Finite Group Databasebeta

← #6393417

Subgroups — #6393417

Order 512.

3764 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3537C211yesmaximal, characteristic
2C2×#2977C211yesmaximal, characteristic, complemented
3#34200C211yesmaximal, characteristic, complemented
4#22493C214yesmaximal, complemented
5#22488C214yesmaximal, complemented
6#22423C214yesmaximal, complemented
7D4×#102C214yesmaximal, complemented
8#21872C212yesmaximal, complemented
9#21770C212yesmaximal, complemented
10#21761C212yesmaximal, complemented
11#21399C212yesmaximal, complemented
12#20914C212yesmaximal, complemented
13C2×#1874C212yesmaximal, complemented
14#3537—28no—
15#3537C4116yescomplemented
16C2×#560—22no—
17C2×#553C2214yes—
18C2×D42—22no—
19C2×D42C2214yes—
20C2×#540C2212yes—
21C2×#538—22no—
22C2×#535C2212yes—
23C2×#534—22no—
24C2×#525C2211yescharacteristic
25C22×#121C2212yes—
26C2×#520C2211yescharacteristic, complemented
27C23×C4⋊C4C2211yescharacteristic, complemented
28#2977—28no—
29C2×#435C2214yescomplemented
30C2×C8×D4C2214yescomplemented
31C2×#433C2212yescomplemented
32C2×#418C2212yescomplemented
33C22×#102C2212yescomplemented
34C22×#102C2214yescomplemented
35C2×#408C2212yescomplemented
36C22×C22⋊C8C2212yescomplemented
37#2361—22no—
38#2360—22no—
39#2352C2214yescomplemented
40D4×C4⋊C4C2214yescomplemented
41#2322—22no—
42#2317C2211yescharacteristic, complemented
43#2311C2211yescharacteristic, complemented
44C4×#120—22no—
45#2280C2211yescharacteristic, complemented
46#2274C2211yescharacteristic, complemented
47#1992C2212yescomplemented
48#1991C2212yescomplemented
49#1989C2214yescomplemented
50#1982C2214yescomplemented
51#1961C2212yescomplemented
52#1957C2212yescomplemented
53#1895C2214yescomplemented
54#1893C2214yescomplemented
55#1893C2214yescomplemented
56#1893C2214yescomplemented
57#1875C2214yescomplemented
58#1875C2214yescomplemented
59#1875C2214yescomplemented
60#1874C2214yescomplemented
61#1874C2214yescomplemented
62#1874C2214yescomplemented
63#1874C2214yescomplemented
64#1874C2214yescomplemented
65#1825C2214yescomplemented
66#1824C2212yescomplemented
67#1824C2212yescomplemented
68#1804C2214yescomplemented
69#1802C2214yescomplemented
70#1786C2212yescomplemented
71#1786C2212yescomplemented
72#1786C2214yescomplemented
73#1762C2214yescomplemented
74C4×#102C2214yescomplemented
75#1736C2212yescomplemented
76#1735C2212yescomplemented
77C2×#340C2214yescomplemented
78C22×#41C2311yescharacteristic
79C23×D4—22no—
80C23×D4—24no—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C24×C4—21no—
84C24×C4C2311yescharacteristic
85C22×M16C2312yes—
86C23×C8C2314yes—
87C23×C8C2312yes—
88#560—232no—
89#553—216no—
90#553C2×C4132yescomplemented
91D42—232no—
92D42—216no—
93D42C2×C4132yescomplemented
94#540—28no—
95#540C2×C4116yescomplemented
96#538—232no—
97#535—28no—
98#535C2×C4116yescomplemented
99#534—232no—
100C2×#127—22no—
101C2×#127C2312yes—
102C2×#126—24no—
103C2×#125C2312yes—
104C2×#124—22no—
105#525C2×C418yes—
106#525D418yes—
107C2×#123—24no—
108C2×#123—24no—
109C2×#123D412yes—
110C2×C22⋊Q8D412yes—
111C2×#121—24no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121—28no—
115C2×#121—24no—
116C2×#121—24no—
117C2×#121C2×C4116yescomplemented
118C2×#121D412yes—
119C2×#121C2314yes—
120C2×#121C2314yes—
121C2×#121C2314yes—
122C2×#120—22no—
123C2×#120—22no—
124C2×#120—24no—
125C2×#120—24no—
126C2×#120D412yes—
127#520—44no—
128C2×C4×Q8—21no—
129C2×C4×D4—22no—
130C2×C4×D4—21no—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4C2314yes—
134C2×C4×D4C2314yes—
135C2×#117C2312yes—
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4—22no—
142C22×C4⋊C4—44no—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2314yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2311yescharacteristic
148C22×C22⋊C4—22no—
149#435—216no—
150C8×D4—216no—
151#433—28no—
152C2×C4⋊C8C2314yes—
153#418—44no—
154C2×#102—44no—
155C2×#102—48no—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164#408—28no—
165C2×C22⋊C8—28no—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2314yes—
169C2×C22⋊C8C2314yes—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#391—21no—
173#390—21no—
174#389—22no—
175#389—24no—
176#388—24no—
177#387—24no—
178#386—22no—
179#386—24no—
180#385—22no—
181#385—24no—
182#385—24no—
183#384C2312yes—
184#382C2312yes—
185#381—21no—
186#380C2314yes—
187#380C2312yes—
188#380C2312yes—
189#379—21no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2314yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—24no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203C25—22no—
204C25—22no—
205C25C2×D411yescharacteristic
206C2×#41—42no—
207C2×#41—42no—
208C2×#41C22×C414yes—
209C2×#41C2×D414yes—
210C22×Q8C2×D411yescharacteristic
211C22×D4—24no—
212C22×D4—216no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—232no—
216C22×D4—22no—
217C22×D4—24no—
218C22×D4—22no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—44no—
222C22×D4—42no—
223C22×D4—42no—
224C22×D4—44no—
225C22×D4—42no—
226C22×D4—42no—
227C22×D4—88no—
228C22×D4—88no—
229C22×D4—84no—
230C22×D4—84no—
231C22×D4C22×C418yes—
232C22×D4C22×C418yescomplemented
233C22×D4C2×D414yes—
234C22×D4C2×D411yescharacteristic
235C22×D4C2×D412yes—
236C22×D4C2414yes—
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4—84no—
247C23×C4C22×C414yes—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D414yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C2×M16—24no—
255C2×M16—42no—
256C2×M16—42no—
257C22×C8—28no—
258C22×C8—24no—
259C22×C8—42no—
260C22×C8—44no—
261C22×C8—42no—
262C22×C8—44no—
263C22×C8C2414yes—
264C22×C8C2414yes—
265#127—216no—
266#127—24no—
267#127C22×C418yescomplemented
268#126—232no—
269#125—24no—
270#125C22×C418yescomplemented
271#124—216no—
272#123—232no—
273#123—232no—
274#123C22⋊C4116yes—
275C22⋊Q8C22⋊C4116yes—
276#121—232no—
277#121—216no—
278#121—216no—
279#121—28no—
280#121—28no—
281#121—28no—
282#121—232no—
283#121—232no—
284#121C22⋊C4116yes—
285#121C22×C4116yes—
286#121C22×C4116yes—
287#121C22×C4116yescomplemented
288#120—216no—
289#120—216no—
290#120—232no—
291#120—232no—
292#120C22⋊C4116yes—
293C4×Q8—44no—
294C4×D4—232no—
295C4×D4—28no—
296C4×D4—44no—
297C4×D4—48no—
298C4×D4—88no—
299C4×D4C22×C4116yes—
300C4×D4C22×C4116yescomplemented
301C4×D4C2×D4116yes—
302#117—24no—
303#117—24no—
304#117—42no—
305#117—42no—
306#117—42no—
307#117C22×C414yes—
308#117C22×C418yescomplemented
309#117C2×D414yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C22×C418yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—216no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D414yes—
349C2×C22⋊C4C2×D414yes—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—24no—
355C2×C42C2412yes—
356C4⋊C8—28no—
357#102—28no—
358#102—28no—
359#102—44no—
360#102—44no—
361#102—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365C22⋊C8—28no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368C22⋊C8—28no—
369#97—24no—
370C4×C8—24no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—28no—
380C24—28no—
381C24—41no—
382C24—41no—
383C24—44no—
384C24—44no—
385C24—82no—
386C24—82no—
387C24—82no—
388C24—82no—
389C24—82no—
390C24—82no—
391C24—84no—
392C24—84no—
393C24C2×C22⋊C414yes—
394C24C22×D412yes—
395C24C22×D411yescharacteristic
396C24C22×D412yes—
397#41—48no—
398#41—48no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8C2×C22⋊C414yes—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—216no—
405C2×D4—216no—
406C2×D4—216no—
407C2×D4—28no—
408C2×D4—216no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—416no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—41no—
419C2×D4—42no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—88no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4—864no—
438C2×D4—832no—
439C2×D4C2×C22⋊C4116yes—
440C2×D4C2×C22⋊C414yes—
441C2×D4C2×C22⋊C418yes—
442C2×D4C23×C418yes—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—28no—
455C22×C4—216no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4—41no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—48no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—82no—
473C22×C4—84no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—84no—
478C22×C4—816no—
479C22×C4C2×C22⋊C418yes—
480C22×C4C2×C22⋊C414yes—
481C22×C4C23×C414yes—
482C22×C4C23×C412yes—
483C22×C4C23×C414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C2511yescharacteristic
492M16—84no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—84no—
500C2×C8—88no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—216no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—84no—
514C4⋊C4C2×C22⋊C418yes—
515C4⋊C4C2×C22⋊C4116yes—
516C4⋊C4C23×C418yes—
517C4⋊C4H(5,𝔽2)14yes—
518C22⋊C4—28no—
519C22⋊C4—216no—
520C22⋊C4—216no—
521C22⋊C4—216no—
522C22⋊C4—216no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—84no—
529C22⋊C4C2×C22⋊C418yes—
530C22⋊C4C2×C22⋊C4116yes—
531C22⋊C4C2×C22⋊C4116yes—
532C42—28no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42—84no—
537C42C23×C414yes—
538C42C22×D414yes—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—216no—
544C23—21no—
545C23—22no—
546C23—21no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—28no—
553C23—44no—
554C23—44no—
555C23—48no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—416no—
564C23—88no—
565C23—88no—
566C23—88no—
567C23—816no—
568C23—164no—
569C23—164no—
570C23—164no—
571C23—164no—
572C23—168no—
573C23—164no—
574C23—164no—
575C23C22×C22⋊C414yes—
576C23C22×C22⋊C412yes—
577C23C22×C22⋊C414yes—
578C23#57614yes—
579C23#57714yes—
580C23C23×D411yescharacteristic
581Q8—44no—
582D4—232no—
583D4—28no—
584D4—28no—
585D4—44no—
586D4—48no—
587D4—432no—
588D4—44no—
589D4—44no—
590D4—88no—
591D4—88no—
592D4—816no—
593D4—816no—
594D4—816no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—816no—
628C2×C4—88no—
629C2×C4—84no—
630C2×C4—816no—
631C2×C4—84no—
632C2×C4—164no—
633C2×C4C22×C22⋊C414yes—
634C2×C4C22×C22⋊C418yes—
635C2×C4C22×C22⋊C412yes—
636C2×C4C22×C22⋊C414yes—
637C2×C4C24×C411yescharacteristic
638C2×C4C23×D411yescharacteristic
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4#58512yes—
641C8—44no—
642C8—44no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—28no—
650C22—24no—
651C22—22no—
652C22—24no—
653C22—416no—
654C22—42no—
655C22—44no—
656C22—41no—
657C22—42no—
658C22—41no—
659C22—42no—
660C22—48no—
661C22—48no—
662C22—48no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22—1616no—
668C22—1616no—
669C22#289718yes—
670C22C23×C22⋊C411yescharacteristic
671C22#348411yescharacteristic
672C22#348511yescharacteristic
673C22C2×#57612yes—
674C22C2×#57712yes—
675C4—28no—
676C4—22no—
677C4—24no—
678C4—24no—
679C4—48no—
680C4—44no—
681C4—82no—
682C4—82no—
683C4—81no—
684C4—81no—
685C4C4×H(5,𝔽2)12yes—
686C4#349612yes—
687C2—28no—
688C2—42no—
689C2—44no—
690C2—88no—
691C2#3425611yescharacteristic
692C2C2×#289712yes—
693C2#6163011yescharacteristic
694C2#6163311yescharacteristic
695C2#6176612yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3537C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2977C2semidirect
3. #34200C2semidirect
4. #22493C2semidirect
5. #22488C2semidirect
6. #22423C2semidirect
7. D4×#102C2semidirect
8. #21872C2semidirect
9. #21770C2semidirect
10. #21761C2semidirect
11. #21399C2semidirect
12. #20914C2semidirect
13. C2×#1874C2semidirect
15. #3537C4semidirect
26. C2×#520C22semidirect
27. C23×C4⋊C4C22semidirect
29. C2×#435C22semidirect
30. C2×C8×D4C22semidirect
31. C2×#433C22semidirect
32. C2×#418C22semidirect
33. C22×#102C22semidirect
34. C22×#102C22semidirect
35. C2×#408C22semidirect
36. C22×C22⋊C8C22semidirect
39. #2352C22semidirect
40. D4×C4⋊C4C22semidirect
42. #2317C22semidirect
43. #2311C22semidirect
45. #2280C22semidirect
46. #2274C22semidirect
47. #1992C22semidirect
48. #1991C22semidirect
49. #1989C22semidirect
50. #1982C22semidirect
51. #1961C22semidirect
52. #1957C22semidirect
53. #1895C22semidirect
54. #1893C22semidirect
55. #1893C22semidirect
56. #1893C22semidirect
57. #1875C22semidirect
58. #1875C22semidirect
59. #1875C22semidirect
60. #1874C22semidirect
61. #1874C22semidirect
62. #1874C22semidirect
63. #1874C22semidirect
64. #1874C22semidirect
65. #1825C22semidirect
66. #1824C22semidirect
67. #1824C22semidirect
68. #1804C22semidirect
69. #1802C22semidirect
70. #1786C22semidirect
71. #1786C22semidirect
72. #1786C22semidirect
73. #1762C22semidirect
74. C4×#102C22semidirect
75. #1736C22semidirect
76. #1735C22semidirect
77. C2×#340C22semidirect
90. #553C2×C4semidirect
93. D42C2×C4semidirect
95. #540C2×C4semidirect
98. #535C2×C4semidirect
117. C2×#121C2×C4semidirect
232. C22×D4C22×C4semidirect
267. #127C22×C4semidirect
270. #125C22×C4semidirect
287. #121C22×C4semidirect
300. C4×D4C22×C4semidirect
308. #117C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35372
C2×#29772
#342002
#224932
#224882
#224232
D4×#1022
#218722
#217702
#217612
#213992
#209142
C2×#18742