Finite Group Databasebeta

← #10740344

Subgroups — #10740344

Order 768.

1730 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4991C212yesmaximal
2C2×#4670C212yesmaximal, complemented
3C2×#4670C212yesmaximal, complemented
4C3×#1444C211yesmaximal, characteristic, complemented
5#71570C212yesmaximal, complemented
6#71563C212yesmaximal, complemented
7#69427C212yesmaximal, complemented
8#69426C211yesmaximal, characteristic
9#69415C211yesmaximal, characteristic, complemented
10#15780—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×(C2×D3)⋊C4C2212yes—
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2212yes—
22C22×C4×Dic3C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32#4991—28no—
33#4670—28no—
34#4670—28no—
35C2×#394—32no—
36C2×#388—32no—
37C2×#388—32no—
38#1444D311yescharacteristic, complemented
39#1442—31no—
40#1439—32no—
41#1439—32no—
42#1438—31no—
43C4⋊#95—32no—
44C24×D3C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C23×Dic3C2312yes—
53C22×D12—24no—
54C22×D12—22no—
55C22×D12—42no—
56C22×D12—41no—
57C22×D12—41no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—42no—
61C2×C3×C4⋊C4—41no—
62C2×C3×C4⋊C4—42no—
63C2×(C2×D3)⋊C4—24no—
64C2×(C2×D3)⋊C4—24no—
65C2×(C2×D3)⋊C4—24no—
66C2×(C2×D3)⋊C4—24no—
67C2×(C2×D3)⋊C4—24no—
68C2×(C2×D3)⋊C4—24no—
69C2×(C2×D3)⋊C4—24no—
70C2×(C2×D3)⋊C4—24no—
71C2×(C2×D3)⋊C4—24no—
72C2×(C2×D3)⋊C4—24no—
73C2×(C2×D3)⋊C4—24no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—42no—
77C2×(C2×D3)⋊C4—42no—
78C2×(C2×D3)⋊C4—42no—
79C2×(C2×D3)⋊C4—42no—
80C2×(C2×D3)⋊C4D418yes—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3—24no—
84C2×C4⋊Dic3—41no—
85C2×C4⋊Dic3—42no—
86C2×C4×Dic3—24no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—24no—
89C3×#95—24no—
90C3×#95—41no—
91C3×#95D414yescomplemented
92C3⋊#95—24no—
93C3⋊#95—22no—
94C3⋊#95—24no—
95C3⋊#95—24no—
96C3⋊#95—41no—
97C3⋊#95—42no—
98C23×D4—31no—
99C22×C4⋊C4—32no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4C2×D312yes—
102C22×C4⋊C4C2×D311yescharacteristic
103C22×C22⋊C4—32no—
104C22×C22⋊C4—32no—
105C22×C22⋊C4—32no—
106C22×C22⋊C4—32no—
107C22×C22⋊C4—32no—
108C22×C22⋊C4—32no—
109C22×C42—32no—
110#394—68no—
111#388—68no—
112#388—68no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—31no—
116C2×#95—32no—
117C2×#95—31no—
118C2×#95—32no—
119C2×#95C2×D312yes—
120C2×#95C2×D311yescharacteristic
121C2×#95C2×D311yescharacteristic
122C24×C3C2411yescharacteristic
123C23×D3—22no—
124C23×D3—22no—
125C23×D3—22no—
126C23×D3—22no—
127C23×D3—22no—
128C23×D3—22no—
129C23×D3—22no—
130C23×D3—22no—
131C23×D3—22no—
132C23×D3—22no—
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—22no—
136C23×D3—22no—
137C23×D3C2×D414yes—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D414yes—
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4114yes—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3C2×D414yes—
187C22×Dic3#4114yes—
188C2×D12—48no—
189C2×D12—44no—
190C2×D12—44no—
191C2×D12—48no—
192C2×D12—44no—
193C3×C4⋊C4—48no—
194C3×C4⋊C4—44no—
195(C2×D3)⋊C4—216no—
196(C2×D3)⋊C4—216no—
197(C2×D3)⋊C4—48no—
198(C2×D3)⋊C4—48no—
199(C2×D3)⋊C4—48no—
200(C2×D3)⋊C4—48no—
201C4⋊Dic3—44no—
202C4⋊Dic3—48no—
203C4×Dic3—48no—
204C25—32no—
205C22×D4—64no—
206C22×D4—62no—
207C22×D4—121no—
208C22×D4—122no—
209C22×D4—121no—
210C23×C4—32no—
211C23×C4—32no—
212C23×C4—32no—
213C23×C4C22×D312yes—
214C23×C4C22×D311yescharacteristic
215C23×C4C22×D311yescharacteristic
216C23×C4C22×D311yescharacteristic
217C23×C4C22×D312yes—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—64no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—64no—
226C2×C4⋊C4—122no—
227C2×C4⋊C4—121no—
228C2×C22⋊C4—38no—
229C2×C22⋊C4—64no—
230C2×C22⋊C4—64no—
231C2×C22⋊C4—64no—
232C2×C22⋊C4—64no—
233C2×C22⋊C4—64no—
234C2×C22⋊C4—64no—
235C2×C22⋊C4—64no—
236C2×C22⋊C4—64no—
237C2×C22⋊C4—64no—
238C2×C22⋊C4—64no—
239C2×C22⋊C4—64no—
240C2×C22⋊C4—64no—
241C2×C22⋊C4—64no—
242C2×C22⋊C4—122no—
243C2×C22⋊C4—122no—
244C2×C22⋊C4—122no—
245C2×C22⋊C4—122no—
246C2×C42—64no—
247C2×C42—64no—
248C2×C42—64no—
249#95—24no—
250#95—41no—
251#95—64no—
252#95—64no—
253#95—64no—
254#95—62no—
255#95—121no—
256#95—122no—
257#95D1214yescomplemented
258C23×C3C22×D411yescharacteristic
259C23×C3C22×D412yes—
260C23×C3C2×#4111yescharacteristic
261C23×C3C2×#4112yes—
262C23×C3C2×#4111yescharacteristic
263C23×C3C2×#4112yes—
264C23×C3H(5,𝔽2)12yes—
265C23×C3H(5,𝔽2)11yescharacteristic
266C23×C3H(5,𝔽2)11yescharacteristic
267C23×C32−1+411yescharacteristic
268C23×C32−1+411yescharacteristic
269C22×D3—24no—
270C22×D3—24no—
271C22×D3—24no—
272C22×D3—24no—
273C22×D3—24no—
274C22×D3—24no—
275C22×D3—24no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—42no—
303C22×D3—42no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4#12118yes—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354D12—416no—
355C24—34no—
356C24—62no—
357C24—62no—
358C24—62no—
359C24—62no—
360C24—62no—
361C24—62no—
362C24—62no—
363C24—62no—
364C24—62no—
365C24—62no—
366C24—62no—
367C24—62no—
368C24—62no—
369C24—62no—
370C24C23×D311yescharacteristic
371C2×D4—128no—
372C2×D4—124no—
373C2×D4—124no—
374C2×D4—128no—
375C2×D4—124no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—34no—
400C22×C4—34no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4C2×D1212yes—
421C22×C4C2×D1214yes—
422C22×C4D3×D414yes—
423C22×C4#23712yes—
424C22×C4#23912yes—
425C22×C4#23914yes—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—128no—
429C4⋊C4—124no—
430C22⋊C4—616no—
431C22⋊C4—616no—
432C22⋊C4—128no—
433C22⋊C4—128no—
434C22⋊C4—128no—
435C22⋊C4—128no—
436C42—128no—
437C22×C3C2×#12112yes—
438C22×C3#52512yes—
439C22×C3#52912yes—
440C22×C3#53512yes—
441C22×C3#53711yescharacteristic
442C22×C3#53811yescharacteristic
443C22×C3#53912yes—
444C22×C3#53911yescharacteristic
445C22×C3#54212yes—
446C22×C3D4211yescharacteristic
447C22×C3#54612yes—
448C22×C3#54612yes—
449C22×C3#54712yes—
450C22×C3#54712yes—
451C22×C3#54711yescharacteristic
452C22×C3#55011yescharacteristic
453C22×C3#55312yes—
454C22×C3#55511yescharacteristic
455C22×C3#55512yes—
456C22×C3#55611yescharacteristic
457C22×C3#55911yescharacteristic
458C22×C3H(3,𝔽4)11yescharacteristic
459C22×C3#56311yescharacteristic
460C2×D3—44no—
461C2×D3—44no—
462C2×D3—44no—
463C2×D3—44no—
464C2×D3—44no—
465C2×D3—44no—
466C2×D3—44no—
467C2×D3—82no—
468C2×D3—82no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3—82no—
474C2×D3—82no—
475C3×C4—28no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—82no—
480Dic3—44no—
481Dic3—44no—
482Dic3—82no—
483C23—64no—
484C23—64no—
485C23—64no—
486C23—64no—
487C23—64no—
488C23—64no—
489C23—64no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23—122no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23—122no—
502C23—122no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—122no—
510C23—122no—
511C23—122no—
512C23—122no—
513C23—122no—
514C23—122no—
515C23—122no—
516C23—122no—
517C23—122no—
518C23C22×D1211yescharacteristic
519C23C2×D3×D412yes—
520C23C2×#23711yescharacteristic
521C23#100412yes—
522C23#100411yescharacteristic
523C23C2×#23912yes—
524C23C2×#23911yescharacteristic
525C23#100711yescharacteristic
526C23D3×#4112yes—
527C23#100911yescharacteristic
528C23#101011yescharacteristic
529D4—1216no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4#89518yes—
580C2×C3#241412yes—
581C2×C3#242512yes—
582C2×C3#245511yescharacteristic
583C2×C3#247412yes—
584C2×C3#253711yescharacteristic
585C2×C3#259311yescharacteristic
586C2×C3#267011yescharacteristic
587C2×C3#271211yescharacteristic
588C2×C3#271812yes—
589C2×C3#272711yescharacteristic
590C2×C3#278511yescharacteristic
591D3—84no—
592C22—124no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—124no—
597C22—124no—
598C22—124no—
599C22—242no—
600C22—242no—
601C22—242no—
602C22—242no—
603C22—242no—
604C22—242no—
605C22—242no—
606C22—242no—
607C22C2×#89512yes—
608C22#551011yescharacteristic
609C22#555412yes—
610C22#555511yescharacteristic
611C22#557711yescharacteristic
612C22#559112yes—
613C22#559411yescharacteristic
614C22#560712yes—
615C22#561711yescharacteristic
616C22#562412yes—
617C22#562712yes—
618C22#564012yes—
619C22#564511yescharacteristic
620C22#564512yes—
621C22#565512yes—
622C22#566411yescharacteristic
623C22#569512yes—
624C22#569911yescharacteristic
625C22#571711yescharacteristic
626C22#572612yes—
627C22#573211yescharacteristic
628C22#573412yes—
629C22#573411yescharacteristic
630C4—28no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—124no—
636C4—124no—
637C4—242no—
638C3#1578011yescharacteristic, complemented, Sylow
639C2—244no—
640C2#7436811yescharacteristic
641C2#7441012yes—
642C2#7452911yescharacteristic
643C2#7613111yescharacteristic
644C2#7620912yes—
645C2#7676711yescharacteristic
646C2#7718212yes—
647C2#7723311yescharacteristic
648C2#7780811yescharacteristic
649C2#7798312yes—
650C2#7802411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1444C2
prime radicalC23×D12C22
hypercenter#1444D3
hypocenterC3#15780
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C2×#4670C2semidirect
3. C2×#4670C2semidirect
4. C3×#1444C2semidirect
5. #71570C2semidirect
6. #71563C2semidirect
7. #69427C2semidirect
9. #69415C2semidirect
38. #1444D3semidirect
91. C3×#95D4semidirect
257. #95D12semidirect
638. C3#15780semidirect

Maximal subgroups

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

subgroupindex
C2×#49912
C2×#46702
C3×#14442
#715702
#715632
#694272
#694262
#694152
#157803