Finite Group Databasebeta

← #10728214

Subgroups — #10728214

Order 768.

2046 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C4⋊C4C211yesmaximal, characteristic
2C2×#4668C211yesmaximal, characteristic
3C2×#4660C212yesmaximal, complemented
4C3×#1442C211yesmaximal, characteristic, complemented
5#71566C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#95C211yesmaximal, characteristic
7(C2×D3)⋊#95C212yesmaximal
8(C2×D3)⋊#95C212yesmaximal
9#69421C212yesmaximal, complemented
10#69415C211yesmaximal, characteristic, complemented
11#69410C211yesmaximal, characteristic, complemented
12#14843—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×C4×D3C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C2×D3×C4⋊C4—24no—
24C2×D3×C4⋊C4—28no—
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4668—24no—
39#4660—28no—
40C23×C4⋊C4—31no—
41C2×#400—32no—
42C2×#385—31no—
43#1445—32no—
44#1442—31no—
45#1442D311yescharacteristic, complemented
46#1441—31no—
47#1441—31no—
48C22⋊#95—32no—
49C22⋊#95—31no—
50C22⋊#95—32no—
51C24×D3C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2312yes—
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C22×C4×D3—24no—
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3—24no—
66C22×C4×D3—24no—
67C22×C4×D3—24no—
68C22×C4×D3—24no—
69C22×C4×D3—24no—
70C22×C4×D3—24no—
71C22×C4×D3—42no—
72C22×C4×D3—42no—
73C22×C4×D3D414yes—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4Q814yes—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—24no—
86C2×C4⋊Dic3—24no—
87C2×C4⋊Dic3—22no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4D418yes—
94D3×C4⋊C4—232no—
95C3×#95—24no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—42no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—22no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C3⋊#95—42no—
108C24×C4—31no—
109C24×C4—32no—
110C22×C4⋊C4—32no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4—68no—
115C22×C4⋊C4—64no—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C22⋊C4—31no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—31no—
120#400—68no—
121#385—64no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95C2×D312yes—
132C2×#95C2×D311yescharacteristic
133C2×#95C2×D311yescharacteristic
134C2×#95C2×D312yes—
135C24×C3C2411yescharacteristic
136C23×D3—22no—
137C23×D3—21no—
138C23×D3—21no—
139C23×D3—21no—
140C23×D3—21no—
141C23×D3—22no—
142C23×D3C2×D412yes—
143C23×D3C2×Q812yes—
144C23×D3#4112yes—
145C23×D3#4114yes—
146C23×D3#4114yes—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×Q812yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3C2×D414yes—
204C22×Dic3C2×D412yes—
205C22×Dic3C2×D414yes—
206C22×Dic3C2×Q812yes—
207C2×C4×D3—216no—
208C2×C4×D3—216no—
209C2×C4×D3—28no—
210C2×C4×D3—28no—
211C2×C4×D3—28no—
212C2×C4×D3—216no—
213C2×C4×D3—48no—
214C2×C4×D3—48no—
215C2×C4×D3—48no—
216C2×C4×D3—48no—
217C2×C4×D3—48no—
218C2×C4×D3—48no—
219C3×C4⋊C4—28no—
220(C2×D3)⋊C4—28no—
221(C2×D3)⋊C4—44no—
222(C2×D3)⋊C4—48no—
223C4⋊Dic3—28no—
224C4⋊Dic3—44no—
225Dic3⋊C4—216no—
226Dic3⋊C4—44no—
227C25—31no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—34no—
232C23×C4—31no—
233C23×C4—32no—
234C23×C4—64no—
235C23×C4—64no—
236C23×C4—64no—
237C23×C4—64no—
238C23×C4—64no—
239C23×C4—64no—
240C23×C4—62no—
241C23×C4—62no—
242C23×C4—64no—
243C23×C4—122no—
244C23×C4—122no—
245C23×C4C22×D312yes—
246C23×C4C22×D311yescharacteristic
247C23×C4C22×D312yes—
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D311yescharacteristic
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—38no—
253C2×C4⋊C4—632no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C22⋊C4—34no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—64no—
271#95—22no—
272#95—22no—
273#95—24no—
274#95—42no—
275#95—62no—
276#95—64no—
277#95—64no—
278#95—64no—
279#95—64no—
280#95—62no—
281#95—122no—
282#95—122no—
283#95—122no—
284C23×C3C22×D411yescharacteristic
285C23×C3C22×D412yes—
286C23×C3C22×Q811yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3C2×#4111yescharacteristic
289C23×C3C2×#4111yescharacteristic
290C23×C3C2×#4112yes—
291C23×C3C2×#4112yes—
292C23×C3H(5,𝔽2)11yescharacteristic
293C23×C3H(5,𝔽2)11yescharacteristic
294C23×C32−1+412yes—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—24no—
298C22×D3—24no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—24no—
302C22×D3—24no—
303C22×D3—24no—
304C22×D3—24no—
305C22×D3—24no—
306C22×D3—24no—
307C22×D3—24no—
308C22×D3—24no—
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3C22⋊Q814yes—
314C22×D3#12314yes—
315C22×D3#12414yes—
316C22×D3#12518yes—
317C22×D3#12618yes—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4#12314yes—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—22no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—41no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3#12014yes—
378C4×D3—232no—
379C4×D3—432no—
380C24—32no—
381C24—32no—
382C24—32no—
383C24—34no—
384C24—34no—
385C24—61no—
386C24—62no—
387C24—61no—
388C24—61no—
389C24—61no—
390C24—62no—
391C24C23×D311yescharacteristic
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—34no—
417C22×C4—32no—
418C22×C4—34no—
419C22×C4—32no—
420C22×C4—68no—
421C22×C4—68no—
422C22×C4—616no—
423C22×C4—616no—
424C22×C4—616no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—68no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—128no—
454C22×C4—128no—
455C22×C4—128no—
456C22×C4—128no—
457C22×C4—128no—
458C22×C4—128no—
459C22×C4#23512yes—
460C22×C4D3×D414yes—
461C22×C4D3×D412yes—
462C22×C4D3×Q812yes—
463C22×C4#23912yes—
464C4⋊C4—28no—
465C4⋊C4—616no—
466C4⋊C4—68no—
467C4⋊C4—124no—
468C4⋊C4—124no—
469C22⋊C4—68no—
470C22⋊C4—124no—
471C22⋊C4—128no—
472C22×C3C2×#12011yescharacteristic
473C22×C3C2×C22⋊Q811yescharacteristic
474C22×C3C2×#12311yescharacteristic
475C22×C3C2×#12311yescharacteristic
476C22×C3#52512yes—
477C22×C3#52512yes—
478C22×C3C2×#12411yescharacteristic
479C22×C3C2×#12512yes—
480C22×C3C2×#12612yes—
481C22×C3#53511yescharacteristic
482C22×C3#53611yescharacteristic
483C22×C3#53912yes—
484C22×C3#53912yes—
485C22×C3#54612yes—
486C22×C3#54612yes—
487C22×C3#54812yes—
488C22×C3D4×Q812yes—
489C22×C3#55212yes—
490C22×C3#55212yes—
491C22×C3#55311yescharacteristic
492C22×C3#55311yescharacteristic
493C22×C3#55411yescharacteristic
494C22×C3#55411yescharacteristic
495C2×D3—24no—
496C2×D3—24no—
497C2×D3—24no—
498C2×D3—24no—
499C2×D3—28no—
500C2×D3—28no—
501C2×D3—28no—
502C2×D3—28no—
503C2×D3—24no—
504C2×D3—24no—
505C2×D3#40018yescomplemented
506C3×C4—24no—
507C3×C4—42no—
508C3×C4—44no—
509C3×C4—81no—
510C3×C4—82no—
511Dic3—24no—
512Dic3—44no—
513Dic3—42no—
514Dic3—81no—
515Dic3—82no—
516C23—34no—
517C23—34no—
518C23—38no—
519C23—38no—
520C23—34no—
521C23—64no—
522C23—64no—
523C23—64no—
524C23—62no—
525C23—64no—
526C23—64no—
527C23—62no—
528C23—64no—
529C23—62no—
530C23—62no—
531C23—64no—
532C23—64no—
533C23—62no—
534C23—62no—
535C23—62no—
536C23—62no—
537C23—64no—
538C23—64no—
539C23C2×#23511yescharacteristic
540C23C2×D3×D412yes—
541C23C2×D3×D411yescharacteristic
542C23#100411yescharacteristic
543C23C2×D3×Q811yescharacteristic
544C23C2×#23911yescharacteristic
545C23D3×#4112yes—
546C23D3×#4112yes—
547C23D3×#4111yescharacteristic
548C23#100911yescharacteristic
549C23#101012yes—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—34no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—632no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—62no—
590C2×C4—1232no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—122no—
597C2×C4—121no—
598C2×C4—122no—
599C2×C4—121no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—121no—
607C2×C4—122no—
608C2×C4—121no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4#89414yes—
612C2×C3C2×#40011yescharacteristic
613C2×C3#240911yescharacteristic
614C2×C3#243511yescharacteristic
615C2×C3#247312yes—
616C2×C3#247312yes—
617C2×C3#248112yes—
618C2×C3#248112yes—
619C2×C3#248711yescharacteristic
620C2×C3#250311yescharacteristic
621C2×C3#262311yescharacteristic
622C2×C3#284911yescharacteristic
623D3—28no—
624C22—38no—
625C22—68no—
626C22—68no—
627C22—68no—
628C22—64no—
629C22—64no—
630C22—64no—
631C22—64no—
632C22—68no—
633C22—64no—
634C22—64no—
635C22C2×#89411yescharacteristic
636C22#552912yes—
637C22#554712yes—
638C22#555811yescharacteristic
639C22#557911yescharacteristic
640C22D3×#12011yescharacteristic
641C22#560711yescharacteristic
642C22#561012yes—
643C22D3×C22⋊Q811yescharacteristic
644C22#562712yes—
645C22#563212yes—
646C22D3×#12311yescharacteristic
647C22#565412yes—
648C22#566012yes—
649C22D3×#12411yescharacteristic
650C22#567511yescharacteristic
651C22#567712yes—
652C22D3×#12512yes—
653C22#569412yes—
654C22#569511yescharacteristic
655C22D3×#12612yes—
656C22#570712yes—
657C22#573011yescharacteristic
658C4—24no—
659C4—44no—
660C4—42no—
661C4—64no—
662C4—82no—
663C4—81no—
664C4—124no—
665C4—122no—
666C4—242no—
667C4—241no—
668C3#1484311yescharacteristic, complemented, Sylow
669C2—68no—
670C2#7416711yescharacteristic
671C2#7465611yescharacteristic
672C2#7500812yes—
673C2#7554212yes—
674C2#7623211yescharacteristic
675C2#7678011yescharacteristic
676C2#7690511yescharacteristic
677C2#7768611yescharacteristic
678C2#7775812yes—
679C2#7814512yes—
680C2D3×#40011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1442C2
prime radicalC24×D3C23
hypercenter#1442D3
hypocenterC3#14843
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
3. C2×#4660C2semidirect
4. C3×#1442C2semidirect
5. #71566C2semidirect
9. #69421C2semidirect
10. #69415C2semidirect
11. #69410C2semidirect
45. #1442D3semidirect
505. C2×D3#400semidirect
668. C3#14843semidirect

Maximal subgroups

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

subgroupindex
C22×D3×C4⋊C42
C2×#46682
C2×#46602
C3×#14422
#715662
(C2×D3)⋊#952
#694212
#694152
#694102
#148433