Finite Group Databasebeta

← #10720107

Subgroups — #10720107

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#892C211yesmaximal, characteristic
2C2×#4668C212yesmaximal
3C2×#4664C211yesmaximal, characteristic
4C2×D3×#95C211yesmaximal, characteristic
5C3×#1442C211yesmaximal, characteristic, complemented
6#71567C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#95C212yesmaximal
8#69420C211yesmaximal, characteristic, complemented
9#69419C211yesmaximal, characteristic, complemented
10#69407C212yesmaximal, complemented
11#69406C212yesmaximal, complemented
12#14128—31nomaximal, Sylow
13C23×C4×D3C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×C4×Dic3C2212yes—
23C2×#892—24no—
24C2×#892—28no—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38#4668—28no—
39#4664C418yescomplemented
40D3×#95—24no—
41C22×#117—31no—
42C2×#385—32no—
43C2×#381—31no—
44C22×#95—31no—
45#1444—31no—
46#1444—31no—
47#1442—31no—
48#1442D311yescharacteristic, complemented
49#1436—32no—
50#1436—32no—
51C22⋊#95—32no—
52C24×D3C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C22×C4×D3—24no—
64C22×C4×D3—24no—
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×D3—24no—
72C22×C4×D3—24no—
73C22×C4×D3—42no—
74C22×C4×D3—42no—
75C22×C4×D3C2×C414yes—
76C22×C4×D3D414yes—
77C22×C4×D3Q814yes—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—22no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4C2×C418yes—
85C2×(C2×D3)⋊C4D418yes—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3—42no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—42no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3—24no—
97#892—232no—
98C3×#95—22no—
99C3×#95—24no—
100C3×#95—24no—
101C3×#95C2×C414yescomplemented
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106C3⋊#95—24no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C3⋊#95C2×C414yescomplemented
110C3⋊#95C2×C418yescomplemented
111C24×C4—31no—
112C24×C4—31no—
113C24×C4—31no—
114C2×#117—68no—
115C2×#117—64no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—32no—
118C22×C4⋊C4—32no—
119C22×C4⋊C4C2×D311yescharacteristic
120C22×C22⋊C4—32no—
121C22×C22⋊C4—32no—
122C22×C42—32no—
123#385—68no—
124#381—38no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—64no—
135C2×#95C2×D312yes—
136C2×#95C2×D312yes—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D311yescharacteristic
139C24×C3C2411yescharacteristic
140C23×D3—21no—
141C23×D3—21no—
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—22no—
145C23×D3—22no—
146C23×D3C22×C412yes—
147C23×D3C2×D412yes—
148C23×D3C2×D414yes—
149C23×D3C2×Q812yes—
150C23×D3#4114yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4C22×C414yes—
174C22×C3×C4C22×C412yes—
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×Q812yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3C22×C414yes—
203C22×Dic3C22×C412yes—
204C22×Dic3C2×D414yes—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×Q812yes—
207C22×Dic3#4112yes—
208C22×Dic3#4112yes—
209C2×C4×D3—216no—
210C2×C4×D3—28no—
211C2×C4×D3—216no—
212C2×C4×D3—28no—
213C2×C4×D3—28no—
214C2×C4×D3—28no—
215C2×C4×D3—48no—
216C2×C4×D3—48no—
217C2×C4×D3—48no—
218C2×C4×D3—48no—
219C2×C4×D3—44no—
220C2×C4×D3—44no—
221C2×C4×D3—44no—
222C2×C4×D3—44no—
223C2×C4×D3C4⋊C4116yescomplemented
224C3×C4⋊C4—28no—
225(C2×D3)⋊C4—216no—
226(C2×D3)⋊C4—216no—
227C4⋊Dic3—28no—
228C4⋊Dic3—48no—
229Dic3⋊C4—48no—
230C4×Dic3—216no—
231C25—31no—
232C23×C4—31no—
233C23×C4—34no—
234C23×C4—34no—
235C23×C4—34no—
236C23×C4—32no—
237C23×C4—32no—
238C23×C4—31no—
239C23×C4—31no—
240C23×C4—62no—
241C23×C4—62no—
242C23×C4—64no—
243C23×C4—64no—
244C23×C4—62no—
245C23×C4—62no—
246C23×C4—64no—
247C23×C4—64no—
248C23×C4—62no—
249C23×C4—62no—
250C23×C4—122no—
251C23×C4—122no—
252C23×C4C22×D312yes—
253C23×C4C22×D312yes—
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257#117—632no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4—122no—
267C2×C4⋊C4—122no—
268C2×C22⋊C4—38no—
269C2×C22⋊C4—38no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—64no—
274C2×C42—64no—
275C2×C42—64no—
276C2×C42—64no—
277#95—22no—
278#95—24no—
279#95—24no—
280#95—38no—
281#95—34no—
282#95—64no—
283#95—62no—
284#95—62no—
285#95—62no—
286#95—64no—
287#95—122no—
288#95—122no—
289#95C4×D314yescomplemented
290C23×C3C23×C411yescharacteristic
291C23×C3C22×D411yescharacteristic
292C23×C3C22×D412yes—
293C23×C3C22×Q811yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4111yescharacteristic
296C23×C3C2×#4112yes—
297C23×C3C2×#4111yescharacteristic
298C23×C3C2×#4111yescharacteristic
299C23×C3H(5,𝔽2)12yes—
300C23×C32−1+412yes—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—24no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—24no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—24no—
317C22×D3—24no—
318C22×D3—24no—
319C22×D3C2×C4⋊C414yes—
320C22×D3C4×D418yes—
321C22×D3#12014yes—
322C22×D3C22⋊Q818yes—
323C22×D3#12314yes—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4C2×C4⋊C414yes—
353C2×Dic3—24no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—42no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C2×Dic3—42no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3C2×C4⋊C414yes—
382C4×D3—232no—
383C4×D3—416no—
384C4×D3—416no—
385C24—32no—
386C24—34no—
387C24—32no—
388C24—32no—
389C24—34no—
390C24—61no—
391C24—61no—
392C24—61no—
393C24—61no—
394C24—62no—
395C24—62no—
396C24C23×D311yescharacteristic
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—32no—
422C22×C4—316no—
423C22×C4—34no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—68no—
428C22×C4—68no—
429C22×C4—616no—
430C22×C4—616no—
431C22×C4—68no—
432C22×C4—68no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—128no—
456C22×C4—128no—
457C22×C4—128no—
458C22×C4—128no—
459C22×C4—124no—
460C22×C4—124no—
461C22×C4—124no—
462C22×C4—124no—
463C22×C4C2×C4×D314yes—
464C22×C4C2×C4×D312yes—
465C22×C4D3×D414yes—
466C22×C4D3×D412yes—
467C22×C4#23712yes—
468C22×C4D3×Q812yes—
469C22×C4#23912yes—
470C4⋊C4—28no—
471C4⋊C4—68no—
472C4⋊C4—128no—
473C4⋊C4—128no—
474C22⋊C4—616no—
475C22⋊C4—616no—
476C42—616no—
477C22×C3C22×C4⋊C411yescharacteristic
478C22×C3C2×C4×D412yes—
479C22×C3C4×#4111yescharacteristic
480C22×C3C4×#4111yescharacteristic
481C22×C3#52012yes—
482C22×C3C2×#12011yescharacteristic
483C22×C3C2×C22⋊Q812yes—
484C22×C3C2×#12311yescharacteristic
485C22×C3#52511yescharacteristic
486C22×C3#52511yescharacteristic
487C22×C3#53311yescharacteristic
488C22×C3#53311yescharacteristic
489C22×C3#53511yescharacteristic
490C22×C3#53611yescharacteristic
491C22×C3#54412yes—
492C22×C3#54612yes—
493C22×C3#54612yes—
494C22×C3#54712yes—
495C22×C3#54712yes—
496C22×C3#55112yes—
497C22×C3#55112yes—
498C22×C3#55212yes—
499C22×C3#55212yes—
500C2×D3—24no—
501C2×D3—28no—
502C2×D3—28no—
503C2×D3—24no—
504C2×D3—24no—
505C2×D3—28no—
506C2×D3—28no—
507C2×D3—24no—
508C2×D3—24no—
509C2×D3—24no—
510C2×D3#38518yes—
511C3×C4—24no—
512C3×C4—44no—
513C3×C4—42no—
514C3×C4—44no—
515C3×C4—42no—
516Dic3—24no—
517Dic3—44no—
518Dic3—42no—
519Dic3—44no—
520Dic3—42no—
521C23—34no—
522C23—34no—
523C23—38no—
524C23—34no—
525C23—38no—
526C23—64no—
527C23—64no—
528C23—62no—
529C23—62no—
530C23—64no—
531C23—62no—
532C23—64no—
533C23—64no—
534C23—64no—
535C23—64no—
536C23—62no—
537C23—64no—
538C23—62no—
539C23—62no—
540C23—62no—
541C23—64no—
542C23—62no—
543C23—64no—
544C23C22×C4×D311yescharacteristic
545C23C2×D3×D411yescharacteristic
546C23C2×D3×D412yes—
547C23C2×#23711yescharacteristic
548C23C2×D3×Q811yescharacteristic
549C23C2×#23911yescharacteristic
550C23D3×#4112yes—
551C23D3×#4111yescharacteristic
552C23D3×#4111yescharacteristic
553C23#100912yes—
554C23#101012yes—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—34no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—632no—
585C2×C4—62no—
586C2×C4—64no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—1216no—
600C2×C4—1216no—
601C2×C4—121no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—121no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4D3×C4⋊C414yes—
616C2×C3C2×#38511yescharacteristic
617C2×C3#229611yescharacteristic
618C2×C3#231711yescharacteristic
619C2×C3#235912yes—
620C2×C3#235912yes—
621C2×C3#240411yescharacteristic
622C2×C3#240611yescharacteristic
623C2×C3#243812yes—
624C2×C3#243812yes—
625C2×C3#248211yescharacteristic
626C2×C3#248411yescharacteristic
627D3—28no—
628C22—38no—
629C22—68no—
630C22—64no—
631C22—68no—
632C22—64no—
633C22—68no—
634C22—64no—
635C22—64no—
636C22—68no—
637C22—64no—
638C22—64no—
639C22C2×D3×C4⋊C411yescharacteristic
640C22#552312yes—
641C22C4×#23711yescharacteristic
642C22C4×D3×D412yes—
643C22C4×#23911yescharacteristic
644C22D3×#12011yescharacteristic
645C22#560611yescharacteristic
646C22#561112yes—
647C22D3×C22⋊Q812yes—
648C22#562711yescharacteristic
649C22#563012yes—
650C22D3×#12311yescharacteristic
651C22#565512yes—
652C22#565812yes—
653C22#567411yescharacteristic
654C22#567511yescharacteristic
655C22#567612yes—
656C22#568811yescharacteristic
657C22#568912yes—
658C22#569412yes—
659C22#570512yes—
660C22#570812yes—
661C22#572811yescharacteristic
662C4—24no—
663C4—44no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—64no—
668C4—124no—
669C4—122no—
670C4—124no—
671C4—122no—
672C3#1412811yescharacteristic, complemented, Sylow
673C2—68no—
674C2#7422511yescharacteristic
675C2D3×#38511yescharacteristic
676C2#7530812yes—
677C2#7583812yes—
678C2#7596311yescharacteristic
679C2#7686311yescharacteristic
680C2#7710711yescharacteristic
681C2#7767611yescharacteristic
682C2#7794912yes—
683C2#7834212yes—
684C2#7834511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1442C2
socle, cosocleC24×C3C24
prime radicalC24×D3C23
hypercenter#1442D3
hypocenterC3#14128
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
5. C3×#1442C2semidirect
6. #71567C2semidirect
8. #69420C2semidirect
9. #69419C2semidirect
10. #69407C2semidirect
11. #69406C2semidirect
39. #4664C4semidirect
48. #1442D3semidirect
101. C3×#95C2×C4semidirect
109. C3⋊#95C2×C4semidirect
110. C3⋊#95C2×C4semidirect
223. C2×C4×D3C4⋊C4semidirect
289. #95C4×D3semidirect
672. C3#14128semidirect

Maximal subgroups

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

subgroupindex
C22×#8922
C2×#46682
C2×#46642
C2×D3×#952
C3×#14422
#715672
(C2×D3)⋊#952
#694202
#694192
#694072
#694062
#141283