Finite Group Databasebeta

← #10740396

Subgroups — #10740396

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×C3×#124C211yesmaximal, characteristic
2C2×#5229C211yesmaximal, characteristic, complemented
3C2×#4949C211yesmaximal, characteristic, complemented
4C2×#4953C211yesmaximal, characteristic, complemented
5#72134C211yesmaximal, characteristic, complemented
6#72131C212yesmaximal
7#71549C212yesmaximal, complemented
8#71547C212yesmaximal, complemented
9#71545C211yesmaximal, characteristic, complemented
10#71545C211yesmaximal, characteristic, complemented
11#71536C212yesmaximal, complemented
12#15785—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×D4C2211yescharacteristic
15C2×C3×#124—24no—
16C2×C3×#124—24no—
17C2×C3×#124—24no—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×Dic3⋊C4C2212yes—
29C22×C4×Dic3C2211yescharacteristic
30#5229—24no—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4953—24no—
41#4949—24no—
42C22×#124D311yescharacteristic
43C2×#399—31no—
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
54C25×C3C2311yescharacteristic
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—24no—
58C22×C3×Q8—42no—
59C22×C3×Q8D414yes—
60C22×C3×D4—22no—
61C22×C3×D4—22no—
62C22×C3×D4—22no—
63C22×C3×D4—24no—
64C22×C3×D4—42no—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×Dic3C2312yes—
71C23×Dic3C2312yes—
72C23×Dic3C2312yes—
73C23×Dic3C2312yes—
74C3×#124—232no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4D414yes—
86C2×C3×C22⋊C4D414yes—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×#217—24no—
91C2×#217—24no—
92C2×#217—24no—
93C2×#217—24no—
94C2×#217—24no—
95C2×#217—24no—
96C2×#217—42no—
97C2×#217—42no—
98C2×#217—42no—
99C2×#217—42no—
100C2×#217—42no—
101C2×#217—42no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—22no—
105C2×Dic3⋊C4—24no—
106C2×Dic3⋊C4—24no—
107C2×Dic3⋊C4—42no—
108C2×C4×Dic3—22no—
109C2×C4×Dic3—42no—
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×Q8C2×D311yescharacteristic
120C23×D4C2×D311yescharacteristic
121C2×#124—24no—
122C2×#124—24no—
123C2×#124—24no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4—32no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—32no—
130C22×C22⋊C4C2×D311yescharacteristic
131C22×C22⋊C4C2×D311yescharacteristic
132C22×C22⋊C4C2×D311yescharacteristic
133C22×C22⋊C4C2×D311yescharacteristic
134C22×C42—31no—
135C22×C42C2×D311yescharacteristic
136#399—64no—
137#397—64no—
138#391—64no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—31no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—32no—
148C24×C3—21no—
149C24×C3—21no—
150C24×C3—21no—
151C24×C3—21no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—22no—
157C24×C3—22no—
158C24×C3—22no—
159C24×C3—22no—
160C24×C3—22no—
161C24×C3—22no—
162C24×C3—22no—
163C24×C3—22no—
164C24×C3C2×D412yes—
165C24×C3C2×D412yes—
166C24×C3C2411yescharacteristic
167C2×C3×Q8—28no—
168C2×C3×Q8—28no—
169C2×C3×Q8—44no—
170C2×C3×Q8—48no—
171C2×C3×Q8—48no—
172C2×C3×D4—28no—
173C2×C3×D4—48no—
174C2×C3×D4—48no—
175C2×C3×D4—44no—
176C2×C3×D4—44no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4#4114yes—
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×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—22no—
221C22×Dic3—22no—
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—22no—
225C22×Dic3—22no—
226C22×Dic3—22no—
227C22×Dic3—22no—
228C22×Dic3#4114yes—
229C22×Dic3#4114yes—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—28no—
233C3×C22⋊C4—28no—
234C3×C42—28no—
235#217—48no—
236#217—48no—
237#217—48no—
238#217—48no—
239C4⋊Dic3—44no—
240Dic3⋊C4—48no—
241C4×Dic3—44no—
242C25C22×D311yescharacteristic
243C25C22×D311yescharacteristic
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—42no—
248C22×Q8#6714yes—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—24no—
252C22×D4—22no—
253C22×D4—42no—
254C23×C4—32no—
255C23×C4—32no—
256C23×C4—32no—
257C23×C4—32no—
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C23×C4C22×D311yescharacteristic
262C23×C4C22×D311yescharacteristic
263#124—232no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—64no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—122no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—122no—
287C2×C22⋊C4—122no—
288C2×C22⋊C4—122no—
289C2×C22⋊C4—122no—
290C2×C22⋊C4—122no—
291C2×C22⋊C4—122no—
292C2×C22⋊C4#6714yes—
293C2×C22⋊C4#6714yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—62no—
298C2×C42—122no—
299#95—64no—
300#95—64no—
301#95—64no—
302#95—64no—
303#95—62no—
304#95—64no—
305#95—62no—
306#95—122no—
307#95—122no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—42no—
340C23×C3—42no—
341C23×C3—42no—
342C23×C3—42no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3—42no—
346C23×C3—42no—
347C23×C3—42no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3C22×D411yescharacteristic
351C23×C3C22×D411yescharacteristic
352C23×C3C22×D411yescharacteristic
353C23×C3C2×#4112yes—
354C23×C3C2×#4112yes—
355C23×C3C2×#4112yes—
356C23×C3H(5,𝔽2)12yes—
357C23×C3H(5,𝔽2)12yes—
358C23×C32−1+412yes—
359C3×Q8—416no—
360C3×D4—416no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4#12014yes—
385C2×C3×C4#12314yes—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—24no—
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—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—42no—
413C2×Dic3—42no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—21no—
427C24—21no—
428C24—21no—
429C24—21no—
430C24C2×#6712yes—
431C24C2×#6712yes—
432C24C23×D311yescharacteristic
433C2×Q8—28no—
434C2×Q8—28no—
435C2×Q8—44no—
436C2×Q8—48no—
437C2×Q8—48no—
438C2×D4—28no—
439C2×D4—48no—
440C2×D4—48no—
441C2×D4—44no—
442C2×D4—44no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—34no—
461C22×C4—34no—
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—62no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4#23714yes—
489C22×C4C2×#6712yes—
490C22×C4C2×#6712yes—
491C22×C4C2×#6712yes—
492C22×C4C2×#6712yes—
493C22×C4C2×#6712yes—
494C22×C4C2×#6712yes—
495C22×C4C2×#6712yes—
496C4⋊C4—128no—
497C4⋊C4—124no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—128no—
503C22⋊C4—128no—
504C22⋊C4—128no—
505C22⋊C4—128no—
506C42—28no—
507C42—124no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3—44no—
517C22×C3—44no—
518C22×C3—82no—
519C22×C3—82no—
520C22×C3—82no—
521C22×C3—82no—
522C22×C3—82no—
523C22×C3—82no—
524C22×C3—82no—
525C22×C3—82no—
526C22×C3C2×#12011yescharacteristic
527C22×C3C2×#12311yescharacteristic
528C22×C3#52912yes—
529C22×C3#53211yescharacteristic
530C22×C3#53211yescharacteristic
531C22×C3#53411yescharacteristic
532C22×C3#53511yescharacteristic
533C22×C3#53611yescharacteristic
534C22×C3#53812yes—
535C22×C3#54212yes—
536C22×C3#54212yes—
537C22×C3#54612yes—
538C22×C3#54612yes—
539C22×C3#54612yes—
540C22×C3#54712yes—
541C22×C3#54812yes—
542C22×C3#55012yes—
543C22×C3#55112yes—
544C22×C3#55612yes—
545C22×C3#55912yes—
546C22×C3#56212yes—
547C3×C4—24no—
548C3×C4—24no—
549C3×C4—42no—
550C3×C4—42no—
551C3×C4—42no—
552Dic3—44no—
553Dic3—44no—
554Dic3—82no—
555Dic3—82no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—22no—
560C23—24no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—41no—
590C23—42no—
591C23—41no—
592C23—42no—
593C23—42no—
594C23—41no—
595C23—41no—
596C23—42no—
597C23—42no—
598C23C2×#23712yes—
599C23#100412yes—
600C23D3×#4112yes—
601C23D3×#4112yes—
602C23#100912yes—
603C23#101012yes—
604C23C22×#6711yescharacteristic
605C23C22×#6711yescharacteristic
606C23C22×#6711yescharacteristic
607Q8—416no—
608D4—416no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—64no—
633C2×C4—64no—
634C2×C4—64no—
635C2×C4—64no—
636C2×C4—64no—
637C2×C4—64no—
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×C4—122no—
658C2×C4—122no—
659C2×C4—122no—
660C2×C4#93514yes—
661C2×C4#95314yes—
662C2×C3—82no—
663C2×C3—82no—
664C2×C3#240511yescharacteristic
665C2×C3#243111yescharacteristic
666C2×C3#243211yescharacteristic
667C2×C3#244812yes—
668C2×C3#252112yes—
669C2×C3#255712yes—
670C2×C3#270012yes—
671C2×C3#271012yes—
672C2×C3#273012yes—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22#566912yes—
692C22#567112yes—
693C22#567812yes—
694C22#567912yes—
695C22#568012yes—
696C22#568112yes—
697C22#568212yes—
698C22#568412yes—
699C22C2×#93511yescharacteristic
700C22#580411yescharacteristic
701C22#580612yes—
702C22#580612yes—
703C22#582412yes—
704C22#582511yescharacteristic
705C22#582912yes—
706C22#583011yescharacteristic
707C22#583112yes—
708C22#583211yescharacteristic
709C22#583211yescharacteristic
710C22#583412yes—
711C22C2×#95311yescharacteristic
712C4—24no—
713C4—24no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—124no—
718C4—124no—
719C4—242no—
720C4—242no—
721C3#1578511yescharacteristic, complemented, Sylow
722C2—82no—
723C2—82no—
724C2#8296312yes—
725C2#8296812yes—
726C2#8298512yes—
727C2#8298612yes—
728C2#8299912yes—
729C2#8300612yes—
730C2#8404711yescharacteristic
731C2#8409211yescharacteristic
732C2#8409311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#124C2
prime radicalC23×C3×D4C22
hypercenterC22×#124D3
hypocenterC3#15785
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
2. C2×#5229C2semidirect
3. C2×#4949C2semidirect
4. C2×#4953C2semidirect
5. #72134C2semidirect
7. #71549C2semidirect
8. #71547C2semidirect
9. #71545C2semidirect
10. #71545C2semidirect
11. #71536C2semidirect
721. C3#15785semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1242
C2×#52292
C2×#49492
C2×#49532
#721342
#721312
#715492
#715472
#715452
#715362
#157853