Finite Group Databasebeta

← C3×#24216 #10642823

Subgroups — C3×#24216 #10642823

Order 768.

1482 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#464C212yesmaximal, complemented
3C2×C3×#452C212yesmaximal, complemented
4C3×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
5C3×#2043C211yesmaximal, characteristic, complemented
6C3×#2041C211yesmaximal, characteristic, complemented
7C3×#2016C211yesmaximal, characteristic, complemented
8C3×#2014C211yesmaximal, characteristic, complemented
9C3×#1983C212yesmaximal, complemented
10C3×#1797C211yesmaximal, characteristic, complemented
11C3×#1788C212yesmaximal, complemented
12#24216C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C3×#532—28no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121—21no—
22C2×C3×#121—21no—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4—21no—
26C22×C3×C42C2211yescharacteristic
27C3×#464—28no—
28C3×#452—28no—
29C2×C3×C4⋊C8C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×C22⋊C8C2212yes—
37C3×#391C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#387—21no—
40C3×#385—21no—
41C3×#380C2211yescharacteristic
42C3×C4×C4⋊C4—21no—
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×C4×C22⋊C4—21no—
45C3×#336C2211yescharacteristic
46C3×#336C2211yescharacteristic
47C2×#532C2×C311yescharacteristic
48C2×#464C2×C312yescomplemented
49C2×#452C2×C312yescomplemented
50C4×C22⋊Q8C2×C311yescharacteristic, complemented
51#2043C2×C311yescharacteristic, complemented
52#2041C2×C311yescharacteristic, complemented
53#2016C2×C311yescharacteristic, complemented
54#2014C2×C311yescharacteristic, complemented
55#1983C2×C312yescomplemented
56#1797C2×C311yescharacteristic, complemented
57#1788C2×C312yescomplemented
58C2×C3×#41—24no—
59C2×C3×#41—24no—
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41—42no—
63C2×C3×#41—42no—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2312yes—
66C22×C3×D4—22no—
67C22×C3×D4—21no—
68C22×C3×D4—21no—
69C22×C3×D4C2312yes—
70C23×C3×C4—22no—
71C23×C3×C4D411yescharacteristic
72C23×C3×C4D411yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C2×C3×Q16—24no—
75C2×C3×Q16—24no—
76C2×C3×Q16—42no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C2×C3×SD16—42no—
80C22×C3×C8C2312yes—
81C22×C3×C8C2312yes—
82C3×C4⋊Q8—24no—
83C3×#127—24no—
84C3×#124—24no—
85C3×#124D418yes—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8D414yescomplemented
88C3×#121—28no—
89C3×#121—28no—
90C3×C4×Q8—24no—
91C3×C4×D4—28no—
92C3×C4×D4—28no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C42—24no—
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D411yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×C4⋊C8—24no—
113C3×#103—24no—
114C3×#103—24no—
115C3×#103—42no—
116C3×#102—24no—
117C3×#102—24no—
118C3×#102—42no—
119C3×C22⋊C8—24no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C22×#41C22×C312yes—
126C22×Q16C22×C312yes—
127C22×SD16C22×C312yes—
128#532—28no—
129C2×C4⋊Q8C22×C311yescharacteristic
130C2×#127C22×C311yescharacteristic
131C2×#124C22×C312yes—
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×#121—21no—
134C2×#121—21no—
135C2×C4×Q8C22×C311yescharacteristic
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C22×C42C22×C311yescharacteristic
139#464—28no—
140#452—28no—
141C2×C4⋊C8C22×C312yes—
142C2×#103C22×C312yes—
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×#102C22×C312yes—
146C2×#102C22×C312yes—
147C2×#102C22×C312yes—
148C2×C22⋊C8C22×C312yes—
149#391C22×C311yescharacteristic
150#389C22×C311yescharacteristic
151#387—21no—
152#385—21no—
153#380C22×C311yescharacteristic
154C4×C4⋊C4—21no—
155C4×C4⋊C4C22×C311yescharacteristic
156C4×C22⋊C4—21no—
157#336C22×C311yescharacteristic
158#336C22×C311yescharacteristic
159C24×C3—22no—
160C24×C3C2×D411yescharacteristic
161C3×#41—48no—
162C3×#41—48no—
163C3×#41—48no—
164C3×#41—48no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8C2×D414yes—
174C2×C3×Q8C2×D412yes—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—28no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4C2×D414yes—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—28no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—41no—
203C22×C3×C4—41no—
204C22×C3×C4—44no—
205C22×C3×C4—44no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D412yes—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4#4112yes—
213C22×C3×C4C2411yescharacteristic
214C3×Q16—48no—
215C3×SD16—48no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4C2×D412yes—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—28no—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—42no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—42no—
239C3×C42—42no—
240C3×C42C2×D414yes—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41—42no—
246C2×#41—42no—
247C22×Q8C23×C311yescharacteristic
248C22×Q8C23×C312yes—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C23×C312yes—
253C23×C4—22no—
254C23×C4C3×D411yescharacteristic
255C23×C4C3×D411yescharacteristic
256C23×C4C23×C311yescharacteristic
257C2×Q16—24no—
258C2×Q16—24no—
259C2×Q16—42no—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—42no—
263C22×C8C23×C312yes—
264C22×C8C23×C312yes—
265C4⋊Q8—24no—
266#127—24no—
267#124—24no—
268#124C3×D418yes—
269C22⋊Q8—22no—
270C22⋊Q8C3×D414yescomplemented
271#121—28no—
272#121—28no—
273C4×Q8—24no—
274C4×D4—28no—
275C4×D4—28no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C42—24no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C3×D411yescharacteristic
292C2×C42C3×D411yescharacteristic
293C2×C42C23×C311yescharacteristic
294C2×C42C23×C311yescharacteristic
295C4⋊C8—24no—
296#103—24no—
297#103—24no—
298#103—42no—
299#102—24no—
300#102—24no—
301#102—42no—
302C22⋊C8—24no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—24no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3#12112yes—
322C23×C3C22×D411yescharacteristic
323C3×Q8—24no—
324C3×Q8—24no—
325C3×Q8—24no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—48no—
332C3×D4—48no—
333C3×D4—416no—
334C3×D4—42no—
335C3×D4—42no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—44no—
352C2×C3×C4—44no—
353C2×C3×C4—416no—
354C2×C3×C4—41no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4#12012yes—
367C2×C3×C4#12012yes—
368C2×C3×C4#12112yes—
369C2×C3×C4#13512yes—
370C2×C3×C4#13712yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C22×D412yes—
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)12yes—
375C3×C8—42no—
376C3×C8—42no—
377C24—22no—
378C24C2×C3×D411yescharacteristic
379#41—48no—
380#41—48no—
381#41—48no—
382#41—48no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8C2×C3×D414yes—
392C2×Q8C2×C3×D412yes—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4C2×C3×D414yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4C2×C3×D411yescharacteristic
425C22×C4C2×C3×D411yescharacteristic
426C22×C4C2×C3×D411yescharacteristic
427C22×C4C2×C3×D412yes—
428C22×C4C2×C3×D411yescharacteristic
429C22×C4C2×C3×D411yescharacteristic
430C22×C4C3×#4112yes—
431C22×C4C24×C311yescharacteristic
432Q16—48no—
433SD16—48no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4C2×C3×D412yes—
449C22⋊C4—24no—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—42no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—42no—
457C42—42no—
458C42C2×C3×D414yes—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—82no—
470C22×C3—82no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3C2×#12011yescharacteristic
474C22×C3C2×#12111yescharacteristic
475C22×C3#53511yescharacteristic
476C22×C3C2×#13511yescharacteristic
477C22×C3C2×#13711yescharacteristic
478C22×C3#57511yescharacteristic
479C22×C3#57511yescharacteristic
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—44no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4D4212yes—
491C3×C4#54612yes—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23C3×#12112yes—
506C23C22×C3×D411yescharacteristic
507Q8—24no—
508Q8—24no—
509Q8—24no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
513D4—24no—
514D4—24no—
515D4—48no—
516D4—48no—
517D4—416no—
518D4—42no—
519D4—42no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—416no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4C3×#12012yes—
551C2×C4C3×#12012yes—
552C2×C4C3×#12112yes—
553C2×C4C3×#13512yes—
554C2×C4C3×#13712yes—
555C2×C4C22×C3×D411yescharacteristic
556C2×C4C22×C3×D412yes—
557C2×C4C2×C3×#4111yescharacteristic
558C2×C4C3×H(5,𝔽2)12yes—
559C8—42no—
560C8—42no—
561C2×C3—42no—
562C2×C3—84no—
563C2×C3#240811yescharacteristic
564C2×C3#300211yescharacteristic
565C2×C3#300711yescharacteristic
566C2×C3#303511yescharacteristic
567C2×C3#303711yescharacteristic
568C2×C3#321011yescharacteristic
569C2×C3#321911yescharacteristic
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22C2×C3×#12011yescharacteristic
585C22C2×C3×#12111yescharacteristic
586C22C3×#53511yescharacteristic
587C22C2×C3×#13511yescharacteristic
588C22C2×C3×#13711yescharacteristic
589C22C3×#57511yescharacteristic
590C22C3×#57511yescharacteristic
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4C3×D4212yes—
602C4C3×#54612yes—
603C3#2421611yescharacteristic, direct product factor, Sylow
604C2—42no—
605C2—84no—
606C2C3×#240811yescharacteristic
607C2C3×#300211yescharacteristic
608C2C3×#300711yescharacteristic
609C2C3×#303511yescharacteristic
610C2C3×#303711yescharacteristic
611C2C3×#321011yescharacteristic
612C2C3×#321911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#464C2semidirect
3. C2×C3×#452C2semidirect
4. C3×C4×C22⋊Q8C2semidirect
5. C3×#2043C2semidirect
6. C3×#2041C2semidirect
7. C3×#2016C2semidirect
8. C3×#2014C2semidirect
9. C3×#1983C2semidirect
10. C3×#1797C2semidirect
11. C3×#1788C2semidirect
12. #24216C3direct (N × H)
48. C2×#464C2×C3semidirect
49. C2×#452C2×C3semidirect
50. C4×C22⋊Q8C2×C3semidirect
51. #2043C2×C3semidirect
52. #2041C2×C3semidirect
53. #2016C2×C3semidirect
54. #2014C2×C3semidirect
55. #1983C2×C3semidirect
56. #1797C2×C3semidirect
57. #1788C2×C3semidirect
87. C3×C22⋊Q8D4semidirect
270. C22⋊Q8C3×D4semidirect
603. C3#24216direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4642
C2×C3×#4522
C3×C4×C22⋊Q82
C3×#20432
C3×#20412
C3×#20162
C3×#20142
C3×#19832
C3×#17972
C3×#17882
#242163