Finite Group Databasebeta

← #10830941

Subgroups — #10830941

Order 768.

1776 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#213C211yesmaximal, characteristic, complemented
2C3×#2408C211yesmaximal, characteristic, complemented
3C2×#5039C212yesmaximal, complemented
4#82403C211yesmaximal, characteristic, complemented
5C2×#5011C211yesmaximal, characteristic, complemented
6#72128C211yesmaximal, characteristic, complemented
7#71684C212yesmaximal, complemented
8#69650C212yesmaximal, complemented
9#69648C212yesmaximal, complemented
10#69521C211yesmaximal, characteristic, complemented
11#69511C211yesmaximal, characteristic, complemented
12#24272—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C2×C3×#121C2212yescomplemented
16C2×C3×#120—21no—
17C2×C3×C4×D4C2211yescharacteristic
18C2×#937—21no—
19C22×#213—28no—
20C22×#213—24no—
21C22×#213C2212yes—
22C22×#213C2212yes—
23C2×#930C2212yescomplemented
24C22×#208C2211yescharacteristic, complemented
25C23×C3⋊C8C2211yescharacteristic
26C3×#394—21no—
27C3×#392—21no—
28C3×#390C2212yescomplemented
29C3×#380C2211yescharacteristic
30C3×#379—21no—
31C2×#832C2212yescomplemented
32C2×#832C2212yescomplemented
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic, complemented
36#5039—28no—
37#5011—24no—
38#4991—21no—
39C2×#809C2212yescomplemented
40C2×#809C2212yescomplemented
41C2×#808C2212yescomplemented
42#4960—21no—
43#4955—21no—
44#4939C2212yescomplemented
45C2×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
46C2×#803C2211yescharacteristic, complemented
47#4529C2212yescomplemented
48C23×D8—31no—
49C2×#469—32no—
50C2×#435—31no—
51#2408—31no—
52#2408D311yescharacteristic, complemented
53#2054—32no—
54#1999—32no—
55#1972—32no—
56#1807—31no—
57#1786—31no—
58#1786—31no—
59C25×C3—21no—
60C24×D3—21no—
61C22×C3×D4—24no—
62C22×C3×D4—22no—
63C22×C3×D4—22no—
64C22×C3×D4—21no—
65C22×C3×D4—22no—
66C22×C3×D4—42no—
67C22×C3×D4D411yescharacteristic
68C22×C3×D4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C22×#67—22no—
72C22×D12—22no—
73C22×D12—24no—
74C22×D12—22no—
75C22×D12—21no—
76C22×D12—42no—
77C22×D12C2312yes—
78C3×#121—24no—
79C3×#121—42no—
80C3×#120—44no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93#937—44no—
94C2×#213—232no—
95C2×#213—24no—
96C2×#213—24no—
97C2×#213—24no—
98C2×#213—24no—
99C2×#213—24no—
100C2×#213D418yescomplemented
101#930—24no—
102#930—42no—
103C2×(C2×D3)⋊C4—22no—
104C2×(C2×D3)⋊C4—22no—
105C2×(C2×D3)⋊C4—22no—
106C2×C4⋊Dic3C2312yes—
107C2×#208—22no—
108C2×#208—24no—
109C22×C3⋊C8—24no—
110C22×C3⋊C8D414yescomplemented
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C3×#95—21no—
116#832—24no—
117#832—42no—
118C3⋊#95—21no—
119#830—22no—
120#830—22no—
121#809—24no—
122#809—42no—
123#808—24no—
124C4⋊(C3⋊C8)—22no—
125#803—22no—
126C23×D4—31no—
127C23×D4C2×D311yescharacteristic
128C22×D8—32no—
129C22×D8—32no—
130C22×D8—64no—
131C22×D8—68no—
132C22×M16—31no—
133C23×C8—31no—
134C2×#121—32no—
135C2×#121C2×D312yescomplemented
136C2×#120—21no—
137C2×#120—61no—
138C2×C4×D4C2×D311yescharacteristic
139#469—68no—
140#435—64no—
141C2×#106—32no—
142C2×C4⋊C8—31no—
143C2×#102—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×#102—32no—
147C2×C22⋊C8—31no—
148C2×C22⋊C8—31no—
149C2×#97—31no—
150#394—21no—
151#394—61no—
152#392—21no—
153#392—61no—
154#390—32no—
155#390C2×D312yescomplemented
156#380—31no—
157#380C2×D311yescharacteristic
158#379—21no—
159#379—61no—
160#340—32no—
161C24×C3—22no—
162C24×C3—44no—
163C24×C3—42no—
164C24×C3—42no—
165C24×C3—42no—
166C24×C3—82no—
167C24×C3C2×D411yescharacteristic
168C24×C3C2×D411yescharacteristic
169C23×D3—22no—
170C23×D3—44no—
171C23×D3—42no—
172C23×D3—42no—
173C23×D3—42no—
174C23×D3—82no—
175C2×C3×D4—28no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—48no—
185C2×C3×D4—44no—
186C2×C3×D4—44no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4—42no—
190C2×C3×D4—82no—
191C2×C3×D4C2×D414yes—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4#4112yes—
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×D12C2×D414yes—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—44no—
227C3×C22⋊C4—44no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—82no—
230C3×C42—21no—
231C3×C42—21no—
232#217—42no—
233#213—216no—
234#213—216no—
235(C2×D3)⋊C4—44no—
236(C2×D3)⋊C4—44no—
237(C2×D3)⋊C4—82no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240#208—44no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—28no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8C2×D412yes—
248C2×C3⋊C8C2×D412yes—
249C2×C3⋊C8C2×D414yes—
250C25—21no—
251C25—61no—
252C22×D4—24no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—32no—
258C22×D4—42no—
259C22×D4—64no—
260C22×D4—62no—
261C22×D4—62no—
262C22×D4—62no—
263C22×D4—61no—
264C22×D4—122no—
265C22×D4#6711yescharacteristic
266C22×D4C22×D312yes—
267C23×C4C22×D311yescharacteristic
268C23×C4C22×D311yescharacteristic
269C2×D8—38no—
270C2×D8—64no—
271C2×D8—64no—
272C2×D8—64no—
273C2×D8—64no—
274C2×D8—632no—
275C2×D8—64no—
276C2×M16—62no—
277C2×M16—64no—
278C22×C8—31no—
279C22×C8—34no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—64no—
284#121—24no—
285#121—42no—
286#121—64no—
287#121—122no—
288#120—44no—
289#120—124no—
290C4×D4—22no—
291C4×D4—22no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4#6711yescharacteristic
294C2×C4⋊C4C22×D312yes—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—62no—
301C2×C22⋊C4—62no—
302C2×C22⋊C4—62no—
303C2×C22⋊C4#6711yescharacteristic
304C2×C22⋊C4#6711yescharacteristic
305C2×C42C22×D311yescharacteristic
306#106—64no—
307C4⋊C8—62no—
308#102—64no—
309#102—64no—
310#102—122no—
311#102—122no—
312C22⋊C8—62no—
313C22⋊C8—62no—
314#97—62no—
315#95—21no—
316#95—61no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—21no—
320C23×C3—24no—
321C23×C3—22no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—22no—
325C23×C3—48no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—88no—
333C23×C3—88no—
334C23×C3—84no—
335C23×C3—84no—
336C23×C3#12112yes—
337C23×C3AGL(1,ℤ/8)12yes—
338C23×C3AGL(1,ℤ/8)12yes—
339C23×C3C22×D411yescharacteristic
340C22×D3—24no—
341C22×D3—48no—
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—88no—
349C22×D3—88no—
350C22×D3—84no—
351C22×D3—84no—
352C3×D4—24no—
353C3×D4—24no—
354C3×D4—48no—
355C3×D4—44no—
356C3×D4—42no—
357C3×D4—42no—
358C3×D4—88no—
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—21no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4#12112yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D412yes—
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)12yes—
381#67—88no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385D12—24no—
386D12—24no—
387D12—48no—
388D12—42no—
389D12—42no—
390C3⋊C8—22no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C3⋊C8#12014yescomplemented
394C24—22no—
395C24—44no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—62no—
400C24—82no—
401C24—124no—
402C24—122no—
403C24—122no—
404C24—122no—
405C24—242no—
406C24C2×#6711yescharacteristic
407C24C2×#6711yescharacteristic
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—34no—
418C2×D4—48no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—68no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—82no—
431C2×D4—128no—
432C2×D4—124no—
433C2×D4—124no—
434C2×D4—124no—
435C2×D4—124no—
436C2×D4—122no—
437C2×D4—242no—
438C2×D4D3×D414yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—62no—
447C22×C4D3×D412yes—
448C22×C4#23712yes—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C23×D311yescharacteristic
454D8—616no—
455D8—616no—
456M16—124no—
457C2×C8—32no—
458C2×C8—34no—
459C2×C8—32no—
460C2×C8—68no—
461C2×C8—62no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—62no—
465C2×C8—62no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—42no—
476C22⋊C4—82no—
477C22⋊C4—124no—
478C22⋊C4—124no—
479C22⋊C4—122no—
480C22⋊C4—242no—
481C42—21no—
482C42—21no—
483C22×C3—22no—
484C22×C3—24no—
485C22×C3—24no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—44no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—816no—
496C22×C3—82no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—82no—
500C22×C3#46914yescomplemented
501C22×C3C2×#12111yescharacteristic
502C22×C3D4212yes—
503C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
505C22×C3#57612yes—
506C2×D3—44no—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—816no—
510C2×D3—82no—
511C2×D3—82no—
512C2×D3—82no—
513C2×D3—82no—
514C3×C4—22no—
515C3×C4—41no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—82no—
519C3×C4C2×#12011yescharacteristic
520C3×C4#53511yescharacteristic
521C3×C4#54612yes—
522Dic3—82no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—24no—
527C23—22no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—64no—
539C23—88no—
540C23—88no—
541C23—84no—
542C23—84no—
543C23—128no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—122no—
548C23—122no—
549C23—122no—
550C23—248no—
551C23—248no—
552C23—244no—
553C23—244no—
554C23#93212yes—
555C23#93912yes—
556C23#94912yes—
557C23C22×#6711yescharacteristic
558D4—24no—
559D4—24no—
560D4—48no—
561D4—44no—
562D4—42no—
563D4—42no—
564D4—64no—
565D4—64no—
566D4—88no—
567D4—128no—
568D4—122no—
569D4—122no—
570D4—248no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4#93912yes—
592C2×C4C2×D3×D412yes—
593C2×C4C2×D3×D411yescharacteristic
594C2×C4C2×#23711yescharacteristic
595C2×C4#100412yes—
596C8—34no—
597C8—62no—
598C8—121no—
599C8—121no—
600C2×C3—24no—
601C2×C3—81no—
602C2×C3—84no—
603C2×C3#240811yescharacteristic
604C2×C3C2×#46911yescharacteristic
605C2×C3#305811yescharacteristic
606C2×C3#326412yes—
607C2×C3#326412yes—
608D3—84no—
609C22—24no—
610C22—24no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—44no—
617C22—41no—
618C22—44no—
619C22—41no—
620C22—41no—
621C22—816no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—124no—
627C22—124no—
628C22—124no—
629C22—2416no—
630C22—242no—
631C22—242no—
632C22—242no—
633C22—242no—
634C22#503914yescomplemented
635C22C2×#93211yescharacteristic
636C22D4×#6712yes—
637C22C2×#93911yescharacteristic
638C22C2×#94911yescharacteristic
639C22#583512yes—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4—242no—
646C4D3×#12011yescharacteristic
647C4#559112yes—
648C4#559211yescharacteristic
649C3#2427211yescharacteristic, complemented, Sylow
650C2—24no—
651C2—81no—
652C2—84no—
653C2—244no—
654C2#8258211yescharacteristic
655C2C2×#503911yescharacteristic
656C2#8337811yescharacteristic
657C2#8348812yes—
658C2#8350212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2408C2
socleC23×C3C22×D4
hypercenter#2408D3
hypocenterC3#24272
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C23×#213C2semidirect
2. C3×#2408C2semidirect
3. C2×#5039C2semidirect
4. #82403C2semidirect
5. C2×#5011C2semidirect
6. #72128C2semidirect
7. #71684C2semidirect
8. #69650C2semidirect
9. #69648C2semidirect
10. #69521C2semidirect
11. #69511C2semidirect
15. C2×C3×#121C22semidirect
23. C2×#930C22semidirect
24. C22×#208C22semidirect
28. C3×#390C22semidirect
31. C2×#832C22semidirect
32. C2×#832C22semidirect
35. C2×#830C22semidirect
39. C2×#809C22semidirect
40. C2×#809C22semidirect
41. C2×#808C22semidirect
44. #4939C22semidirect
45. C2×C4⋊(C3⋊C8)C22semidirect
46. C2×#803C22semidirect
47. #4529C22semidirect
52. #2408D3semidirect
100. C2×#213D4semidirect
110. C22×C3⋊C8D4semidirect
135. C2×#121C2×D3semidirect
155. #390C2×D3semidirect
393. C3⋊C8#120semidirect
500. C22×C3#469semidirect
634. C22#5039semidirect
649. C3#24272semidirect

Maximal subgroups

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

subgroupindex
C23×#2132
C3×#24082
C2×#50392
#824032
C2×#50112
#721282
#716842
#696502
#696482
#695212
#695112
#242723