Finite Group Databasebeta

← #11635875

Subgroups — #11635875

Order 768.

5070 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5717C211yesmaximal, characteristic, complemented
2C2×D3×#127C211yesmaximal, characteristic, complemented
3C2×D4×D12C214yesmaximal, complemented
4C3×#2366C211yesmaximal, characteristic, complemented
5#82409C214yesmaximal, complemented
6#76436C214yesmaximal, complemented
7#76314C214yesmaximal, complemented
8#74912C214yesmaximal, complemented
9#74884C214yesmaximal, complemented
10#73799C212yesmaximal, complemented
11C4×#895C212yesmaximal, complemented
12#59126—31nomaximal, Sylow
13C22×D3×D4C2214yescomplemented
14C22×D3×D4C2214yescomplemented
15C22×D3×D4C2212yescomplemented
16C23×D12C2214yescomplemented
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×D4C2214yescomplemented
19C2×#940C2214yescomplemented
20C2×#940C2212yescomplemented
21C2×#938C2214yescomplemented
22C2×#937C2214yescomplemented
23C2×#930C2218yescomplemented
24#5717—28no—
25D3×#127—28no—
26D4×D12—232no—
27C2×#895C2212yes—
28C2×#895C2214yescomplemented
29C2×#895C2214yescomplemented
30C2×#882C2218yescomplemented
31C2×#882C2218yescomplemented
32C2×#882C2214yescomplemented
33C2×#874C2214yescomplemented
34C2×#874C2211yescharacteristic, complemented
35C2×C4×D12C2214yescomplemented
36C2×C4×D12C2212yescomplemented
37C2×C42×D3C2211yescharacteristic, complemented
38C3×#390C2214yescomplemented
39C3×#386C2214yescomplemented
40C3×C4×C4⋊C4C2212yescomplemented
41#5213C2214yescomplemented
42#4988C2212yescomplemented
43#4987C2212yescomplemented
44#4984C2212yescomplemented
45#4961C2218yescomplemented
46#4960C2218yescomplemented
47#4960C2218yescomplemented
48#4955C2214yescomplemented
49#4939C2214yescomplemented
50C4×(C2×D3)⋊C4C2214yescomplemented
51#4671C2218yescomplemented
52#4670C2214yescomplemented
53#4665C2214yescomplemented
54C2×D42—31no—
55C2×D42—34no—
56C22×#127—31no—
57#2669—34no—
58#2541—32no—
59#2433—34no—
60#2428—34no—
61#2408—34no—
62#2403—34no—
63#2366D311yescharacteristic, complemented
64C4×#121—32no—
65C24×D3C2314yes—
66C24×D3C2314yes—
67C22×C3×D4C2314yes—
68C22×C3×D4C2312yes—
69C23×C3×C4C2314yes—
70C22×#67C2314yes—
71C22×#67C2318yes—
72C22×#67C2318yes—
73C2×D3×D4—216no—
74C2×D3×D4—28no—
75C2×D3×D4—216no—
76C2×D3×D4—216no—
77C2×D3×D4—216no—
78C2×D3×D4—48no—
79C2×D3×D4—48no—
80C2×D3×D4—44no—
81C2×D3×D4—44no—
82C22×D12—232no—
83C22×D12—48no—
84C22×D12C2314yes—
85C22×D12C2318yes—
86C22×D12C2314yes—
87C22×D12C2312yes—
88C22×D12C2312yes—
89C22×D12C2318yes—
90C22×C4×D3C2311yescharacteristic
91C22×C4×D3C2314yes—
92C22×C4×D3C2312yes—
93C3×#127D418yescomplemented
94C3×C4×D4—216no—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C22⋊C4C2318yes—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C42C2314yes—
100C2×C3×C42C2311yescharacteristic
101C2×#217C2314yes—
102#940—216no—
103#940—28no—
104#938—216no—
105#937—216no—
106#930—232no—
107C2×(C2×D3)⋊C4C2314yes—
108C2×(C2×D3)⋊C4C2314yes—
109C2×(C2×D3)⋊C4C2314yes—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2318yes—
112C2×(C2×D3)⋊C4C2318yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×C4⋊Dic3C2314yes—
115C2×C4⋊Dic3C2312yes—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2312yes—
118#895—216no—
119#895—48no—
120#895D4116yescomplemented
121#882—232no—
122#882—232no—
123#882—48no—
124#874—216no—
125#874—24no—
126C4×D12—216no—
127C4×D12—28no—
128C42×D3—24no—
129C3×#95C2314yes—
130C3⋊#95C2314yes—
131C3⋊#95C2314yes—
132C23×D4—34no—
133C23×D4—34no—
134C23×D4—32no—
135C23×D4—34no—
136D42—632no—
137D42—68no—
138C2×#127—32no—
139C2×#127—34no—
140C2×#127—31no—
141C2×#127—34no—
142C2×#127—68no—
143C2×#127C2×D311yescharacteristic
144C2×#121—34no—
145C2×#121—38no—
146C2×#121—32no—
147C2×#121—34no—
148C2×#121—34no—
149C2×#120—34no—
150C2×#120—38no—
151C2×#120—38no—
152C2×#120—34no—
153C2×C4×D4—34no—
154C2×C4×D4—32no—
155C2×C4×D4C2×D314yescomplemented
156C22×C42—31no—
157#394—38no—
158#392—38no—
159#392—38no—
160#392—38no—
161#390—32no—
162#390—34no—
163#390C2×D314yescomplemented
164#389—32no—
165#388—34no—
166#386—34no—
167#386—34no—
168#386C2×D314yescomplemented
169#380—32no—
170#379—34no—
171C4×C4⋊C4C2×D312yescomplemented
172C4×C22⋊C4—34no—
173C24×C3C2414yes—
174C23×D3—28no—
175C23×D3—28no—
176C23×D3—216no—
177C23×D3—216no—
178C23×D3—28no—
179C23×D3—44no—
180C23×D3—48no—
181C23×D3—44no—
182C23×D3—48no—
183C23×D3—44no—
184C23×D3C2414yes—
185C23×D3C2414yes—
186C23×D3C2411yescharacteristic
187C2×C3×D4—28no—
188C2×C3×D4—28no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4C2×D4116yescomplemented
192C2×C3×D4C2×D418yescomplemented
193C22×C3×C4—216no—
194C22×C3×C4—44no—
195C22×C3×C4C2414yes—
196C22×C3×C4C2414yes—
197C22×C3×C4C2411yescharacteristic
198C22×C3×C4C2412yes—
199C2×#67—216no—
200C2×#67—28no—
201C2×#67—216no—
202C2×#67—216no—
203C2×#67—216no—
204C2×#67—28no—
205C2×#67—48no—
206C2×#67—44no—
207C2×#67—48no—
208C22×Dic3C2414yes—
209C22×Dic3C2412yes—
210C22×Dic3C2411yescharacteristic
211D3×D4—464no—
212D3×D4—432no—
213D3×D4—464no—
214C2×D12—28no—
215C2×D12—28no—
216C2×D12—24no—
217C2×D12—24no—
218C2×D12—216no—
219C2×D12—28no—
220C2×D12—24no—
221C2×D12—28no—
222C2×D12—216no—
223C2×D12—28no—
224C2×D12—216no—
225C2×D12—464no—
226C2×D12—48no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12—48no—
230C2×D12—48no—
231C2×D12C2×D418yescomplemented
232C2×D12C2×D4116yescomplemented
233C2×C4×D3—22no—
234C2×C4×D3—22no—
235C2×C4×D3—24no—
236C2×C4×D3—24no—
237C2×C4×D3—22no—
238C2×C4×D3—28no—
239C2×C4×D3—28no—
240C2×C4×D3—28no—
241C2×C4×D3C2×D418yescomplemented
242C3×C4⋊C4—24no—
243C3×C4⋊C4C2×D418yescomplemented
244C3×C22⋊C4—216no—
245C3×C22⋊C4—44no—
246C3×C42—28no—
247C3×C42C2×D414yescomplemented
248#217—28no—
249(C2×D3)⋊C4—28no—
250(C2×D3)⋊C4—28no—
251(C2×D3)⋊C4—216no—
252(C2×D3)⋊C4—28no—
253(C2×D3)⋊C4—28no—
254(C2×D3)⋊C4—48no—
255(C2×D3)⋊C4C2×D4116yescomplemented
256C4⋊Dic3—24no—
257C4⋊Dic3—28no—
258C4×Dic3—22no—
259C4×Dic3—24no—
260C25—34no—
261C25—34no—
262C22×D4—38no—
263C22×D4—38no—
264C22×D4—34no—
265C22×D4—34no—
266C22×D4—32no—
267C22×D4—34no—
268C22×D4—38no—
269C22×D4—38no—
270C22×D4—32no—
271C22×D4—632no—
272C22×D4—616no—
273C22×D4—616no—
274C22×D4—616no—
275C22×D4—68no—
276C22×D4—616no—
277C22×D4—128no—
278C22×D4—128no—
279C22×D4—128no—
280C22×D4—124no—
281C22×D4—124no—
282C22×D4C22×D314yes—
283C22×D4C22×D312yes—
284C23×C4—32no—
285C23×C4—34no—
286C23×C4—31no—
287C23×C4C22×D314yes—
288#127—616no—
289#127—616no—
290#127—68no—
291#127—64no—
292#127D1218yescomplemented
293#121—316no—
294#121—632no—
295#121—616no—
296#121—616no—
297#121—128no—
298#120—632no—
299#120—632no—
300#120—616no—
301#120—128no—
302C4×D4—216no—
303C4×D4—616no—
304C4×D4—68no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4C22×D312yes—
308C2×C4⋊C4C22×D312yes—
309C2×C22⋊C4—34no—
310C2×C22⋊C4—34no—
311C2×C22⋊C4—34no—
312C2×C22⋊C4—34no—
313C2×C22⋊C4—34no—
314C2×C22⋊C4—34no—
315C2×C22⋊C4—38no—
316C2×C22⋊C4—38no—
317C2×C22⋊C4C22×D318yes—
318C2×C22⋊C4C22×D314yes—
319C2×C42—32no—
320C2×C42—31no—
321C2×C42—64no—
322C2×C42C22×D314yes—
323C2×C42C22×D311yescharacteristic
324#95—34no—
325#95—34no—
326#95C22×D314yes—
327C23×C3—28no—
328C23×C3—24no—
329C23×C3—28no—
330C23×C3—24no—
331C23×C3C22×D418yes—
332C23×C3C2511yescharacteristic
333C22×D3—21no—
334C22×D3—22no—
335C22×D3—21no—
336C22×D3—24no—
337C22×D3—24no—
338C22×D3—24no—
339C22×D3—24no—
340C22×D3—24no—
341C22×D3—24no—
342C22×D3—28no—
343C22×D3—24no—
344C22×D3—28no—
345C22×D3—24no—
346C22×D3—24no—
347C22×D3—416no—
348C22×D3—432no—
349C22×D3—416no—
350C22×D3—432no—
351C22×D3—416no—
352C22×D3—816no—
353C22×D3—88no—
354C22×D3—816no—
355C22×D3—88no—
356C22×D3—88no—
357C22×D3C22×D418yes—
358C22×D3C22×D414yes—
359C22×D3H(5,𝔽2)12yes—
360C3×D4—232no—
361C3×D4—48no—
362C2×C3×C4—24no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—416no—
370C2×C3×C4C22×D412yes—
371C2×C3×C4C22×D414yes—
372C2×C3×C4C22×D418yes—
373C2×C3×C4C22×D414yes—
374C2×C3×C4H(5,𝔽2)14yes—
375#67—432no—
376#67—416no—
377#67—432no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—21no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3C22×D414yes—
386C2×Dic3H(5,𝔽2)12yes—
387D12—232no—
388D12—216no—
389D12—432no—
390D12—48no—
391D12—416no—
392D12—816no—
393C4×D3—28no—
394C4×D3—216no—
395C4×D3—416no—
396C24—31no—
397C24—34no—
398C24—34no—
399C24—616no—
400C24—616no—
401C24—68no—
402C24—68no—
403C24—68no—
404C24—128no—
405C24—128no—
406C24—124no—
407C24—124no—
408C24—124no—
409C24C23×D314yes—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—316no—
413C2×D4—38no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—68no—
417C2×D4—68no—
418C2×D4—616no—
419C2×D4—616no—
420C2×D4—616no—
421C2×D4—68no—
422C2×D4—616no—
423C2×D4—616no—
424C2×D4—616no—
425C2×D4—68no—
426C2×D4—64no—
427C2×D4—64no—
428C2×D4—68no—
429C2×D4—616no—
430C2×D4—68no—
431C2×D4—64no—
432C2×D4—68no—
433C2×D4—128no—
434C2×D4—122no—
435C2×D4—122no—
436C2×D4—128no—
437C2×D4—128no—
438C2×D4—1264no—
439C2×D4—1264no—
440C2×D4—1232no—
441C2×D4—128no—
442C2×D4—124no—
443C2×D4—128no—
444C2×D4—1264no—
445C2×D4C2×D12116yescomplemented
446C2×D4C2×D1218yescomplemented
447C22×C4—216no—
448C22×C4—31no—
449C22×C4—32no—
450C22×C4—38no—
451C22×C4—34no—
452C22×C4—44no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—64no—
457C22×C4—64no—
458C22×C4—68no—
459C22×C4—68no—
460C22×C4—68no—
461C22×C4C23×D314yes—
462C22×C4C23×D311yescharacteristic
463C22×C4C23×D314yes—
464C22×C4C23×D312yes—
465C4⋊C4—24no—
466C4⋊C4—68no—
467C4⋊C4—64no—
468C4⋊C4D3×D418yescomplemented
469C22⋊C4—216no—
470C22⋊C4—316no—
471C22⋊C4—44no—
472C22⋊C4—68no—
473C22⋊C4—68no—
474C22⋊C4—68no—
475C22⋊C4—68no—
476C22⋊C4—68no—
477C22⋊C4—616no—
478C22⋊C4—128no—
479C42—28no—
480C42—62no—
481C42—64no—
482C42C2×D1214yescomplemented
483C22×C3—216no—
484C22×C3—28no—
485C22×C3—44no—
486C22×C3—48no—
487C22×C3—44no—
488C22×C3C23×D411yescharacteristic
489C22×C3C23×D412yes—
490C22×C3C2×H(5,𝔽2)12yes—
491C22×C3C2×H(5,𝔽2)11yescharacteristic
492C22×C3C2×H(5,𝔽2)11yescharacteristic
493C2×D3—24no—
494C2×D3—22no—
495C2×D3—24no—
496C2×D3—22no—
497C2×D3—28no—
498C2×D3—28no—
499C2×D3—28no—
500C2×D3—44no—
501C2×D3—44no—
502C2×D3—44no—
503C2×D3—44no—
504C2×D3—44no—
505C2×D3—48no—
506C2×D3—44no—
507C2×D3—48no—
508C2×D3—44no—
509C2×D3—832no—
510C2×D3—832no—
511C2×D3D4214yescomplemented
512C3×C4—28no—
513C3×C4—22no—
514C3×C4—44no—
515C3×C4D4218yescomplemented
516Dic3—24no—
517Dic3—22no—
518Dic3—44no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—34no—
524C23—32no—
525C23—38no—
526C23—61no—
527C23—62no—
528C23—61no—
529C23—64no—
530C23—64no—
531C23—64no—
532C23—64no—
533C23—64no—
534C23—64no—
535C23—64no—
536C23—68no—
537C23—68no—
538C23—64no—
539C23—64no—
540C23—1232no—
541C23—1232no—
542C23—1216no—
543C23—1216no—
544C23—1216no—
545C23—248no—
546C23—248no—
547C23—2416no—
548C23—2416no—
549C23—248no—
550C23C22×D1218yes—
551C23C24×D311yescharacteristic
552D4—232no—
553D4—48no—
554D4—632no—
555D4—616no—
556D4—1232no—
557D4—128no—
558D4—1232no—
559D4—1216no—
560D4—1232no—
561D4—1216no—
562D4—2416no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—34no—
571C2×C4—32no—
572C2×C4—416no—
573C2×C4—68no—
574C2×C4—62no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—616no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—1216no—
583C2×C4C22×D1212yes—
584C2×C4C22×D1214yes—
585C2×C4C2×D3×D418yes—
586C2×C4C2×D3×D414yes—
587C2×C4#100914yes—
588C2×C3—48no—
589C2×C3C2×D4212yes—
590C2×C3C2×D4211yescharacteristic
591C2×C3#348411yescharacteristic
592C2×C3#349812yes—
593C2×C3#354311yescharacteristic
594D3—24no—
595D3—48no—
596D3—84no—
597C22—216no—
598C22—28no—
599C22—34no—
600C22—44no—
601C22—44no—
602C22—48no—
603C22—64no—
604C22—62no—
605C22—62no—
606C22—64no—
607C22—68no—
608C22—68no—
609C22—68no—
610C22—124no—
611C22—124no—
612C22—124no—
613C22—124no—
614C22—128no—
615C22—124no—
616C22—124no—
617C22—124no—
618C22—128no—
619C22—2432no—
620C22—2432no—
621C22C23×D1211yescharacteristic
622C22C22×D3×D412yes—
623C22C2×#100912yes—
624C22D3×H(5,𝔽2)11yescharacteristic
625C22D3×H(5,𝔽2)11yescharacteristic
626C4—28no—
627C4—22no—
628C4—44no—
629C4—64no—
630C4—62no—
631C4—124no—
632C4D4×D1218yescomplemented
633C3#5912611yescharacteristic, complemented, Sylow
634C2—48no—
635C2—64no—
636C2—128no—
637C2—244no—
638C2C2×D4×D1212yes—
639C2#8586711yescharacteristic
640C2D3×D4211yescharacteristic
641C2#8677012yes—
642C2#8706611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2366C2
hypercenter#2366D3
hypocenterC3#59126
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. C2×#5717C2semidirect
2. C2×D3×#127C2semidirect
3. C2×D4×D12C2semidirect
4. C3×#2366C2semidirect
5. #82409C2semidirect
6. #76436C2semidirect
7. #76314C2semidirect
8. #74912C2semidirect
9. #74884C2semidirect
10. #73799C2semidirect
11. C4×#895C2semidirect
13. C22×D3×D4C22semidirect
14. C22×D3×D4C22semidirect
15. C22×D3×D4C22semidirect
16. C23×D12C22semidirect
18. C2×C3×C4×D4C22semidirect
19. C2×#940C22semidirect
20. C2×#940C22semidirect
21. C2×#938C22semidirect
22. C2×#937C22semidirect
23. C2×#930C22semidirect
28. C2×#895C22semidirect
29. C2×#895C22semidirect
30. C2×#882C22semidirect
31. C2×#882C22semidirect
32. C2×#882C22semidirect
33. C2×#874C22semidirect
34. C2×#874C22semidirect
35. C2×C4×D12C22semidirect
36. C2×C4×D12C22semidirect
37. C2×C42×D3C22semidirect
38. C3×#390C22semidirect
39. C3×#386C22semidirect
40. C3×C4×C4⋊C4C22semidirect
41. #5213C22semidirect
42. #4988C22semidirect
43. #4987C22semidirect
44. #4984C22semidirect
45. #4961C22semidirect
46. #4960C22semidirect
47. #4960C22semidirect
48. #4955C22semidirect
49. #4939C22semidirect
50. C4×(C2×D3)⋊C4C22semidirect
51. #4671C22semidirect
52. #4670C22semidirect
53. #4665C22semidirect
63. #2366D3semidirect
93. C3×#127D4semidirect
120. #895D4semidirect
155. C2×C4×D4C2×D3semidirect
163. #390C2×D3semidirect
168. #386C2×D3semidirect
171. C4×C4⋊C4C2×D3semidirect
191. C2×C3×D4C2×D4semidirect
192. C2×C3×D4C2×D4semidirect
231. C2×D12C2×D4semidirect
232. C2×D12C2×D4semidirect
241. C2×C4×D3C2×D4semidirect
243. C3×C4⋊C4C2×D4semidirect
247. C3×C42C2×D4semidirect
255. (C2×D3)⋊C4C2×D4semidirect
292. #127D12semidirect
445. C2×D4C2×D12semidirect
446. C2×D4C2×D12semidirect
468. C4⋊C4D3×D4semidirect
482. C42C2×D12semidirect
511. C2×D3D42semidirect
515. C3×C4D42semidirect
632. C4D4×D12semidirect
633. C3#59126semidirect

Maximal subgroups

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

subgroupindex
C2×#57172
C2×D3×#1272
C2×D4×D122
C3×#23662
#824092
#764362
#763142
#749122
#748842
#737992
C4×#8952
#591263