Finite Group Databasebeta

← C3×#24356 #10642963

Subgroups — C3×#24356 #10642963

Order 768.

1656 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#460C212yesmaximal, complemented
3C2×C3×#448C211yesmaximal, characteristic, complemented
4C3×#2433C211yesmaximal, characteristic
5C3×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2014C212yesmaximal, complemented
8C3×#1893C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1850C211yesmaximal, characteristic, complemented
11C3×#1786C211yesmaximal, characteristic, complemented
12#24356C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C22×C3×SD16—24no—
16C22×C3×SD16—28no—
17C22×C3×SD16C2212yes—
18C22×C3×SD16C2212yes—
19C22×C3×SD16C2211yescharacteristic, complemented
20C22×C3×SD16C2211yescharacteristic, complemented
21C23×C3×C8C2211yescharacteristic
22C2×C3×#127C2212yes—
23C2×C3×#121C2211yescharacteristic
24C2×C3×#120—21no—
25C2×C3×#120—21no—
26C2×C3×C4×Q8C2212yescomplemented
27C3×#460—28no—
28C3×#448—24no—
29C2×C3×C4⋊C8C2212yescomplemented
30C2×C3×#103C2211yescharacteristic, complemented
31C2×C3×#103C2211yescharacteristic, complemented
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yescomplemented
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#392—22no—
37C3×#391—22no—
38C3×#391C2212yescomplemented
39C3×#390C2212yes—
40C3×#380—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C22⋊C4—21no—
43C3×C4×C22⋊C4C2211yescharacteristic
44C3×#340C2212yescomplemented
45C3×#336C2212yescomplemented
46C23×SD16C2×C311yescharacteristic, complemented
47C2×#460C2×C312yescomplemented
48C2×#448C2×C311yescharacteristic, complemented
49#2433C2×C311yescharacteristic
50Q8×C22⋊C4C2×C311yescharacteristic, complemented
51#2055C2×C312yescomplemented
52#2014C2×C312yescomplemented
53#1893C2×C311yescharacteristic, complemented
54#1880C2×C312yescomplemented
55#1850C2×C311yescharacteristic, complemented
56#1786C2×C311yescharacteristic, complemented
57C25×C3—21no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—24no—
61C22×C3×Q8—42no—
62C22×C3×Q8D414yescomplemented
63C22×C3×Q8C2311yescharacteristic
64C22×C3×Q8C2311yescharacteristic
65C22×C3×D4—24no—
66C22×C3×D4—21no—
67C22×C3×D4—22no—
68C22×C3×D4—21no—
69C22×C3×D4—22no—
70C22×C3×D4—22no—
71C22×C3×D4—22no—
72C22×C3×D4—42no—
73C22×C3×D4C2311yescharacteristic
74C22×C3×D4C2312yes—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C2×C3×SD16—232no—
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C2×C3×SD16—24no—
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—24no—
85C2×C3×SD16—24no—
86C2×C3×SD16—42no—
87C2×C3×SD16D418yescomplemented
88C22×C3×C8—22no—
89C22×C3×C8—24no—
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2312yes—
93C3×#127—24no—
94C3×#127—42no—
95C3×#121—22no—
96C3×#121—22no—
97C3×#120—44no—
98C3×#120—44no—
99C3×C4×Q8—24no—
100C3×C4×Q8—42no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4D411yescharacteristic
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C42—22no—
111C2×C3×C42C2312yes—
112C3×C4⋊C8—24no—
113C3×#103—22no—
114C3×#103—22no—
115C3×#102—24no—
116C3×#102—24no—
117C3×#102—42no—
118C3×C22⋊C8—22no—
119C3×#95—22no—
120C3×#95—22no—
121C3×#95—21no—
122C3×#95D412yes—
123C23×Q8C22×C311yescharacteristic
124C23×D4C22×C311yescharacteristic
125C22×SD16—24no—
126C22×SD16—28no—
127C22×SD16C22×C312yes—
128C22×SD16C22×C312yes—
129C22×SD16C22×C311yescharacteristic, complemented
130C22×SD16C22×C311yescharacteristic, complemented
131C23×C8C22×C311yescharacteristic
132C2×#127C22×C312yes—
133C2×#121C22×C311yescharacteristic
134C2×#120—21no—
135C2×#120—21no—
136C2×C4×Q8C22×C312yescomplemented
137#460—28no—
138#448—24no—
139C2×C4⋊C8C22×C312yescomplemented
140C2×#103C22×C311yescharacteristic, complemented
141C2×#103C22×C311yescharacteristic, complemented
142C2×#102C22×C312yes—
143C2×#102C22×C312yes—
144C2×#102C22×C312yescomplemented
145C2×C22⋊C8C22×C311yescharacteristic
146#392—22no—
147#391—22no—
148#391C22×C312yescomplemented
149#390C22×C312yes—
150#380—21no—
151#380C22×C311yescharacteristic
152C4×C22⋊C4—21no—
153C4×C22⋊C4C22×C311yescharacteristic
154#340C22×C312yescomplemented
155#336C22×C312yescomplemented
156C24×C3—22no—
157C24×C3—21no—
158C24×C3—44no—
159C24×C3—42no—
160C24×C3—42no—
161C24×C3—42no—
162C24×C3—82no—
163C24×C3C2×D411yescharacteristic
164C2×C3×Q8—24no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—28no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—44no—
172C2×C3×Q8—44no—
173C2×C3×Q8—48no—
174C2×C3×Q8C2×D412yes—
175C2×C3×Q8C2×D412yes—
176C2×C3×Q8C2×D414yes—
177C2×C3×D4—28no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—21no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—48no—
188C2×C3×D4—44no—
189C2×C3×D4—44no—
190C2×C3×D4—44no—
191C2×C3×D4—44no—
192C2×C3×D4—44no—
193C2×C3×D4—42no—
194C2×C3×D4—42no—
195C2×C3×D4—42no—
196C2×C3×D4—42no—
197C2×C3×D4—44no—
198C2×C3×D4—82no—
199C2×C3×D4—82no—
200C2×C3×D4C2×D414yes—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—22no—
204C22×C3×C4—24no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—21no—
208C22×C3×C4—44no—
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D412yes—
213C22×C3×C4#4112yes—
214C22×C3×C4C2411yescharacteristic
215C3×SD16—216no—
216C3×SD16—216no—
217C3×SD16—44no—
218C3×SD16—44no—
219C2×C3×C8—28no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8C2×D414yes—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—44no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—44no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—44no—
237C3×C22⋊C4SD1614yescomplemented
238C3×C42—22no—
239C3×C42—22no—
240C3×C42—44no—
241C25—21no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8C3×D414yescomplemented
247C22×Q8C23×C311yescharacteristic
248C22×Q8C23×C311yescharacteristic
249C22×D4—24no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—42no—
257C22×D4C23×C311yescharacteristic
258C22×D4C23×C312yes—
259C23×C4—21no—
260C23×C4C23×C311yescharacteristic
261C2×SD16—232no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—22no—
267C2×SD16—22no—
268C2×SD16—24no—
269C2×SD16—24no—
270C2×SD16—42no—
271C2×SD16C3×D418yescomplemented
272C22×C8—22no—
273C22×C8—24no—
274C22×C8C23×C311yescharacteristic
275C22×C8C23×C311yescharacteristic
276C22×C8C23×C312yes—
277#127—24no—
278#127—42no—
279#121—22no—
280#121—22no—
281#120—44no—
282#120—44no—
283C4×Q8—24no—
284C4×Q8—42no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4C23×C312yes—
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4C3×D411yescharacteristic
293C2×C22⋊C4C3×D411yescharacteristic
294C2×C42—22no—
295C2×C42C23×C312yes—
296C4⋊C8—24no—
297#103—22no—
298#103—22no—
299#102—24no—
300#102—24no—
301#102—42no—
302C22⋊C8—22no—
303#95—22no—
304#95—22no—
305#95—21no—
306#95C3×D412yes—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—24no—
310C23×C3—48no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—41no—
318C23×C3—42no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—42no—
322C23×C3—88no—
323C23×C3—88no—
324C23×C3—84no—
325C23×C3—84no—
326C23×C3#12112yes—
327C23×C3C2×SD1612yes—
328C23×C3AGL(1,ℤ/8)12yes—
329C23×C3C22×D411yescharacteristic
330C3×Q8—22no—
331C3×Q8—24no—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—48no—
335C3×Q8—42no—
336C3×Q8#12014yescomplemented
337C3×D4—24no—
338C3×D4—24no—
339C3×D4—48no—
340C3×D4—41no—
341C3×D4—42no—
342C3×D4—41no—
343C3×D4—42no—
344C3×D4—42no—
345C3×D4—88no—
346C3×D4—88no—
347C3×D4—84no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—48no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4#12112yes—
369C2×C3×C4C2×SD1614yescomplemented
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D412yes—
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)12yes—
374C3×C8—24no—
375C3×C8—41no—
376C3×C8—41no—
377C24—22no—
378C24—21no—
379C24—44no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—82no—
384C24C2×C3×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—28no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—48no—
395C2×Q8C2×C3×D412yes—
396C2×Q8C2×C3×D412yes—
397C2×Q8C2×C3×D414yes—
398C2×D4—28no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—48no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4C2×C3×D414yes—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—44no—
430C22×C4C2×C3×D412yes—
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D412yes—
433C22×C4C2×C3×D412yes—
434C22×C4C3×#4112yes—
435C22×C4C24×C311yescharacteristic
436SD16—216no—
437SD16—216no—
438SD16—44no—
439SD16—44no—
440C2×C8—28no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8C2×C3×D414yes—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—44no—
453C22⋊C4—22no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4C3×SD1614yescomplemented
459C42—22no—
460C42—22no—
461C42—44no—
462C22×C3—22no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—22no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—816no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3—82no—
474C22×C3—81no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3#46014yescomplemented
480C22×C3C2×#12111yescharacteristic
481C22×C3D4212yes—
482C22×C3C22×SD1611yescharacteristic
483C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
484C22×C3#57712yes—
485C3×C4—22no—
486C3×C4—24no—
487C3×C4—24no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—82no—
491C3×C4—81no—
492C3×C4C2×#12011yescharacteristic
493C3×C4#53411yescharacteristic
494C3×C4#54612yes—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—48no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—88no—
511C23—88no—
512C23—84no—
513C23—84no—
514C23C3×#12112yes—
515C23C2×C3×SD1612yes—
516C23C3×AGL(1,ℤ/8)12yes—
517C23C22×C3×D411yescharacteristic
518Q8—22no—
519Q8—24no—
520Q8—22no—
521Q8—22no—
522Q8—48no—
523Q8—42no—
524Q8C3×#12014yescomplemented
525D4—24no—
526D4—24no—
527D4—48no—
528D4—41no—
529D4—42no—
530D4—41no—
531D4—42no—
532D4—42no—
533D4—88no—
534D4—88no—
535D4—84no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4C3×#12112yes—
557C2×C4C2×C3×SD1614yescomplemented
558C2×C4C22×C3×D411yescharacteristic
559C2×C4C22×C3×D412yes—
560C2×C4C2×C3×#4111yescharacteristic
561C2×C4C3×H(5,𝔽2)12yes—
562C8—24no—
563C8—41no—
564C8—41no—
565C2×C3—24no—
566C2×C3—84no—
567C2×C3—161no—
568C2×C3#241011yescharacteristic
569C2×C3C2×#46011yescharacteristic
570C2×C3#304311yescharacteristic
571C2×C3#326512yes—
572C2×C3D4×SD1612yes—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—22no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—816no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—81no—
586C22—82no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22C3×#46014yescomplemented
591C22C2×C3×#12111yescharacteristic
592C22C3×D4212yes—
593C22C22×C3×SD1611yescharacteristic
594C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
595C22C3×#57712yes—
596C4—22no—
597C4—24no—
598C4—24no—
599C4—42no—
600C4—42no—
601C4—82no—
602C4—81no—
603C4C2×C3×#12011yescharacteristic
604C4C3×#53411yescharacteristic
605C4C3×#54612yes—
606C3#2435611yescharacteristic, direct product factor, Sylow
607C2—24no—
608C2—84no—
609C2—161no—
610C2C3×#241011yescharacteristic
611C2C2×C3×#46011yescharacteristic
612C2C3×#304311yescharacteristic
613C2C3×#326512yes—
614C2C3×D4×SD1612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2433C2
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. C23×C3×SD16C2semidirect
2. C2×C3×#460C2semidirect
3. C2×C3×#448C2semidirect
5. C3×Q8×C22⋊C4C2semidirect
6. C3×#2055C2semidirect
7. C3×#2014C2semidirect
8. C3×#1893C2semidirect
9. C3×#1880C2semidirect
10. C3×#1850C2semidirect
11. C3×#1786C2semidirect
12. #24356C3direct (N × H)
19. C22×C3×SD16C22semidirect
20. C22×C3×SD16C22semidirect
26. C2×C3×C4×Q8C22semidirect
29. C2×C3×C4⋊C8C22semidirect
30. C2×C3×#103C22semidirect
31. C2×C3×#103C22semidirect
34. C2×C3×#102C22semidirect
38. C3×#391C22semidirect
44. C3×#340C22semidirect
45. C3×#336C22semidirect
46. C23×SD16C2×C3semidirect
47. C2×#460C2×C3semidirect
48. C2×#448C2×C3semidirect
50. Q8×C22⋊C4C2×C3semidirect
51. #2055C2×C3semidirect
52. #2014C2×C3semidirect
53. #1893C2×C3semidirect
54. #1880C2×C3semidirect
55. #1850C2×C3semidirect
56. #1786C2×C3semidirect
62. C22×C3×Q8D4semidirect
87. C2×C3×SD16D4semidirect
129. C22×SD16C22×C3semidirect
130. C22×SD16C22×C3semidirect
136. C2×C4×Q8C22×C3semidirect
139. C2×C4⋊C8C22×C3semidirect
140. C2×#103C22×C3semidirect
141. C2×#103C22×C3semidirect
144. C2×#102C22×C3semidirect
148. #391C22×C3semidirect
154. #340C22×C3semidirect
155. #336C22×C3semidirect
237. C3×C22⋊C4SD16semidirect
246. C22×Q8C3×D4semidirect
271. C2×SD16C3×D4semidirect
336. C3×Q8#120semidirect
369. C2×C3×C4C2×SD16semidirect
458. C22⋊C4C3×SD16semidirect
479. C22×C3#460semidirect
524. Q8C3×#120semidirect
557. C2×C4C2×C3×SD16semidirect
590. C22C3×#460semidirect
606. C3#24356direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SD162
C2×C3×#4602
C2×C3×#4482
C3×#24332
C3×Q8×C22⋊C42
C3×#20552
C3×#20142
C3×#18932
C3×#18802
C3×#18502
C3×#17862
#243563