Finite Group Databasebeta

← #10732144

Subgroups — #10732144

Order 768.

1746 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4972C211yesmaximal, characteristic, complemented
2C2×#4959C212yesmaximal
3C2×#4957C211yesmaximal, characteristic
4C2×#4658C212yesmaximal, complemented
5C3×#1439C211yesmaximal, characteristic, complemented
6#71570C211yesmaximal, characteristic, complemented
7#71546C211yesmaximal, characteristic, complemented
8#71545C212yesmaximal, complemented
9#69427C211yesmaximal, characteristic, complemented
10#69408C212yesmaximal, complemented
11Dic3×#95C211yesmaximal, characteristic, complemented
12#15170—31nomaximal, Sylow
13C23×#67C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2212yes—
26C22×C4×Dic3C2212yes—
27C22×C4×Dic3C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36#4972—24no—
37#4959—28no—
38#4957—24no—
39#4658—28no—
40C2×#400—32no—
41C2×#394—32no—
42C2×#390—31no—
43C2×#389—31no—
44#1439—31no—
45#1439—31no—
46#1439—31no—
47#1439D311yescharacteristic, complemented
48#1437—32no—
49#1437—32no—
50C4×#95—31no—
51C25×C3C2311yescharacteristic
52C24×D3C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C22×#67—22no—
58C22×#67—22no—
59C22×#67—22no—
60C22×#67—24no—
61C22×#67—42no—
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—42no—
75C2×#217—22no—
76C2×#217—24no—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—42no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—42no—
90C2×(C2×D3)⋊C4—42no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×C4×Dic3—24no—
99C2×C4×Dic3—42no—
100C2×C4×Dic3—42no—
101C2×C4×Dic3—42no—
102C2×C4×Dic3D414yes—
103C3×#95—24no—
104C3×#95—22no—
105C3⋊#95—24no—
106C3⋊#95—22no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—24no—
110C3⋊#95D418yes—
111C23×D4—31no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—31no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4C2×D311yescharacteristic
116C22×C22⋊C4—31no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4—31no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4—32no—
121C22×C22⋊C4—32no—
122C22×C22⋊C4C2×D311yescharacteristic
123C22×C22⋊C4C2×D312yes—
124C22×C42—31no—
125C22×C42—32no—
126#400—68no—
127#394—68no—
128#390—64no—
129#389—64no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—31no—
135C2×#95—32no—
136C2×#95C2×D312yes—
137C2×#95C2×D311yescharacteristic
138C24×C3—21no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—21no—
143C24×C3—21no—
144C24×C3—21no—
145C24×C3—21no—
146C24×C3—22no—
147C24×C3—21no—
148C24×C3#4112yes—
149C24×C3C2411yescharacteristic
150C23×D3—21no—
151C23×D3—22no—
152C23×D3—21no—
153C23×D3—21no—
154C23×D3—22no—
155C23×D3—22no—
156C23×D3—21no—
157C23×D3—21no—
158C23×D3—22no—
159C23×D3—21no—
160C23×D3#4112yes—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
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—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4#4112yes—
184C22×C3×C4#4112yes—
185C2×#67—28no—
186C2×#67—48no—
187C2×#67—48no—
188C2×#67—44no—
189C2×#67—44no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
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—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—22no—
213C22×Dic3C2×D414yes—
214C22×Dic3C2×D412yes—
215C22×Dic3C2×D414yes—
216C22×Dic3C2×D412yes—
217C22×Dic3#4114yes—
218C22×Dic3#4114yes—
219C3×C4⋊C4—44no—
220C3×C22⋊C4—48no—
221C3×C22⋊C4—44no—
222#217—48no—
223#217—44no—
224(C2×D3)⋊C4—48no—
225(C2×D3)⋊C4—48no—
226(C2×D3)⋊C4—44no—
227(C2×D3)⋊C4—44no—
228C4⋊Dic3—28no—
229Dic3⋊C4—48no—
230Dic3⋊C4—44no—
231C4×Dic3—44no—
232C4×Dic3—48no—
233C25—31no—
234C25C22×D311yescharacteristic
235C22×D4—62no—
236C22×D4—64no—
237C22×D4—62no—
238C22×D4—62no—
239C22×D4—122no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4—32no—
243C23×C4—32no—
244C23×C4—31no—
245C23×C4C22×D312yes—
246C23×C4C22×D311yescharacteristic
247C23×C4C22×D312yes—
248C23×C4C22×D311yescharacteristic
249C2×C4⋊C4—22no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—64no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—122no—
278C2×C22⋊C4—122no—
279C2×C22⋊C4—122no—
280C2×C42—34no—
281C2×C42—64no—
282C2×C42—122no—
283C2×C42—122no—
284C2×C42—122no—
285#95—24no—
286#95—22no—
287#95—38no—
288#95—64no—
289#95—64no—
290#95—64no—
291#95—62no—
292#95—64no—
293C23×C3—22no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—41no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C22×D411yescharacteristic
317C23×C3C22×D412yes—
318C23×C3C2×#4111yescharacteristic
319C23×C3C2×#4111yescharacteristic
320C23×C3C2×#4111yescharacteristic
321C23×C3C2×#4111yescharacteristic
322C23×C3C2×#4112yes—
323C23×C3C2×#4112yes—
324C23×C3H(5,𝔽2)12yes—
325C23×C3H(5,𝔽2)11yescharacteristic
326C23×C32−1+411yescharacteristic
327C22×D3—22no—
328C22×D3—24no—
329C22×D3—24no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—42no—
336C22×D3—41no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—41no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—41no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4#12414yes—
374#67—416no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—24no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3#12414yes—
403C2×Dic3#12714yes—
404C24—21no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24—22no—
409C24—21no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24—21no—
414C24—32no—
415C24—61no—
416C24—61no—
417C24—62no—
418C24—61no—
419C24—61no—
420C24—62no—
421C24—61no—
422C24—62no—
423C24—62no—
424C24—61no—
425C24#23712yes—
426C24C23×D311yescharacteristic
427C2×D4—68no—
428C2×D4—124no—
429C2×D4—128no—
430C2×D4—128no—
431C2×D4—124no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—34no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—34no—
456C22×C4—34no—
457C22×C4—34no—
458C22×C4—61no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
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×C4—62no—
476C22×C4—61no—
477C22×C4—62no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—62no—
481C22×C4#23512yes—
482C22×C4D3×D414yes—
483C22×C4D3×D412yes—
484C22×C4#23712yes—
485C4⋊C4—44no—
486C4⋊C4—68no—
487C4⋊C4—128no—
488C4⋊C4—124no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—128no—
492C22⋊C4—124no—
493C22⋊C4—124no—
494C22⋊C4—128no—
495C22⋊C4—128no—
496C22⋊C4—124no—
497C42—124no—
498C42—128no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—44no—
503C22×C3—44no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3—82no—
509C22×C3—82no—
510C22×C3#52511yescharacteristic
511C22×C3#52511yescharacteristic
512C22×C3C2×#12411yescharacteristic
513C22×C3C2×#12411yescharacteristic
514C22×C3#52912yes—
515C22×C3#52912yes—
516C22×C3C2×#12711yescharacteristic
517C22×C3#53212yes—
518C22×C3#53212yes—
519C22×C3#53411yescharacteristic
520C22×C3#53711yescharacteristic
521C22×C3#53812yes—
522C22×C3#53912yes—
523C22×C3#54612yes—
524C22×C3#54612yes—
525C22×C3#54612yes—
526C22×C3#54812yes—
527C22×C3#55211yescharacteristic
528C22×C3#55211yescharacteristic
529C22×C3#55311yescharacteristic
530C22×C3#55312yes—
531C22×C3#55312yes—
532C22×C3#55311yescharacteristic
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—44no—
537C2×D3—44no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—82no—
543C2×D3—82no—
544C3×C4—24no—
545C3×C4—44no—
546C3×C4—81no—
547C3×C4—82no—
548Dic3—24no—
549Dic3—24no—
550Dic3—44no—
551Dic3—44no—
552Dic3—81no—
553C23—24no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—62no—
577C23—64no—
578C23—62no—
579C23—62no—
580C23—64no—
581C23—122no—
582C23—122no—
583C23—122no—
584C23—122no—
585C23—121no—
586C23—121no—
587C23—122no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—122no—
592C23—122no—
593C23—121no—
594C23—122no—
595C23—122no—
596C23—122no—
597C23—121no—
598C23—121no—
599C23C2×#23511yescharacteristic
600C23C2×D3×D412yes—
601C23C2×D3×D411yescharacteristic
602C23C2×#23711yescharacteristic
603C23C2×#23711yescharacteristic
604C23#100412yes—
605C23#100711yescharacteristic
606C23D3×#4112yes—
607C23D3×#4112yes—
608C23D3×#4111yescharacteristic
609C23#100911yescharacteristic
610D4—1216no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—34no—
619C2×C4—34no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—64no—
637C2×C4—64no—
638C2×C4—62no—
639C2×C4—62no—
640C2×C4—64no—
641C2×C4—62no—
642C2×C4—64no—
643C2×C4—62no—
644C2×C4—64no—
645C2×C4—64no—
646C2×C4—64no—
647C2×C4—64no—
648C2×C4—64no—
649C2×C4—64no—
650C2×C4—122no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4—122no—
658C2×C4—121no—
659C2×C4—121no—
660C2×C4—121no—
661C2×C4—122no—
662C2×C4—122no—
663C2×C4#88514yes—
664C2×C3—82no—
665C2×C3#239811yescharacteristic
666C2×C3#253911yescharacteristic
667C2×C3#254011yescharacteristic
668C2×C3#255812yes—
669C2×C3#255812yes—
670C2×C3#255912yes—
671C2×C3#255912yes—
672C2×C3#256211yescharacteristic
673C2×C3#259111yescharacteristic
674C2×C3#263911yescharacteristic
675C2×C3#285811yescharacteristic
676D3—82no—
677C22—42no—
678C22—44no—
679C22—42no—
680C22—42no—
681C22—44no—
682C22—82no—
683C22—81no—
684C22—81no—
685C22—82no—
686C22—81no—
687C22—81no—
688C22—124no—
689C22—124no—
690C22—122no—
691C22—122no—
692C22—122no—
693C22—241no—
694C22—241no—
695C22—241no—
696C22—242no—
697C22—241no—
698C22—242no—
699C22C2×#88511yescharacteristic
700C22#551311yescharacteristic
701C22#553011yescharacteristic
702C22#554012yes—
703C22#554812yes—
704C22#558911yescharacteristic
705C22#559112yes—
706C22#559612yes—
707C22#560112yes—
708C22#560611yescharacteristic
709C22#561212yes—
710C22#561811yescharacteristic
711C22#562911yescharacteristic
712C22#563312yes—
713C22#563812yes—
714C22#563911yescharacteristic
715C22#564912yes—
716C22#565412yes—
717C22#565611yescharacteristic
718C22D3×#12411yescharacteristic
719C22#567812yes—
720C22#570612yes—
721C22D3×#12711yescharacteristic
722C4—24no—
723C4—44no—
724C4—64no—
725C4—64no—
726C4—81no—
727C4—82no—
728C4—124no—
729C4—124no—
730C4—241no—
731C3#1517011yescharacteristic, complemented, Sylow
732C2—82no—
733C2—242no—
734C2#7394011yescharacteristic
735C2#7447512yes—
736C2#7477211yescharacteristic
737C2#7588812yes—
738C2#7642411yescharacteristic
739C2#7675811yescharacteristic
740C2#7687211yescharacteristic
741C2#7695112yes—
742C2#7731912yes—
743C2#7746011yescharacteristic
744C2#7803611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1439C2
prime radicalC23×#67C22
hypercenter#1439D3
hypocenterC3#15170
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C2×#4972C2semidirect
4. C2×#4658C2semidirect
5. C3×#1439C2semidirect
6. #71570C2semidirect
7. #71546C2semidirect
8. #71545C2semidirect
9. #69427C2semidirect
10. #69408C2semidirect
11. Dic3×#95C2semidirect
47. #1439D3semidirect
731. C3#15170semidirect

Maximal subgroups

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

subgroupindex
C2×#49722
C2×#49592
C2×#49572
C2×#46582
C3×#14392
#715702
#715462
#715452
#694272
#694082
Dic3×#952
#151703