Finite Group Databasebeta

← C3×#23917 #10642524

Subgroups — C3×#23917 #10642524

Order 768.

1682 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3486C211yesmaximal, characteristic, complemented
2C2×C3×#536C211yesmaximal, characteristic, complemented
3C2×C3×#456C211yesmaximal, characteristic, complemented
4C2×C3×#454C211yesmaximal, characteristic, complemented
5C3×#3005C212yesmaximal, complemented
6C3×#2005C212yesmaximal, complemented
7C3×#2003C212yesmaximal, complemented
8C3×#1897C212yesmaximal, complemented
9C3×#1888C211yesmaximal, characteristic, complemented
10C3×#1886C211yesmaximal, characteristic, complemented
11C3×#1784C211yesmaximal, characteristic, complemented
12#23917C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C23×C3×Q8C2211yescharacteristic
16C2×C3×#137C2212yes—
17C2×C3×AGL(1,ℤ/8)C2212yes—
18C3×D4×Q8—24no—
19C3×#548—24no—
20C3×#536—22no—
21C3×#536—24no—
22C3×#536—24no—
23C3×#536C2212yes—
24C2×C3×C4⋊Q8C2211yescharacteristic
25C2×C3×C4⋊Q8C2211yescharacteristic
26C2×C3×#124C2211yescharacteristic
27C2×C3×#124C2211yescharacteristic
28C3×#525—22no—
29C3×#525C2212yes—
30C2×C3×#123—22no—
31C2×C3×C22⋊Q8—22no—
32C3×#519C2211yescharacteristic
33C2×C3×#117C2211yescharacteristic
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×#101C2212yes—
45C3×#407C2212yes—
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3486C2×C311yescharacteristic, complemented
49C2×#536C2×C311yescharacteristic, complemented
50C2×#456C2×C311yescharacteristic, complemented
51C2×#454C2×C311yescharacteristic, complemented
52#3005C2×C312yescomplemented
53#2005C2×C312yescomplemented
54#2003C2×C312yescomplemented
55#1897C2×C312yescomplemented
56#1888C2×C311yescharacteristic, complemented
57#1886C2×C311yescharacteristic, complemented
58#1784C2×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—24no—
67C2×C3×#41—42no—
68C2×C3×#41C2312yes—
69C22×C3×Q8—22no—
70C22×C3×Q8—22no—
71C22×C3×Q8—22no—
72C22×C3×Q8—24no—
73C22×C3×Q8—42no—
74C22×C3×Q8D412yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×Q8C2311yescharacteristic
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×C3×C4D412yes—
82C23×C3×C4C2311yescharacteristic
83C3×#137—24no—
84C3×#137—42no—
85C3×AGL(1,ℤ/8)—24no—
86C3×AGL(1,ℤ/8)—42no—
87C2×C3×Q16—22no—
88C2×C3×SD16—22no—
89C2×C3×SD16—22no—
90C2×C3×D8—22no—
91C2×C3×M16C2312yes—
92C2×C3×M16C2312yes—
93C3×C4⋊Q8—24no—
94C3×C4⋊Q8—22no—
95C3×C4⋊Q8—22no—
96C3×C4⋊Q8—22no—
97C3×C4⋊Q8—22no—
98C3×C4⋊Q8D414yescomplemented
99C3×#124—24no—
100C3×#124—22no—
101C3×#124—22no—
102C3×#124—22no—
103C3×#124—22no—
104C3×#124D414yescomplemented
105C3×#123—216no—
106C3×#123—24no—
107C3×#123—22no—
108C3×C22⋊Q8—216no—
109C3×C22⋊Q8—24no—
110C3×C22⋊Q8—28no—
111C3×C22⋊Q8—24no—
112C3×C22⋊Q8D414yes—
113C3×#121—24no—
114C3×#121D414yes—
115C3×#120—22no—
116C3×C4×Q8—24no—
117C3×C4×D4—28no—
118C3×C4×D4—24no—
119C3×#117—22no—
120C3×#117—22no—
121C3×#117—22no—
122C3×#117C2312yes—
123C2×C3×C4⋊C4—21no—
124C2×C3×C4⋊C4—22no—
125C2×C3×C4⋊C4—22no—
126C2×C3×C22⋊C4—21no—
127C2×C3×C22⋊C4—22no—
128C2×C3×C22⋊C4—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×#101—42no—
138C3×C22⋊C8—24no—
139C2×2−1+4C22×C311yescharacteristic
140C22×#41C22×C311yescharacteristic
141C23×Q8C22×C311yescharacteristic
142C2×#137C22×C312yes—
143C2×AGL(1,ℤ/8)C22×C312yes—
144D4×Q8—24no—
145#548—24no—
146#536—22no—
147#536—24no—
148#536—24no—
149#536C22×C312yes—
150C2×C4⋊Q8C22×C311yescharacteristic
151C2×C4⋊Q8C22×C311yescharacteristic
152C2×#124C22×C311yescharacteristic
153C2×#124C22×C311yescharacteristic
154#525—22no—
155#525C22×C312yes—
156C2×#123—22no—
157C2×C22⋊Q8—22no—
158#519C22×C311yescharacteristic
159C2×#117C22×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×#101C22×C312yes—
171#407C22×C312yes—
172#339C22×C312yes—
173#339C22×C312yes—
174C24×C3—21no—
175C24×C3C2×D411yescharacteristic
176C3×#41—22no—
177C3×#41—22no—
178C3×#41—22no—
179C3×#41—22no—
180C3×#41—48no—
181C3×#41—48no—
182C3×#41—44no—
183C3×#41—44no—
184C3×#41—44no—
185C3×#41—44no—
186C2×C3×Q8—22no—
187C2×C3×Q8—24no—
188C2×C3×Q8—24no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—42no—
199C2×C3×Q8—48no—
200C2×C3×Q8—48no—
201C2×C3×Q8—44no—
202C2×C3×Q8—44no—
203C2×C3×Q8C2×D412yes—
204C2×C3×Q8C2×D412yes—
205C2×C3×Q8C2×D412yes—
206C2×C3×Q8C2×D412yes—
207C2×C3×D4—22no—
208C2×C3×D4—24no—
209C2×C3×D4—24no—
210C2×C3×D4—22no—
211C2×C3×D4—22no—
212C2×C3×D4—21no—
213C2×C3×D4—22no—
214C2×C3×D4—42no—
215C2×C3×D4—42no—
216C2×C3×D4C2×D412yes—
217C2×C3×D4C2×D412yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—22no—
221C22×C3×C4—24no—
222C22×C3×C4—24no—
223C22×C3×C4—24no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—24no—
228C22×C3×C4—22no—
229C22×C3×C4—42no—
230C22×C3×C4—42no—
231C22×C3×C4—42no—
232C22×C3×C4C2×D412yes—
233C22×C3×C4C2×D412yes—
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D414yes—
236C22×C3×C4C2411yescharacteristic
237C3×Q16—44no—
238C3×SD16—44no—
239C3×SD16—44no—
240C3×D8—44no—
241C3×M16—22no—
242C3×M16—22no—
243C3×M16—22no—
244C3×M16—22no—
245C2×C3×C8—22no—
246C2×C3×C8—22no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—28no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—28no—
251C3×C4⋊C4—28no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4C2×D414yes—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—28no—
256C3×C22⋊C4—28no—
257C3×C22⋊C4—24no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—28no—
260C3×C22⋊C4—24no—
261C3×C42—24no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—22no—
265C3×C42C2×D412yes—
266C3×C42C2×D412yes—
2672−1+4—22no—
2682−1+4—22no—
2692−1+4—42no—
270C2×#41—22no—
271C2×#41—22no—
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—24no—
275C2×#41—42no—
276C2×#41C23×C312yes—
277C22×Q8—22no—
278C22×Q8—22no—
279C22×Q8—22no—
280C22×Q8—24no—
281C22×Q8—42no—
282C22×Q8C3×D412yes—
283C22×Q8C23×C311yescharacteristic
284C22×Q8C23×C311yescharacteristic
285C22×Q8C23×C311yescharacteristic
286C22×D4—21no—
287C22×D4C23×C311yescharacteristic
288C23×C4—21no—
289C23×C4C3×D412yes—
290C23×C4C23×C311yescharacteristic
291#137—24no—
292#137—42no—
293AGL(1,ℤ/8)—24no—
294AGL(1,ℤ/8)—42no—
295C2×Q16—22no—
296C2×SD16—22no—
297C2×SD16—22no—
298C2×D8—22no—
299C2×M16C23×C312yes—
300C2×M16C23×C312yes—
301C4⋊Q8—24no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8C3×D414yescomplemented
307#124—24no—
308#124—22no—
309#124—22no—
310#124—22no—
311#124—22no—
312#124C3×D414yescomplemented
313#123—216no—
314#123—24no—
315#123—22no—
316C22⋊Q8—216no—
317C22⋊Q8—24no—
318C22⋊Q8—28no—
319C22⋊Q8—24no—
320C22⋊Q8C3×D414yes—
321#121—24no—
322#121C3×D414yes—
323#120—22no—
324C4×Q8—24no—
325C4×D4—28no—
326C4×D4—24no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117C23×C312yes—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C42C23×C311yescharacteristic
338C2×C42C23×C311yescharacteristic
339#104—24no—
340#104—24no—
341#103—24no—
342#102—24no—
343#101—22no—
344#101—22no—
345#101—42no—
346C22⋊C8—24no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—42no—
352C23×C3—44no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3#12012yes—
356C23×C3C22×D411yescharacteristic
357C3×Q8—24no—
358C3×Q8—24no—
359C3×Q8—22no—
360C3×Q8—24no—
361C3×Q8—22no—
362C3×Q8—48no—
363C3×Q8—44no—
364C3×Q8—42no—
365C3×Q8—42no—
366C3×Q8—42no—
367C3×D4—22no—
368C3×D4—24no—
369C3×D4—22no—
370C3×D4—44no—
371C3×D4—48no—
372C3×D4—48no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—42no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—24no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—48no—
391C2×C3×C4—48no—
392C2×C3×C4—48no—
393C2×C3×C4—42no—
394C2×C3×C4—42no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—42no—
398C2×C3×C4—42no—
399C2×C3×C4#12014yes—
400C2×C3×C4#12012yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D412yes—
404C2×C3×C4H(5,𝔽2)12yes—
405C3×C8—42no—
406C3×C8—42no—
407C24—21no—
408C24C2×C3×D411yescharacteristic
409#41—22no—
410#41—22no—
411#41—22no—
412#41—22no—
413#41—48no—
414#41—48no—
415#41—44no—
416#41—44no—
417#41—44no—
418#41—44no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—42no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8C2×C3×D412yes—
437C2×Q8C2×C3×D412yes—
438C2×Q8C2×C3×D412yes—
439C2×Q8C2×C3×D412yes—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4C2×C3×D412yes—
450C2×D4C2×C3×D412yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4C2×C3×D412yes—
466C22×C4C2×C3×D412yes—
467C22×C4C2×C3×D411yescharacteristic
468C22×C4C2×C3×D414yes—
469C22×C4C24×C311yescharacteristic
470Q16—44no—
471SD16—44no—
472SD16—44no—
473D8—44no—
474M16—22no—
475M16—22no—
476M16—22no—
477M16—22no—
478C2×C8—22no—
479C2×C8—22no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4C2×C3×D414yes—
487C22⋊C4—24no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—22no—
492C22⋊C4—28no—
493C22⋊C4—24no—
494C42—24no—
495C42—21no—
496C42—21no—
497C42—22no—
498C42C2×C3×D412yes—
499C42C2×C3×D412yes—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—44no—
510C22×C3—82no—
511C22×C3—82no—
512C22×C3C2×#12011yescharacteristic
513C22×C3C2×#12011yescharacteristic
514C22×C3#53411yescharacteristic
515C3×C4—22no—
516C3×C4—24no—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—44no—
523C3×C4D4212yes—
524C3×C4D4212yes—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—42no—
530C23—44no—
531C23—41no—
532C23—41no—
533C23C3×#12012yes—
534C23C22×C3×D411yescharacteristic
535Q8—24no—
536Q8—24no—
537Q8—22no—
538Q8—24no—
539Q8—22no—
540Q8—48no—
541Q8—44no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545D4—22no—
546D4—24no—
547D4—22no—
548D4—44no—
549D4—48no—
550D4—48no—
551D4—42no—
552D4—42no—
553D4—42no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4C3×#12014yes—
578C2×C4C3×#12012yes—
579C2×C4C22×C3×D411yescharacteristic
580C2×C4C22×C3×D411yescharacteristic
581C2×C4C22×C3×D412yes—
582C2×C4C3×H(5,𝔽2)12yes—
583C8—42no—
584C8—42no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—42no—
588C2×C3—82no—
589C2×C3—82no—
590C2×C3#240311yescharacteristic
591C2×C3#302012yes—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—44no—
602C22—82no—
603C22—82no—
604C22C2×C3×#12011yescharacteristic
605C22C2×C3×#12011yescharacteristic
606C22C3×#53411yescharacteristic
607C4—22no—
608C4—24no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4C3×D4212yes—
616C4C3×D4212yes—
617C3#2391711yescharacteristic, direct product factor, Sylow
618C2—21no—
619C2—21no—
620C2—42no—
621C2—82no—
622C2—82no—
623C2C3×#240311yescharacteristic
624C2C3×#302012yes—
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×#3486C2semidirect
2. C2×C3×#536C2semidirect
3. C2×C3×#456C2semidirect
4. C2×C3×#454C2semidirect
5. C3×#3005C2semidirect
6. C3×#2005C2semidirect
7. C3×#2003C2semidirect
8. C3×#1897C2semidirect
9. C3×#1888C2semidirect
10. C3×#1886C2semidirect
11. C3×#1784C2semidirect
12. #23917C3direct (N × H)
48. #3486C2×C3semidirect
49. C2×#536C2×C3semidirect
50. C2×#456C2×C3semidirect
51. C2×#454C2×C3semidirect
52. #3005C2×C3semidirect
53. #2005C2×C3semidirect
54. #2003C2×C3semidirect
55. #1897C2×C3semidirect
56. #1888C2×C3semidirect
57. #1886C2×C3semidirect
58. #1784C2×C3semidirect
98. C3×C4⋊Q8D4semidirect
104. C3×#124D4semidirect
306. C4⋊Q8C3×D4semidirect
312. #124C3×D4semidirect
617. C3#23917direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34862
C2×C3×#5362
C2×C3×#4562
C2×C3×#4542
C3×#30052
C3×#20052
C3×#20032
C3×#18972
C3×#18882
C3×#18862
C3×#17842
#239173