Finite Group Databasebeta

← #354265

Subgroups — #354265

Order 512.

2372 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#442C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic, complemented
3C2×#1879C212yesmaximal
4C2×C4×#102C211yesmaximal, characteristic, complemented
5C2×C4×#102C211yesmaximal, characteristic, complemented
6#13785C211yesmaximal, characteristic, complemented
7#10931C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10#10703C211yesmaximal, characteristic, complemented
11#10694C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#442—24no—
17C2×#442—24no—
18C2×#442C418yescomplemented
19C22×#106C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#390C2212yes—
27#2275—24no—
28#2275—24no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—22no—
33C2×C4×C22⋊C4—21no—
34C2×#376C2211yescharacteristic
35C22×#95—21no—
36#1879C4116yescomplemented
37C4×#102—24no—
38C4×#102—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1437—21no—
46#1437—21no—
47C4⋊#95C2212yes—
48C22⋊#95—22no—
49C23×D4C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—22no—
52C24×C4—21no—
53C24×C4—21no—
54C22×D8—22no—
55C22×D8—22no—
56C22×D8—24no—
57C22×D8C2×C418yes—
58C22×D8C2×C414yes—
59C23×C8C2312yes—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—24no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4C2×C414yes—
76C2×C4×D4C2×C418yescomplemented
77C2×C4×D4C2×C414yescomplemented
78C2×C4×D4D414yes—
79C2×C4×D4D414yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C42—22no—
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94#442—232no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106C2×C414yescomplemented
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102C2×C414yes—
107C2×#102C2×C418yescomplemented
108C2×#102C2×C418yescomplemented
109C2×#102C2×C414yescomplemented
110C2×#97—22no—
111C2×#97—22no—
112C2×#97C2×C414yescomplemented
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116#390—24no—
117#390C2×C418yescomplemented
118C4×C4⋊C4—24no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4D414yes—
124C4×C22⋊C4—216no—
125C4×C22⋊C4—28no—
126#376—22no—
127#376—22no—
128C2×#95—28no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D412yes—
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137#340—24no—
138#340—24no—
139#340—24no—
140#340C2×C418yescomplemented
141#336—24no—
142#336C2×C418yescomplemented
143C25—21no—
144C25—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4C22×C412yes—
159C22×D4C22×C414yes—
160C22×D4C22×C412yes—
161C22×D4C22×C414yes—
162C22×D4C2×D412yes—
163C22×D4C2×D412yes—
164C22×D4#4114yes—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—24no—
170C23×C4—28no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—24no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—82no—
179C23×C4—82no—
180C23×C4—82no—
181C23×C4—82no—
182C23×C4—82no—
183C23×C4—82no—
184C23×C4—82no—
185C23×C4—82no—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×D8—28no—
192C2×D8—28no—
193C2×D8—216no—
194C2×D8—216no—
195C2×D8C42116yescomplemented
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8C22×C414yes—
210C22×C8C22×C412yes—
211C22×C8C22×C414yes—
212C22×C8C22×C412yes—
213C4×D4—216no—
214C4×D4—216no—
215C4×D4—216no—
216C4×D4—216no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—44no—
223C4×D4—44no—
224C4×D4C22⋊C4116yescomplemented
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4#4114yes—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—82no—
261C2×C22⋊C4—82no—
262C2×C22⋊C4—82no—
263C2×C22⋊C4—82no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—28no—
269C2×C42—24no—
270C2×C42—42no—
271C2×C42—82no—
272C2×C42—82no—
273C2×C42—82no—
274C2×C42—82no—
275C2×C42C22×C414yes—
276C2×C42C22×C412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42#4114yes—
281C2×C42#4112yes—
282C2×C42#4112yes—
283#106—28no—
284#102—28no—
285#102—216no—
286#102—28no—
287#102—216no—
288#97—28no—
289C4×C8—44no—
290#95—28no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—42no—
296#95—82no—
297#95—82no—
298C24—22no—
299C24—24no—
300C24—22no—
301C24—22no—
302C24—24no—
303C24—24no—
304C24—22no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—41no—
310C24—41no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24C22×D411yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4C2×C4214yes—
348C2×D4C2×C4214yes—
349C2×D4C2×C22⋊C414yes—
350C2×D4#11714yes—
351C2×D4C4×D418yes—
352C2×D4C4×D418yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—232no—
376C22×C4—216no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—48no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4C4×D414yes—
411C22×C4C4×D414yes—
412C22×C4C4×D412yes—
413C22×C4C4×D412yes—
414C22×C4#12012yes—
415C22×C4#12012yes—
416C22×C4#12112yes—
417C22×C4#12112yes—
418C22×C4C23×C411yescharacteristic
419C22×C4C23×C412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4112yes—
428D8—232no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8C2×C4214yes—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4C2×C22⋊C414yes—
461C4⋊C4C4×D418yes—
462C22⋊C4—232no—
463C22⋊C4—216no—
464C22⋊C4—48no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—24no—
471C42—84no—
472C42—84no—
473C42—88no—
474C42C2×C22⋊C414yes—
475C42C4×D414yes—
476C42#12114yes—
477C42#12314yes—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—81no—
515C23—81no—
516C23—82no—
517C23—81no—
518C23—81no—
519C23—82no—
520C23C2×C4×D411yescharacteristic
521C23C2×C4×D412yes—
522C23C2×#12011yescharacteristic
523C23C2×#12111yescharacteristic
524C23#52512yes—
525C23#57612yes—
526C23#57612yes—
527C23#57712yes—
528C23#57712yes—
529D4—24no—
530D4—24no—
531D4—24no—
532D4—24no—
533D4—24no—
534D4—28no—
535D4—28no—
536D4—42no—
537D4—42no—
538D4—44no—
539D4—44no—
540D4C4×C22⋊C418yes—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—816no—
581C2×C4—816no—
582C2×C4—832no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4#38614yes—
591C2×C4#38614yes—
592C2×C4C22×C4211yescharacteristic
593C2×C4C22×C22⋊C411yescharacteristic
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×C4×D412yes—
596C2×C4C2×C4×D412yes—
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D412yes—
599C2×C4C4×#4111yescharacteristic
600C2×C4C4×#4112yes—
601C2×C4C4×#4111yescharacteristic
602C2×C4C4×#4112yes—
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×#12311yescharacteristic
605C2×C4#52512yes—
606C2×C4#52512yes—
607C2×C4#52912yes—
608C2×C4#52912yes—
609C2×C4#53212yes—
610C8—24no—
611C8—42no—
612C8—44no—
613C22—28no—
614C22—48no—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—48no—
623C22—44no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—82no—
628C22—82no—
629C22—84no—
630C22—84no—
631C22C42×D411yescharacteristic
632C22C2×#38611yescharacteristic
633C22#229011yescharacteristic
634C22C4×#12012yes—
635C22C4×#12112yes—
636C22#294612yes—
637C22#294612yes—
638C22#294612yes—
639C22#294612yes—
640C22#294812yes—
641C22#295212yes—
642C22#300811yescharacteristic
643C22#301011yescharacteristic
644C22#301212yes—
645C22#304111yescharacteristic
646C22#304211yescharacteristic
647C22#304512yes—
648C22#311812yes—
649C22#311912yes—
650C22#312212yes—
651C22#312512yes—
652C4—24no—
653C4—24no—
654C4—44no—
655C4—44no—
656C4—48no—
657C4—44no—
658C4—84no—
659C4—82no—
660C4C2×C4×C22⋊C411yescharacteristic
661C4C42×D411yescharacteristic
662C4#229011yescharacteristic
663C4#229711yescharacteristic
664C4C4×#12112yes—
665C4C4×#12312yes—
666C2—48no—
667C2—84no—
668C2C4×#38611yescharacteristic
669C2#2050312yes—
670C2#2092911yescharacteristic
671C2#2093211yescharacteristic
672C2#2112612yes—
673C2#2145812yes—
674C2#2146412yes—
675C2#2151012yes—
676C2#2151312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2275C2semidirect
4. C2×C4×#102C2semidirect
5. C2×C4×#102C2semidirect
6. #13785C2semidirect
7. #10931C2semidirect
8. #10908C2semidirect
9. #10788C2semidirect
10. #10703C2semidirect
11. #10694C2semidirect
18. C2×#442C4semidirect
36. #1879C4semidirect
76. C2×C4×D4C2×C4semidirect
77. C2×C4×D4C2×C4semidirect
97. C2×#106C2×C4semidirect
107. C2×#102C2×C4semidirect
108. C2×#102C2×C4semidirect
109. C2×#102C2×C4semidirect
112. C2×#97C2×C4semidirect
117. #390C2×C4semidirect
140. #340C2×C4semidirect
142. #336C2×C4semidirect
195. C2×D8C42semidirect
224. C4×D4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4422
C2×#22752
C2×#18792
C2×C4×#1022
#137852
#109312
#109082
#107882
#107032
#106942