Finite Group Databasebeta

← C3×#24372 #10642979

Subgroups — C3×#24372 #10642979

Order 768.

1698 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C2×C3×#459C212yesmaximal, complemented
3C2×C3×#449C211yesmaximal, characteristic, complemented
4C3×#2433C211yesmaximal, characteristic, complemented
5C3×#2339C211yesmaximal, characteristic, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2014C212yesmaximal, complemented
8C3×#1895C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1851C211yesmaximal, characteristic, complemented
11C3×#1786C211yesmaximal, characteristic, complemented
12#24372C311yesmaximal, 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×SD16C2212yes—
18C22×C3×SD16C2211yescharacteristic, complemented
19C22×C3×D8C2212yes—
20C22×C3×D8C2211yescharacteristic, complemented
21C22×C3×M16C2211yescharacteristic
22C2×C3×#127C2212yes—
23C2×C3×#121C2211yescharacteristic
24C2×C3×#120—22no—
25C2×C3×C4×D4C2212yescomplemented
26C3×#459—28no—
27C3×#449—24no—
28C2×C3×C4⋊C8C2212yescomplemented
29C2×C3×#103C2211yescharacteristic, complemented
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C2×C3×#102C2212yescomplemented
33C2×C3×#102C2211yescharacteristic, complemented
34C2×C3×C22⋊C8C2211yescharacteristic
35C3×#392—21no—
36C3×#392—21no—
37C3×#391C2212yescomplemented
38C3×#390C2212yes—
39C3×#386—22no—
40C3×#381—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C22⋊C4C2211yescharacteristic
43C2×C3×#95—21no—
44C3×#340C2212yescomplemented
45C3×#336C2212yescomplemented
46C22×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
47C2×#459C2×C312yescomplemented
48C2×#449C2×C311yescharacteristic, complemented
49#2433C2×C311yescharacteristic, complemented
50#2339C2×C311yescharacteristic, complemented
51#2055C2×C312yescomplemented
52#2014C2×C312yescomplemented
53#1895C2×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×#41D414yescomplemented
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4—24no—
64C22×C3×D4—21no—
65C22×C3×D4—21no—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4—22no—
71C22×C3×D4—42no—
72C22×C3×D4C2311yescharacteristic
73C22×C3×D4C2312yes—
74C22×C3×D4C2311yescharacteristic
75C23×C3×C4—21no—
76C23×C3×C4—22no—
77C23×C3×C4C2311yescharacteristic
78C3×AGL(1,ℤ/8)—232no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—22no—
82C2×C3×SD16—24no—
83C2×C3×SD16—42no—
84C2×C3×D8—24no—
85C2×C3×D8—24no—
86C2×C3×D8—22no—
87C2×C3×D8—24no—
88C2×C3×D8D418yescomplemented
89C2×C3×M16—22no—
90C2×C3×M16—24no—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2312yes—
94C3×#127—24no—
95C3×#127—42no—
96C3×#121—22no—
97C3×#121—22no—
98C3×#120—48no—
99C3×C4×D4—24no—
100C3×C4×D4—42no—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4D412yes—
109C2×C3×C42C2312yes—
110C3×C4⋊C8—24no—
111C3×#103—22no—
112C3×#102—24no—
113C3×#102—22no—
114C3×#102—24no—
115C3×#102—42no—
116C3×C22⋊C8—22no—
117C3×#95—21no—
118C3×#95—22no—
119C3×#95—22no—
120C3×#95D411yescharacteristic
121C3×#95D411yescharacteristic
122C22×#41C22×C311yescharacteristic
123C23×D4C22×C311yescharacteristic
124C2×AGL(1,ℤ/8)—24no—
125C2×AGL(1,ℤ/8)—28no—
126C22×SD16C22×C312yes—
127C22×SD16C22×C311yescharacteristic, complemented
128C22×D8C22×C312yes—
129C22×D8C22×C311yescharacteristic, complemented
130C22×M16C22×C311yescharacteristic
131C2×#127C22×C312yes—
132C2×#121C22×C311yescharacteristic
133C2×#120—22no—
134C2×C4×D4C22×C312yescomplemented
135#459—28no—
136#449—24no—
137C2×C4⋊C8C22×C312yescomplemented
138C2×#103C22×C311yescharacteristic, complemented
139C2×#102C22×C312yes—
140C2×#102C22×C312yes—
141C2×#102C22×C311yescharacteristic, complemented
142C2×#102C22×C312yescomplemented
143C2×C22⋊C8C22×C311yescharacteristic
144#392—21no—
145#392—21no—
146#391C22×C312yescomplemented
147#390C22×C312yes—
148#386—22no—
149#381—21no—
150#380C22×C311yescharacteristic
151C4×C22⋊C4C22×C311yescharacteristic
152C2×#95—21no—
153#340C22×C312yescomplemented
154#336C22×C312yescomplemented
155C24×C3—21no—
156C24×C3—22no—
157C24×C3—21no—
158C24×C3—44no—
159C24×C3—44no—
160C24×C3—42no—
161C24×C3—81no—
162C24×C3—81no—
163C24×C3C2×D411yescharacteristic
164C3×#41—28no—
165C3×#41—44no—
166C3×#41—48no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8C2×D412yes—
171C2×C3×D4—28no—
172C2×C3×D4—24no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
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—48no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—44no—
192C2×C3×D4—42no—
193C2×C3×D4—42no—
194C2×C3×D4—44no—
195C2×C3×D4—82no—
196C2×C3×D4—82no—
197C2×C3×D4C2×D414yes—
198C2×C3×D4C2×D412yes—
199C2×C3×D4C2×D414yes—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4—21no—
203C22×C3×C4—24no—
204C22×C3×C4—22no—
205C22×C3×C4—21no—
206C22×C3×C4—44no—
207C22×C3×C4—44no—
208C22×C3×C4—44no—
209C22×C3×C4—82no—
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D412yes—
214C22×C3×C4#4112yes—
215C22×C3×C4C2411yescharacteristic
216C3×SD16—216no—
217C3×SD16—44no—
218C3×D8—216no—
219C3×D8—44no—
220C3×M16—28no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8C2×D414yes—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—22no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—44no—
234C3×C22⋊C4—44no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—44no—
237C3×C22⋊C4—81no—
238C3×C42—22no—
239C3×C42—22no—
240C25—21no—
241C2×#41—22no—
242C2×#41—24no—
243C2×#41—42no—
244C2×#41C3×D414yescomplemented
245C22×Q8C23×C311yescharacteristic
246C22×D4—24no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—42no—
255C22×D4C23×C311yescharacteristic
256C22×D4C23×C312yes—
257C22×D4C23×C311yescharacteristic
258C23×C4—21no—
259C23×C4—22no—
260C23×C4C23×C311yescharacteristic
261AGL(1,ℤ/8)—232no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—22no—
265C2×SD16—24no—
266C2×SD16—42no—
267C2×D8—24no—
268C2×D8—24no—
269C2×D8—22no—
270C2×D8—24no—
271C2×D8C3×D418yescomplemented
272C2×M16—22no—
273C2×M16—24no—
274C22×C8C23×C311yescharacteristic
275C22×C8C23×C311yescharacteristic
276C22×C8C23×C312yes—
277#127—24no—
278#127—42no—
279#121—22no—
280#121—22no—
281#120—48no—
282C4×D4—24no—
283C4×D4—42no—
284C2×C4⋊C4C23×C312yes—
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4C3×D412yes—
292C2×C42C23×C312yes—
293C4⋊C8—24no—
294#103—22no—
295#102—24no—
296#102—22no—
297#102—24no—
298#102—42no—
299C22⋊C8—22no—
300#95—21no—
301#95—22no—
302#95—22no—
303#95C3×D411yescharacteristic
304#95C3×D411yescharacteristic
305C23×C3—22no—
306C23×C3—24no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—24no—
312C23×C3—48no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—84no—
328C23×C3—84no—
329C23×C3—88no—
330C23×C3—88no—
331C23×C3#12112yes—
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—42no—
336C3×D4—24no—
337C3×D4—22no—
338C3×D4—24no—
339C3×D4—24no—
340C3×D4—48no—
341C3×D4—48no—
342C3×D4—41no—
343C3×D4—42no—
344C3×D4—41no—
345C3×D4—42no—
346C3×D4—42no—
347C3×D4—88no—
348C3×D4—88no—
349C3×D4—84no—
350C3×D4#12014yescomplemented
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—48no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—88no—
369C2×C3×C4#12112yes—
370C2×C3×C4AGL(1,ℤ/8)14yescomplemented
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D412yes—
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)12yes—
375C3×C8—24no—
376C3×C8—41no—
377C3×C8—41no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—44no—
382C24—44no—
383C24—42no—
384C24—81no—
385C24—81no—
386C24C2×C3×D411yescharacteristic
387#41—28no—
388#41—44no—
389#41—48no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8C2×C3×D412yes—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—48no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4C2×C3×D414yes—
421C2×D4C2×C3×D412yes—
422C2×D4C2×C3×D414yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—82no—
433C22×C4C2×C3×D412yes—
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D412yes—
436C22×C4C2×C3×D412yes—
437C22×C4C3×#4112yes—
438C22×C4C24×C311yescharacteristic
439SD16—216no—
440SD16—44no—
441D8—216no—
442D8—44no—
443M16—28no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8C2×C3×D414yes—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—81no—
461C42—22no—
462C42—22no—
463C22×C3—22no—
464C22×C3—24no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—44no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—44no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—816no—
477C22×C3—82no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—82no—
481C22×C3—81no—
482C22×C3—82no—
483C22×C3—82no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3C2×#12111yescharacteristic
487C22×C3D4212yes—
488C22×C3C2×AGL(1,ℤ/8)12yes—
489C22×C3#57611yescharacteristic
490C22×C3#57711yescharacteristic
491C3×C4—22no—
492C3×C4—24no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—82no—
496C3×C4—81no—
497C3×C4C2×#12011yescharacteristic
498C3×C4#53411yescharacteristic
499C3×C4#54612yes—
500C23—22no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—48no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—84no—
523C23—84no—
524C23—88no—
525C23—88no—
526C23C3×#12112yes—
527C23C22×C3×D411yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—42no—
531D4—24no—
532D4—22no—
533D4—24no—
534D4—24no—
535D4—48no—
536D4—48no—
537D4—41no—
538D4—42no—
539D4—41no—
540D4—42no—
541D4—42no—
542D4—88no—
543D4—88no—
544D4—84no—
545D4C3×#12014yescomplemented
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—88no—
564C2×C4C3×#12112yes—
565C2×C4C3×AGL(1,ℤ/8)14yescomplemented
566C2×C4C22×C3×D411yescharacteristic
567C2×C4C22×C3×D412yes—
568C2×C4C2×C3×#4111yescharacteristic
569C2×C4C3×H(5,𝔽2)12yes—
570C8—24no—
571C8—41no—
572C8—41no—
573C2×C3—42no—
574C2×C3—44no—
575C2×C3—84no—
576C2×C3—161no—
577C2×C3#241011yescharacteristic
578C2×C3#303811yescharacteristic
579C2×C3#304511yescharacteristic
580C2×C3#326412yes—
581C2×C3#327212yes—
582C22—22no—
583C22—24no—
584C22—22no—
585C22—22no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—816no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—81no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22C2×C3×#12111yescharacteristic
606C22C3×D4212yes—
607C22C2×C3×AGL(1,ℤ/8)12yes—
608C22C3×#57611yescharacteristic
609C22C3×#57711yescharacteristic
610C4—22no—
611C4—24no—
612C4—42no—
613C4—42no—
614C4—82no—
615C4—81no—
616C4C2×C3×#12011yescharacteristic
617C4C3×#53411yescharacteristic
618C4C3×#54612yes—
619C3#2437211yescharacteristic, direct product factor, Sylow
620C2—42no—
621C2—44no—
622C2—84no—
623C2—161no—
624C2C3×#241011yescharacteristic
625C2C3×#303811yescharacteristic
626C2C3×#304511yescharacteristic
627C2C3×#326412yes—
628C2C3×#327212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C22×C3×AGL(1,ℤ/8)C2semidirect
2. C2×C3×#459C2semidirect
3. C2×C3×#449C2semidirect
4. C3×#2433C2semidirect
5. C3×#2339C2semidirect
6. C3×#2055C2semidirect
7. C3×#2014C2semidirect
8. C3×#1895C2semidirect
9. C3×#1879C2semidirect
10. C3×#1851C2semidirect
11. C3×#1786C2semidirect
12. #24372C3direct (N × H)
18. C22×C3×SD16C22semidirect
20. C22×C3×D8C22semidirect
25. C2×C3×C4×D4C22semidirect
28. C2×C3×C4⋊C8C22semidirect
29. C2×C3×#103C22semidirect
32. C2×C3×#102C22semidirect
33. C2×C3×#102C22semidirect
37. C3×#391C22semidirect
44. C3×#340C22semidirect
45. C3×#336C22semidirect
46. C22×AGL(1,ℤ/8)C2×C3semidirect
47. C2×#459C2×C3semidirect
48. C2×#449C2×C3semidirect
49. #2433C2×C3semidirect
50. #2339C2×C3semidirect
51. #2055C2×C3semidirect
52. #2014C2×C3semidirect
53. #1895C2×C3semidirect
54. #1879C2×C3semidirect
55. #1851C2×C3semidirect
56. #1786C2×C3semidirect
61. C2×C3×#41D4semidirect
88. C2×C3×D8D4semidirect
127. C22×SD16C22×C3semidirect
129. C22×D8C22×C3semidirect
134. C2×C4×D4C22×C3semidirect
137. C2×C4⋊C8C22×C3semidirect
138. C2×#103C22×C3semidirect
141. C2×#102C22×C3semidirect
142. C2×#102C22×C3semidirect
146. #391C22×C3semidirect
153. #340C22×C3semidirect
154. #336C22×C3semidirect
244. C2×#41C3×D4semidirect
271. C2×D8C3×D4semidirect
350. C3×D4#120semidirect
370. C2×C3×C4AGL(1,ℤ/8)semidirect
545. D4C3×#120semidirect
565. C2×C4C3×AGL(1,ℤ/8)semidirect
619. C3#24372direct (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×#4592
C2×C3×#4492
C3×#24332
C3×#23392
C3×#20552
C3×#20142
C3×#18952
C3×#18792
C3×#18512
C3×#17862
#243723