Finite Group Databasebeta

← #10728989

Subgroups — #10728989

Order 768.

1886 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#872C211yesmaximal, characteristic
2C2×#4928C211yesmaximal, characteristic, complemented
3C2×#4668C212yesmaximal
4C2×#4661C211yesmaximal, characteristic, complemented
5C2×#4674C211yesmaximal, characteristic
6C3×#1434C211yesmaximal, characteristic, complemented
7#69429C212yesmaximal
8#69427C211yesmaximal, characteristic
9#69411C212yesmaximal, complemented
10#69410C211yesmaximal, characteristic, complemented
11#69405C212yesmaximal, complemented
12#14907—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×C4×Dic3C2211yescharacteristic
26C2×#872—24no—
27C2×#872—24no—
28C2×#872—24no—
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4928—24no—
40#4674—24no—
41#4668—28no—
42#4661—24no—
43C22×#117—31no—
44C2×#398—31no—
45C2×#397—31no—
46C2×#389—31no—
47C2×#385—32no—
48#1441—31no—
49#1441—32no—
50#1440—32no—
51#1439—31no—
52#1436—32no—
53#1434D311yescharacteristic, complemented
54C24×D3C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2311yescharacteristic
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3—22no—
70C22×C4×D3—22no—
71C22×C4×D3D414yes—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42D414yes—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4D414yes—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3—24no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4D414yes—
104C2×C4×Dic3—22no—
105C2×C4×Dic3—22no—
106C2×C4×Dic3—22no—
107#872—232no—
108C3×#95—22no—
109C3×#95—24no—
110C3×#95—22no—
111C3×#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—24no—
116C3⋊#95—24no—
117C3⋊#95—24no—
118C24×C4—31no—
119C2×#117—64no—
120C2×#117—64no—
121C2×#117—64no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4—31no—
128C22×C22⋊C4—31no—
129C22×C22⋊C4—32no—
130C22×C22⋊C4—31no—
131C22×C22⋊C4—32no—
132C22×C42—31no—
133C22×C42C2×D311yescharacteristic
134#398—64no—
135#397—64no—
136#389—64no—
137#385—68no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—32no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95C2×D311yescharacteristic
145C2×#95C2×D311yescharacteristic
146C2×#95C2×D312yes—
147C2×#95C2×D312yes—
148C24×C3C2411yescharacteristic
149C23×D3—21no—
150C23×D3—21no—
151C23×D3—22no—
152C23×D3—22no—
153C23×D3—21no—
154C23×D3—21no—
155C23×D3—22no—
156C23×D3—22no—
157C23×D3C2×D412yes—
158C23×D3C2×D412yes—
159C23×D3#4112yes—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4#4114yes—
190C22×C3×C4#4112yes—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3C2×D412yes—
217C22×Dic3C2×D412yes—
218C22×Dic3C2×D412yes—
219C22×Dic3C2×D412yes—
220C22×Dic3#4114yes—
221C22×Dic3#4112yes—
222C2×C4×D3—28no—
223C2×C4×D3—28no—
224C2×C4×D3—28no—
225C2×C4×D3—28no—
226C2×C4×D3—28no—
227C2×C4×D3—28no—
228C2×C4×D3—28no—
229C3×C42—28no—
230(C2×D3)⋊C4—28no—
231(C2×D3)⋊C4—216no—
232(C2×D3)⋊C4—28no—
233(C2×D3)⋊C4—48no—
234C4⋊Dic3—28no—
235C4⋊Dic3—44no—
236C4⋊Dic3—48no—
237Dic3⋊C4—28no—
238Dic3⋊C4—28no—
239Dic3⋊C4—48no—
240C4×Dic3—28no—
241C25—31no—
242C23×C4—31no—
243C23×C4—32no—
244C23×C4—31no—
245C23×C4—31no—
246C23×C4—34no—
247C23×C4—32no—
248C23×C4—62no—
249C23×C4—62no—
250C23×C4—62no—
251C23×C4—62no—
252C23×C4—62no—
253C23×C4—62no—
254C23×C4C22×D312yes—
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D312yes—
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259#117—632no—
260C2×C4⋊C4—34no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—64no—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—62no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—64no—
288C2×C22⋊C4—64no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—62no—
292C2×C42—62no—
293C2×C42—62no—
294C2×C42D1214yes—
295#95—24no—
296#95—24no—
297#95—22no—
298#95—22no—
299#95—64no—
300#95—64no—
301#95—64no—
302#95—64no—
303#95—64no—
304#95—64no—
305C23×C3C22×D411yescharacteristic
306C23×C3C22×D411yescharacteristic
307C23×C3C22×D411yescharacteristic
308C23×C3C22×D411yescharacteristic
309C23×C3C2×#4112yes—
310C23×C3C2×#4111yescharacteristic
311C23×C3C2×#4111yescharacteristic
312C23×C3C2×#4111yescharacteristic
313C23×C3C2×#4112yes—
314C23×C32−1+412yes—
315C23×C32−1+412yes—
316C22×D3—22no—
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—24no—
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—22no—
323C22×D3—24no—
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—24no—
327C22×D3—24no—
328C22×D3—41no—
329C22×D3—42no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3#12114yes—
339C2×C3×C4—22no—
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—24no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4#12014yes—
369C2×C3×C4#12114yes—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—42no—
384C2×Dic3—41no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3#12114yes—
401C4×D3—232no—
402C24—32no—
403C24—32no—
404C24—32no—
405C24—61no—
406C24—62no—
407C24—61no—
408C24—61no—
409C24—62no—
410C24—61no—
411C24—62no—
412C24—62no—
413C24C23×D311yescharacteristic
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—32no—
442C22×C4—34no—
443C22×C4—68no—
444C22×C4—68no—
445C22×C4—68no—
446C22×C4—68no—
447C22×C4—68no—
448C22×C4—68no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—68no—
454C22×C4—62no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4C2×D1212yes—
476C22×C4C2×D1212yes—
477C22×C4C2×D1212yes—
478C22×C4#23514yes—
479C22×C4D3×D412yes—
480C22×C4D3×D412yes—
481C22×C4D3×D412yes—
482C22×C4#23912yes—
483C4⋊C4—68no—
484C4⋊C4—68no—
485C4⋊C4—68no—
486C4⋊C4—128no—
487C4⋊C4—124no—
488C4⋊C4—128no—
489C22⋊C4—68no—
490C22⋊C4—616no—
491C22⋊C4—68no—
492C22⋊C4—128no—
493C42—28no—
494C42—68no—
495C22×C3C2×#12011yescharacteristic
496C22×C3C2×#12111yescharacteristic
497C22×C3C2×#12111yescharacteristic
498C22×C3C2×#12111yescharacteristic
499C22×C3#52511yescharacteristic
500C22×C3#52511yescharacteristic
501C22×C3#52512yes—
502C22×C3#52511yescharacteristic
503C22×C3#52511yescharacteristic
504C22×C3#52912yes—
505C22×C3#53211yescharacteristic
506C22×C3#53612yes—
507C22×C3#53611yescharacteristic
508C22×C3#53611yescharacteristic
509C22×C3#54112yes—
510C22×C3#54712yes—
511C22×C3#54712yes—
512C22×C3#54712yes—
513C22×C3#54812yes—
514C22×C3#55212yes—
515C22×C3#55212yes—
516C22×C3#55212yes—
517C22×C3#55612yes—
518C2×D3—24no—
519C2×D3—28no—
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—44no—
526C2×D3—42no—
527C2×D3—44no—
528C2×D3—44no—
529C3×C4—24no—
530C3×C4—24no—
531C3×C4—44no—
532C3×C4—42no—
533C3×C4—82no—
534Dic3—24no—
535Dic3—44no—
536Dic3—42no—
537Dic3—42no—
538Dic3—82no—
539C23—34no—
540C23—64no—
541C23—64no—
542C23—64no—
543C23—62no—
544C23—62no—
545C23—62no—
546C23—64no—
547C23—64no—
548C23—64no—
549C23—62no—
550C23—62no—
551C23—62no—
552C23—122no—
553C23—122no—
554C23—122no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—122no—
560C23—122no—
561C23—122no—
562C23C22×D1211yescharacteristic
563C23C2×#23512yes—
564C23C2×D3×D411yescharacteristic
565C23C2×D3×D411yescharacteristic
566C23C2×D3×D411yescharacteristic
567C23C2×#23911yescharacteristic
568C23D3×#4112yes—
569C23D3×#4111yescharacteristic
570C23D3×#4111yescharacteristic
571C23#101012yes—
572C23#101012yes—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—34no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—632no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—64no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—64no—
614C2×C4—64no—
615C2×C4—64no—
616C2×C4—64no—
617C2×C4—122no—
618C2×C4—121no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4#88214yes—
635C2×C4#89514yes—
636C2×C3#239211yescharacteristic
637C2×C3#240611yescharacteristic
638C2×C3#241312yes—
639C2×C3#241512yes—
640C2×C3#243211yescharacteristic
641C2×C3#247512yes—
642C2×C3#247812yes—
643C2×C3#259911yescharacteristic
644C2×C3#261111yescharacteristic
645C2×C3#262111yescharacteristic
646C2×C3#284711yescharacteristic
647D3—44no—
648C22—64no—
649C22—68no—
650C22—124no—
651C22—124no—
652C22—122no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—122no—
657C22—122no—
658C22—124no—
659C22#547711yescharacteristic
660C22C2×#88211yescharacteristic
661C22C2×#89511yescharacteristic
662C22#552411yescharacteristic
663C22#552411yescharacteristic
664C22#552611yescharacteristic
665C22#552611yescharacteristic
666C22#552912yes—
667C22#553112yes—
668C22#554712yes—
669C22#555112yes—
670C22#555212yes—
671C22#558012yes—
672C22D3×#12111yescharacteristic
673C22D3×#12111yescharacteristic
674C22#562711yescharacteristic
675C22#562711yescharacteristic
676C22#565512yes—
677C22#565512yes—
678C22#567512yes—
679C22#569812yes—
680C22#570212yes—
681C22#570812yes—
682C4—24no—
683C4—24no—
684C4—42no—
685C4—44no—
686C4—64no—
687C4—82no—
688C4—122no—
689C4—124no—
690C4—122no—
691C4—242no—
692C3#1490711yescharacteristic, complemented, Sylow
693C2—124no—
694C2#7342711yescharacteristic
695C2#7348911yescharacteristic
696C2#7377611yescharacteristic
697C2#7404811yescharacteristic
698C2#7406311yescharacteristic
699C2#7419311yescharacteristic
700C2#7430811yescharacteristic
701C2#7501312yes—
702C2#7501712yes—
703C2#7549612yes—
704C2#7584312yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1434C2
prime radicalC24×D3C23
hypercenter#1434D3
hypocenterC3#14907
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#4928C2semidirect
4. C2×#4661C2semidirect
6. C3×#1434C2semidirect
9. #69411C2semidirect
10. #69410C2semidirect
11. #69405C2semidirect
53. #1434D3semidirect
692. C3#14907semidirect

Maximal subgroups

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

subgroupindex
C22×#8722
C2×#49282
C2×#46682
C2×#46612
C2×#46742
C3×#14342
#694292
#694272
#694112
#694102
#694052
#149073