Finite Group Databasebeta

← #10742712

Subgroups — #10742712

Order 768.

2354 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#894C212yesmaximal
2C2×#4989C212yesmaximal, complemented
3C2×#4668C212yesmaximal, complemented
4C3×#1444C211yesmaximal, characteristic, complemented
5#71568C212yesmaximal, complemented
6#69429C212yesmaximal, complemented
7#69429C212yesmaximal, complemented
8#69426C211yesmaximal, characteristic
9#69420C211yesmaximal, characteristic, complemented
10#15949—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C23×C4×D3C2212yes—
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×(C2×D3)⋊C4C2212yes—
16C22×(C2×D3)⋊C4C2212yes—
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2212yes—
23C2×#894—28no—
24C2×#894—28no—
25C2×#894—28no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35#4989—28no—
36#4668—28no—
37C22×#123—32no—
38C2×#385—32no—
39C2×#385—32no—
40#1444—31no—
41#1444—32no—
42#1444D311yescharacteristic, complemented
43#1440—32no—
44#1440—32no—
45#1438—31no—
46C24×D3C2312yes—
47C23×C3×C4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C22×D12—24no—
56C22×D12—24no—
57C22×D12—41no—
58C22×D12—41no—
59C22×D12D414yes—
60C22×C4×D3—24no—
61C22×C4×D3—24no—
62C22×C4×D3—24no—
63C22×C4×D3—24no—
64C22×C4×D3—24no—
65C22×C4×D3—42no—
66C22×C4×D3—42no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—41no—
72C2×(C2×D3)⋊C4—24no—
73C2×(C2×D3)⋊C4—24no—
74C2×(C2×D3)⋊C4—24no—
75C2×(C2×D3)⋊C4—24no—
76C2×(C2×D3)⋊C4—24no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4D418yes—
86C2×(C2×D3)⋊C4D418yes—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3—41no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—42no—
95#894—264no—
96C3×#95—24no—
97C3×#95—22no—
98C3×#95—41no—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—41no—
105C23×D4—31no—
106C24×C4—32no—
107C2×#123—68no—
108C2×#123—68no—
109C2×#123—68no—
110C22×C4⋊C4—32no—
111C22×C4⋊C4—32no—
112C22×C4⋊C4—31no—
113C22×C4⋊C4C2×D312yes—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C22⋊C4—32no—
116C22×C22⋊C4—32no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120#385—68no—
121#385—68no—
122C2×#95—32no—
123C2×#95—31no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—31no—
128C2×#95C2×D311yescharacteristic
129C2×#95C2×D312yes—
130C2×#95C2×D311yescharacteristic
131C24×C3C2411yescharacteristic
132C23×D3—22no—
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—22no—
136C23×D3—22no—
137C23×D3—22no—
138C23×D3—22no—
139C23×D3—22no—
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—22no—
144C23×D3C2×D414yes—
145C23×D3C2×D414yes—
146C23×D3C2×D414yes—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
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×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D414yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
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×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3C2×D414yes—
195C22×Dic3C2×D414yes—
196C22×Dic3#4114yes—
197C2×D12—216no—
198C2×D12—44no—
199C2×D12—44no—
200C2×D12—44no—
201C2×D12—48no—
202C2×C4×D3—216no—
203C2×C4×D3—48no—
204C2×C4×D3—48no—
205C2×C4×D3—48no—
206C2×C4×D3—48no—
207C2×C4×D3—48no—
208C2×C4×D3—48no—
209C3×C4⋊C4—216no—
210C3×C4⋊C4—44no—
211(C2×D3)⋊C4—216no—
212(C2×D3)⋊C4—216no—
213(C2×D3)⋊C4—216no—
214(C2×D3)⋊C4—216no—
215(C2×D3)⋊C4—216no—
216C4⋊Dic3—44no—
217Dic3⋊C4—216no—
218Dic3⋊C4—48no—
219C25—32no—
220C22×D4—34no—
221C22×D4—64no—
222C22×D4—64no—
223C22×D4—121no—
224C22×D4—121no—
225C23×C4—32no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4—64no—
229C23×C4—64no—
230C23×C4—64no—
231C23×C4—64no—
232C23×C4—64no—
233C23×C4—122no—
234C23×C4—122no—
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D312yes—
237C23×C4C22×D312yes—
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D311yescharacteristic
240#123—664no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—64no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—122no—
253C2×C4⋊C4—121no—
254C2×C22⋊C4—38no—
255C2×C22⋊C4—38no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—64no—
258C2×C22⋊C4—64no—
259C2×C22⋊C4—64no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—64no—
269#95—22no—
270#95—24no—
271#95—41no—
272#95—64no—
273#95—64no—
274#95—64no—
275#95—64no—
276#95—62no—
277#95—121no—
278C23×C3C22×D411yescharacteristic
279C23×C3C22×D412yes—
280C23×C3C22×D412yes—
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C23×C3C2×#4112yes—
284C23×C3H(5,𝔽2)11yescharacteristic
285C23×C3H(5,𝔽2)11yescharacteristic
286C23×C32−1+412yes—
287C23×C32−1+411yescharacteristic
288C23×C32−1+411yescharacteristic
289C22×D3—24no—
290C22×D3—24no—
291C22×D3—24no—
292C22×D3—24no—
293C22×D3—24no—
294C22×D3—24no—
295C22×D3—24no—
296C22×D3—24no—
297C22×D3—24no—
298C22×D3—24no—
299C22×D3—24no—
300C22×D3—24no—
301C22×D3—24no—
302C22×D3—24no—
303C22×D3—24no—
304C22×D3—24no—
305C22×D3—24no—
306C22×D3—24no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—42no—
316C22×D3—42no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3#12018yes—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374D12—416no—
375C4×D3—432no—
376C24—34no—
377C24—34no—
378C24—34no—
379C24—62no—
380C24—62no—
381C24—62no—
382C24—62no—
383C24—62no—
384C24—62no—
385C24—62no—
386C24—62no—
387C24—62no—
388C24—62no—
389C24—62no—
390C24—62no—
391C24C23×D311yescharacteristic
392C2×D4—616no—
393C2×D4—124no—
394C2×D4—124no—
395C2×D4—124no—
396C2×D4—128no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—34no—
422C22×C4—34no—
423C22×C4—34no—
424C22×C4—616no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—128no—
444C22×C4—128no—
445C22×C4—128no—
446C22×C4—128no—
447C22×C4—128no—
448C22×C4—128no—
449C22×C4#23512yes—
450C22×C4D3×D414yes—
451C22×C4D3×D414yes—
452C22×C4D3×D412yes—
453C22×C4#23912yes—
454C4⋊C4—216no—
455C4⋊C4—44no—
456C4⋊C4—616no—
457C4⋊C4—128no—
458C4⋊C4—124no—
459C22⋊C4—616no—
460C22⋊C4—616no—
461C22⋊C4—616no—
462C22⋊C4—616no—
463C22⋊C4—616no—
464C22×C3C2×#12012yes—
465C22×C3#52512yes—
466C22×C3#52512yes—
467C22×C3#53411yescharacteristic
468C22×C3#53612yes—
469C22×C3#53612yes—
470C22×C3#53911yescharacteristic
471C22×C3#53912yes—
472C22×C3#53911yescharacteristic
473C22×C3D4211yescharacteristic
474C22×C3D4211yescharacteristic
475C22×C3D4212yes—
476C22×C3#54712yes—
477C22×C3#54711yescharacteristic
478C22×C3#54712yes—
479C22×C3#54712yes—
480C22×C3#54711yescharacteristic
481C22×C3#54712yes—
482C22×C3#54712yes—
483C22×C3#55911yescharacteristic
484C22×C3#55911yescharacteristic
485C22×C3#56311yescharacteristic
486C22×C3#56311yescharacteristic
487C2×D3—28no—
488C2×D3—28no—
489C2×D3—28no—
490C2×D3—44no—
491C2×D3—44no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—44no—
495C2×D3—44no—
496C2×D3—44no—
497C2×D3—44no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—44no—
502C3×C4—44no—
503C3×C4—42no—
504C3×C4—44no—
505C3×C4—42no—
506C3×C4—42no—
507Dic3—44no—
508Dic3—44no—
509Dic3—44no—
510C23—38no—
511C23—64no—
512C23—64no—
513C23—64no—
514C23—64no—
515C23—64no—
516C23—64no—
517C23—64no—
518C23—64no—
519C23—64no—
520C23—64no—
521C23—64no—
522C23—64no—
523C23—64no—
524C23—64no—
525C23—64no—
526C23—64no—
527C23—64no—
528C23—64no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23—122no—
533C23—122no—
534C23—122no—
535C23—122no—
536C23—122no—
537C23—122no—
538C23—122no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23C2×#23511yescharacteristic
546C23C2×D3×D412yes—
547C23C2×D3×D412yes—
548C23C2×D3×D411yescharacteristic
549C23#100411yescharacteristic
550C23C2×#23911yescharacteristic
551C23#100712yes—
552C23#100711yescharacteristic
553C23D3×#4112yes—
554C23#100911yescharacteristic
555C23#101011yescharacteristic
556D4—1216no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—64no—
587C2×C4—64no—
588C2×C4—64no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—1232no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C3#240311yescharacteristic
609C2×C3#240612yes—
610C2×C3#240612yes—
611C2×C3#245511yescharacteristic
612C2×C3#245512yes—
613C2×C3#245511yescharacteristic
614C2×C3#268011yescharacteristic
615C2×C3#268012yes—
616C2×C3#268011yescharacteristic
617C2×C3#272411yescharacteristic
618C2×C3#272411yescharacteristic
619D3—48no—
620C22—68no—
621C22—68no—
622C22—68no—
623C22—124no—
624C22—124no—
625C22—124no—
626C22—124no—
627C22—124no—
628C22—124no—
629C22—124no—
630C22—124no—
631C22—124no—
632C22—124no—
633C22—124no—
634C22—124no—
635C22#550712yes—
636C22#551511yescharacteristic
637C22#554712yes—
638C22#555111yescharacteristic
639C22#558511yescharacteristic
640C22D3×#12012yes—
641C22#559011yescharacteristic
642C22#560511yescharacteristic
643C22#560912yes—
644C22#562712yes—
645C22#562812yes—
646C22#563112yes—
647C22#563112yes—
648C22#564511yescharacteristic
649C22#565512yes—
650C22#566411yescharacteristic
651C22#569212yes—
652C22#569911yescharacteristic
653C22#571711yescharacteristic
654C22#572412yes—
655C22#572611yescharacteristic
656C22#572612yes—
657C22#573511yescharacteristic
658C4—44no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—124no—
664C4—124no—
665C4—124no—
666C3#1594911yescharacteristic, complemented, Sylow
667C2—128no—
668C2#7437111yescharacteristic
669C2#7460012yes—
670C2#7461211yescharacteristic
671C2#7622612yes—
672C2#7627111yescharacteristic
673C2#7675311yescharacteristic
674C2#7724912yes—
675C2#7726211yescharacteristic
676C2#7796912yes—
677C2#7798411yescharacteristic
678C2#7802411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1444C2
prime radicalC23×D12C22
hypercenter#1444D3
hypocenterC3#15949
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C2×#4989C2semidirect
3. C2×#4668C2semidirect
4. C3×#1444C2semidirect
5. #71568C2semidirect
6. #69429C2semidirect
7. #69429C2semidirect
9. #69420C2semidirect
42. #1444D3semidirect
666. C3#15949semidirect

Maximal subgroups

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

subgroupindex
C22×#8942
C2×#49892
C2×#46682
C3×#14442
#715682
#694292
#694262
#694202
#159493