirect
4. C2×C3×#448C2semidirect
5. C2×C3×#450C2semidirect
7. C3×#2850C2semidirect
8. C3×#2015C2semidirect
9. C3×#2012C2semidirect
10. C3×#1958C2semidirect
11. C3×C4×C22⋊C8C2semidirect
12. #25724C3direct (N × H)
15. C22×C3×Q16C22semidirect
17. C22×C3×SD16C22semidirect
23. C2×C3×#124C22semidirect
33. C2×C3×#103C22semidirect
34. C2×C3×#103C22semidirect
39. C2×C3×C4×C8C22semidirect
40. C3×#391C22semidirect
41. C3×#390C22semidirect
44. C3×#382C22semidirect
45. C3×#336C22semidirect
46. C22×#124C2×C3semidirect
47. C2×#495C2×C3semidirect
48. C2×#451C2×C3semidirect
49. C2×#448C2×C3semidirect
50. C2×#450C2×C3semidirect
52. #2850C2×C3semidirect
53. #2015C2×C3semidirect
54. #2012C2×C3semidirect
55. #1958C2×C3semidirect
56. C4×C22⋊C8C2×C3semidirect
95. C3×#124D4semidirect
125. C22×Q16C22×C3semidirect
127. C22×SD16C22×C3semidirect
133. C2×#124C22×C3semidirect
143. C2×#103C22×C3semidirect
144. C2×#103C22×C3semidirect
149. C2×C4×C8C22×C3semidirect
150. #391C22×C3semidirect
151. #390C22×C3semidirect
154. #382C22×C3semidirect
155. #336C22×C3semidirect
276. #124C3×D4semidirect
466. C22×C3#495semidirect
569. C22C3×#495semidirect
584. C3#25724direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1242
C2×C3×#4952
C2×C3×#4512
C2×C3×#4482
C2×C3×#4502
C2×C3×#4472
C3×#28502
C3×#20152
C3×#20122
C3×#19582
C3×C4×C22⋊C82
#257243