Finite Group Databasebeta

← #10796702

Subgroups — #10796702

Order 768.

1778 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5748C211yesmaximal, characteristic
2C2×#5504C211yesmaximal, characteristic, complemented
3C2×#5139C211yesmaximal, characteristic, complemented
4#83768C211yesmaximal, characteristic, complemented
5#82186C211yesmaximal, characteristic, complemented
6C2×C3×#343C211yesmaximal, characteristic, complemented
7#71982C211yesmaximal, characteristic, complemented
8#69005C214yesmaximal, complemented
9#69004C214yesmaximal, complemented
10#20943—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×C4×#67—22no—
17C2×#926C2211yescharacteristic
18C2×#926C2211yescharacteristic
19#5748—24no—
20#5748C418yescomplemented
21C2×#909—21no—
22C2×#909C2212yes—
23C2×#908—21no—
24C2×#908C2212yes—
25C2×#904—22no—
26#5504—22no—
27#5504—22no—
28#5504—22no—
29#5504C2214yes—
30C2×#893—21no—
31C2×#892—21no—
32C2×D3×C4⋊C4—21no—
33C2×#887—21no—
34#5213—22no—
35#5139—24no—
36C2×#825C2214yes—
37#5133—21no—
38#5133—21no—
39#5131C2212yescomplemented
40#5120C2212yescomplemented
41#5113—22no—
42#4988C2212yescomplemented
43#4987—22no—
44#4973C2212yescomplemented
45C3×#343C2214yescomplemented
46#4555C2214yescomplemented
47C2×#575—31no—
48C2×#520—31no—
49C2×#421—31no—
50#2896—31no—
51#2311—31no—
52#1883—34no—
53#1882—34no—
54C2×#343D311yescharacteristic, complemented
55#1733—31no—
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C22×#67—21no—
60C2×#235—21no—
61C2×#235—21no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—21no—
66C2×#235—21no—
67C2×#235—21no—
68C2×#235—22no—
69C2×#235—21no—
70C2×#235—22no—
71C2×#235C2×C414yes—
72C2×#235C2×C414yescomplemented
73C2×#235D414yes—
74C2×#235D412yes—
75C2×#235C2312yes—
76C22×D12D411yescharacteristic
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C4×D3—22no—
81C22×Dic6D411yescharacteristic
82C22×Dic6C2311yescharacteristic
83C2×C3×M16—21no—
84C2×C3×M16—22no—
85C2×C3×M16—21no—
86C2×C3×M16—22no—
87C2×C3×M16—22no—
88C2×C3×M16C2×C414yescomplemented
89C2×C3×M16C2312yes—
90C2×C3×M16C2312yes—
91C22×C3×C8—21no—
92C22×C3×C8—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4D412yes—
98C2×C3×C4⋊C4C2312yes—
99C2×#217—21no—
100C2×#217—21no—
101C4×#67—28no—
102C4×#67—28no—
103C2×(C2×D3)⋊C4—22no—
104C2×(C2×D3)⋊C4—24no—
105#926—21no—
106#926—21no—
107#926—22no—
108#926—22no—
109#926D412yes—
110#926C2312yes—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3D411yescharacteristic
114C2×Dic3⋊C4—22no—
115C2×C4×Dic3—22no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2312yes—
118#909—24no—
119#909—28no—
120#909C2×C418yescomplemented
121#908—24no—
122#908—28no—
123#908C2×C418yescomplemented
124C2×Dic12—22no—
125#904—216no—
126C2×D24—22no—
127C2×#204—22no—
128C2×#204—22no—
129#893—24no—
130#893—24no—
131#892—24no—
132#892—24no—
133D3×C4⋊C4—24no—
134D3×C4⋊C4—24no—
135#887—24no—
136#887—24no—
137C3⋊#95—24no—
138#825—28no—
139#821—22no—
140#821—22no—
141#816—22no—
142#816—22no—
143C22×#41—31no—
144C22×#41—31no—
145#575—38no—
146#575—64no—
147C2×#137—32no—
148C2×#137—61no—
149C2×AGL(1,ℤ/8)—32no—
150C2×AGL(1,ℤ/8)—61no—
151C2×#135—62no—
152C22×M16C2×D311yescharacteristic
153C22×M16C2×D311yescharacteristic
154#520—34no—
155#520—62no—
156#520—62no—
157#520—62no—
158C2×C4×Q8—61no—
159C2×C4×D4—61no—
160C2×C4×D4—62no—
161C2×#117—31no—
162C2×#117—31no—
163C2×#117—61no—
164C22×C4⋊C4—61no—
165C22×C4⋊C4C2×D311yescharacteristic
166#421—64no—
167C2×#104—34no—
168#419—32no—
169#418—32no—
170#417—61no—
171#417—61no—
172#416—62no—
173#391—32no—
174#390—32no—
175#386—62no—
176#380—62no—
177#343C2×D314yescomplemented
178#339—34no—
179C24×C3C2×D411yescharacteristic
180C23×D3—21no—
181C23×D3—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—82no—
185C22×C3×C4C22×C412yes—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4#4112yes—
194C22×C3×C4#4112yes—
195C22×C3×C4C2411yescharacteristic
196C2×#67—24no—
197C2×#67—22no—
198C2×#67—24no—
199C2×#67—22no—
200C2×#67—24no—
201C2×#67—41no—
202C2×#67—41no—
203C2×#67—42no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—22no—
209#235—24no—
210#235—24no—
211#235—24no—
212#235—24no—
213#235—24no—
214#235—22no—
215#235—22no—
216#235—24no—
217#235—24no—
218#235—24no—
219#235—24no—
220#235—42no—
221#235—42no—
222#235—42no—
223#235—42no—
224#235C22⋊C418yescomplemented
225#235C22×C414yes—
226#235C2×D414yes—
227C2×D12—21no—
228C2×D12—22no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12C22⋊C414yescomplemented
235C2×D12C22×C412yescomplemented
236C2×D12C2×D412yes—
237C2×C4×D3—24no—
238C2×C4×D3—22no—
239C2×C4×D3—24no—
240C2×C4×D3—22no—
241C2×C4×D3—24no—
242C2×C4×D3—28no—
243C2×C4×D3—41no—
244C2×C4×D3—41no—
245C2×C4×D3—42no—
246C2×C4×D3—84no—
247C2×Dic6—21no—
248C2×Dic6—22no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6C22⋊C414yescomplemented
255C2×Dic6C22×C412yescomplemented
256C2×Dic6C2×D412yes—
257C3×M16—24no—
258C3×M16—22no—
259C3×M16—22no—
260C3×M16—22no—
261C3×M16—42no—
262C3×M16C22×C414yescomplemented
263C2×C3×C8—24no—
264C2×C3×C8—22no—
265C2×C3×C8—22no—
266C2×C3×C8—42no—
267C3×C4⋊C4—22no—
268C3×C4⋊C4—22no—
269C3×C4⋊C4—22no—
270C3×C4⋊C4—22no—
271#217—22no—
272#217—22no—
273#217—42no—
274(C2×D3)⋊C4—28no—
275(C2×D3)⋊C4—84no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—42no—
279C4⋊Dic3C2×D412yes—
280C4⋊Dic3C2×D412yes—
281Dic3⋊C4—28no—
282C4×Dic3—28no—
283C4×Dic3—22no—
284C4×Dic3—22no—
285C4×Dic3—22no—
286C4×Dic3—22no—
287Dic12—28no—
288D24—28no—
289#204—28no—
290#204—28no—
291C2×#41—34no—
292C2×#41—34no—
293C2×#41—32no—
294C2×#41—34no—
295C2×#41—32no—
296C2×#41—61no—
297C2×#41—62no—
298C2×#41—62no—
299C2×#41—61no—
300C2×#41—62no—
301C2×#41—61no—
302C2×#41—61no—
303C2×#41—62no—
304C2×#41—61no—
305C2×#41—61no—
306C2×#41—62no—
307C22×Q8—31no—
308C22×Q8—31no—
309C22×D4—31no—
310C22×D4—31no—
311C22×D4—61no—
312C22×D4—61no—
313C23×C4—62no—
314C23×C4—61no—
315C23×C4—61no—
316C23×C4D1212yes—
317C23×C4C22×D311yescharacteristic
318#137—38no—
319#137—68no—
320#137—64no—
321AGL(1,ℤ/8)—38no—
322AGL(1,ℤ/8)—68no—
323AGL(1,ℤ/8)—64no—
324#135—616no—
325C2×Q16—62no—
326C2×SD16—62no—
327C2×SD16—62no—
328C2×D8—62no—
329C2×M16—21no—
330C2×M16—22no—
331C2×M16—21no—
332C2×M16—22no—
333C2×M16—22no—
334C2×M16C4×D314yescomplemented
335C2×M16C22×D312yes—
336C2×M16C22×D312yes—
337C22×C8—21no—
338C22×C8—21no—
339C4×Q8—64no—
340C4×Q8—64no—
341C4×D4—64no—
342C4×D4—64no—
343C4×D4—68no—
344C4×D4—68no—
345#117—32no—
346#117—32no—
347#117—64no—
348#117—64no—
349#117—61no—
350#117—61no—
351#117—62no—
352#117—62no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—31no—
358C2×C4⋊C4—31no—
359C2×C4⋊C4—64no—
360C2×C4⋊C4—64no—
361C2×C4⋊C4—62no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4D1212yes—
364C2×C4⋊C4C22×D312yes—
365C2×C22⋊C4—62no—
366C2×C22⋊C4—61no—
367C2×C22⋊C4—61no—
368C2×C22⋊C4—64no—
369C2×C42—32no—
370C2×C42—62no—
371C2×C42—61no—
372#104—68no—
373#103—62no—
374#103—62no—
375#102—62no—
376#102—62no—
377#95—64no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3C4×D412yes—
381C23×C3#12011yescharacteristic
382C23×C3#12011yescharacteristic
383C23×C3#12112yes—
384C23×C3C22×D411yescharacteristic
385C22×D3—22no—
386C22×D3—24no—
387C22×D3—22no—
388C22×D3—42no—
389C22×D3—42no—
390C22×D3—42no—
391C22×D3—42no—
392C22×D3—41no—
393C22×D3—41no—
394C2×C3×C4—24no—
395C2×C3×C4—44no—
396C2×C3×C4—44no—
397C2×C3×C4—82no—
398C2×C3×C4C2×C22⋊C412yes—
399C2×C3×C4C2×C22⋊C412yes—
400C2×C3×C4C4×D412yes—
401C2×C3×C4C4×D412yes—
402C2×C3×C4C4×D412yes—
403C2×C3×C4#12011yescharacteristic
404C2×C3×C4#12011yescharacteristic
405C2×C3×C4#12112yes—
406C2×C3×C4#12112yes—
407C2×C3×C4#12112yes—
408C2×C3×C4#12312yes—
409C2×C3×C4#12312yes—
410C2×C3×C4C23×C411yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416#67—24no—
417#67—24no—
418#67—28no—
419#67—44no—
420#67—44no—
421C2×Dic3—22no—
422C2×Dic3—24no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—42no—
427C2×Dic3—42no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—42no—
431C2×Dic3—42no—
432C2×Dic3—42no—
433D12—22no—
434D12—22no—
435D12—22no—
436D12—24no—
437D12—41no—
438D12—41no—
439D12C2×C22⋊C414yes—
440C4×D3—24no—
441C4×D3—24no—
442C4×D3—28no—
443C4×D3—44no—
444C4×D3—44no—
445C4×D3—88no—
446Dic6—22no—
447Dic6—24no—
448Dic6—22no—
449Dic6—22no—
450Dic6—41no—
451Dic6—41no—
452Dic6C2×C22⋊C414yes—
453C3×C8—24no—
454C3×C8—42no—
455C24—61no—
456C24—61no—
457C24C2×D1211yescharacteristic
458#41—38no—
459#41—34no—
460#41—34no—
461#41—64no—
462#41—64no—
463#41—64no—
464#41—64no—
465#41—64no—
466#41—62no—
467#41—62no—
468#41—64no—
469#41—64no—
470#41—64no—
471#41—64no—
472#41—122no—
473#41—122no—
474#41—122no—
475#41—122no—
476C2×Q8—34no—
477C2×Q8—32no—
478C2×Q8—32no—
479C2×Q8—61no—
480C2×Q8—62no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8—61no—
484C2×Q8—61no—
485C2×Q8—61no—
486C2×D4—34no—
487C2×D4—32no—
488C2×D4—32no—
489C2×D4—64no—
490C2×D4—62no—
491C2×D4—64no—
492C2×D4—62no—
493C2×D4—64no—
494C2×D4—61no—
495C2×D4—62no—
496C2×D4—61no—
497C2×D4—61no—
498C2×D4—61no—
499C2×D4—61no—
500C2×D4—61no—
501C2×D4—121no—
502C2×D4—121no—
503C2×D4—122no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—68no—
507C22×C4—61no—
508C22×C4—62no—
509C22×C4—64no—
510C22×C4—64no—
511C22×C4—62no—
512C22×C4—61no—
513C22×C4—61no—
514C22×C4—61no—
515C22×C4—64no—
516C22×C4—62no—
517C22×C4—82no—
518C22×C4—121no—
519C22×C4—121no—
520C22×C4—122no—
521C22×C4—244no—
522C22×C4C2×C4×D312yes—
523C22×C4C2×D1214yes—
524C22×C4C2×D1211yescharacteristic
525C22×C4#23512yes—
526C22×C4D3×D411yescharacteristic
527C22×C4D3×D411yescharacteristic
528C22×C4D3×D412yes—
529C22×C4D3×D411yescharacteristic
530C22×C4D3×D411yescharacteristic
531C22×C4#23912yes—
532C22×C4C23×D311yescharacteristic
533Q16—68no—
534SD16—68no—
535SD16—68no—
536D8—68no—
537M16—24no—
538M16—22no—
539M16—22no—
540M16—22no—
541M16—42no—
542M16C2×C4×D314yescomplemented
543C2×C8—24no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—42no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—32no—
552C4⋊C4—32no—
553C4⋊C4—68no—
554C4⋊C4—61no—
555C4⋊C4—61no—
556C4⋊C4—122no—
557C22⋊C4—68no—
558C22⋊C4—62no—
559C22⋊C4—62no—
560C22⋊C4—122no—
561C22⋊C4—244no—
562C42—68no—
563C42—62no—
564C42—62no—
565C42—62no—
566C42—62no—
567C22×C3—21no—
568C22×C3—21no—
569C22×C3—21no—
570C22×C3—21no—
571C22×C3—21no—
572C22×C3—21no—
573C22×C3—42no—
574C22×C3—42no—
575C22×C3#38612yes—
576C22×C3#38612yes—
577C22×C3C2×C4×D411yescharacteristic
578C22×C3C2×#12011yescharacteristic
579C22×C3C2×#12111yescharacteristic
580C2×D3—24no—
581C2×D3—48no—
582C2×D3—42no—
583C2×D3—42no—
584C2×D3—44no—
585C2×D3—42no—
586C2×D3—82no—
587C2×D3—81no—
588C2×D3—81no—
589C3×C4—44no—
590C3×C4#38612yes—
591C3×C4#38612yes—
592C3×C4C22×C22⋊C411yescharacteristic
593C3×C4C2×C4×D411yescharacteristic
594C3×C4C2×#12111yescharacteristic
595C3×C4C2×#12311yescharacteristic
596Dic3—24no—
597Dic3—42no—
598Dic3—42no—
599Dic3—42no—
600Dic3—82no—
601C23—41no—
602C23—41no—
603C23—64no—
604C23—62no—
605C23—62no—
606C23—122no—
607C23—122no—
608C23—122no—
609C23—122no—
610C23—121no—
611C23—121no—
612C23C4×D1212yes—
613C23#88211yescharacteristic
614C23#88211yescharacteristic
615C23#89512yes—
616C23C22×D1211yescharacteristic
617Q8—34no—
618Q8—64no—
619Q8—62no—
620Q8—62no—
621Q8—62no—
622Q8—121no—
623Q8—121no—
624D4—34no—
625D4—68no—
626D4—64no—
627D4—64no—
628D4—64no—
629D4—62no—
630D4—62no—
631D4—62no—
632D4—124no—
633D4—124no—
634D4—121no—
635D4—121no—
636C2×C4—24no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—64no—
640C2×C4—62no—
641C2×C4—68no—
642C2×C4—64no—
643C2×C4—64no—
644C2×C4—62no—
645C2×C4—62no—
646C2×C4—62no—
647C2×C4—82no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—124no—
653C2×C4—124no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4—248no—
658C2×C4C4×D1212yes—
659C2×C4D3×C22⋊C412yes—
660C2×C4D3×C22⋊C412yes—
661C2×C4#88211yescharacteristic
662C2×C4#88211yescharacteristic
663C2×C4#88412yes—
664C2×C4#88412yes—
665C2×C4#89312yes—
666C2×C4#89312yes—
667C2×C4#89412yes—
668C2×C4#89412yes—
669C2×C4#89512yes—
670C2×C4C22×C4×D311yescharacteristic
671C2×C4C2×#23511yescharacteristic
672C2×C4C2×D3×D411yescharacteristic
673C2×C4C2×D3×D411yescharacteristic
674C2×C4C2×D3×D411yescharacteristic
675C2×C4C2×#23911yescharacteristic
676C8—24no—
677C8—42no—
678C2×C3—21no—
679C2×C3—21no—
680C2×C3—22no—
681C2×C3—22no—
682C2×C3C2×#38611yescharacteristic
683C2×C3#301911yescharacteristic
684C2×C3#302011yescharacteristic
685D3—44no—
686D3—82no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—42no—
694C22—42no—
695C22—64no—
696C22—128no—
697C22—124no—
698C22—122no—
699C22—122no—
700C22—122no—
701C22—242no—
702C22—241no—
703C22—241no—
704C22#466512yes—
705C22#466512yes—
706C22C2×C4×D1211yescharacteristic
707C22C2×#88211yescharacteristic
708C22C2×#89511yescharacteristic
709C4—44no—
710C4—64no—
711C4—122no—
712C4—122no—
713C4—122no—
714C4—242no—
715C4#466512yes—
716C4#466512yes—
717C4C2×D3×C22⋊C411yescharacteristic
718C4C2×#88411yescharacteristic
719C4C2×#89311yescharacteristic
720C4C2×#89411yescharacteristic
721C3#2094311yescharacteristic, complemented, Sylow
722C2—21no—
723C2—22no—
724C2—21no—
725C2—22no—
726C2—124no—
727C2—242no—
728C2C2×#466511yescharacteristic
729C2#7870011yescharacteristic
730C2#7870911yescharacteristic
subgroupG/H
centerC22C2×#882
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#343C2
socleC22×C3C2×#120
prime radicalC2×#5748C2
cosocleC22×C3×C4C24
hypercenterC2×#343D3
hypocenterC3#20943
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#5504C2semidirect
3. C2×#5139C2semidirect
4. #83768C2semidirect
5. #82186C2semidirect
6. C2×C3×#343C2semidirect
7. #71982C2semidirect
8. #69005C2semidirect
9. #69004C2semidirect
20. #5748C4semidirect
39. #5131C22semidirect
40. #5120C22semidirect
42. #4988C22semidirect
44. #4973C22semidirect
45. C3×#343C22semidirect
46. #4555C22semidirect
54. C2×#343D3semidirect
72. C2×#235C2×C4semidirect
88. C2×C3×M16C2×C4semidirect
120. #909C2×C4semidirect
123. #908C2×C4semidirect
177. #343C2×D3semidirect
224. #235C22⋊C4semidirect
234. C2×D12C22⋊C4semidirect
235. C2×D12C22×C4semidirect
254. C2×Dic6C22⋊C4semidirect
255. C2×Dic6C22×C4semidirect
262. C3×M16C22×C4semidirect
334. C2×M16C4×D3semidirect
542. M16C2×C4×D3semidirect
721. C3#20943semidirect

Maximal subgroups

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

subgroupindex
C2×#57482
C2×#55042
C2×#51392
#837682
#821862
C2×C3×#3432
#719822
#690052
#690042
#209433