Finite Group Databasebeta

← #10827617

Subgroups — #10827617

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5541C211yesmaximal, characteristic, complemented
2C2×#5131C211yesmaximal, characteristic, complemented
3#82403C211yesmaximal, characteristic
4C2×#4731C211yesmaximal, characteristic, complemented
5C3×#1870C211yesmaximal, characteristic, complemented
6#71989C211yesmaximal, characteristic, complemented
7#69962C212yesmaximal, complemented
8#69932C212yesmaximal, complemented
9#69926C211yesmaximal, characteristic, complemented
10#68686C212yesmaximal, complemented
11#68679C212yesmaximal, complemented
12#23982—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#937—21no—
17C2×#935—21no—
18C2×#930C2212yes—
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic, complemented
21C2×#924C2211yescharacteristic
22C2×#924C2211yescharacteristic, complemented
23C22×#204C2212yes—
24#5541—24no—
25#5541—24no—
26C2×#889—21no—
27C2×#879—21no—
28C2×#877—21no—
29C2×#870C2211yescharacteristic, complemented
30C2×C4×Dic6C2211yescharacteristic, complemented
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2212yescomplemented
33#5212C2211yescharacteristic
34#5131—24no—
35C2×#821C2212yes—
36C2×#821C2212yes—
37C2×#821C2212yes—
38C2×#816C2212yescomplemented
39#4991—21no—
40#4960—21no—
41#4955—21no—
42#4939C2212yes—
43#4731—24no—
44C3×#336C2212yescomplemented
45#4551C2212yescomplemented
46#4551C2212yescomplemented
47#4551C2212yescomplemented
48C2×#552—31no—
49C2×#450—31no—
50C2×#418—31no—
51#2408—31no—
52#2025—32no—
53#2000—32no—
54#1894—31no—
55#1880—32no—
56#1874—32no—
57#1870D311yescharacteristic, complemented
58#1825—31no—
59C24×D3—21no—
60C22×C3×D4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×#67—22no—
64C23×Dic3—21no—
65C22×D12—22no—
66C22×D12—24no—
67C22×D12—22no—
68C22×D12—21no—
69C22×D12—42no—
70C22×D12C2312yes—
71C22×Dic6C2311yescharacteristic
72C2×C3×M16—22no—
73C2×C3×M16—42no—
74C22×C3×C8C2312yes—
75C22×C3×C8C2312yes—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C22⋊C4D411yescharacteristic
80C2×C3×C22⋊C4D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#217—21no—
83C2×#217—21no—
84C2×#217—21no—
85C2×#217—21no—
86#937—44no—
87#935—28no—
88#930—24no—
89#930—42no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93#926—22no—
94#926D414yescomplemented
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102#924—22no—
103#924—22no—
104#924—22no—
105#924D414yescomplemented
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×#204—24no—
114C2×#204—24no—
115C2×#204—42no—
116#889—28no—
117#879—28no—
118#877—28no—
119#870—22no—
120#870—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×C22⋊C8—22no—
124C3×C22⋊C8—42no—
125C3⋊#95—21no—
126#821—24no—
127#821—24no—
128#821—24no—
129#816—42no—
130C23×D4—31no—
131C22×SD16—32no—
132C22×M16C2×D311yescharacteristic
133#552—64no—
134#552—64no—
135C2×#126—61no—
136C2×#125—31no—
137C2×#125—61no—
138C2×#123—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—31no—
141C2×#121—32no—
142C2×#120—61no—
143C2×C4×Q8—31no—
144C2×C4×D4C2×D311yescharacteristic
145C2×#117—31no—
146C2×#117—61no—
147C22×C4⋊C4—31no—
148#450—64no—
149#418—64no—
150C2×#103—32no—
151C2×#102—32no—
152C2×#102—32no—
153C2×#102—32no—
154C2×C22⋊C8C2×D311yescharacteristic
155C2×C22⋊C8C2×D312yescomplemented
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#336C2×D312yescomplemented
165C24×C3C2×D411yescharacteristic
166C24×C3C2×D411yescharacteristic
167C23×D3—22no—
168C23×D3—44no—
169C23×D3—42no—
170C23×D3—42no—
171C23×D3—42no—
172C23×D3—82no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—41no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2411yescharacteristic
189C2×#67—44no—
190C2×#67—44no—
191C2×#67—82no—
192C22×Dic3—24no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—42no—
199C22×Dic3—42no—
200C2×D12—28no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—48no—
207C2×D12—44no—
208C2×D12—44no—
209C2×D12—42no—
210C2×D12#4114yes—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C2×Dic6—21no—
214C2×Dic6C2×D412yes—
215C3×M16—44no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C3×C4⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C42—21no—
226C3×C42—21no—
227#217—24no—
228#217—24no—
229#217—24no—
230#217—42no—
231(C2×D3)⋊C4—44no—
232(C2×D3)⋊C4—44no—
233(C2×D3)⋊C4—82no—
234C4⋊Dic3—24no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240C4⋊Dic3C2×D412yes—
241C4⋊Dic3C2×D412yes—
242C4⋊Dic3C2×D412yes—
243Dic3⋊C4—24no—
244Dic3⋊C4—24no—
245Dic3⋊C4—24no—
246Dic3⋊C4—24no—
247Dic3⋊C4—24no—
248C4×Dic3—24no—
249C4×Dic3—24no—
250#204—48no—
251C25—61no—
252C22×Q8—31no—
253C22×D4—32no—
254C22×D4—64no—
255C22×D4—62no—
256C22×D4—61no—
257C22×D4—62no—
258C22×D4—62no—
259C22×D4—122no—
260C22×D4D1211yescharacteristic
261C23×C4—61no—
262C23×C4C22×D311yescharacteristic
263C23×C4C22×D311yescharacteristic
264C2×SD16—64no—
265C2×SD16—64no—
266C2×SD16—122no—
267C2×M16—22no—
268C2×M16—42no—
269C22×C8C22×D312yes—
270C22×C8C22×D312yes—
271#126—68no—
272#125—62no—
273#125—62no—
274#125—68no—
275#123—68no—
276C22⋊Q8—34no—
277C22⋊Q8—62no—
278C22⋊Q8—62no—
279C22⋊Q8—62no—
280#121—64no—
281#121—122no—
282#120—124no—
283C4×Q8—62no—
284C4×Q8—62no—
285C4×D4—22no—
286C4×D4—22no—
287#117—34no—
288#117—68no—
289#117—62no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—32no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4—62no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4D1211yescharacteristic
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—62no—
308C2×C22⋊C4—62no—
309C2×C22⋊C4—62no—
310C2×C22⋊C4D1211yescharacteristic
311C2×C22⋊C4D1211yescharacteristic
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#103—122no—
316#102—64no—
317#102—64no—
318#102—64no—
319C22⋊C8—22no—
320C22⋊C8—42no—
321#95—61no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3#12012yes—
331C23×C3AGL(1,ℤ/8)12yes—
332C23×C3C22×D411yescharacteristic
333C22×D3—24no—
334C22×D3—48no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—88no—
342C22×D3—88no—
343C22×D3—84no—
344C22×D3—84no—
345C3×D4—44no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4#12012yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C2×#4112yes—
364C2×C3×C4H(5,𝔽2)12yes—
365#67—88no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—48no—
371C2×Dic3—42no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380D12—24no—
381D12—24no—
382D12—48no—
383D12—42no—
384D12—42no—
385Dic6—24no—
386Dic6—41no—
387Dic6—41no—
388C3×C8—42no—
389C3×C8—42no—
390C24—62no—
391C24—124no—
392C24—122no—
393C24—122no—
394C24—122no—
395C24—242no—
396C24C2×D1211yescharacteristic
397C24C2×D1211yescharacteristic
398C2×Q8—32no—
399C2×Q8—61no—
400C2×Q8—62no—
401C2×Q8—62no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—34no—
405C2×D4—41no—
406C2×D4—68no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—128no—
413C2×D4—124no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—122no—
418C2×D4—242no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—64no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—122no—
432C22×C4—122no—
433C22×C4C2×D1211yescharacteristic
434C22×C4C2×D1211yescharacteristic
435C22×C4C2×D1211yescharacteristic
436C22×C4C2×D1211yescharacteristic
437C22×C4D3×D412yes—
438C22×C4D3×D412yes—
439C22×C4C23×D311yescharacteristic
440SD16—128no—
441M16—44no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C4⋊C4—22no—
449C4⋊C4—32no—
450C4⋊C4—32no—
451C4⋊C4—32no—
452C4⋊C4—64no—
453C4⋊C4—64no—
454C4⋊C4—64no—
455C4⋊C4—64no—
456C4⋊C4—64no—
457C4⋊C4—64no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—64no—
466C22⋊C4—64no—
467C22⋊C4—64no—
468C22⋊C4—124no—
469C22⋊C4—124no—
470C22⋊C4—122no—
471C22⋊C4—242no—
472C42—21no—
473C42—21no—
474C42—64no—
475C42—64no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—24no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3C2×#12011yescharacteristic
487C22×C3#54612yes—
488C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
489C22×C3#57511yescharacteristic
490C22×C3#57712yes—
491C2×D3—44no—
492C2×D3—44no—
493C2×D3—44no—
494C2×D3—816no—
495C2×D3—82no—
496C2×D3—82no—
497C2×D3—82no—
498C2×D3—82no—
499C3×C4—22no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4#52511yescharacteristic
504C3×C4#53511yescharacteristic
505C3×C4#54612yes—
506Dic3—42no—
507Dic3—42no—
508Dic3—42no—
509Dic3—42no—
510Dic3—82no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—64no—
520C23—128no—
521C23—122no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—244no—
528C23—244no—
529C23—248no—
530C23—248no—
531C23#88212yes—
532C23#90812yes—
533C23C22×D1211yescharacteristic
534Q8—64no—
535Q8—121no—
536Q8—121no—
537D4—44no—
538D4—64no—
539D4—64no—
540D4—128no—
541D4—122no—
542D4—122no—
543D4—248no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—128no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4#88212yes—
573C2×C4C2×D3×D411yescharacteristic
574C2×C4C2×D3×D411yescharacteristic
575C2×C4#100412yes—
576C2×C4D3×#4112yes—
577C8—42no—
578C8—42no—
579C2×C3—42no—
580C2×C3—42no—
581C2×C3#240411yescharacteristic
582C2×C3#300611yescharacteristic
583C2×C3#301011yescharacteristic
584C2×C3#331012yes—
585C2×C3#332312yes—
586D3—84no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—24no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—124no—
598C22—124no—
599C22—124no—
600C22—2416no—
601C22—242no—
602C22—242no—
603C22—242no—
604C22—242no—
605C22C2×#88211yescharacteristic
606C22#555412yes—
607C22C2×#90811yescharacteristic
608C22#574811yescharacteristic
609C22#575112yes—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—122no—
615C4—122no—
616C4—122no—
617C4—122no—
618C4—242no—
619C4#559211yescharacteristic
620C4#561012yes—
621C4#565611yescharacteristic
622C3#2398211yescharacteristic, complemented, Sylow
623C2—42no—
624C2—42no—
625C2—244no—
626C2#7488511yescharacteristic
627C2#7863911yescharacteristic
628C2#7867811yescharacteristic
629C2#7950112yes—
630C2#7955412yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1870C2
socleC23×C3C22×D4
prime radical#82403C2
hypercenter#1870D3
hypocenterC3#23982
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1870

Complements

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

NH ≅ G/Ntype
1. C2×#5541C2semidirect
2. C2×#5131C2semidirect
4. C2×#4731C2semidirect
5. C3×#1870C2semidirect
6. #71989C2semidirect
7. #69962C2semidirect
8. #69932C2semidirect
9. #69926C2semidirect
10. #68686C2semidirect
11. #68679C2semidirect
20. C22×C4⋊Dic3C22semidirect
22. C2×#924C22semidirect
29. C2×#870C22semidirect
30. C2×C4×Dic6C22semidirect
32. C2×C3×C22⋊C8C22semidirect
38. C2×#816C22semidirect
44. C3×#336C22semidirect
45. #4551C22semidirect
46. #4551C22semidirect
47. #4551C22semidirect
57. #1870D3semidirect
94. #926D4semidirect
105. #924D4semidirect
155. C2×C22⋊C8C2×D3semidirect
164. #336C2×D3semidirect
622. C3#23982semidirect

Maximal subgroups

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

subgroupindex
C2×#55412
C2×#51312
#824032
C2×#47312
C3×#18702
#719892
#699622
#699322
#699262
#686862
#686792
#239823