Finite Group Databasebeta

← C3×#23909 #10642516

Subgroups — C3×#23909 #10642516

Order 768.

1802 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3487C211yesmaximal, characteristic, complemented
2C2×C3×#532C211yesmaximal, characteristic, complemented
3C2×C3×#456C211yesmaximal, characteristic, complemented
4C2×C3×#454C211yesmaximal, characteristic, complemented
5C3×#3005C212yesmaximal, complemented
6C3×#2004C212yesmaximal, complemented
7C3×#2002C212yesmaximal, complemented
8C3×#1896C212yesmaximal, complemented
9C3×#1888C211yesmaximal, characteristic, complemented
10C3×#1886C211yesmaximal, characteristic, complemented
11C3×#1785C211yesmaximal, characteristic, complemented
12#23909C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C22×C3×#41C2211yescharacteristic
16C2×C3×#137C2212yes—
17C2×C3×AGL(1,ℤ/8)C2212yes—
18C3×#550—24no—
19C3×#548—24no—
20C3×#535—22no—
21C3×#535C2212yes—
22C3×#532—24no—
23C3×#532—24no—
24C2×C3×C4⋊Q8C2211yescharacteristic
25C2×C3×#127C2211yescharacteristic
26C2×C3×#124C2211yescharacteristic
27C2×C3×#124C2211yescharacteristic
28C3×#525—22no—
29C3×#525C2212yes—
30C2×C3×#121—22no—
31C3×#519C2211yescharacteristic
32C2×C3×C4×D4—22no—
33C22×C3×C42C2211yescharacteristic
34C3×#456—24no—
35C3×#454—24no—
36C3×#452—24no—
37C3×#449—24no—
38C2×C3×#104C2212yes—
39C2×C3×#104C2212yes—
40C3×#419C2212yes—
41C3×#418C2212yes—
42C2×C3×#101C2211yescharacteristic
43C2×C3×#101C2211yescharacteristic
44C2×C3×#100C2212yes—
45C3×#407C2212yes—
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3487C2×C311yescharacteristic, complemented
49C2×#532C2×C311yescharacteristic, complemented
50C2×#456C2×C311yescharacteristic, complemented
51C2×#454C2×C311yescharacteristic, complemented
52#3005C2×C312yescomplemented
53#2004C2×C312yescomplemented
54#2002C2×C312yescomplemented
55#1896C2×C312yescomplemented
56#1888C2×C311yescharacteristic, complemented
57#1886C2×C311yescharacteristic, complemented
58#1785C2×C311yescharacteristic, complemented
59C3×2−1+4—22no—
60C3×2−1+4—22no—
61C3×2−1+4—42no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—22no—
66C2×C3×#41—22no—
67C2×C3×#41—24no—
68C2×C3×#41—22no—
69C2×C3×#41—24no—
70C2×C3×#41—42no—
71C2×C3×#41—42no—
72C2×C3×#41C2312yes—
73C22×C3×Q8—22no—
74C22×C3×Q8C2311yescharacteristic
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4—21no—
77C22×C3×D4—22no—
78C22×C3×D4D412yes—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4—21no—
83C23×C3×C4D412yes—
84C23×C3×C4C2311yescharacteristic
85C3×#137—24no—
86C3×#137—42no—
87C3×AGL(1,ℤ/8)—24no—
88C3×AGL(1,ℤ/8)—42no—
89C2×C3×Q16—22no—
90C2×C3×SD16—22no—
91C2×C3×SD16—22no—
92C2×C3×D8—22no—
93C2×C3×M16C2312yes—
94C2×C3×M16C2312yes—
95C3×C4⋊Q8—22no—
96C3×C4⋊Q8D414yescomplemented
97C3×#127—24no—
98C3×#127—22no—
99C3×#127—22no—
100C3×#127—22no—
101C3×#124—24no—
102C3×#124—22no—
103C3×#124—22no—
104C3×#124—22no—
105C3×#124—22no—
106C3×#124D414yescomplemented
107C3×#123—22no—
108C3×C22⋊Q8—24no—
109C3×C22⋊Q8D414yescomplemented
110C3×#121—216no—
111C3×#121—24no—
112C3×#121—28no—
113C3×#121—24no—
114C3×#121D414yescomplemented
115C3×#120—24no—
116C3×#120—22no—
117C3×C4×Q8—24no—
118C3×C4×D4—216no—
119C3×C4×D4—28no—
120C3×C4×D4—24no—
121C3×#117—22no—
122C3×#117C2312yes—
123C2×C3×C4⋊C4—22no—
124C2×C3×C22⋊C4—22no—
125C2×C3×C22⋊C4—22no—
126C2×C3×C42—21no—
127C2×C3×C42—22no—
128C2×C3×C42—22no—
129C2×C3×C42C2311yescharacteristic
130C2×C3×C42C2311yescharacteristic
131C3×#104—24no—
132C3×#104—24no—
133C3×#103—24no—
134C3×#102—24no—
135C3×#101—22no—
136C3×#101—22no—
137C3×#100—42no—
138C3×C22⋊C8—24no—
139C2×2−1+4C22×C311yescharacteristic
140C22×#41C22×C311yescharacteristic
141C22×#41C22×C311yescharacteristic
142C2×#137C22×C312yes—
143C2×AGL(1,ℤ/8)C22×C312yes—
144#550—24no—
145#548—24no—
146#535—22no—
147#535C22×C312yes—
148#532—24no—
149#532—24no—
150C2×C4⋊Q8C22×C311yescharacteristic
151C2×#127C22×C311yescharacteristic
152C2×#124C22×C311yescharacteristic
153C2×#124C22×C311yescharacteristic
154#525—22no—
155#525C22×C312yes—
156C2×#121—22no—
157#519C22×C311yescharacteristic
158C2×C4×D4—22no—
159C22×C42C22×C311yescharacteristic
160#456—24no—
161#454—24no—
162#452—24no—
163#449—24no—
164C2×#104C22×C312yes—
165C2×#104C22×C312yes—
166#419C22×C312yes—
167#418C22×C312yes—
168C2×#101C22×C311yescharacteristic
169C2×#101C22×C311yescharacteristic
170C2×#100C22×C312yes—
171#407C22×C312yes—
172#339C22×C312yes—
173#339C22×C312yes—
174C24×C3—21no—
175C24×C3C2×D412yes—
176C24×C3C2×D411yescharacteristic
177C3×#41—22no—
178C3×#41—22no—
179C3×#41—22no—
180C3×#41—22no—
181C3×#41—48no—
182C3×#41—48no—
183C3×#41—48no—
184C3×#41—48no—
185C3×#41—44no—
186C3×#41—44no—
187C3×#41—44no—
188C3×#41—44no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—21no—
192C2×C3×Q8—22no—
193C2×C3×Q8—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8—44no—
196C2×C3×Q8—44no—
197C2×C3×Q8C2×D412yes—
198C2×C3×Q8C2×D412yes—
199C2×C3×Q8C2×D412yes—
200C2×C3×Q8C2×D412yes—
201C2×C3×D4—28no—
202C2×C3×D4—24no—
203C2×C3×D4—22no—
204C2×C3×D4—24no—
205C2×C3×D4—22no—
206C2×C3×D4—28no—
207C2×C3×D4—24no—
208C2×C3×D4—24no—
209C2×C3×D4—22no—
210C2×C3×D4—22no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—22no—
215C2×C3×D4—22no—
216C2×C3×D4—42no—
217C2×C3×D4—48no—
218C2×C3×D4—42no—
219C2×C3×D4—42no—
220C2×C3×D4C2×D412yes—
221C2×C3×D4C2×D412yes—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—22no—
226C22×C3×C4—24no—
227C22×C3×C4—24no—
228C22×C3×C4—24no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—24no—
232C22×C3×C4—24no—
233C22×C3×C4—22no—
234C22×C3×C4—42no—
235C22×C3×C4—44no—
236C22×C3×C4—42no—
237C22×C3×C4—42no—
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2×D412yes—
240C22×C3×C4C2×D414yes—
241C22×C3×C4C2411yescharacteristic
242C3×Q16—44no—
243C3×SD16—44no—
244C3×SD16—44no—
245C3×D8—44no—
246C3×M16—22no—
247C3×M16—22no—
248C3×M16—22no—
249C3×M16—22no—
250C2×C3×C8—22no—
251C2×C3×C8—22no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—28no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4C2×D414yes—
256C3×C22⋊C4—28no—
257C3×C22⋊C4—28no—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—28no—
261C3×C22⋊C4—24no—
262C3×C42—24no—
263C3×C42—24no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—22no—
267C3×C42C2×D412yes—
268C3×C42C2×D412yes—
2692−1+4—22no—
2702−1+4—22no—
2712−1+4—42no—
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—22no—
277C2×#41—22no—
278C2×#41—24no—
279C2×#41—24no—
280C2×#41—42no—
281C2×#41—42no—
282C2×#41C23×C312yes—
283C22×Q8—22no—
284C22×Q8C23×C311yescharacteristic
285C22×Q8C23×C311yescharacteristic
286C22×D4—22no—
287C22×D4—21no—
288C22×D4C3×D412yes—
289C22×D4C23×C311yescharacteristic
290C22×D4C23×C311yescharacteristic
291C23×C4—21no—
292C23×C4—21no—
293C23×C4C3×D412yes—
294C23×C4C23×C311yescharacteristic
295#137—24no—
296#137—42no—
297AGL(1,ℤ/8)—24no—
298AGL(1,ℤ/8)—42no—
299C2×Q16—22no—
300C2×SD16—22no—
301C2×SD16—22no—
302C2×D8—22no—
303C2×M16C23×C312yes—
304C2×M16C23×C312yes—
305C4⋊Q8—22no—
306C4⋊Q8C3×D414yescomplemented
307#127—24no—
308#127—22no—
309#127—22no—
310#127—22no—
311#124—24no—
312#124—22no—
313#124—22no—
314#124—22no—
315#124—22no—
316#124C3×D414yescomplemented
317#123—22no—
318C22⋊Q8—24no—
319C22⋊Q8C3×D414yescomplemented
320#121—216no—
321#121—24no—
322#121—28no—
323#121—24no—
324#121C3×D414yescomplemented
325#120—24no—
326#120—22no—
327C4×Q8—24no—
328C4×D4—216no—
329C4×D4—28no—
330C4×D4—24no—
331#117—22no—
332#117C23×C312yes—
333C2×C4⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42C23×C311yescharacteristic
340C2×C42C23×C311yescharacteristic
341#104—24no—
342#104—24no—
343#103—24no—
344#102—24no—
345#101—22no—
346#101—22no—
347#100—42no—
348C22⋊C8—24no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—42no—
357C23×C3—44no—
358C23×C3—44no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3#12012yes—
362C23×C3C22×D411yescharacteristic
363C3×Q8—24no—
364C3×Q8—24no—
365C3×Q8—22no—
366C3×Q8—24no—
367C3×Q8—22no—
368C3×Q8—44no—
369C3×Q8—42no—
370C3×D4—22no—
371C3×D4—24no—
372C3×D4—22no—
373C3×D4—48no—
374C3×D4—416no—
375C3×D4—44no—
376C3×D4—48no—
377C3×D4—42no—
378C3×D4—42no—
379C3×D4—42no—
380C3×D4—42no—
381C3×D4—42no—
382C3×D4—84no—
383C3×D4—88no—
384C2×C3×C4—24no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—24no—
391C2×C3×C4—24no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—24no—
395C2×C3×C4—22no—
396C2×C3×C4—48no—
397C2×C3×C4—48no—
398C2×C3×C4—48no—
399C2×C3×C4—48no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—42no—
404C2×C3×C4—42no—
405C2×C3×C4#12014yes—
406C2×C3×C4#12012yes—
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C22×D412yes—
410C2×C3×C4H(5,𝔽2)12yes—
411C3×C8—42no—
412C3×C8—42no—
413C24—21no—
414C24C2×C3×D412yes—
415C24C2×C3×D411yescharacteristic
416#41—22no—
417#41—22no—
418#41—22no—
419#41—22no—
420#41—48no—
421#41—48no—
422#41—48no—
423#41—48no—
424#41—44no—
425#41—44no—
426#41—44no—
427#41—44no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8C2×C3×D412yes—
437C2×Q8C2×C3×D412yes—
438C2×Q8C2×C3×D412yes—
439C2×Q8C2×C3×D412yes—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—28no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—42no—
456C2×D4—48no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4C2×C3×D412yes—
460C2×D4C2×C3×D412yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4C2×C3×D411yescharacteristic
478C22×C4C2×C3×D412yes—
479C22×C4C2×C3×D414yes—
480C22×C4C24×C311yescharacteristic
481Q16—44no—
482SD16—44no—
483SD16—44no—
484D8—44no—
485M16—22no—
486M16—22no—
487M16—22no—
488M16—22no—
489C2×C8—22no—
490C2×C8—22no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4C2×C3×D414yes—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—24no—
498C22⋊C4—22no—
499C22⋊C4—28no—
500C22⋊C4—24no—
501C42—24no—
502C42—24no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42C2×C3×D412yes—
507C42C2×C3×D412yes—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—24no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—44no—
514C22×C3—44no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—44no—
519C22×C3—44no—
520C22×C3—82no—
521C22×C3—82no—
522C22×C3—82no—
523C22×C3C2×#12011yescharacteristic
524C22×C3C2×#12011yescharacteristic
525C22×C3#53411yescharacteristic
526C3×C4—24no—
527C3×C4—22no—
528C3×C4—24no—
529C3×C4—42no—
530C3×C4—42no—
531C3×C4—42no—
532C3×C4—44no—
533C3×C4D4212yes—
534C3×C4D4212yes—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—41no—
546C23—41no—
547C23C3×#12012yes—
548C23C22×C3×D411yescharacteristic
549Q8—24no—
550Q8—24no—
551Q8—22no—
552Q8—24no—
553Q8—22no—
554Q8—44no—
555Q8—42no—
556D4—22no—
557D4—24no—
558D4—22no—
559D4—48no—
560D4—416no—
561D4—44no—
562D4—48no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—84no—
569D4—88no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4C3×#12012yes—
592C2×C4C3×#12014yes—
593C2×C4C22×C3×D411yescharacteristic
594C2×C4C22×C3×D411yescharacteristic
595C2×C4C22×C3×D412yes—
596C2×C4C3×H(5,𝔽2)12yes—
597C8—42no—
598C8—42no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—42no—
602C2×C3—82no—
603C2×C3—82no—
604C2×C3—82no—
605C2×C3#240311yescharacteristic
606C2×C3#301812yes—
607C22—22no—
608C22—22no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22C2×C3×#12011yescharacteristic
623C22C2×C3×#12011yescharacteristic
624C22C3×#53411yescharacteristic
625C4—24no—
626C4—22no—
627C4—24no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4C3×D4212yes—
633C4C3×D4212yes—
634C3#2390911yescharacteristic, direct product factor, Sylow
635C2—21no—
636C2—21no—
637C2—42no—
638C2—82no—
639C2—82no—
640C2—82no—
641C2C3×#240311yescharacteristic
642C2C3×#301812yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3487C2semidirect
2. C2×C3×#532C2semidirect
3. C2×C3×#456C2semidirect
4. C2×C3×#454C2semidirect
5. C3×#3005C2semidirect
6. C3×#2004C2semidirect
7. C3×#2002C2semidirect
8. C3×#1896C2semidirect
9. C3×#1888C2semidirect
10. C3×#1886C2semidirect
11. C3×#1785C2semidirect
12. #23909C3direct (N × H)
48. #3487C2×C3semidirect
49. C2×#532C2×C3semidirect
50. C2×#456C2×C3semidirect
51. C2×#454C2×C3semidirect
52. #3005C2×C3semidirect
53. #2004C2×C3semidirect
54. #2002C2×C3semidirect
55. #1896C2×C3semidirect
56. #1888C2×C3semidirect
57. #1886C2×C3semidirect
58. #1785C2×C3semidirect
96. C3×C4⋊Q8D4semidirect
106. C3×#124D4semidirect
109. C3×C22⋊Q8D4semidirect
114. C3×#121D4semidirect
306. C4⋊Q8C3×D4semidirect
316. #124C3×D4semidirect
319. C22⋊Q8C3×D4semidirect
324. #121C3×D4semidirect
634. C3#23909direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34872
C2×C3×#5322
C2×C3×#4562
C2×C3×#4542
C3×#30052
C3×#20042
C3×#20022
C3×#18962
C3×#18882
C3×#18862
C3×#17852
#239093