Finite Group Databasebeta

← #361779

Subgroups — #361779

Order 512.

2348 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2C2×#1826C212yesmaximal, complemented
3C2×#1789C212yesmaximal
4C4×#407C211yesmaximal, characteristic, complemented
5C4×#380C211yesmaximal, characteristic, complemented
6#11075C214yesmaximal, complemented
7#10933C214yesmaximal, complemented
8C23×#41C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C22×#104C2218yes—
12C2×#407C2212yes—
13C2×C4×M16C2212yescomplemented
14#2297—21no—
15#2297—21no—
16#2297—21no—
17#2297—21no—
18#2297C412yescomplemented
19#2297C412yescomplemented
20#2297C2212yes—
21#2297C2212yes—
22C2×#391C411yescharacteristic, complemented
23C2×#391C411yescharacteristic, complemented
24C2×#390C411yescharacteristic, complemented
25C2×#390C411yescharacteristic, complemented
26C2×#386—21no—
27C2×#386—21no—
28C2×#380—21no—
29C2×C4×C4⋊C4C2212yescomplemented
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4C2212yes—
33#1826—28no—
34#1789—28no—
35#1749—21no—
36#1749—21no—
37C4×C22⋊C8—21no—
38C4×C22⋊C8—21no—
39C2×#339C2218yescomplemented
40#1728C2212yescomplemented
41C4⋊#95—21no—
42#1434—21no—
43C22⋊#95—21no—
44C4×#95C2212yescomplemented
45C22×#41—28no—
46C22×#41—21no—
47C22×#41—21no—
48C22×#41—24no—
49C22×#41—24no—
50C22×#41—44no—
51C22×#41C2×C412yes—
52C22×#41C2312yes—
53C23×Q8C2×C411yescharacteristic
54C23×D4—21no—
55C23×D4C2×C411yescharacteristic
56C24×C4—21no—
57C24×C4D411yescharacteristic
58C24×C4D411yescharacteristic
59C24×C4C2311yescharacteristic
60C22×M16C2314yes—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—24no—
64C2×#117—24no—
65C2×#117C2×C412yes—
66C2×#117C2312yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—21no—
77C22×C42—44no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2314yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#104—216no—
92C2×#104—216no—
93C2×#104—48no—
94#407—24no—
95#407D418yes—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C4×M16—44no—
99C2×#97—22no—
100C2×C4×C8—22no—
101#391—21no—
102#391—21no—
103#391—21no—
104#391—21no—
105#391C2×C412yes—
106#391C2×C412yes—
107#391C2×C412yes—
108#391C2×C412yes—
109#390—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#390C2×C412yes—
114#390C2×C412yes—
115#390C2×C412yes—
116#390C2×C412yes—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#380—28no—
126C4×C4⋊C4—44no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—24no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145#339—216no—
146#336—22no—
147#336—22no—
148C25—21no—
149C25C2×D411yescharacteristic
150C2×#41—28no—
151C2×#41—28no—
152C2×#41—24no—
153C2×#41—24no—
154C2×#41—24no—
155C2×#41—28no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—28no—
159C2×#41—44no—
160C2×#41—432no—
161C2×#41—432no—
162C2×#41—44no—
163C2×#41—44no—
164C2×#41—44no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41#4118yes—
168C22×Q8—24no—
169C22×Q8—21no—
170C22×Q8—24no—
171C22×Q8—42no—
172C22×Q8C22×C412yes—
173C22×D4—28no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—21no—
178C22×D4—24no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4C22×C412yes—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—21no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—24no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
205C23×C4C22⋊C411yescharacteristic
206C23×C4C22⋊C411yescharacteristic
207C23×C4C22⋊C411yescharacteristic
208C23×C4C22⋊C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D414yes—
218C23×C4C2×D411yescharacteristic
219C23×C4#4114yes—
220C23×C4C2411yescharacteristic
221C2×M16—24no—
222C2×M16—28no—
223C2×M16—28no—
224C2×M16C2×D418yes—
225C22×C8—24no—
226#117—28no—
227#117—24no—
228#117—24no—
229#117—44no—
230#117—42no—
231#117—42no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
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—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
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×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295#104—432no—
296C22⋊C8—28no—
297C22⋊C8—28no—
298#97—44no—
299C4×C8—44no—
300#95—28no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—44no—
310#95—44no—
311#95—44no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24C2×C22⋊C411yescharacteristic
327C24C2×C22⋊C411yescharacteristic
328C24#12412yes—
329C24#12712yes—
330C24C22×D411yescharacteristic
331#41—216no—
332#41—216no—
333#41—416no—
334#41—416no—
335#41—416no—
336#41—416no—
337#41—432no—
338#41—48no—
339#41—48no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—416no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8#11714yes—
348C2×D4—24no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—416no—
355C2×D4—42no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4#11714yes—
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—216no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
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—416no—
420C22×C4—44no—
421C22×C4—432no—
422C22×C4—42no—
423C22×C4—416no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4#10414yescomplemented
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#12712yes—
457C22×C4C23×C411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)14yes—
463M16—216no—
464M16—48no—
465C2×C8—28no—
466C2×C8—44no—
467C2×C8—44no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—416no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—416no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—44no—
502C42—44no—
503C42—48no—
504C42—48no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—24no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—24no—
520C23—24no—
521C23—21no—
522C23—24no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23#39012yes—
544C23#39012yes—
545C23C2×#10412yes—
546C23#42112yes—
547C23C22×C22⋊C411yescharacteristic
548C23#52512yes—
549C23C2×#12411yescharacteristic
550C23C2×#12711yescharacteristic
551C23#53412yes—
552Q8—28no—
553Q8—48no—
554Q8—44no—
555D4—28no—
556D4—48no—
557D4—416no—
558D4—48no—
559D4—48no—
560D4—44no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
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—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—24no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—416no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—416no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4#39012yes—
625C2×C4#39012yes—
626C2×C4C2×#10412yes—
627C2×C4C2×#10412yes—
628C2×C4#42112yes—
629C2×C4#42112yes—
630C2×C4C2×#11712yes—
631C2×C4C2×C4×D412yes—
632C2×C4#51812yes—
633C2×C4C2×#12112yes—
634C2×C4#52512yes—
635C2×C4#53412yes—
636C2×C4#54618yes—
637C2×C4#55118yes—
638C8—48no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—22no—
643C22—21no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—48no—
657C22—48no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—816no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22#198014yes—
671C22#228011yescharacteristic
672C22C2×#39011yescharacteristic
673C22#230811yescharacteristic
674C22#253912yes—
675C22#256112yes—
676C22C22×#10411yescharacteristic
677C22C2×#42111yescharacteristic
678C22C2×#42111yescharacteristic
679C22C2×#42111yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—44no—
691C4—88no—
692C4—82no—
693C4—82no—
694C4#234214yes—
695C4#253414yes—
696C2—22no—
697C2—24no—
698C2—22no—
699C2—42no—
700C2—84no—
701C2#1390311yescharacteristic
702C2#2074711yescharacteristic
703C2#2074811yescharacteristic
704C2C2×#198011yescharacteristic
705C2#2125211yescharacteristic
706C2#2177311yescharacteristic
707C2#2177411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1826C2semidirect
4. C4×#407C2semidirect
5. C4×#380C2semidirect
6. #11075C2semidirect
7. #10933C2semidirect
13. C2×C4×M16C22semidirect
18. #2297C4semidirect
19. #2297C4semidirect
22. C2×#391C4semidirect
23. C2×#391C4semidirect
24. C2×#390C4semidirect
25. C2×#390C4semidirect
29. C2×C4×C4⋊C4C22semidirect
39. C2×#339C22semidirect
40. #1728C22semidirect
44. C4×#95C22semidirect
440. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
C2×#22972
C2×#18262
C2×#17892
C4×#4072
C4×#3802
#110752
#109332