Finite Group Databasebeta

← #204227

Subgroups — #204227

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4C2×#1465C214yesmaximal, complemented
5#9652C214yesmaximal, complemented
6#9650C212yesmaximal, complemented
7#9649C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4C411yescharacteristic
10C22×C4×D4C2212yes—
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C2×#407C2212yes—
14C2×#407C2212yes—
15C2×C4×M16—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121—21no—
21C4×#121—21no—
22C4×#121—21no—
23C4×#121—21no—
24C2×#390C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388C2212yes—
27C2×#386C412yes—
28C2×#380C411yescharacteristic
29C2×C4×C4⋊C4C411yescharacteristic
30C2×C4×C22⋊C4C412yes—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1749—21no—
36#1749C2212yes—
37C4×C22⋊C8—21no—
38C4×C22⋊C8C2212yes—
39#1728—21no—
40#1728—21no—
41#1465—216no—
42C2×#324C2218yes—
43C2×#323C2218yes—
44C23×D4C2×C412yes—
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C2×#121—24no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—24no—
58C2×C4×D4—28no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—28no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—44no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2312yes—
78C22×C42—44no—
79C22×C42C2×C412yes—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#423—44no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95#407—24no—
96#407—28no—
97#407D418yescomplemented
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C4×M16—44no—
103C2×#97—22no—
104C2×C4×C8—22no—
105#390—21no—
106#390—21no—
107#390—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#324—216no—
141#323—216no—
142C25C22×C412yes—
143C25C22×C411yescharacteristic
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—24no—
147C22×D4—28no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—44no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4M1614yescomplemented
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—28no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C412yes—
184C23×C4C22×C412yes—
185C23×C4C22×C411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D414yes—
191C23×C4#4114yes—
192C23×C4C2411yescharacteristic
193C2×M16—28no—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16—44no—
197C2×M16—44no—
198C2×M16—44no—
199C22×C8—24no—
200C22×C8—24no—
201#121—28no—
202#121—28no—
203#121—28no—
204#121—28no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—48no—
209C4×D4—48no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4M1614yescomplemented
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—28no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C4⋊C8—44no—
278C4⋊C8—44no—
279C22⋊C8—28no—
280C22⋊C8—24no—
281C22⋊C8—28no—
282C22⋊C8—44no—
283C22⋊C8C2×D418yes—
284#97—44no—
285C4×C8—44no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24C2×C22⋊C411yescharacteristic
308C24C2×C22⋊C411yescharacteristic
309C24C4×D414yes—
310C24C2×M1614yes—
311C24C23×C411yescharacteristic
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—416no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—416no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—42no—
327C2×D4—42no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—416no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—816no—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C4×D414yes—
411C22×C4#12114yes—
412C22×C4#12114yes—
413C22×C4#12412yes—
414C22×C4#12412yes—
415C22×C4#12712yes—
416C22×C4#12712yes—
417C22×C4C2×M1614yes—
418C22×C4C2×M1612yes—
419C22×C4C2×M1612yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C2×#4112yes—
424M16—416no—
425M16—88no—
426C2×C8—28no—
427C2×C8—44no—
428C2×C8—44no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—44no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—44no—
468C42—44no—
469C42#9914yes—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23#39012yes—
514C23#39012yes—
515C23C22×C22⋊C411yescharacteristic
516C23C2×C4×D412yes—
517C23C22×M1611yescharacteristic
518C23C22×M1611yescharacteristic
519C23#56812yes—
520D4—48no—
521D4—416no—
522D4—416no—
523D4—48no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—88no—
583C2×C4—88no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4#39012yes—
587C2×C4#39012yes—
588C2×C4#40712yes—
589C2×C4#40712yes—
590C2×C4C2×#9912yes—
591C2×C4C2×#9912yes—
592C2×C4C2×#9912yes—
593C2×C4#41012yes—
594C2×C4#41312yes—
595C2×C4#41312yes—
596C2×C4C2×#12112yes—
597C2×C4C2×#12411yescharacteristic
598C2×C4C2×#12711yescharacteristic
599C8—48no—
600C8—84no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—48no—
616C22—48no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—88no—
629C22—88no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×#39011yescharacteristic
635C22C2×#40711yescharacteristic
636C22#287911yescharacteristic
637C22C22×#9911yescharacteristic
638C22C2×#41011yescharacteristic
639C22C2×#41311yescharacteristic
640C22C2×#41311yescharacteristic
641C22D4×M1612yes—
642C22D4×M1612yes—
643C4—22no—
644C4—22no—
645C4—21no—
646C4—22no—
647C4—21no—
648C4—22no—
649C4—22no—
650C4—44no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—82no—
656C4—82no—
657C4#146514yescomplemented
658C4#196414yes—
659C2—22no—
660C2—22no—
661C2—42no—
662C2—42no—
663C2—84no—
664C2C2×#146511yescharacteristic
665C2#1736611yescharacteristic
666C2#1740011yescharacteristic
667C2#1740111yescharacteristic
668C2#2119011yescharacteristic
669C2C2×#196411yescharacteristic
670C2#2121811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
3. C4×#407C2semidirect
4. C2×#1465C2semidirect
5. #9652C2semidirect
6. #9650C2semidirect
7. #9649C2semidirect
97. #407D4semidirect
155. C22×D4M16semidirect
228. C2×C4⋊C4M16semidirect
657. C4#1465semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#206262
C4×#4072
C2×#14652
#96522
#96502
#96492