Finite Group Databasebeta

← C3×#24373 #10642980

Subgroups — C3×#24373 #10642980

Order 768.

1568 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×C3×#463C212yesmaximal, complemented
3C2×C3×#452C211yesmaximal, characteristic, complemented
4C3×#2435C211yesmaximal, characteristic, complemented
5C3×#2339C211yesmaximal, characteristic, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2015C212yesmaximal, complemented
8C3×#1894C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1851C211yesmaximal, characteristic, complemented
11C3×#1786C211yesmaximal, characteristic, complemented
12#24373C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C2×C3×AGL(1,ℤ/8)—24no—
16C2×C3×AGL(1,ℤ/8)—28no—
17C22×C3×Q16C2211yescharacteristic, complemented
18C22×C3×SD16C2212yes—
19C22×C3×SD16C2211yescharacteristic, complemented
20C22×C3×D8C2212yes—
21C22×C3×M16C2211yescharacteristic
22C2×C3×#124C2212yes—
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×C4×D4C2212yescomplemented
25C3×#463—28no—
26C3×#452—24no—
27C2×C3×C4⋊C8C2212yescomplemented
28C2×C3×#103C2212yescomplemented
29C2×C3×#103C2211yescharacteristic, complemented
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2211yescharacteristic, complemented
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#394—21no—
35C3×#394—21no—
36C3×#391C2212yescomplemented
37C3×#390C2212yes—
38C3×#386—22no—
39C3×#381—21no—
40C3×#380C2211yescharacteristic
41C3×#379—22no—
42C3×C4×C22⋊C4C2211yescharacteristic
43C2×C3×#95—21no—
44C3×#340C2212yescomplemented
45C3×#336C2212yescomplemented
46C22×AGL(1,ℤ/8)C2×C311yescharacteristic
47C2×#463C2×C312yescomplemented
48C2×#452C2×C311yescharacteristic, complemented
49#2435C2×C311yescharacteristic, complemented
50#2339C2×C311yescharacteristic, complemented
51#2055C2×C312yescomplemented
52#2015C2×C312yescomplemented
53#1894C2×C311yescharacteristic, complemented
54#1879C2×C312yescomplemented
55#1851C2×C311yescharacteristic, complemented
56#1786C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C2×C3×#41—22no—
59C2×C3×#41—24no—
60C2×C3×#41—42no—
61C2×C3×#41D414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4—24no—
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—22no—
69C22×C3×D4—42no—
70C22×C3×D4C2312yes—
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—21no—
73C23×C3×C4—22no—
74C23×C3×C4C2311yescharacteristic
75C3×AGL(1,ℤ/8)—232no—
76C2×C3×Q16—22no—
77C2×C3×Q16—42no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—22no—
81C2×C3×SD16—24no—
82C2×C3×SD16—24no—
83C2×C3×D8—24no—
84C2×C3×D8—24no—
85C2×C3×D8D418yes—
86C2×C3×M16—22no—
87C2×C3×M16—24no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2312yes—
91C3×#124—24no—
92C3×#124—42no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—22no—
95C3×C4×D4—24no—
96C3×C4×D4—42no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4D412yes—
108C2×C3×C42C2312yes—
109C3×C4⋊C8—24no—
110C3×#103—22no—
111C3×#103—24no—
112C3×#102—24no—
113C3×#102—22no—
114C3×#102—42no—
115C3×C22⋊C8—22no—
116C3×#95—21no—
117C3×#95—22no—
118C3×#95—22no—
119C3×#95D411yescharacteristic
120C3×#95D411yescharacteristic
121C22×#41C22×C311yescharacteristic
122C23×D4C22×C311yescharacteristic
123C2×AGL(1,ℤ/8)—24no—
124C2×AGL(1,ℤ/8)—28no—
125C22×Q16C22×C311yescharacteristic, complemented
126C22×SD16C22×C312yes—
127C22×SD16C22×C311yescharacteristic, complemented
128C22×D8C22×C312yes—
129C22×M16C22×C311yescharacteristic
130C2×#124C22×C312yes—
131C2×C22⋊Q8C22×C311yescharacteristic
132C2×C4×D4C22×C312yescomplemented
133#463—28no—
134#452—24no—
135C2×C4⋊C8C22×C312yescomplemented
136C2×#103C22×C311yescharacteristic, complemented
137C2×#103C22×C312yescomplemented
138C2×#102C22×C312yes—
139C2×#102C22×C312yes—
140C2×#102C22×C311yescharacteristic, complemented
141C2×C22⋊C8C22×C311yescharacteristic
142#394—21no—
143#394—21no—
144#391C22×C312yescomplemented
145#390C22×C312yes—
146#386—22no—
147#381—21no—
148#380C22×C311yescharacteristic
149#379—22no—
150C4×C22⋊C4C22×C311yescharacteristic
151C2×#95—21no—
152#340C22×C312yescomplemented
153#336C22×C312yescomplemented
154C24×C3—21no—
155C24×C3—22no—
156C24×C3—44no—
157C24×C3—44no—
158C24×C3—42no—
159C24×C3—81no—
160C24×C3—81no—
161C24×C3C2×D411yescharacteristic
162C3×#41—28no—
163C3×#41—44no—
164C3×#41—48no—
165C2×C3×Q8—21no—
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×Q8C2×D412yes—
173C2×C3×D4—28no—
174C2×C3×D4—24no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—48no—
183C2×C3×D4—44no—
184C2×C3×D4—44no—
185C2×C3×D4—44no—
186C2×C3×D4—42no—
187C2×C3×D4C2×D414yes—
188C2×C3×D4C2×D412yes—
189C2×C3×D4C2×D414yes—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—24no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—44no—
198C22×C3×C4—44no—
199C22×C3×C4—44no—
200C22×C3×C4—82no—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D412yes—
205C22×C3×C4#4112yes—
206C22×C3×C4C2411yescharacteristic
207C3×Q16—44no—
208C3×SD16—216no—
209C3×SD16—44no—
210C3×D8—216no—
211C3×M16—28no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8C2×D414yes—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—42no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—44no—
226C3×C22⋊C4—44no—
227C3×C22⋊C4—44no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—82no—
230C3×C22⋊C4—82no—
231C3×C22⋊C4—81no—
232C3×C42—22no—
233C3×C42—22no—
234C25—21no—
235C2×#41—22no—
236C2×#41—24no—
237C2×#41—42no—
238C2×#41C3×D414yes—
239C22×Q8C23×C311yescharacteristic
240C22×Q8C23×C311yescharacteristic
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—42no—
247C22×D4C23×C312yes—
248C22×D4C23×C311yescharacteristic
249C23×C4—21no—
250C23×C4—22no—
251C23×C4C23×C311yescharacteristic
252AGL(1,ℤ/8)—232no—
253C2×Q16—22no—
254C2×Q16—42no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—22no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×D8—24no—
261C2×D8—24no—
262C2×D8C3×D418yes—
263C2×M16—22no—
264C2×M16—24no—
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C312yes—
268#124—24no—
269#124—42no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C4×D4—24no—
273C4×D4—42no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4C23×C312yes—
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4C3×D412yes—
285C2×C42C23×C312yes—
286C4⋊C8—24no—
287#103—22no—
288#103—24no—
289#102—24no—
290#102—22no—
291#102—42no—
292C22⋊C8—22no—
293#95—21no—
294#95—22no—
295#95—22no—
296#95C3×D411yescharacteristic
297#95C3×D411yescharacteristic
298C23×C3—22no—
299C23×C3—24no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—24no—
305C23×C3—48no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—84no—
316C23×C3—84no—
317C23×C3—88no—
318C23×C3—88no—
319C23×C3#12112yes—
320C23×C3C22×D411yescharacteristic
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—42no—
324C3×Q8—41no—
325C3×Q8—41no—
326C3×Q8—42no—
327C3×D4—24no—
328C3×D4—22no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—48no—
332C3×D4—48no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4#12014yes—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—48no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—41no—
357C2×C3×C4—88no—
358C2×C3×C4#12112yes—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D412yes—
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4H(5,𝔽2)12yes—
363C3×C8—24no—
364C3×C8—41no—
365C3×C8—41no—
366C24—21no—
367C24—22no—
368C24—44no—
369C24—44no—
370C24—42no—
371C24—81no—
372C24—81no—
373C24C2×C3×D411yescharacteristic
374#41—28no—
375#41—44no—
376#41—48no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8C2×C3×D412yes—
385C2×D4—28no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4C2×C3×D414yes—
400C2×D4C2×C3×D412yes—
401C2×D4C2×C3×D414yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—82no—
413C22×C4C2×C3×D412yes—
414C22×C4C2×C3×D412yes—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D412yes—
417C22×C4C3×#4112yes—
418C22×C4C24×C311yescharacteristic
419Q16—44no—
420SD16—216no—
421SD16—44no—
422D8—216no—
423M16—28no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8C2×C3×D414yes—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—42no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4—81no—
444C42—22no—
445C42—22no—
446C22×C3—22no—
447C22×C3—24no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—42no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—816no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3C2×#12111yescharacteristic
465C22×C3D4212yes—
466C22×C3#57512yes—
467C22×C3#57611yescharacteristic
468C22×C3#57711yescharacteristic
469C3×C4—22no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—81no—
474C3×C4—82no—
475C3×C4—81no—
476C3×C4C2×#12011yescharacteristic
477C3×C4#53411yescharacteristic
478C3×C4#54612yes—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—48no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—84no—
497C23—84no—
498C23—88no—
499C23—88no—
500C23C3×#12112yes—
501C23C22×C3×D411yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—42no—
505Q8—41no—
506Q8—41no—
507Q8—42no—
508D4—24no—
509D4—22no—
510D4—24no—
511D4—24no—
512D4—48no—
513D4—48no—
514D4—42no—
515D4—42no—
516D4C3×#12014yes—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—88no—
539C2×C4C3×#12112yes—
540C2×C4C22×C3×D411yescharacteristic
541C2×C4C22×C3×D412yes—
542C2×C4C2×C3×#4111yescharacteristic
543C2×C4C3×H(5,𝔽2)12yes—
544C8—24no—
545C8—41no—
546C8—41no—
547C2×C3—42no—
548C2×C3—44no—
549C2×C3—84no—
550C2×C3#241011yescharacteristic
551C2×C3#304011yescharacteristic
552C2×C3#304511yescharacteristic
553C2×C3#326712yes—
554C2×C3#327312yes—
555C22—22no—
556C22—24no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—816no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22C2×C3×#12111yescharacteristic
574C22C3×D4212yes—
575C22C3×#57512yes—
576C22C3×#57611yescharacteristic
577C22C3×#57711yescharacteristic
578C4—22no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—82no—
584C4—81no—
585C4C2×C3×#12011yescharacteristic
586C4C3×#53411yescharacteristic
587C4C3×#54612yes—
588C3#2437311yescharacteristic, direct product factor, Sylow
589C2—42no—
590C2—44no—
591C2—84no—
592C2C3×#241011yescharacteristic
593C2C3×#304011yescharacteristic
594C2C3×#304511yescharacteristic
595C2C3×#326712yes—
596C2C3×#327312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#463C2semidirect
3. C2×C3×#452C2semidirect
4. C3×#2435C2semidirect
5. C3×#2339C2semidirect
6. C3×#2055C2semidirect
7. C3×#2015C2semidirect
8. C3×#1894C2semidirect
9. C3×#1879C2semidirect
10. C3×#1851C2semidirect
11. C3×#1786C2semidirect
12. #24373C3direct (N × H)
17. C22×C3×Q16C22semidirect
19. C22×C3×SD16C22semidirect
24. C2×C3×C4×D4C22semidirect
27. C2×C3×C4⋊C8C22semidirect
28. C2×C3×#103C22semidirect
29. C2×C3×#103C22semidirect
32. C2×C3×#102C22semidirect
36. C3×#391C22semidirect
44. C3×#340C22semidirect
45. C3×#336C22semidirect
47. C2×#463C2×C3semidirect
48. C2×#452C2×C3semidirect
49. #2435C2×C3semidirect
50. #2339C2×C3semidirect
51. #2055C2×C3semidirect
52. #2015C2×C3semidirect
53. #1894C2×C3semidirect
54. #1879C2×C3semidirect
55. #1851C2×C3semidirect
56. #1786C2×C3semidirect
125. C22×Q16C22×C3semidirect
127. C22×SD16C22×C3semidirect
132. C2×C4×D4C22×C3semidirect
135. C2×C4⋊C8C22×C3semidirect
136. C2×#103C22×C3semidirect
137. C2×#103C22×C3semidirect
140. C2×#102C22×C3semidirect
144. #391C22×C3semidirect
152. #340C22×C3semidirect
153. #336C22×C3semidirect
588. C3#24373direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,ℤ/8)2
C2×C3×#4632
C2×C3×#4522
C3×#24352
C3×#23392
C3×#20552
C3×#20152
C3×#18942
C3×#18792
C3×#18512
C3×#17862
#243733