Finite Group Databasebeta

← #10740391

Subgroups — #10740391

Order 768.

1890 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#885C211yesmaximal, characteristic
2C2×C3×#399C211yesmaximal, characteristic, complemented
3C2×#4992C211yesmaximal, characteristic, complemented
4C2×#4973C211yesmaximal, characteristic, complemented
5#71570C212yesmaximal, complemented
6#71564C212yesmaximal, complemented
7#71550C212yesmaximal
8#71548C212yesmaximal, complemented
9#71545C211yesmaximal, characteristic, complemented
10#69424C211yesmaximal, characteristic, complemented
11#69408C211yesmaximal, characteristic, complemented
12#15785—31nomaximal, Sylow
13C23×#67C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×#217C2212yes—
21C22×(C2×D3)⋊C4C2211yescharacteristic
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×(C2×D3)⋊C4C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C2×#885—24no—
28C2×#885—24no—
29C2×#885—24no—
30C3×#399—24no—
31C2×C3×#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4992—24no—
41#4973—24no—
42C22×#124—31no—
43C2×#399D311yescharacteristic, complemented
44C2×#397—31no—
45C2×#391—31no—
46#1446—31no—
47#1440—32no—
48#1439—32no—
49#1438—32no—
50#1437—31no—
51#1437—31no—
52#1436—32no—
53C25×C3C2311yescharacteristic
54C24×D3C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67—22no—
60C22×#67—22no—
61C22×#67—22no—
62C22×#67—24no—
63C22×#67—42no—
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2312yes—
68C23×Dic3C2311yescharacteristic
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—24no—
72C22×Dic6—42no—
73C22×Dic6D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—42no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—42no—
85C2×C3×C22⋊C4D414yes—
86C2×#217—22no—
87C2×#217—22no—
88C2×#217—22no—
89C2×#217—24no—
90C2×#217—42no—
91C2×#217—42no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—24no—
99C2×(C2×D3)⋊C4—24no—
100C2×(C2×D3)⋊C4—42no—
101C2×(C2×D3)⋊C4—42no—
102C2×(C2×D3)⋊C4—42no—
103C2×(C2×D3)⋊C4D414yes—
104C2×C4×Dic3—22no—
105C2×C4×Dic3—22no—
106C2×C4×Dic3—22no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3—42no—
109#885—232no—
110C3×#95—22no—
111C3⋊#95—22no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—24no—
116C3⋊#95—24no—
117C3⋊#95—42no—
118C3⋊#95—42no—
119C23×Q8—31no—
120C23×D4—31no—
121C2×#124—64no—
122C2×#124—64no—
123C2×#124—64no—
124C22×C4⋊C4C2×D311yescharacteristic
125C22×C4⋊C4C2×D312yes—
126C22×C22⋊C4—31no—
127C22×C22⋊C4—31no—
128C22×C22⋊C4—31no—
129C22×C22⋊C4—32no—
130C22×C22⋊C4—32no—
131C22×C22⋊C4—32no—
132C22×C22⋊C4C2×D312yes—
133C22×C22⋊C4C2×D311yescharacteristic
134C22×C42—31no—
135C22×C42—31no—
136#399—24no—
137#397—64no—
138#391—64no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95C2×D311yescharacteristic
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3C2×D412yes—
157C24×C3C2411yescharacteristic
158C23×D3—21no—
159C23×D3—21no—
160C23×D3—22no—
161C23×D3—22no—
162C23×D3—22no—
163C23×D3—22no—
164C23×D3—22no—
165C23×D3—22no—
166C23×D3C2×D412yes—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
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×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4#4114yes—
191C2×#67—28no—
192C2×#67—48no—
193C2×#67—48no—
194C2×#67—44no—
195C2×#67—44no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
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×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3C2×D412yes—
221C22×Dic3C2×D412yes—
222C22×Dic3C2×D412yes—
223C22×Dic3#4114yes—
224C22×Dic3#4114yes—
225C2×Dic6—28no—
226C2×Dic6—28no—
227C2×Dic6—44no—
228C2×Dic6—48no—
229C2×Dic6—48no—
230C3×C4⋊C4—48no—
231C3×C4⋊C4—44no—
232C3×C22⋊C4—28no—
233C3×C22⋊C4—48no—
234#217—28no—
235#217—48no—
236(C2×D3)⋊C4—28no—
237(C2×D3)⋊C4—28no—
238(C2×D3)⋊C4—48no—
239(C2×D3)⋊C4—48no—
240C4×Dic3—28no—
241C4×Dic3—44no—
242C25—31no—
243C25C22×D311yescharacteristic
244C22×Q8—34no—
245C22×Q8—64no—
246C22×Q8—62no—
247C22×Q8—62no—
248C22×Q8—122no—
249C22×D4—62no—
250C22×D4—64no—
251C22×D4—62no—
252C22×D4—62no—
253C22×D4—122no—
254C23×C4—31no—
255C23×C4—31no—
256C23×C4—31no—
257C23×C4—32no—
258C23×C4—32no—
259C23×C4C22×D312yes—
260C23×C4C22×D312yes—
261C23×C4C22×D311yescharacteristic
262C23×C4C22×D311yescharacteristic
263#124—632no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—42no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—34no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—64no—
288C2×C22⋊C4—122no—
289C2×C22⋊C4—122no—
290C2×C22⋊C4—122no—
291C2×C22⋊C4—122no—
292C2×C22⋊C4—122no—
293C2×C22⋊C4#6714yes—
294C2×C42—62no—
295C2×C42—62no—
296C2×C42—62no—
297C2×C42—62no—
298C2×C42—122no—
299#95—22no—
300#95—64no—
301#95—64no—
302#95—64no—
303#95—64no—
304#95—64no—
305#95—62no—
306#95—122no—
307#95—122no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—41no—
314C23×C3—42no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3C22×D411yescharacteristic
330C23×C3C22×D411yescharacteristic
331C23×C3C22×D411yescharacteristic
332C23×C3C2×#4112yes—
333C23×C3C2×#4112yes—
334C23×C3C2×#4112yes—
335C23×C3H(5,𝔽2)12yes—
336C23×C3H(5,𝔽2)12yes—
337C23×C32−1+412yes—
338C22×D3—22no—
339C22×D3—22no—
340C22×D3—22no—
341C22×D3—24no—
342C22×D3—24no—
343C22×D3—41no—
344C22×D3—42no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—42no—
351C22×D3—42no—
352C22×D3—42no—
353C22×D3—42no—
354C22×D3—42no—
355C22×D3—42no—
356C22×D3—42no—
357C22×D3—42no—
358C22×D3—42no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4#12014yes—
383#67—416no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—24no—
396C2×Dic3—41no—
397C2×Dic3—42no—
398C2×Dic3—41no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407C2×Dic3—42no—
408C2×Dic3—42no—
409C2×Dic3—42no—
410C2×Dic3—42no—
411C2×Dic3—42no—
412C2×Dic3#12314yes—
413Dic6—416no—
414C24—21no—
415C24—21no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—32no—
423C24—61no—
424C24—62no—
425C24—62no—
426C24—62no—
427C24—62no—
428C24—62no—
429C24—62no—
430C24—61no—
431C24C2×#6712yes—
432C24C23×D311yescharacteristic
433C2×Q8—68no—
434C2×Q8—68no—
435C2×Q8—124no—
436C2×Q8—128no—
437C2×Q8—128no—
438C2×D4—68no—
439C2×D4—124no—
440C2×D4—128no—
441C2×D4—128no—
442C2×D4—124no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—32no—
463C22×C4—34no—
464C22×C4—32no—
465C22×C4—32no—
466C22×C4—34no—
467C22×C4—61no—
468C22×C4—62no—
469C22×C4—61no—
470C22×C4—62no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4D3×D412yes—
492C22×C4D3×D412yes—
493C22×C4#23714yes—
494C22×C4C2×#6712yes—
495C22×C4C2×#6712yes—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—68no—
501C22⋊C4—68no—
502C22⋊C4—68no—
503C22⋊C4—128no—
504C22⋊C4—128no—
505C22⋊C4—128no—
506C42—68no—
507C42—124no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—44no—
512C22×C3—44no—
513C22×C3—82no—
514C22×C3—82no—
515C22×C3—82no—
516C22×C3—82no—
517C22×C3C2×#12011yescharacteristic
518C22×C3C2×#12311yescharacteristic
519C22×C3#52912yes—
520C22×C3#53211yescharacteristic
521C22×C3#53211yescharacteristic
522C22×C3#53411yescharacteristic
523C22×C3#53511yescharacteristic
524C22×C3#53611yescharacteristic
525C22×C3#53812yes—
526C22×C3#54212yes—
527C22×C3#54212yes—
528C22×C3#54612yes—
529C22×C3#54612yes—
530C22×C3#54612yes—
531C22×C3#54712yes—
532C22×C3#54812yes—
533C22×C3#55012yes—
534C22×C3#55112yes—
535C22×C3#55612yes—
536C22×C3#55912yes—
537C22×C3#56212yes—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—44no—
542C2×D3—44no—
543C2×D3—82no—
544C2×D3—82no—
545C2×D3—82no—
546C2×D3—82no—
547C3×C4—24no—
548C3×C4—42no—
549C3×C4—44no—
550C3×C4—82no—
551Dic3—24no—
552Dic3—42no—
553Dic3—42no—
554Dic3—44no—
555Dic3—82no—
556C23—22no—
557C23—24no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23—42no—
576C23—42no—
577C23—64no—
578C23—62no—
579C23—62no—
580C23—62no—
581C23—64no—
582C23—122no—
583C23—122no—
584C23—122no—
585C23—122no—
586C23—122no—
587C23—122no—
588C23—121no—
589C23—121no—
590C23—122no—
591C23—122no—
592C23—122no—
593C23—122no—
594C23—122no—
595C23—121no—
596C23—122no—
597C23—121no—
598C23C2×D3×D411yescharacteristic
599C23C2×D3×D411yescharacteristic
600C23C2×#23712yes—
601C23#100412yes—
602C23#100712yes—
603C23D3×#4112yes—
604C23D3×#4112yes—
605C23#100912yes—
606C23C22×#6711yescharacteristic
607Q8—1216no—
608D4—1216no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—34no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—62no—
634C2×C4—64no—
635C2×C4—62no—
636C2×C4—64no—
637C2×C4—64no—
638C2×C4—62no—
639C2×C4—62no—
640C2×C4—64no—
641C2×C4—64no—
642C2×C4—62no—
643C2×C4—62no—
644C2×C4—64no—
645C2×C4—122no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—121no—
652C2×C4—122no—
653C2×C4—121no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4—122no—
658C2×C4—122no—
659C2×C4—122no—
660C2×C4—122no—
661C2×C4#93714yes—
662C2×C3—82no—
663C2×C3#240511yescharacteristic
664C2×C3#243111yescharacteristic
665C2×C3#243211yescharacteristic
666C2×C3#244812yes—
667C2×C3#252112yes—
668C2×C3#255712yes—
669C2×C3#270012yes—
670C2×C3#271012yes—
671C2×C3#273012yes—
672D3—82no—
673C22—42no—
674C22—44no—
675C22—44no—
676C22—42no—
677C22—42no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—124no—
683C22—122no—
684C22—122no—
685C22—124no—
686C22—122no—
687C22—242no—
688C22—242no—
689C22—242no—
690C22—242no—
691C22#563112yes—
692C22#563312yes—
693C22#564012yes—
694C22#564112yes—
695C22#564312yes—
696C22#564712yes—
697C22#564912yes—
698C22D3×#12311yescharacteristic
699C22#565311yescharacteristic
700C22#565412yes—
701C22#565712yes—
702C22#565812yes—
703C22#566412yes—
704C22#571511yescharacteristic
705C22#571611yescharacteristic
706C22#571812yes—
707C22C2×#93711yescharacteristic
708C22#580612yes—
709C22#581611yescharacteristic
710C22#582912yes—
711C22#583211yescharacteristic
712C4—24no—
713C4—44no—
714C4—42no—
715C4—64no—
716C4—82no—
717C4—122no—
718C4—122no—
719C4—124no—
720C4—242no—
721C3#1578511yescharacteristic, complemented, Sylow
722C2—82no—
723C2—242no—
724C2#7802212yes—
725C2#7803412yes—
726C2#7807312yes—
727C2#7811712yes—
728C2#8275911yescharacteristic
729C2#8280012yes—
730C2#8286212yes—
731C2#8289311yescharacteristic
732C2#8318711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#399C2
prime radicalC23×#67C22
hypercenterC2×#399D3
hypocenterC3#15785
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. C2×C3×#399C2semidirect
3. C2×#4992C2semidirect
4. C2×#4973C2semidirect
5. #71570C2semidirect
6. #71564C2semidirect
8. #71548C2semidirect
9. #71545C2semidirect
10. #69424C2semidirect
11. #69408C2semidirect
43. C2×#399D3semidirect
721. C3#15785semidirect

Maximal subgroups

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

subgroupindex
C22×#8852
C2×C3×#3992
C2×#49922
C2×#49732
#715702
#715642
#715502
#715482
#715452
#694242
#694082
#157853