Finite Group Databasebeta

← #375887

Subgroups — #375887

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1862C212yesmaximal
4C4×#390C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11324C211yesmaximal, characteristic, complemented
7#11049C211yesmaximal, characteristic
8#10931C212yesmaximal
9#10907C212yesmaximal, complemented
10#10884C212yesmaximal, complemented
11#10795C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C4×C4⋊Q8—28no—
19C2×#391—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4—21no—
30C2×C43C2211yescharacteristic
31#2023C418yescomplemented
32#1862—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1437—21no—
43#1437—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C23×Q8—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yescomplemented
55C2×C4×Q8—28no—
56C2×C4×Q8—44no—
57C2×C4×Q8—44no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—42no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—22no—
77C22×C42—21no—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×#102C2×C418yes—
94#391—28no—
95#390—22no—
96#390—22no—
97#390C2×C414yes—
98#390Q814yescomplemented
99#389—28no—
100#389—24no—
101#389D418yes—
102#384—24no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—42no—
107C4×C22⋊C4—44no—
108C4×C22⋊C4—44no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C43—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#340C2×C418yescomplemented
122#336—24no—
123#336—22no—
124#336—24no—
125#336C2×C414yescomplemented
126C25—21no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C22×C412yes—
142C22×D4C2×Q812yes—
143C22×D4#4112yes—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C22×C414yes—
177C4⋊Q8—216no—
178C4⋊Q8—216no—
179C4×Q8—432no—
180C4×D4—48no—
181C4×D4—48no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×Q812yes—
216C2×C4⋊C4#4112yes—
217C2×C4⋊C4#4114yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—28no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42D814yes—
258C2×C42C22×C412yes—
259C2×C42C2×D414yes—
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263#107—48no—
264#102—216no—
265#102—48no—
266#95—28no—
267#95—28no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24C22×D411yescharacteristic
289C2×Q8—216no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—48no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4C4×Q814yescomplemented
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4C4×D412yes—
383C22×C4C4×D412yes—
384C22×C4#12012yes—
385C22×C4#12012yes—
386C22×C4C22⋊Q812yes—
387C22×C4C22⋊Q812yes—
388C22×C4C2×D812yes—
389C22×C4C2×D812yes—
390C22×C4C2×D812yes—
391C22×C4#13512yes—
392C22×C4AGL(1,ℤ/8)12yes—
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4#13712yes—
395C22×C4#13712yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4H(5,𝔽2)12yes—
405C22×C42−1+412yes—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C4⋊C4—216no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—416no—
437C4⋊C4C4×D418yes—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C42—28no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—28no—
451C42—44no—
452C42—44no—
453C42—416no—
454C42—84no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23C2×C4×D411yescharacteristic
491C23C2×#12011yescharacteristic
492C23C2×C22⋊Q811yescharacteristic
493C23#52511yescharacteristic
494C23#52511yescharacteristic
495C23#53511yescharacteristic
496C23#53611yescharacteristic
497C23C22×D811yescharacteristic
498C23C2×#13511yescharacteristic
499C23C2×AGL(1,ℤ/8)11yescharacteristic
500C23C2×AGL(1,ℤ/8)11yescharacteristic
501C23C2×#13711yescharacteristic
502C23C2×#13711yescharacteristic
503C23#57511yescharacteristic
504C23#57511yescharacteristic
505Q8—432no—
506D4—28no—
507D4—44no—
508D4—44no—
509D4—44no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—816no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4C4×D814yes—
585C2×C4#44014yes—
586C2×C4#44714yes—
587C2×C4#47414yes—
588C2×C4C2×C4×D412yes—
589C2×C4C2×C4×Q811yescharacteristic
590C2×C4C4×#4111yescharacteristic
591C2×C4#52012yes—
592C2×C4#52512yes—
593C2×C4#52911yescharacteristic
594C2×C4#53311yescharacteristic
595C2×C4#54212yes—
596C2×C4#54612yes—
597C2×C4#54612yes—
598C2×C4#54812yes—
599C2×C4D4×Q812yes—
600C2×C4#55212yes—
601C2×C4#55412yes—
602C2×C4#55612yes—
603C2×C4#55912yes—
604C8—44no—
605C8—82no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22C4×#12011yescharacteristic
622C22C4×C22⋊Q811yescharacteristic
623C22#232611yescharacteristic
624C22#240411yescharacteristic
625C22#240911yescharacteristic
626C22#258011yescharacteristic
627C22#259911yescharacteristic
628C22C2×C4×D811yescharacteristic
629C22C2×#44011yescharacteristic
630C22C4×AGL(1,ℤ/8)11yescharacteristic
631C22C4×#13711yescharacteristic
632C22C2×#44711yescharacteristic
633C22#300211yescharacteristic
634C22#300411yescharacteristic
635C22#300611yescharacteristic
636C22C2×#47411yescharacteristic
637C22#307611yescharacteristic
638C22#307711yescharacteristic
639C22#307911yescharacteristic
640C22#310011yescharacteristic
641C22#310011yescharacteristic
642C22#310511yescharacteristic
643C22#310711yescharacteristic
644C22#310911yescharacteristic
645C22#311211yescharacteristic
646C22#311611yescharacteristic
647C22#311711yescharacteristic
648C22#320511yescharacteristic
649C22#321011yescharacteristic
650C22#321311yescharacteristic
651C22#321611yescharacteristic
652C22#322611yescharacteristic
653C22#322811yescharacteristic
654C22#323911yescharacteristic
655C22#324011yescharacteristic
656C4—24no—
657C4—24no—
658C4—24no—
659C4—24no—
660C4—42no—
661C4—48no—
662C4—42no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4#236012yes—
668C4#236712yes—
669C4#245812yes—
670C4#246512yes—
671C2—84no—
672C2#1417211yescharacteristic
673C2C4×#44711yescharacteristic
674C2#2145411yescharacteristic
675C2C4×#47411yescharacteristic
676C2#2161411yescharacteristic
677C2#2222011yescharacteristic
678C2#2222311yescharacteristic
679C2#2393011yescharacteristic
680C2#2393511yescharacteristic
681C2#2429811yescharacteristic
682C2#2430411yescharacteristic
683C2#2925411yescharacteristic
684C2#2926011yescharacteristic
685C2#2938811yescharacteristic
686C2#2938911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×C4⋊Q8C2semidirect
5. #11384C2semidirect
6. #11324C2semidirect
9. #10907C2semidirect
10. #10884C2semidirect
11. #10795C2semidirect
31. #2023C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
98. #390Q8semidirect
121. #340C2×C4semidirect
125. #336C2×C4semidirect
311. C2×D4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20232
C2×#18622
C4×#3902
#113842
#113242
#110492
#109312
#109072
#108842
#107952