Finite Group Databasebeta

← #10724450

Subgroups — #10724450

Order 768.

1918 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic
2C2×#5220C211yesmaximal, characteristic
3C2×#5220C211yesmaximal, characteristic
4#71567C212yesmaximal, complemented
5#71547C212yesmaximal, complemented
6#71547C212yesmaximal, complemented
7#71544C212yesmaximal, complemented
8#71544C212yesmaximal, complemented
9#71536C212yesmaximal, complemented
10#14513—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×Dic3C2211yescharacteristic
13C24×Dic3C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C42C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×#217C2212yes—
21C22×#217C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C3×#390—24no—
24#5220—24no—
25#5220C418yescomplemented
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×#390D311yescharacteristic
36C2×#386—31no—
37C2×#386—31no—
38#1442—32no—
39#1439—32no—
40#1439—32no—
41#1436—32no—
42C22⋊#95—32no—
43C22⋊#95—32no—
44C25×C3C2311yescharacteristic
45C25×C3C2311yescharacteristic
46C22×C3×D4—41no—
47C22×C3×D4—41no—
48C22×C3×D4—41no—
49C22×C3×D4—41no—
50C22×C3×D4C2×C414yes—
51C22×C3×D4D414yes—
52C22×C3×D4Q814yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3—22no—
57C23×Dic3—24no—
58C23×Dic3—41no—
59C23×Dic3—41no—
60C23×Dic3—41no—
61C23×Dic3—41no—
62C23×Dic3—42no—
63C23×Dic3—42no—
64C23×Dic3—42no—
65C23×Dic3C2×C414yescomplemented
66C23×Dic3C2312yes—
67C23×Dic3C2312yes—
68C23×Dic3C2312yes—
69C23×Dic3C2312yes—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—41no—
72C2×C3×C4⋊C4—41no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—42no—
78C2×C3×C22⋊C4—42no—
79C2×C3×C42—22no—
80C2×C3×C42—41no—
81C2×C3×C42—41no—
82C2×#217—24no—
83C2×#217—22no—
84C2×#217—24no—
85C2×#217—42no—
86C2×#217—42no—
87C2×#217—41no—
88C2×#217—41no—
89C2×#217—42no—
90C2×#217—41no—
91C2×#217—41no—
92C2×#217C2×C418yes—
93C2×#217C2×C414yescomplemented
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—42no—
96C2×Dic3⋊C4—42no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—42no—
100C3⋊#95—42no—
101C3⋊#95—42no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95C2×C418yescomplemented
106C23×D4C2×D311yescharacteristic
107C24×C4—31no—
108C24×C4—31no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C22⋊C4—32no—
112C22×C22⋊C4—31no—
113C22×C22⋊C4—32no—
114C22×C22⋊C4—31no—
115C22×C22⋊C4C2×D312yes—
116C22×C22⋊C4C2×D312yes—
117C22×C42C2×D311yescharacteristic
118#390—24no—
119#386—38no—
120#386—64no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C24×C3—22no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—22no—
137C24×C3—21no—
138C24×C3—21no—
139C24×C3—21no—
140C24×C3—21no—
141C24×C3—22no—
142C24×C3—21no—
143C24×C3—21no—
144C24×C3—21no—
145C24×C3—21no—
146C24×C3C22×C412yes—
147C24×C3C22×C412yes—
148C24×C3C2×D412yes—
149C24×C3C2×D412yes—
150C24×C3C2×Q812yes—
151C24×C3C2×Q812yes—
152C24×C3C2411yescharacteristic
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—44no—
156C2×C3×D4—44no—
157C2×C3×D4—44no—
158C2×C3×D4—44no—
159C2×C3×D4C4⋊C4116yescomplemented
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
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—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4C22×C412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×Q812yes—
180C22×C3×C4#4114yes—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
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—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
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—48no—
207C22×Dic3—44no—
208C22×Dic3—44no—
209C22×Dic3—48no—
210C22×Dic3—48no—
211C22×Dic3—44no—
212C22×Dic3—44no—
213C22×Dic3—48no—
214C22×Dic3—44no—
215C22×Dic3—44no—
216C22×Dic3C22×C414yes—
217C22×Dic3C22×C414yes—
218C22×Dic3#4114yes—
219C3×C4⋊C4—44no—
220C3×C22⋊C4—48no—
221C3×C22⋊C4—48no—
222C3×C42—44no—
223#217—48no—
224#217—44no—
225#217—48no—
226#217—44no—
227Dic3⋊C4—48no—
228C25C22×D311yescharacteristic
229C25C22×D311yescharacteristic
230C22×D4—41no—
231C22×D4—41no—
232C22×D4—41no—
233C22×D4—41no—
234C22×D4Dic614yes—
235C22×D4C4×D314yes—
236C22×D4#6714yes—
237C23×C4—32no—
238C23×C4—32no—
239C23×C4—32no—
240C23×C4—34no—
241C23×C4—32no—
242C23×C4—64no—
243C23×C4—62no—
244C23×C4—121no—
245C23×C4—121no—
246C23×C4—121no—
247C23×C4—121no—
248C23×C4—122no—
249C23×C4—122no—
250C23×C4—122no—
251C23×C4C22×D312yes—
252C23×C4C22×D312yes—
253C23×C4C22×D311yescharacteristic
254C2×C4⋊C4—22no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—64no—
258C2×C4⋊C4—122no—
259C2×C4⋊C4—122no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—38no—
265C2×C22⋊C4—34no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—121no—
272C2×C22⋊C4—122no—
273C2×C22⋊C4—121no—
274C2×C22⋊C4—122no—
275C2×C22⋊C4—122no—
276C2×C22⋊C4—121no—
277C2×C22⋊C4—121no—
278C2×C42—22no—
279C2×C42—41no—
280C2×C42—41no—
281#95—38no—
282#95—64no—
283#95—64no—
284#95—122no—
285#95—122no—
286#95—122no—
287#95—122no—
288#95—122no—
289#95—122no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—22no—
293C23×C3—24no—
294C23×C3—22no—
295C23×C3—24no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—42no—
330C23×C3—41no—
331C23×C3—42no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3C2×C4⋊C414yes—
335C23×C3C2×C4⋊C414yes—
336C23×C3C23×C411yescharacteristic
337C23×C3C22×D411yescharacteristic
338C23×C3C22×Q811yescharacteristic
339C23×C3C2×#4112yes—
340C23×C3C2×#4112yes—
341C23×C3H(5,𝔽2)11yescharacteristic
342C23×C3H(5,𝔽2)11yescharacteristic
343C23×C3H(5,𝔽2)11yescharacteristic
344C23×C3H(5,𝔽2)11yescharacteristic
345C23×C3H(5,𝔽2)12yes—
346C23×C3H(5,𝔽2)12yes—
347C3×D4—416no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
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—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4C2×C4⋊C414yes—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—416no—
379C2×Dic3—416no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
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—42no—
398C2×Dic3—42no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—21no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24C2×Dic612yes—
416C24C2×Dic612yes—
417C24C2×C4×D312yes—
418C24C2×C4×D312yes—
419C24C2×#6712yes—
420C24C2×#6712yes—
421C24C23×D311yescharacteristic
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4Dic3⋊C4116yescomplemented
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—34no—
447C22×C4—34no—
448C22×C4—34no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
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—124no—
475C22×C4—128no—
476C22×C4—124no—
477C22×C4—128no—
478C22×C4—128no—
479C22×C4—124no—
480C22×C4—124no—
481C22×C4—124no—
482C22×C4—124no—
483C22×C4—128no—
484C22×C4C2×Dic612yes—
485C22×C4C2×C4×D312yes—
486C22×C4#23714yes—
487C22×C4C2×#6712yes—
488C4⋊C4—44no—
489C4⋊C4—128no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—124no—
493C22⋊C4—128no—
494C22⋊C4—128no—
495C22⋊C4—124no—
496C42—44no—
497C22×C3—24no—
498C22×C3—24no—
499C22×C3—28no—
500C22×C3—28no—
501C22×C3—44no—
502C22×C3—44no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—44no—
510C22×C3—44no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—44no—
516C22×C3—44no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3C22×C4⋊C411yescharacteristic
520C22×C3C4×#4112yes—
521C22×C3#51811yescharacteristic
522C22×C3#51811yescharacteristic
523C22×C3#51812yes—
524C22×C3#53411yescharacteristic
525C22×C3#53411yescharacteristic
526C22×C3#53812yes—
527C22×C3#53812yes—
528C22×C3#54012yes—
529C22×C3#54012yes—
530C22×C3#54311yescharacteristic
531C22×C3#54311yescharacteristic
532C22×C3#54612yes—
533C22×C3#54612yes—
534C22×C3#55111yescharacteristic
535C22×C3#55312yes—
536C22×C3#55412yes—
537C22×C3#55412yes—
538C22×C3#55911yescharacteristic
539C22×C3#56011yescharacteristic
540C22×C3#56012yes—
541C22×C3H(3,𝔽4)11yescharacteristic
542C3×C4—24no—
543C3×C4—44no—
544C3×C4—82no—
545Dic3—44no—
546Dic3—44no—
547Dic3—44no—
548Dic3—82no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—42no—
582C23—42no—
583C23—41no—
584C23—41no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23C2×Dic3⋊C414yes—
594C23C2×Dic3⋊C414yes—
595C23C22×Dic611yescharacteristic
596C23C22×C4×D311yescharacteristic
597C23C2×#23712yes—
598C23#100411yescharacteristic
599C23#100411yescharacteristic
600C23#100412yes—
601C23#100411yescharacteristic
602C23#100411yescharacteristic
603C23D3×#4112yes—
604C23#100912yes—
605C23C22×#6711yescharacteristic
606D4—416no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—64no—
628C2×C4—64no—
629C2×C4—64no—
630C2×C4—64no—
631C2×C4—64no—
632C2×C4—64no—
633C2×C4—64no—
634C2×C4—64no—
635C2×C4—64no—
636C2×C4—1216no—
637C2×C4—1216no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—122no—
645C2×C4—122no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4C2×Dic3⋊C414yes—
658C2×C3—44no—
659C2×C3—44no—
660C2×C3#231511yescharacteristic
661C2×C3#233312yes—
662C2×C3#234912yes—
663C2×C3#237411yescharacteristic
664C2×C3#237511yescharacteristic
665C2×C3#268411yescharacteristic
666C2×C3#268812yes—
667C2×C3#270911yescharacteristic
668C2×C3#274811yescharacteristic
669C2×C3#278012yes—
670C2×C3#280411yescharacteristic
671C22—28no—
672C22—28no—
673C22—24no—
674C22—24no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22#548311yescharacteristic
694C22#548311yescharacteristic
695C22#548611yescharacteristic
696C22#548611yescharacteristic
697C22C4×#23712yes—
698C22#553912yes—
699C22#554312yes—
700C22#554512yes—
701C22#560112yes—
702C22#561512yes—
703C22#561712yes—
704C22#562012yes—
705C22#567812yes—
706C22#567911yescharacteristic
707C22#568311yescharacteristic
708C22#571311yescharacteristic
709C22#571912yes—
710C22#572011yescharacteristic
711C22C22×Dic3⋊C411yescharacteristic
712C22#580411yescharacteristic
713C22#580411yescharacteristic
714C22#580612yes—
715C22#582912yes—
716C4—24no—
717C4—44no—
718C4—82no—
719C4—124no—
720C4—124no—
721C4—124no—
722C4—242no—
723C3#1451311yescharacteristic, complemented, Sylow
724C2—44no—
725C2—44no—
726C2#7629912yes—
727C2#7630811yescharacteristic
728C2#7640511yescharacteristic
729C2#7641312yes—
730C2#7642111yescharacteristic
731C2#7643411yescharacteristic
732C2#8209511yescharacteristic
733C2#8233412yes—
734C2#8268912yes—
735C2#8300511yescharacteristic
736C2#8318311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#390C2
socle, cosocleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3#14513
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
4. #71567C2semidirect
5. #71547C2semidirect
6. #71547C2semidirect
7. #71544C2semidirect
8. #71544C2semidirect
9. #71536C2semidirect
25. #5220C4semidirect
65. C23×Dic3C2×C4semidirect
93. C2×#217C2×C4semidirect
105. C3⋊#95C2×C4semidirect
159. C2×C3×D4C4⋊C4semidirect
428. C2×D4Dic3⋊C4semidirect
723. C3#14513semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3902
C2×#52202
#715672
#715472
#715442
#715362
#145133