Finite Group Databasebeta

← C3×#29632 #10648239

Subgroups — C3×#29632 #10648239

Order 768.

1452 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic
2C2×C3×#452C211yesmaximal, characteristic, complemented
3C2×C3×#450C211yesmaximal, characteristic, complemented
4C3×#2625C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2041C212yesmaximal, complemented
7C3×#2015C212yesmaximal, complemented
8C3×#2014C212yesmaximal, complemented
9C3×#1818C211yesmaximal, characteristic, complemented
10C3×#1788C211yesmaximal, characteristic, complemented
11C3×#1786C211yesmaximal, characteristic, complemented
12#29632C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×Q16C2211yescharacteristic, complemented
16C22×C3×SD16C2212yes—
17C22×C3×SD16C2211yescharacteristic, complemented
18C3×#535—24no—
19C3×#535—24no—
20C2×C3×#127C2212yes—
21C2×C3×#124C2212yes—
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×#121—21no—
24C2×C3×#121—21no—
25C2×C3×#120—22no—
26C2×C3×#117C2211yescharacteristic
27C3×#452—24no—
28C3×#450—24no—
29C2×C3×#103C2211yescharacteristic, complemented
30C2×C3×#103C2212yescomplemented
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2211yescharacteristic, complemented
35C2×C3×#102C2212yescomplemented
36C2×C3×C22⋊C8C2211yescharacteristic
37C2×C3×C22⋊C8C2211yescharacteristic
38C3×#400—21no—
39C3×#398—21no—
40C3×#397—22no—
41C3×#391C2212yescomplemented
42C3×#389C2212yescomplemented
43C3×#380C2211yescharacteristic
44C3×#336C2212yescomplemented
45C3×#336C2212yescomplemented
46C2×#535C2×C311yescharacteristic
47C2×#452C2×C311yescharacteristic, complemented
48C2×#450C2×C311yescharacteristic, complemented
49#2625C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2041C2×C312yescomplemented
52#2015C2×C312yescomplemented
53#2014C2×C312yescomplemented
54#1818C2×C311yescharacteristic, complemented
55#1788C2×C311yescharacteristic, complemented
56#1786C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C2×C3×#41—22no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C2×C3×#41D414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4—21no—
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4—42no—
70C22×C3×D4—42no—
71C22×C3×D4D414yes—
72C22×C3×D4C2312yes—
73C22×C3×D4C2311yescharacteristic
74C23×C3×C4—21no—
75C23×C3×C4C2311yescharacteristic
76C2×C3×Q16—22no—
77C2×C3×Q16—42no—
78C2×C3×SD16—24no—
79C2×C3×SD16—22no—
80C2×C3×SD16—42no—
81C2×C3×SD16—42no—
82C2×C3×SD16—42no—
83C22×C3×C8C2312yes—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#127—24no—
87C3×#127—24no—
88C3×#124—24no—
89C3×#124—24no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—22no—
92C3×#121—28no—
93C3×#121—28no—
94C3×#120—216no—
95C3×#117—22no—
96C3×#117—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4D411yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4D412yes—
108C2×C3×C42C2312yes—
109C3×#103—22no—
110C3×#103—42no—
111C3×#102—24no—
112C3×#102—22no—
113C3×#102—42no—
114C3×#102—42no—
115C3×#102—42no—
116C3×C22⋊C8—22no—
117C3×C22⋊C8—22no—
118C3×#95—21no—
119C3×#95—22no—
120C3×#95—22no—
121C3×#95—22no—
122C22×#41C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C22×Q16C22×C311yescharacteristic, complemented
125C22×SD16C22×C312yes—
126C22×SD16C22×C311yescharacteristic, complemented
127#535—24no—
128#535—24no—
129C2×#127C22×C312yes—
130C2×#124C22×C312yes—
131C2×C22⋊Q8C22×C311yescharacteristic
132C2×#121—21no—
133C2×#121—21no—
134C2×#120—22no—
135C2×#117C22×C311yescharacteristic
136#452—24no—
137#450—24no—
138C2×#103C22×C311yescharacteristic, complemented
139C2×#103C22×C312yescomplemented
140C2×#102C22×C312yes—
141C2×#102C22×C312yes—
142C2×#102C22×C312yes—
143C2×#102C22×C311yescharacteristic, complemented
144C2×#102C22×C312yescomplemented
145C2×C22⋊C8C22×C311yescharacteristic
146C2×C22⋊C8C22×C311yescharacteristic
147#400—21no—
148#398—21no—
149#397—22no—
150#391C22×C312yescomplemented
151#389C22×C312yescomplemented
152#380C22×C311yescharacteristic
153#336C22×C312yescomplemented
154#336C22×C312yescomplemented
155C24×C3—22no—
156C24×C3—21no—
157C24×C3—24no—
158C24×C3—44no—
159C24×C3—81no—
160C24×C3—82no—
161C24×C3—81no—
162C24×C3C2×D411yescharacteristic
163C3×#41—44no—
164C3×#41—44no—
165C3×#41—48no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8C2×D412yes—
174C2×C3×D4—24no—
175C2×C3×D4—28no—
176C2×C3×D4—28no—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—48no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—82no—
192C2×C3×D4—82no—
193C2×C3×D4—82no—
194C2×C3×D4—82no—
195C2×C3×D4C2×D414yes—
196C2×C3×D4C2×D412yes—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—21no—
200C22×C3×C4—24no—
201C22×C3×C4—22no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—44no—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2411yescharacteristic
211C3×Q16—44no—
212C3×SD16—48no—
213C3×SD16—44no—
214C2×C3×C8—22no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4—82no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—81no—
234C3×C42—22no—
235C3×C42—22no—
236C25—21no—
237C2×#41—22no—
238C2×#41—42no—
239C2×#41—42no—
240C2×#41C3×D414yes—
241C22×Q8C23×C311yescharacteristic
242C22×Q8C23×C311yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4C3×D414yes—
251C22×D4C23×C312yes—
252C22×D4C23×C311yescharacteristic
253C23×C4—21no—
254C23×C4C23×C311yescharacteristic
255C2×Q16—22no—
256C2×Q16—42no—
257C2×SD16—24no—
258C2×SD16—22no—
259C2×SD16—42no—
260C2×SD16—42no—
261C2×SD16—42no—
262C22×C8C23×C312yes—
263C22×C8C23×C311yescharacteristic
264C22×C8C23×C311yescharacteristic
265#127—24no—
266#127—24no—
267#124—24no—
268#124—24no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271#121—28no—
272#121—28no—
273#120—216no—
274#117—22no—
275#117—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4C3×D411yescharacteristic
279C2×C4⋊C4C3×D411yescharacteristic
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C312yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4C3×D412yes—
287C2×C42C23×C312yes—
288#103—22no—
289#103—42no—
290#102—24no—
291#102—22no—
292#102—42no—
293#102—42no—
294#102—42no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297#95—21no—
298#95—22no—
299#95—22no—
300#95—22no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—24no—
306C23×C3—22no—
307C23×C3—48no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—41no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—84no—
319C23×C3—84no—
320C23×C3—88no—
321C23×C3—88no—
322C23×C3#12012yes—
323C23×C3C22×D411yescharacteristic
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×D4—24no—
331C3×D4—22no—
332C3×D4—24no—
333C3×D4—22no—
334C3×D4—48no—
335C3×D4—48no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—88no—
340C3×D4—88no—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—48no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4#12012yes—
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4H(5,𝔽2)12yes—
366C2×C3×C4H(5,𝔽2)12yes—
367C3×C8—42no—
368C3×C8—41no—
369C3×C8—41no—
370C24—22no—
371C24—21no—
372C24—24no—
373C24—44no—
374C24—81no—
375C24—82no—
376C24—81no—
377C24C2×C3×D411yescharacteristic
378#41—44no—
379#41—44no—
380#41—48no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8C2×C3×D412yes—
389C2×D4—24no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—48no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4C2×C3×D414yes—
411C2×D4C2×C3×D412yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—44no—
420C22×C4C2×C3×D412yes—
421C22×C4C2×C3×D412yes—
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D412yes—
424C22×C4C2×C3×D412yes—
425C22×C4C24×C311yescharacteristic
426Q16—44no—
427SD16—48no—
428SD16—44no—
429C2×C8—22no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—82no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—24no—
447C22⋊C4—42no—
448C22⋊C4—81no—
449C42—22no—
450C42—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—44no—
458C22×C3—42no—
459C22×C3—44no—
460C22×C3—42no—
461C22×C3—44no—
462C22×C3—816no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3—82no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3C2×#12011yescharacteristic
470C22×C3#53512yes—
471C22×C3#57611yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57712yes—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—81no—
479C3×C4—82no—
480C3×C4—81no—
481C3×C4#53412yes—
482C3×C4#53411yescharacteristic
483C3×C4#53411yescharacteristic
484C23—21no—
485C23—22no—
486C23—21no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—48no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—84no—
502C23—84no—
503C23—88no—
504C23—88no—
505C23C3×#12012yes—
506C23C22×C3×D411yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—42no—
510Q8—42no—
511Q8—41no—
512Q8—41no—
513D4—24no—
514D4—22no—
515D4—24no—
516D4—22no—
517D4—48no—
518D4—48no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—88no—
523D4—88no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4C3×#12012yes—
546C2×C4C22×C3×D411yescharacteristic
547C2×C4C22×C3×D411yescharacteristic
548C2×C4C3×H(5,𝔽2)12yes—
549C2×C4C3×H(5,𝔽2)12yes—
550C8—42no—
551C8—41no—
552C8—41no—
553C2×C3—42no—
554C2×C3—82no—
555C2×C3—84no—
556C2×C3#261311yescharacteristic
557C2×C3#300911yescharacteristic
558C2×C3#301211yescharacteristic
559C2×C3#322212yes—
560C2×C3#322412yes—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—44no—
572C22—816no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22C2×C3×#12011yescharacteristic
580C22C3×#53512yes—
581C22C3×#57611yescharacteristic
582C22C3×#57711yescharacteristic
583C22C3×#57712yes—
584C4—22no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—82no—
589C4—81no—
590C4—81no—
591C4C3×#53412yes—
592C4C3×#53411yescharacteristic
593C4C3×#53411yescharacteristic
594C3#2963211yescharacteristic, direct product factor, Sylow
595C2—42no—
596C2—84no—
597C2—82no—
598C2C3×#261311yescharacteristic
599C2C3×#300911yescharacteristic
600C2C3×#301211yescharacteristic
601C2C3×#322212yes—
602C2C3×#322412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#452C2semidirect
3. C2×C3×#450C2semidirect
4. C3×#2625C2semidirect
5. C3×#2042C2semidirect
6. C3×#2041C2semidirect
7. C3×#2015C2semidirect
8. C3×#2014C2semidirect
9. C3×#1818C2semidirect
10. C3×#1788C2semidirect
11. C3×#1786C2semidirect
12. #29632C3direct (N × H)
15. C22×C3×Q16C22semidirect
17. C22×C3×SD16C22semidirect
29. C2×C3×#103C22semidirect
30. C2×C3×#103C22semidirect
34. C2×C3×#102C22semidirect
35. C2×C3×#102C22semidirect
41. C3×#391C22semidirect
42. C3×#389C22semidirect
44. C3×#336C22semidirect
45. C3×#336C22semidirect
47. C2×#452C2×C3semidirect
48. C2×#450C2×C3semidirect
49. #2625C2×C3semidirect
50. #2042C2×C3semidirect
51. #2041C2×C3semidirect
52. #2015C2×C3semidirect
53. #2014C2×C3semidirect
54. #1818C2×C3semidirect
55. #1788C2×C3semidirect
56. #1786C2×C3semidirect
124. C22×Q16C22×C3semidirect
126. C22×SD16C22×C3semidirect
138. C2×#103C22×C3semidirect
139. C2×#103C22×C3semidirect
143. C2×#102C22×C3semidirect
144. C2×#102C22×C3semidirect
150. #391C22×C3semidirect
151. #389C22×C3semidirect
153. #336C22×C3semidirect
154. #336C22×C3semidirect
594. C3#29632direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C2×C3×#4522
C2×C3×#4502
C3×#26252
C3×#20422
C3×#20412
C3×#20152
C3×#20142
C3×#18182
C3×#17882
C3×#17862
#296323