Finite Group Databasebeta

← #10826806

Subgroups — #10826806

Order 768.

2032 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3485C211yesmaximal, characteristic, complemented
2C2×#5830C211yesmaximal, characteristic, complemented
3C2×#5244C212yesmaximal, complemented
4#82903C212yesmaximal, complemented
5#72139C211yesmaximal, characteristic, complemented
6#71851C214yesmaximal, complemented
7#71705C212yesmaximal, complemented
8#71654C212yesmaximal, complemented
9#23904—31nomaximal, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic, complemented
13C3×#547—24no—
14C3×#546—24no—
15C3×#536—21no—
16C3×#534—21no—
17C3×#534C2212yes—
18C3×#525—21no—
19C2×C3×#123—21no—
20C2×C3×#123C2212yes—
21C3×#518C2211yescharacteristic
22#5830—28no—
23C2×#947—21no—
24C2×#940—21no—
25C2×#937C2214yes—
26C2×#937C2212yes—
27C2×#937C2212yescomplemented
28C2×#930—21no—
29C2×#930—21no—
30C2×#926C2211yescharacteristic, complemented
31#5244—28no—
32C2×#834C2214yes—
33C2×#834C2212yes—
34C2×#834C2212yescomplemented
35#5211C2212yescomplemented
36#4991—24no—
37#4960—24no—
38#4955C2214yescomplemented
39#4526C2214yescomplemented
40#3485D311yescharacteristic, complemented
41C2×#535—31no—
42C2×B(4,𝔽2)—32no—
43#2613—32no—
44#2021—34no—
45#1899—32no—
46#1889—32no—
47#1782—31no—
48C24×D3C2312yes—
49C3×H(5,𝔽2)—24no—
50C3×H(5,𝔽2)—42no—
51C2×C3×#41—24no—
52C2×C3×#41—24no—
53C2×C3×#41—21no—
54C2×C3×#41—22no—
55C2×C3×#41—42no—
56C22×C3×Q8D411yescharacteristic, complemented
57C22×C3×D4—21no—
58C22×C3×D4—22no—
59C22×C3×D4D411yescharacteristic, complemented
60C22×C3×D4C2311yescharacteristic
61C22×C3×D4C2312yes—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4D411yescharacteristic, complemented
64C23×C3×C4D411yescharacteristic, complemented
65C23×C3×C4C2311yescharacteristic
66C22×#67—22no—
67C22×#67—22no—
68C22×#67—24no—
69C22×D12—24no—
70C22×D12—21no—
71C22×D12—21no—
72C22×D12—24no—
73C22×D12—42no—
74C3×C4⋊Q8—22no—
75C3×#124—22no—
76C3×#123—24no—
77C3×#123—22no—
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—22no—
81C3×#123—24no—
82C3×#123—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—24no—
85C3×#121—22no—
86C3×#121—24no—
87C3×#121—24no—
88C3×#120—22no—
89C3×#120—22no—
90C3×#120—22no—
91C3×C4×D4—24no—
92C3×C4×D4—24no—
93C3×#117—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D414yescomplemented
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4D414yescomplemented
99C2×C3×C22⋊C4C2312yes—
100C2×#217C2314yes—
101C2×#217C2312yes—
102#947—28no—
103#940—28no—
104#937—24no—
105#937—24no—
106#937—24no—
107#937—28no—
108#937—44no—
109#937D418yescomplemented
110#930—28no—
111#930—28no—
112C2×(C2×D3)⋊C4—22no—
113C2×(C2×D3)⋊C4—22no—
114C2×(C2×D3)⋊C4—24no—
115C2×(C2×D3)⋊C4—28no—
116#926—24no—
117C2×C4⋊Dic3—21no—
118C2×C4×Dic3—21no—
119#834—24no—
120#834—28no—
121#834—42no—
122C3⋊#95—24no—
123C2×H(5,𝔽2)C2×D311yescharacteristic
124C22×#41C2×D311yescharacteristic
125C23×D4—31no—
126#547—24no—
127#546—24no—
128#536—21no—
129#535—68no—
130#534—21no—
131#534C2×D312yes—
132C2×#127—61no—
133C2×#124—61no—
134#525—21no—
135C2×#123—21no—
136C2×#123C2×D312yes—
137C2×#121—61no—
138C2×#121—61no—
139C2×#120—32no—
140C2×#120—32no—
141C2×#120—34no—
142#518C2×D311yescharacteristic
143C2×#117—31no—
144B(4,𝔽2)—68no—
145C2×#99—32no—
146C2×#99—34no—
147C2×#99—32no—
148#394—64no—
149#392—64no—
150#381—32no—
151#379—34no—
152#342—34no—
153C24×C3—21no—
154C24×C3C2×D412yes—
155C24×C3C2×D411yescharacteristic
156C24×C3C2×D411yescharacteristic
157C24×C3C2411yescharacteristic
158C23×D3—24no—
159C23×D3—22no—
160C23×D3—22no—
161C23×D3—24no—
162C23×D3—84no—
163C23×D3C2×D414yes—
164C3×#41—416no—
165C3×#41—48no—
166C3×#41—42no—
167C3×#41—42no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—41no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—24no—
184C2×C3×D4—24no—
185C2×C3×D4—22no—
186C2×C3×D4—41no—
187C2×C3×D4—48no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4C2×D414yes—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—41no—
201C22×C3×C4—41no—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D414yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D414yes—
209C2×#67—28no—
210C2×#67—28no—
211C2×#67—48no—
212C2×#67—84no—
213C2×#67—84no—
214C2×#67—88no—
215C22×Dic3—22no—
216C22×Dic3—24no—
217C2×D12—24no—
218C2×D12—24no—
219C2×D12—416no—
220C2×D12—42no—
221C2×D12—42no—
222C2×D12—42no—
223C2×D12—42no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—24no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C42—22no—
236#217—22no—
237#217—24no—
238#217—24no—
239#217C2×D414yes—
240(C2×D3)⋊C4—28no—
241(C2×D3)⋊C4—28no—
242(C2×D3)⋊C4—48no—
243(C2×D3)⋊C4—88no—
244C4⋊Dic3—24no—
245C4×Dic3—24no—
246C25—32no—
247H(5,𝔽2)—24no—
248H(5,𝔽2)—42no—
249C2×#41—24no—
250C2×#41—21no—
251C2×#41—24no—
252C2×#41—22no—
253C2×#41—42no—
254C22×Q8#6711yescharacteristic, complemented
255C22×D4—21no—
256C22×D4—22no—
257C22×D4—64no—
258C22×D4—62no—
259C22×D4—62no—
260C22×D4—64no—
261C22×D4—61no—
262C22×D4—61no—
263C22×D4—64no—
264C22×D4—122no—
265C22×D4#6711yescharacteristic, complemented
266C22×D4C22×D312yes—
267C22×D4C22×D311yescharacteristic
268C22×D4C22×D311yescharacteristic
269C23×C4#6711yescharacteristic, complemented
270C23×C4#6711yescharacteristic, complemented
271C23×C4C22×D311yescharacteristic
272C4⋊Q8—22no—
273#127—68no—
274#124—22no—
275#124—68no—
276#123—24no—
277#123—22no—
278#123—22no—
279#123—22no—
280#123—22no—
281#123—24no—
282#123—22no—
283C22⋊Q8—22no—
284C22⋊Q8—24no—
285#121—22no—
286#121—24no—
287#121—24no—
288#121—68no—
289#121—68no—
290#120—22no—
291#120—22no—
292#120—22no—
293#120—38no—
294#120—64no—
295#120—64no—
296#120—64no—
297#120—68no—
298#120—124no—
299C4×D4—24no—
300C4×D4—24no—
301#117—21no—
302#117—64no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4#6714yescomplemented
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—32no—
309C2×C22⋊C4—34no—
310C2×C22⋊C4—62no—
311C2×C22⋊C4—62no—
312C2×C22⋊C4—64no—
313C2×C22⋊C4—68no—
314C2×C22⋊C4#6714yescomplemented
315C2×C22⋊C4C22×D312yes—
316C2×C42—61no—
317#99—68no—
318#99—64no—
319#99—122no—
320#95—64no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—24no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—44no—
333C23×C3—82no—
334C23×C3#12012yes—
335C23×C3C22×D412yes—
336C23×C3C22×D411yescharacteristic
337C23×C3C22×D412yes—
338C23×C3H(5,𝔽2)12yes—
339C22×D3—24no—
340C22×D3—28no—
341C22×D3—48no—
342C22×D3—48no—
343C22×D3—44no—
344C22×D3—44no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—44no—
348C22×D3—84no—
349C22×D3—84no—
350C22×D3—88no—
351C3×Q8—44no—
352C3×Q8—42no—
353C3×D4—28no—
354C3×D4—44no—
355C3×D4—42no—
356C3×D4—44no—
357C3×D4—42no—
358C3×D4—42no—
359C3×D4—44no—
360C3×D4—44no—
361C3×D4—84no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—28no—
369C2×C3×C4—21no—
370C2×C3×C4—24no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—24no—
375C2×C3×C4—44no—
376C2×C3×C4—44no—
377C2×C3×C4—44no—
378C2×C3×C4—42no—
379C2×C3×C4—44no—
380C2×C3×C4—82no—
381C2×C3×C4#12012yes—
382C2×C3×C4#12012yes—
383C2×C3×C4#12012yes—
384#67—88no—
385#67—88no—
386#67—816no—
387C2×Dic3—24no—
388C2×Dic3—44no—
389C2×Dic3—48no—
390C2×Dic3—44no—
391D12—48no—
392D12—48no—
393C24—21no—
394C24—34no—
395C24—64no—
396C24—62no—
397C24—62no—
398C24—64no—
399C24—244no—
400C24C2×#6712yes—
401C24C2×#6711yescharacteristic
402C24C2×#6711yescharacteristic
403C24C23×D311yescharacteristic
404#41—416no—
405#41—48no—
406#41—42no—
407#41—42no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—41no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—41no—
427C2×D4—48no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—64no—
431C2×D4—64no—
432C2×D4—68no—
433C2×D4—68no—
434C2×D4—1216no—
435C2×D4—122no—
436C2×D4—122no—
437C2×D4—128no—
438C2×D4—122no—
439C2×D4—122no—
440C2×D4—244no—
441C2×D4—244no—
442C2×D4—248no—
443C2×D4D3×D414yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—62no—
456C22×C4—64no—
457C22×C4C2×#6712yes—
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6714yes—
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C2×#6714yes—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—64no—
469C22⋊C4—22no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—34no—
477C22⋊C4—68no—
478C22⋊C4—62no—
479C22⋊C4—68no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—128no—
483C22⋊C4—248no—
484C42—22no—
485C42—64no—
486C22×C3—24no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—41no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—44no—
498C22×C3—81no—
499C22×C3—84no—
500C22×C3—81no—
501C22×C3—162no—
502C22×C3C2×#12011yescharacteristic
503C22×C3C2×#12011yescharacteristic
504C22×C3#53411yescharacteristic
505C22×C3D4214yes—
506C2×D3—44no—
507C2×D3—44no—
508C2×D3—48no—
509C2×D3—48no—
510C2×D3—44no—
511C2×D3—88no—
512C2×D3—84no—
513C2×D3—84no—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—22no—
517C3×C4—42no—
518C3×C4—41no—
519C3×C4—44no—
520Dic3—44no—
521Dic3—84no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—21no—
527C23—24no—
528C23—21no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—64no—
535C23—68no—
536C23—82no—
537C23—128no—
538C23—128no—
539C23—124no—
540C23—124no—
541C23—124no—
542C23—122no—
543C23—122no—
544C23—244no—
545C23—244no—
546C23—248no—
547C23#93712yes—
548C23C2×D3×D412yes—
549C23#100412yes—
550C23C22×#6711yescharacteristic
551C23C22×#6712yes—
552Q8—44no—
553Q8—42no—
554D4—28no—
555D4—44no—
556D4—42no—
557D4—44no—
558D4—42no—
559D4—44no—
560D4—44no—
561D4—42no—
562D4—84no—
563D4—128no—
564D4—128no—
565D4—248no—
566D4—248no—
567D4—2416no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—24no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—64no—
587C2×C4—82no—
588C2×C4—124no—
589C2×C4—128no—
590C2×C4—124no—
591C2×C4#93712yes—
592C2×C4#95312yes—
593C2×C4#95312yes—
594C2×C3—21no—
595C2×C3—24no—
596C2×C3—21no—
597C2×C3—42no—
598C2×C3—81no—
599C2×C3—82no—
600C2×C3#240311yescharacteristic
601C2×C3#302712yes—
602D3—84no—
603C22—24no—
604C22—22no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—81no—
616C22—84no—
617C22—81no—
618C22—124no—
619C22—124no—
620C22—128no—
621C22—128no—
622C22—124no—
623C22—162no—
624C22—248no—
625C22—244no—
626C22—244no—
627C22C2×#93711yescharacteristic
628C22D4×#6714yes—
629C22#580411yescharacteristic
630C22C2×#95311yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—41no—
636C4—44no—
637C4—124no—
638C4—244no—
639C3#2390411yescharacteristic, complemented, Sylow
640C2—21no—
641C2—24no—
642C2—21no—
643C2—42no—
644C2—81no—
645C2—82no—
646C2—244no—
647C2#8405211yescharacteristic
648C2#8412612yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#3485C2
socleC22×C3C2×#120
hypercenter#3485D3
hypocenterC3#23904
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. C3×#3485C2semidirect
2. C2×#5830C2semidirect
3. C2×#5244C2semidirect
4. #82903C2semidirect
5. #72139C2semidirect
6. #71851C2semidirect
7. #71705C2semidirect
8. #71654C2semidirect
12. C23×D12C22semidirect
27. C2×#937C22semidirect
30. C2×#926C22semidirect
34. C2×#834C22semidirect
35. #5211C22semidirect
38. #4955C22semidirect
39. #4526C22semidirect
40. #3485D3semidirect
56. C22×C3×Q8D4semidirect
59. C22×C3×D4D4semidirect
63. C23×C3×C4D4semidirect
64. C23×C3×C4D4semidirect
95. C2×C3×C4⋊C4D4semidirect
98. C2×C3×C22⋊C4D4semidirect
109. #937D4semidirect
254. C22×Q8#67semidirect
265. C22×D4#67semidirect
269. C23×C4#67semidirect
270. C23×C4#67semidirect
305. C2×C4⋊C4#67semidirect
314. C2×C22⋊C4#67semidirect
639. C3#23904semidirect

Maximal subgroups

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

subgroupindex
C3×#34852
C2×#58302
C2×#52442
#829032
#721392
#718512
#717052
#716542
#239043