Finite Group Databasebeta

← C3×#33546 #10652153

Subgroups — C3×#33546 #10652153

Order 768.

1236 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3481C211yesmaximal, characteristic, complemented
2C3×#3025C212yesmaximal, complemented
3C3×#3018C211yesmaximal, characteristic, complemented
4C3×#2198C212yesmaximal, complemented
5C3×#2197C214yesmaximal, complemented
6C3×#2196C212yesmaximal, complemented
7C3×#2126C211yesmaximal, characteristic, complemented
8C3×#2120C212yesmaximal, complemented
9#33546C311yesmaximal, characteristic, direct product factor, Sylow
10C3×#585C2211yescharacteristic, complemented
11C3×#575—21no—
12C3×#559C2211yescharacteristic, complemented
13C3×#554—21no—
14C3×#553—22no—
15C3×#551—21no—
16C3×#551C2212yescomplemented
17C3×#546—22no—
18C3×D42C2211yescharacteristic, complemented
19C3×#532C2211yescharacteristic
20C3×#529—22no—
21C3×#525—21no—
22C3×#525—22no—
23C3×#525C2212yes—
24C3×C4×#41—21no—
25C2×C3×C4×D4—22no—
26C3×#458—22no—
27C3×#458C2214yescomplemented
28C3×B(4,𝔽2)—22no—
29C3×B(4,𝔽2)C2214yes—
30C3×#456—21no—
31C3×#455C2212yescomplemented
32C3×#454—21no—
33C3×#453C2212yes—
34C2×C3×#104—21no—
35C3×#413C2211yescharacteristic, complemented
36C3×#410—22no—
37C3×#410C2212yescomplemented
38C3×#354C2212yescomplemented
39C3×#353C2212yescomplemented
40C3×#352C2214yescomplemented
41C3×AΣL(1,𝔽16)C2214yescomplemented
42#3481C2×C311yescharacteristic, complemented
43#3025C2×C312yescomplemented
44#3018C2×C311yescharacteristic, complemented
45#2198C2×C312yescomplemented
46#2197C2×C314yescomplemented
47#2196C2×C312yescomplemented
48#2126C2×C311yescharacteristic, complemented
49#2120C2×C312yescomplemented
50C3×2−1+4—21no—
51C3×2−1+4—41no—
52C3×H(5,𝔽2)—22no—
53C3×H(5,𝔽2)—41no—
54C3×H(5,𝔽2)C2312yes—
55C2×C3×#41—21no—
56C2×C3×#41—22no—
57C2×C3×#41—21no—
58C2×C3×#41—42no—
59C2×C3×#41D411yescharacteristic, complemented
60C2×C3×#41C2311yescharacteristic
61C22×C3×D4—22no—
62C23×C3×C4—21no—
63C23×C3×C4D412yescomplemented
64C3×#137—22no—
65C3×#137—41no—
66C3×AGL(1,ℤ/8)—22no—
67C3×AGL(1,ℤ/8)—41no—
68C3×#135—42no—
69C2×C3×M16—21no—
70C3×C4⋊Q8D411yescharacteristic, complemented
71C3×#127C2311yescharacteristic
72C3×#126—22no—
73C3×#126—22no—
74C3×#125—22no—
75C3×#124—22no—
76C3×#124D411yescharacteristic, complemented
77C3×#124C2311yescharacteristic
78C3×#123—21no—
79C3×#123—22no—
80C3×#123—21no—
81C3×#123—22no—
82C3×#123—24no—
83C3×#123D412yescomplemented
84C3×#123C2312yes—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—42no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—22no—
93C3×#121—42no—
94C3×#120—21no—
95C3×#120—21no—
96C3×#120—24no—
97C3×#120D412yescomplemented
98C3×#120C2312yes—
99C3×C4×Q8—21no—
100C3×C4×Q8—21no—
101C3×C4×D4—21no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—22no—
105C3×C4×D4—22no—
106C3×C4×D4—22no—
107C3×C4×D4—22no—
108C3×C4×D4—22no—
109C3×C4×D4—22no—
110C3×C4×D4—22no—
111C3×C4×D4—21no—
112C3×C4×D4—44no—
113C3×#117—21no—
114C3×#117—21no—
115C3×#117—22no—
116C3×#117—22no—
117C3×#117D412yescomplemented
118C2×C3×C4⋊C4—22no—
119C2×C3×C22⋊C4—22no—
120C2×C3×C22⋊C4—22no—
121C2×C3×C42—22no—
122C2×C3×C42D411yescharacteristic, complemented
123C3×#104—22no—
124C3×#104—22no—
125C3×#101—21no—
126C3×#100C2312yes—
127C3×#99—22no—
128C3×#99—24no—
129C3×#99—24no—
130C3×#99C2314yes—
131#585C22×C311yescharacteristic, complemented
132#575—21no—
133#559C22×C311yescharacteristic, complemented
134#554—21no—
135#553—22no—
136#551—21no—
137#551C22×C312yescomplemented
138#546—22no—
139D42C22×C311yescharacteristic, complemented
140#532C22×C311yescharacteristic
141#529—22no—
142#525—21no—
143#525—22no—
144#525C22×C312yes—
145C4×#41—21no—
146C2×C4×D4—22no—
147#458—22no—
148#458C22×C314yescomplemented
149B(4,𝔽2)—22no—
150B(4,𝔽2)C22×C314yes—
151#456—21no—
152#455C22×C312yescomplemented
153#454—21no—
154#453C22×C312yes—
155C2×#104—21no—
156#413C22×C311yescharacteristic, complemented
157#410—22no—
158#410C22×C312yescomplemented
159#354C22×C312yescomplemented
160#353C22×C312yescomplemented
161#352C22×C314yescomplemented
162AΣL(1,𝔽16)C22×C314yescomplemented
163C24×C3—21no—
164C24×C3C2×D412yes—
165C3×#41—22no—
166C3×#41—22no—
167C3×#41—41no—
168C3×#41—41no—
169C3×#41—42no—
170C3×#41—42no—
171C3×#41—42no—
172C3×#41—41no—
173C3×#41—44no—
174C3×#41—41no—
175C3×#41—41no—
176C3×#41—44no—
177C3×#41—42no—
178C3×#41—42no—
179C3×#41—41no—
180C2×C3×Q8—21no—
181C2×C3×Q8—22no—
182C2×C3×Q8—21no—
183C2×C3×Q8—42no—
184C2×C3×Q8C2×D411yescharacteristic
185C2×C3×Q8C2×D411yescharacteristic
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—22no—
191C2×C3×D4—24no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—21no—
195C2×C3×D4—42no—
196C2×C3×D4—48no—
197C2×C3×D4—44no—
198C2×C3×D4—41no—
199C2×C3×D4—44no—
200C2×C3×D4C2×D412yes—
201C2×C3×D4C2×D411yescharacteristic
202C2×C3×D4C2411yescharacteristic
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—24no—
206C22×C3×C4—21no—
207C22×C3×C4—22no—
208C22×C3×C4—21no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—41no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4—81no—
216C22×C3×C4—81no—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D412yes—
220C3×Q16—42no—
221C3×SD16—42no—
222C3×SD16—42no—
223C3×D8—42no—
224C3×M16—22no—
225C3×M16—41no—
226C2×C3×C8—41no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—44no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—44no—
251C3×C22⋊C4—44no—
252C3×C22⋊C4C2×D412yes—
253C3×C22⋊C4C2×D412yes—
254C3×C42—22no—
255C3×C42—22no—
256C3×C42—21no—
257C3×C42—22no—
258C3×C42—42no—
259C3×C42C2×D411yescharacteristic
260C3×C42C2×D411yescharacteristic
2612−1+4—21no—
2622−1+4—41no—
263H(5,𝔽2)—22no—
264H(5,𝔽2)—41no—
265H(5,𝔽2)C23×C312yes—
266C2×#41—21no—
267C2×#41—22no—
268C2×#41—21no—
269C2×#41—42no—
270C2×#41C3×D411yescharacteristic, complemented
271C2×#41C23×C311yescharacteristic
272C22×D4—22no—
273C23×C4—21no—
274C23×C4C3×D412yescomplemented
275#137—22no—
276#137—41no—
277AGL(1,ℤ/8)—22no—
278AGL(1,ℤ/8)—41no—
279#135—42no—
280C2×M16—21no—
281C4⋊Q8C3×D411yescharacteristic, complemented
282#127C23×C311yescharacteristic
283#126—22no—
284#126—22no—
285#125—22no—
286#124—22no—
287#124C3×D411yescharacteristic, complemented
288#124C23×C311yescharacteristic
289#123—21no—
290#123—22no—
291#123—21no—
292#123—22no—
293#123—24no—
294#123C3×D412yescomplemented
295#123C23×C312yes—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—42no—
300#121—22no—
301#121—22no—
302#121—22no—
303#121—22no—
304#121—42no—
305#120—21no—
306#120—21no—
307#120—24no—
308#120C3×D412yescomplemented
309#120C23×C312yes—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×D4—21no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—21no—
323C4×D4—44no—
324#117—21no—
325#117—21no—
326#117—22no—
327#117—22no—
328#117C3×D412yescomplemented
329C2×C4⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C42—22no—
333C2×C42C3×D411yescharacteristic, complemented
334#104—22no—
335#104—22no—
336#101—21no—
337#100C23×C312yes—
338#99—22no—
339#99—24no—
340#99—24no—
341#99C23×C314yes—
342C23×C3—22no—
343C23×C3—24no—
344C23×C3—21no—
345C23×C3—42no—
346C23×C3—44no—
347C23×C3—44no—
348C23×C3—41no—
349C23×C3—81no—
350C23×C3#12011yescharacteristic
351C23×C3C22×D412yes—
352C3×Q8—22no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×Q8—42no—
356C3×Q8—42no—
357C3×Q8—41no—
358C3×D4—24no—
359C3×D4—22no—
360C3×D4—42no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—44no—
364C3×D4—42no—
365C3×D4—44no—
366C3×D4—44no—
367C3×D4—44no—
368C3×D4—44no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—41no—
372C3×D4—81no—
373C3×D4—88no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—24no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—44no—
386C2×C3×C4—44no—
387C2×C3×C4—42no—
388C2×C3×C4—42no—
389C2×C3×C4—44no—
390C2×C3×C4—41no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—42no—
394C2×C3×C4—42no—
395C2×C3×C4—41no—
396C2×C3×C4—42no—
397C2×C3×C4—44no—
398C2×C3×C4—82no—
399C2×C3×C4—81no—
400C2×C3×C4—82no—
401C2×C3×C4#12012yes—
402C2×C3×C4#12011yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C3×C8—42no—
405C24—21no—
406C24C2×C3×D412yes—
407#41—22no—
408#41—22no—
409#41—41no—
410#41—41no—
411#41—42no—
412#41—42no—
413#41—42no—
414#41—41no—
415#41—44no—
416#41—41no—
417#41—41no—
418#41—44no—
419#41—42no—
420#41—41no—
421#41—42no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—42no—
426C2×Q8C2×C3×D411yescharacteristic
427C2×Q8C2×C3×D411yescharacteristic
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—48no—
439C2×D4—44no—
440C2×D4—41no—
441C2×D4—44no—
442C2×D4C2×C3×D412yes—
443C2×D4C2×C3×D411yescharacteristic
444C2×D4C24×C311yescharacteristic
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4C2×C3×D412yes—
460C22×C4C2×C3×D411yescharacteristic
461C22×C4C2×C3×D412yes—
462Q16—42no—
463SD16—42no—
464SD16—42no—
465D8—42no—
466M16—22no—
467M16—41no—
468C2×C8—41no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—24no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4C2×C3×D412yes—
495C22⋊C4C2×C3×D412yes—
496C42—22no—
497C42—22no—
498C42—21no—
499C42—22no—
500C42—42no—
501C42C2×C3×D411yescharacteristic
502C42C2×C3×D411yescharacteristic
503C22×C3—24no—
504C22×C3—22no—
505C22×C3—21no—
506C22×C3—41no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—44no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—41no—
514C22×C3—82no—
515C22×C3—84no—
516C22×C3—84no—
517C22×C3—161no—
518C22×C3C2×#12011yescharacteristic
519C3×C4—22no—
520C3×C4—21no—
521C3×C4—22no—
522C3×C4—41no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—42no—
526C3×C4—42no—
527C3×C4—41no—
528C3×C4—84no—
529C3×C4B(4,𝔽2)11yescharacteristic
530C3×C4B(4,𝔽2)11yescharacteristic
531C23—22no—
532C23—24no—
533C23—21no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—41no—
538C23—81no—
539C23C3×#12011yescharacteristic
540C23C22×C3×D412yes—
541Q8—22no—
542Q8—41no—
543Q8—41no—
544Q8—42no—
545Q8—42no—
546Q8—41no—
547D4—24no—
548D4—22no—
549D4—42no—
550D4—41no—
551D4—41no—
552D4—42no—
553D4—44no—
554D4—44no—
555D4—44no—
556D4—44no—
557D4—44no—
558D4—42no—
559D4—42no—
560D4—41no—
561D4—81no—
562D4—88no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—82no—
588C2×C4—81no—
589C2×C4—82no—
590C2×C4C3×#12012yes—
591C2×C4C3×#12011yescharacteristic
592C2×C4C22×C3×D411yescharacteristic
593C8—42no—
594C2×C3—21no—
595C2×C3—42no—
596C2×C3—44no—
597C2×C3—41no—
598C2×C3—81no—
599C2×C3—82no—
600C2×C3C2×B(4,𝔽2)11yescharacteristic
601C22—24no—
602C22—22no—
603C22—21no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—82no—
613C22—84no—
614C22—84no—
615C22—161no—
616C22C2×C3×#12011yescharacteristic
617C4—22no—
618C4—21no—
619C4—22no—
620C4—41no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—84no—
627C4C3×B(4,𝔽2)11yescharacteristic
628C4C3×B(4,𝔽2)11yescharacteristic
629C3#3354611yescharacteristic, direct product factor, Sylow
630C2—21no—
631C2—42no—
632C2—44no—
633C2—41no—
634C2—81no—
635C2—82no—
636C2C2×C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
centerC3×C4B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C3
socleC2×C3C2×B(4,𝔽2)
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3481C2semidirect
2. C3×#3025C2semidirect
3. C3×#3018C2semidirect
4. C3×#2198C2semidirect
5. C3×#2197C2semidirect
6. C3×#2196C2semidirect
7. C3×#2126C2semidirect
8. C3×#2120C2semidirect
9. #33546C3direct (N × H)
10. C3×#585C22semidirect
12. C3×#559C22semidirect
16. C3×#551C22semidirect
18. C3×D42C22semidirect
27. C3×#458C22semidirect
31. C3×#455C22semidirect
35. C3×#413C22semidirect
37. C3×#410C22semidirect
38. C3×#354C22semidirect
39. C3×#353C22semidirect
40. C3×#352C22semidirect
41. C3×AΣL(1,𝔽16)C22semidirect
42. #3481C2×C3semidirect
43. #3025C2×C3semidirect
44. #3018C2×C3semidirect
45. #2198C2×C3semidirect
46. #2197C2×C3semidirect
47. #2196C2×C3semidirect
48. #2126C2×C3semidirect
49. #2120C2×C3semidirect
59. C2×C3×#41D4semidirect
63. C23×C3×C4D4semidirect
70. C3×C4⋊Q8D4semidirect
76. C3×#124D4semidirect
83. C3×#123D4semidirect
97. C3×#120D4semidirect
117. C3×#117D4semidirect
122. C2×C3×C42D4semidirect
131. #585C22×C3semidirect
133. #559C22×C3semidirect
137. #551C22×C3semidirect
139. D42C22×C3semidirect
148. #458C22×C3semidirect
152. #455C22×C3semidirect
156. #413C22×C3semidirect
158. #410C22×C3semidirect
159. #354C22×C3semidirect
160. #353C22×C3semidirect
161. #352C22×C3semidirect
162. AΣL(1,𝔽16)C22×C3semidirect
270. C2×#41C3×D4semidirect
274. C23×C4C3×D4semidirect
281. C4⋊Q8C3×D4semidirect
287. #124C3×D4semidirect
294. #123C3×D4semidirect
308. #120C3×D4semidirect
328. #117C3×D4semidirect
333. C2×C42C3×D4semidirect
629. C3#33546direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34812
C3×#30252
C3×#30182
C3×#21982
C3×#21972
C3×#21962
C3×#21262
C3×#21202
#335463