Finite Group Databasebeta

← #375254

Subgroups — #375254

Order 512.

1972 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2C2×#1996C212yesmaximal, complemented
3C2×#1964C212yesmaximal
4C4×#380C211yesmaximal, characteristic, complemented
5C4×#380C211yesmaximal, characteristic, complemented
6#11043C214yesmaximal, complemented
7#10960C214yesmaximal, complemented
8C22×#124C411yescharacteristic, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×#99C2218yes—
13#2850—21no—
14#2850—21no—
15#2850—21no—
16#2850—21no—
17#2850—21no—
18#2850—21no—
19#2850—21no—
20#2850—21no—
21C2×#391C411yescharacteristic, complemented
22C2×#390C411yescharacteristic, complemented
23C2×#388C412yescomplemented
24C2×#388C412yescomplemented
25C2×#386C2212yes—
26C2×#386C2212yes—
27C2×#382C411yescharacteristic, complemented
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2212yes—
34#1996—28no—
35#1964—28no—
36C2×#342C2218yes—
37C4⋊#95—21no—
38C4⋊#95—21no—
39#1434—21no—
40#1434—21no—
41C22⋊#95C2212yes—
42C22⋊#95C2212yes—
43C4×#95—21no—
44C4×#95—21no—
45C23×Q8C2×C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C23×D4C2312yes—
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C2×#124—21no—
52C2×#124—21no—
53C2×#124—21no—
54C2×#124—21no—
55C2×#124—44no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—24no—
67C2×#121—24no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—44no—
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C2×#99—216no—
97C2×#99—216no—
98C2×#99—48no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#382—21no—
124#382—21no—
125#382—21no—
126#382—21no—
127#380—24no—
128#380—28no—
129#380D418yes—
130C4×C4⋊C4—44no—
131C4×C4⋊C4—44no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—44no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95C2×C412yes—
144C2×#95C2×C412yes—
145C2×#95C2×C412yes—
146C2×#95C2×C412yes—
147#342—216no—
148C25C22×C412yes—
149C25C2411yescharacteristic
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—24no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—28no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—24no—
165C22×D4—42no—
166C22×D4—42no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C22×C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C412yes—
185C23×C4C22×C412yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D414yes—
200C23×C4#4114yes—
201#124—44no—
202#124—44no—
203#124—44no—
204#124—44no—
205C22⋊Q8—44no—
206C22⋊Q8—44no—
207C22⋊Q8—42no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210C22⋊Q8—42no—
211#121—44no—
212#121—44no—
213#121—42no—
214#121—42no—
215#121—42no—
216#121—42no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4C2×D418yes—
260C2×C22⋊C4#4118yes—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295#99—432no—
296#95—28no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312C24—22no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24C23×C411yescharacteristic
325C24C22×D411yescharacteristic
326C24C22×D411yescharacteristic
327C24C22×D412yes—
328C24C2×#4112yes—
329C24C2×#4114yes—
330C24H(5,𝔽2)12yes—
331C242−1+412yes—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—88no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—44no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—88no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
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—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C4×D414yes—
452C22×C4C4×D414yes—
453C22×C4#12114yes—
454C22×C4#12114yes—
455C22×C4#12412yes—
456C22×C4#12412yes—
457C22×C4#12712yes—
458C22×C4#12712yes—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C22⋊C4—28no—
468C22⋊C4—216no—
469C22⋊C4—28no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—88no—
502C42—88no—
503C42#9914yescomplemented
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—48no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23C22×C22⋊C411yescharacteristic
541C23C2×C4×D412yes—
542C23C4×#4112yes—
543C23#52012yes—
544C23C2×#12112yes—
545C23#52512yes—
546C23C2×#12411yescharacteristic
547C23C2×#12711yescharacteristic
548C23#53212yes—
549C23#53511yescharacteristic
550C23#53611yescharacteristic
551C23#53712yes—
552C23#54614yes—
553C23#54814yes—
554C23#55614yes—
555C23#55914yes—
556Q8—84no—
557Q8—84no—
558D4—84no—
559D4—84no—
560D4—84no—
561D4—84no—
562D4—84no—
563D4—84no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4#39012yes—
639C2×C4#39012yes—
640C2×C4#39012yes—
641C2×C4#39012yes—
642C2×C4C2×#9912yes—
643C2×C4C2×#9912yes—
644C2×C4C2×#9912yes—
645C2×C4#41012yes—
646C2×C4#41012yes—
647C2×C4#41012yes—
648C22—21no—
649C22—21no—
650C22—22no—
651C22—22no—
652C22—21no—
653C22—22no—
654C22—22no—
655C22—21no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—21no—
664C22—22no—
665C22—21no—
666C22—22no—
667C22—21no—
668C22—22no—
669C22—21no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—168no—
682C22C2×#39011yescharacteristic
683C22#229711yescharacteristic
684C22#231111yescharacteristic
685C22#236714yes—
686C22#254012yes—
687C22#254412yes—
688C22#254512yes—
689C22#259212yes—
690C22C22×#9911yescharacteristic
691C22C2×#41011yescharacteristic
692C22C2×#41011yescharacteristic
693C22C2×#41011yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—88no—
706C4—88no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4#196414yes—
711C2—22no—
712C2—22no—
713C2—24no—
714C2—22no—
715C2—22no—
716C2—162no—
717C2#1415311yescharacteristic
718C2C2×#196411yescharacteristic
719C2#2121211yescharacteristic
720C2#2142011yescharacteristic
721C2#2142111yescharacteristic
722C2#2187411yescharacteristic
723C2#2187511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1996C2semidirect
4. C4×#380C2semidirect
5. C4×#380C2semidirect
6. #11043C2semidirect
7. #10960C2semidirect
8. C22×#124C4semidirect
21. C2×#391C4semidirect
22. C2×#390C4semidirect
23. C2×#388C4semidirect
24. C2×#388C4semidirect
27. C2×#382C4semidirect
503. C42#99semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
C2×#19962
C2×#19642
C4×#3802
#110432
#109602