Finite Group Databasebeta

← #10728203

Subgroups — #10728203

Order 768.

1794 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4992C211yesmaximal, characteristic, complemented
2C2×#4987C211yesmaximal, characteristic, complemented
3C2×#4665C211yesmaximal, characteristic
4C2×#4666C211yesmaximal, characteristic, complemented
5C3×#1445C211yesmaximal, characteristic, complemented
6#69429C212yesmaximal, complemented
7#69427C211yesmaximal, characteristic, complemented
8#69426C212yesmaximal
9#69415C211yesmaximal, characteristic, complemented
10#69408C212yesmaximal, complemented
11#69401C212yesmaximal, complemented
12#14842—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×(C2×D3)⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4×Dic3C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36#4992—24no—
37#4987—24no—
38#4666—24no—
39#4665—24no—
40C2×#397—31no—
41C2×#386—31no—
42C2×#380—31no—
43C2×C4×C22⋊C4—31no—
44#1445D311yescharacteristic, complemented
45#1442—31no—
46#1440—32no—
47#1439—31no—
48#1438—32no—
49#1437—32no—
50#1434—32no—
51C24×D3C2311yescharacteristic
52C24×D3C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C22×D12—22no—
63C22×D12—22no—
64C22×D12—22no—
65C22×D12—24no—
66C22×D12—42no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3—24no—
70C22×C4×D3—42no—
71C22×C4×D3D414yes—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—42no—
90C2×(C2×D3)⋊C4—42no—
91C2×(C2×D3)⋊C4—42no—
92C2×(C2×D3)⋊C4—42no—
93C2×(C2×D3)⋊C4—42no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—42no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3—24no—
98C2×C4×Dic3—42no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—24no—
102C3×#95—42no—
103C3⋊#95—24no—
104C3⋊#95—22no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C3⋊#95D414yescomplemented
110C23×D4—31no—
111C24×C4—31no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C22⋊C4—31no—
115C22×C22⋊C4—31no—
116C22×C22⋊C4—32no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4—32no—
121C22×C42—32no—
122#397—64no—
123#386—64no—
124#380—64no—
125C4×C22⋊C4—64no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95C2×D312yes—
134C2×#95C2×D311yescharacteristic
135C2×#95C2×D311yescharacteristic
136C2×#95C2×D312yes—
137C24×C3C2411yescharacteristic
138C23×D3—21no—
139C23×D3—22no—
140C23×D3—21no—
141C23×D3—22no—
142C23×D3—21no—
143C23×D3—21no—
144C23×D3—21no—
145C23×D3—21no—
146C23×D3—22no—
147C23×D3—21no—
148C23×D3—21no—
149C23×D3—21no—
150C23×D3—22no—
151C23×D3—22no—
152C23×D3—21no—
153C23×D3—22no—
154C23×D3—21no—
155C23×D3—21no—
156C23×D3—22no—
157C23×D3C2×D412yes—
158C23×D3#4112yes—
159C23×D3#4114yes—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
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—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4#4112yes—
187C22×C3×C4#4112yes—
188C22×C3×C4#4114yes—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×D412yes—
212C22×Dic3#4114yes—
213C2×D12—28no—
214C2×D12—48no—
215C2×D12—48no—
216C2×D12—44no—
217C2×D12—44no—
218C2×C4×D3—28no—
219C2×C4×D3—28no—
220C2×C4×D3—44no—
221C2×C4×D3—48no—
222C2×C4×D3—48no—
223C3×C4⋊C4—44no—
224(C2×D3)⋊C4—216no—
225(C2×D3)⋊C4—48no—
226(C2×D3)⋊C4—48no—
227(C2×D3)⋊C4—48no—
228(C2×D3)⋊C4—44no—
229(C2×D3)⋊C4—44no—
230(C2×D3)⋊C4—44no—
231C4⋊Dic3—44no—
232C4×Dic3—48no—
233C25—31no—
234C25—31no—
235C22×D4—62no—
236C22×D4—62no—
237C22×D4—64no—
238C22×D4—62no—
239C22×D4—122no—
240C23×C4—31no—
241C23×C4—32no—
242C23×C4—34no—
243C23×C4—32no—
244C23×C4—31no—
245C23×C4—64no—
246C23×C4—62no—
247C23×C4—62no—
248C23×C4—122no—
249C23×C4C22×D312yes—
250C23×C4C22×D311yescharacteristic
251C23×C4C22×D311yescharacteristic
252C23×C4C22×D312yes—
253C23×C4C22×D311yescharacteristic
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—122no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—122no—
274C2×C22⋊C4—122no—
275C2×C22⋊C4—122no—
276C2×C22⋊C4—122no—
277C2×C22⋊C4—122no—
278C2×C42—64no—
279C2×C42—64no—
280C2×C42—122no—
281#95—22no—
282#95—22no—
283#95—24no—
284#95—34no—
285#95—42no—
286#95—64no—
287#95—62no—
288#95—64no—
289#95—64no—
290#95—122no—
291#95—122no—
292C23×C3C22×D411yescharacteristic
293C23×C3C22×D411yescharacteristic
294C23×C3C2×#4111yescharacteristic
295C23×C3C2×#4111yescharacteristic
296C23×C3C2×#4112yes—
297C23×C3C2×#4112yes—
298C23×C3C2×#4112yes—
299C23×C3C2×#4111yescharacteristic
300C23×C3H(5,𝔽2)11yescharacteristic
301C23×C3H(5,𝔽2)12yes—
302C23×C3H(5,𝔽2)11yescharacteristic
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—24no—
306C22×D3—22no—
307C22×D3—24no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—22no—
316C22×D3—24no—
317C22×D3—22no—
318C22×D3—24no—
319C22×D3—22no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—42no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—42no—
333C22×D3—41no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—41no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—41no—
348C22×D3#12414yes—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
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—41no—
367C2×C3×C4—42no—
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#12114yes—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—41no—
390C2×Dic3—42no—
391C2×Dic3—41no—
392C2×Dic3—42no—
393C2×Dic3—41no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3#12314yes—
402D12—416no—
403C4×D3—416no—
404C24—32no—
405C24—34no—
406C24—32no—
407C24—61no—
408C24—61no—
409C24—61no—
410C24—61no—
411C24—62no—
412C24—62no—
413C24—61no—
414C24—61no—
415C24—62no—
416C24—61no—
417C24—61no—
418C24—61no—
419C24—61no—
420C24—61no—
421C24—62no—
422C24—62no—
423C24—62no—
424C24—61no—
425C24—62no—
426C24C23×D311yescharacteristic
427C2×D4—68no—
428C2×D4—128no—
429C2×D4—124no—
430C2×D4—128no—
431C2×D4—124no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—32no—
457C22×C4—34no—
458C22×C4—34no—
459C22×C4—68no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—68no—
470C22×C4—61no—
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—124no—
483C22×C4—128no—
484C22×C4—128no—
485C22×C4#23512yes—
486C22×C4D3×D412yes—
487C22×C4D3×D412yes—
488C22×C4#23712yes—
489C22×C4#23914yes—
490C4⋊C4—44no—
491C4⋊C4—124no—
492C22⋊C4—616no—
493C22⋊C4—128no—
494C22⋊C4—124no—
495C22⋊C4—128no—
496C22⋊C4—124no—
497C22⋊C4—128no—
498C22⋊C4—124no—
499C42—128no—
500C22×C3C2×#12111yescharacteristic
501C22×C3C2×#12311yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3#52512yes—
504C22×C3C2×#12411yescharacteristic
505C22×C3#52912yes—
506C22×C3#52912yes—
507C22×C3#52912yes—
508C22×C3#53211yescharacteristic
509C22×C3#53411yescharacteristic
510C22×C3#53511yescharacteristic
511C22×C3#53812yes—
512C22×C3#54012yes—
513C22×C3#54612yes—
514C22×C3#54611yescharacteristic
515C22×C3#55011yescharacteristic
516C22×C3#55112yes—
517C22×C3#55112yes—
518C22×C3#55112yes—
519C22×C3#55312yes—
520C22×C3#55312yes—
521C22×C3#55311yescharacteristic
522C22×C3#55511yescharacteristic
523C2×D3—24no—
524C2×D3—24no—
525C2×D3—24no—
526C2×D3—44no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—42no—
531C2×D3—44no—
532C2×D3—44no—
533C2×D3—44no—
534C2×D3—42no—
535C2×D3—44no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—44no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—82no—
543C2×D3—81no—
544C2×D3—82no—
545C3×C4—24no—
546C3×C4—42no—
547C3×C4—44no—
548C3×C4—82no—
549C3×C4—81no—
550Dic3—24no—
551Dic3—44no—
552Dic3—81no—
553Dic3—82no—
554C23—34no—
555C23—62no—
556C23—62no—
557C23—64no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—62no—
562C23—62no—
563C23—64no—
564C23—62no—
565C23—62no—
566C23—64no—
567C23—62no—
568C23—64no—
569C23—64no—
570C23—62no—
571C23—62no—
572C23—122no—
573C23—122no—
574C23—121no—
575C23—122no—
576C23—122no—
577C23—121no—
578C23—122no—
579C23—121no—
580C23—121no—
581C23—122no—
582C23—122no—
583C23—121no—
584C23—121no—
585C23—122no—
586C23—122no—
587C23—122no—
588C23—121no—
589C23—122no—
590C23—122no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—122no—
595C23—121no—
596C23—122no—
597C23—121no—
598C23—122no—
599C23—122no—
600C23C2×#23511yescharacteristic
601C23C2×D3×D411yescharacteristic
602C23C2×D3×D411yescharacteristic
603C23C2×#23711yescharacteristic
604C23#100411yescharacteristic
605C23C2×#23912yes—
606C23D3×#4112yes—
607C23D3×#4112yes—
608C23D3×#4111yescharacteristic
609C23#100912yes—
610C23#100911yescharacteristic
611D4—1216no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—34no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—62no—
642C2×C4—64no—
643C2×C4—62no—
644C2×C4—64no—
645C2×C4—64no—
646C2×C4—64no—
647C2×C4—64no—
648C2×C4—1216no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4—121no—
657C2×C4—122no—
658C2×C4—122no—
659C2×C4—122no—
660C2×C4—121no—
661C2×C4—121no—
662C2×C4—122no—
663C2×C4—122no—
664C2×C4—122no—
665C2×C4#88414yes—
666C2×C3#239511yescharacteristic
667C2×C3#241211yescharacteristic
668C2×C3#241411yescharacteristic
669C2×C3#245712yes—
670C2×C3#246712yes—
671C2×C3#250411yescharacteristic
672C2×C3#253112yes—
673C2×C3#256111yescharacteristic
674C2×C3#257812yes—
675C2×C3#266111yescharacteristic
676C2×C3#285011yescharacteristic
677D3—44no—
678D3—82no—
679C22—64no—
680C22—64no—
681C22—64no—
682C22—122no—
683C22—124no—
684C22—124no—
685C22—124no—
686C22—122no—
687C22—122no—
688C22—122no—
689C22—124no—
690C22—124no—
691C22—124no—
692C22—122no—
693C22—122no—
694C22—122no—
695C22—241no—
696C22—242no—
697C22—241no—
698C22—241no—
699C22—241no—
700C22—242no—
701C22C2×#88411yescharacteristic
702C22#552712yes—
703C22#555811yescharacteristic
704C22#556311yescharacteristic
705C22#558012yes—
706C22#560611yescharacteristic
707C22#560711yescharacteristic
708C22#562412yes—
709C22#562712yes—
710C22#564412yes—
711C22D3×#12311yescharacteristic
712C22#565311yescharacteristic
713C22#565412yes—
714C22#565812yes—
715C22#566112yes—
716C22D3×#12411yescharacteristic
717C22#567611yescharacteristic
718C22#569112yes—
719C22#569712yes—
720C22#570512yes—
721C22#570912yes—
722C22#571811yescharacteristic
723C22#572511yescharacteristic
724C4—24no—
725C4—42no—
726C4—44no—
727C4—64no—
728C4—82no—
729C4—81no—
730C4—124no—
731C4—242no—
732C4—241no—
733C3#1484211yescharacteristic, complemented, Sylow
734C2—124no—
735C2—242no—
736C2#7412611yescharacteristic
737C2#7506712yes—
738C2#7508912yes—
739C2#7643811yescharacteristic
740C2#7666211yescharacteristic
741C2#7749211yescharacteristic
742C2#7768311yescharacteristic
743C2#7773811yescharacteristic
744C2#7783811yescharacteristic
745C2#7802512yes—
746C2#7813812yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1445C2
prime radicalC23×D12C22
hypercenter#1445D3
hypocenterC3#14842
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
1. C2×#4992C2semidirect
2. C2×#4987C2semidirect
4. C2×#4666C2semidirect
5. C3×#1445C2semidirect
6. #69429C2semidirect
7. #69427C2semidirect
9. #69415C2semidirect
10. #69408C2semidirect
11. #69401C2semidirect
44. #1445D3semidirect
109. C3⋊#95D4semidirect
733. C3#14842semidirect

Maximal subgroups

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

subgroupindex
C2×#49922
C2×#49872
C2×#46652
C2×#46662
C3×#14452
#694292
#694272
#694262
#694152
#694082
#694012
#148423