Finite Group Databasebeta

← #10827371

Subgroups — #10827371

Order 768.

1500 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#553C211yesmaximal, characteristic, complemented
2C2×#5036C211yesmaximal, characteristic, complemented
3#82403C211yesmaximal, characteristic, complemented
4C22×#809C211yesmaximal, characteristic, complemented
5#71743C212yesmaximal, complemented
6#71684C212yesmaximal, complemented
7#71681C211yesmaximal, characteristic, complemented
8#71613C212yesmaximal, complemented
9#71610C211yesmaximal, characteristic, complemented
10#71586C212yesmaximal, complemented
11#71554C211yesmaximal, characteristic, complemented
12#23963—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C3×#553—24no—
15C3×#553—24no—
16C2×C3×#126—21no—
17C2×C3×#125C2211yescharacteristic, complemented
18C2×C3×#123—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic, complemented
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic, complemented
25C2×C3×#117C2211yescharacteristic, complemented
26C22×C3×C4⋊C4C2211yescharacteristic
27C2×#937—21no—
28C22×#213C2212yes—
29C2×#930C2212yes—
30C23×C3⋊C8C2211yescharacteristic
31C2×#832C2212yescomplemented
32#5212C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#830C2212yescomplemented
35#5036—24no—
36#4991—21no—
37C2×#809—24no—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#809C2212yes—
41#4960—21no—
42#4955—21no—
43#4939C2212yes—
44#4529C2212yescomplemented
45#4529C2212yescomplemented
46#4529C2212yescomplemented
47#4524C2212yescomplemented
48C2×#553D311yescharacteristic, complemented
49C2×#447—31no—
50C22×#102—31no—
51#2408—31no—
52#2025—32no—
53#1999—32no—
54#1893—31no—
55#1879—32no—
56#1875—32no—
57#1869—31no—
58#1824—31no—
59C24×D3—21no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4D411yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C22×#67—22no—
69C22×D12—22no—
70C22×D12—24no—
71C22×D12—22no—
72C22×D12—21no—
73C22×D12—42no—
74C22×D12C2312yes—
75C3×#126—28no—
76C3×#125—22no—
77C3×#125—22no—
78C3×#123—28no—
79C3×#121—28no—
80C3×#121—22no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121D414yescomplemented
84C3×C4×D4—28no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×#117—22no—
90C3×#117—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D414yescomplemented
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C42—21no—
107C2×C3×C42C2311yescharacteristic
108C2×#217—21no—
109#937—44no—
110C2×#213—24no—
111C2×#213—24no—
112C2×#213—42no—
113#930—24no—
114#930—42no—
115C2×(C2×D3)⋊C4—22no—
116C2×(C2×D3)⋊C4—22no—
117C2×(C2×D3)⋊C4—22no—
118C2×C4⋊Dic3C2312yes—
119C22×C3⋊C8—22no—
120C22×C3⋊C8—42no—
121C22×C3⋊C8C2312yes—
122C22×C3⋊C8C2312yes—
123#832—42no—
124C3⋊#95—21no—
125#830—22no—
126#830—42no—
127#809—24no—
128#809—24no—
129#809—24no—
130C23×D4—31no—
131C22×D8—32no—
132C23×C8—31no—
133#553—24no—
134#553—24no—
135C2×#126—21no—
136C2×#125C2×D311yescharacteristic, complemented
137C2×#123—21no—
138C2×#121—21no—
139C2×#121—32no—
140C2×#121C2×D311yescharacteristic
141C2×#121C2×D311yescharacteristic, complemented
142C2×#120—61no—
143C2×C4×D4—21no—
144C2×C4×D4C2×D311yescharacteristic
145C2×C4×D4C2×D311yescharacteristic, complemented
146C2×#117C2×D311yescharacteristic, complemented
147C22×C4⋊C4C2×D311yescharacteristic
148#447—64no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—64no—
154C2×C22⋊C8—31no—
155C2×C22⋊C8—32no—
156#394—61no—
157#392—61no—
158#390—32no—
159#380—31no—
160#379—61no—
161#340—32no—
162#340—32no—
163#340—32no—
164#336—32no—
165C24×C3—21no—
166C24×C3C2×D411yescharacteristic
167C24×C3C2×D411yescharacteristic
168C23×D3—22no—
169C23×D3—44no—
170C23×D3—42no—
171C23×D3—42no—
172C23×D3—42no—
173C23×D3—82no—
174C2×C3×D4—22no—
175C2×C3×D4—24no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—42no—
183C2×C3×D4—42no—
184C2×C3×D4—82no—
185C2×C3×D4C2×D412yes—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—82no—
198C22×C3×C4—82no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2411yescharacteristic
206C2×#67—44no—
207C2×#67—44no—
208C2×#67—82no—
209C22×Dic3—22no—
210C2×D12—28no—
211C2×D12—22no—
212C2×D12—22no—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—22no—
216C2×D12—48no—
217C2×D12—44no—
218C2×D12—44no—
219C2×D12—42no—
220C2×D12#4114yes—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4C2×D412yes—
229C3×C4⋊C4C2×D412yes—
230C3×C4⋊C4C2×D412yes—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—22no—
237C3×C42—24no—
238C3×C42—21no—
239C3×C42—21no—
240#217—42no—
241#213—48no—
242(C2×D3)⋊C4—44no—
243(C2×D3)⋊C4—44no—
244(C2×D3)⋊C4—82no—
245C4⋊Dic3—22no—
246C4⋊Dic3—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—44no—
254C25—61no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—32no—
258C22×D4—64no—
259C22×D4—62no—
260C22×D4—62no—
261C22×D4—62no—
262C22×D4—61no—
263C22×D4—122no—
264C22×D4#6711yescharacteristic
265C22×D4C22×D311yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D311yescharacteristic
270C2×D8—64no—
271C2×D8—64no—
272C2×D8—122no—
273C22×C8—32no—
274C22×C8—32no—
275C22×C8—62no—
276C22×C8—122no—
277#126—28no—
278#125—22no—
279#125—22no—
280#123—28no—
281#121—28no—
282#121—22no—
283#121—22no—
284#121—22no—
285#121—64no—
286#121—122no—
287#121#6714yescomplemented
288#120—124no—
289C4×D4—28no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294#117—22no—
295#117—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4#6711yescharacteristic
302C2×C4⋊C4#6714yescomplemented
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—62no—
312C2×C22⋊C4—62no—
313C2×C22⋊C4—62no—
314C2×C22⋊C4#6711yescharacteristic
315C2×C22⋊C4#6711yescharacteristic
316C2×C42—21no—
317C2×C42C22×D311yescharacteristic
318#102—64no—
319#102—64no—
320#102—64no—
321#102—122no—
322C22⋊C8—62no—
323C22⋊C8—122no—
324#95—61no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—42no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—42no—
337C23×C3#12012yes—
338C23×C3#13512yes—
339C23×C3AGL(1,ℤ/8)12yes—
340C23×C3C22×D411yescharacteristic
341C22×D3—24no—
342C22×D3—48no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—42no—
349C22×D3—88no—
350C22×D3—88no—
351C22×D3—84no—
352C22×D3—84no—
353C3×D4—24no—
354C3×D4—44no—
355C3×D4—48no—
356C3×D4—41no—
357C3×D4—41no—
358C3×D4—84no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—24no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—42no—
380C2×C3×C4—84no—
381C2×C3×C4—84no—
382C2×C3×C4#12012yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4112yes—
386C2×C3×C4H(5,𝔽2)12yes—
387#67—88no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391D12—24no—
392D12—24no—
393D12—48no—
394D12—42no—
395D12—42no—
396C3⋊C8—42no—
397C3⋊C8—42no—
398C24—21no—
399C24—62no—
400C24—124no—
401C24—122no—
402C24—122no—
403C24—122no—
404C24—242no—
405C24C2×#6711yescharacteristic
406C24C2×#6711yescharacteristic
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—34no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—68no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—82no—
425C2×D4—128no—
426C2×D4—124no—
427C2×D4—124no—
428C2×D4—124no—
429C2×D4—124no—
430C2×D4—122no—
431C2×D4—242no—
432C2×D4C2×#6712yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—62no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6712yes—
452C22×C4C2×#6712yes—
453C22×C4C23×D311yescharacteristic
454D8—128no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—124no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4C2×#6712yes—
472C4⋊C4C2×#6712yes—
473C4⋊C4C2×#6712yes—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—24no—
479C22⋊C4—22no—
480C22⋊C4—124no—
481C22⋊C4—124no—
482C22⋊C4—122no—
483C22⋊C4—242no—
484C42—24no—
485C42—21no—
486C42—21no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—41no—
492C22×C3—42no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—41no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—82no—
500C22×C3—82no—
501C22×C3#44914yes—
502C22×C3C2×#12011yescharacteristic
503C22×C3#54612yes—
504C22×C3C2×#13511yescharacteristic
505C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
506C22×C3#57612yes—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—44no—
510C2×D3—816no—
511C2×D3—82no—
512C2×D3—82no—
513C2×D3—82no—
514C2×D3—82no—
515C3×C4—22no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4#52511yescharacteristic
523C3×C4#53511yescharacteristic
524C3×C4#54612yes—
525Dic3—82no—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—64no—
539C23—128no—
540C23—122no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23—248no—
547C23—248no—
548C23—244no—
549C23—244no—
550C23#94912yes—
551C23#95012yes—
552C23#95312yes—
553C23C22×#6711yescharacteristic
554D4—24no—
555D4—44no—
556D4—48no—
557D4—41no—
558D4—41no—
559D4—64no—
560D4—64no—
561D4—84no—
562D4—128no—
563D4—122no—
564D4—122no—
565D4—248no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4#95312yes—
593C2×C4D3×#4112yes—
594C2×C4#100912yes—
595C2×C4C22×#6711yescharacteristic
596C2×C4C22×#6711yescharacteristic
597C8—122no—
598C8—122no—
599C2×C3—24no—
600C2×C3—81no—
601C2×C3—82no—
602C2×C3#240411yescharacteristic
603C2×C3C2×#44911yescharacteristic
604C2×C3#300811yescharacteristic
605C2×C3#329712yes—
606C2×C3#331812yes—
607D3—84no—
608C22—22no—
609C22—22no—
610C22—24no—
611C22—24no—
612C22—41no—
613C22—42no—
614C22—41no—
615C22—42no—
616C22—42no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—82no—
621C22—82no—
622C22—124no—
623C22—124no—
624C22—124no—
625C22—2416no—
626C22—242no—
627C22—242no—
628C22—242no—
629C22—242no—
630C22#523814yes—
631C22C2×#94911yescharacteristic
632C22C2×#95011yescharacteristic
633C22#582912yes—
634C22#583512yes—
635C22C2×#95311yescharacteristic
636C4—22no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—42no—
643C4—242no—
644C4#582811yescharacteristic
645C4#582912yes—
646C4#583011yescharacteristic
647C3#2396311yescharacteristic, complemented, Sylow
648C2—24no—
649C2—81no—
650C2—82no—
651C2—244no—
652C2#8361412yes—
653C2#8361612yes—
654C2C2×#523811yescharacteristic
655C2#8409611yescharacteristic
656C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#553C2
socleC23×C3C22×D4
hypercenterC2×#553D3
hypocenterC3#23963
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×C3×#553C2semidirect
2. C2×#5036C2semidirect
3. #82403C2semidirect
4. C22×#809C2semidirect
5. #71743C2semidirect
6. #71684C2semidirect
7. #71681C2semidirect
8. #71613C2semidirect
9. #71610C2semidirect
10. #71586C2semidirect
11. #71554C2semidirect
17. C2×C3×#125C22semidirect
21. C2×C3×#121C22semidirect
24. C2×C3×C4×D4C22semidirect
25. C2×C3×#117C22semidirect
31. C2×#832C22semidirect
34. C2×#830C22semidirect
44. #4529C22semidirect
45. #4529C22semidirect
46. #4529C22semidirect
47. #4524C22semidirect
48. C2×#553D3semidirect
83. C3×#121D4semidirect
96. C2×C3×C4⋊C4D4semidirect
136. C2×#125C2×D3semidirect
141. C2×#121C2×D3semidirect
145. C2×C4×D4C2×D3semidirect
146. C2×#117C2×D3semidirect
287. #121#67semidirect
302. C2×C4⋊C4#67semidirect
647. C3#23963semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5532
C2×#50362
#824032
C22×#8092
#717432
#716842
#716812
#716132
#716102
#715862
#715542
#239633