Finite Group Databasebeta

← #358668

Subgroups — #358668

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×C4×#99C214yesmaximal, complemented
3C4×#380C211yesmaximal, characteristic, complemented
4#13753C211yesmaximal, characteristic, complemented
5#11353C212yesmaximal, complemented
6#11043C214yesmaximal, complemented
7#10980C212yesmaximal
8C22×#121C2211yescharacteristic
9C22×C4×D4C411yescharacteristic, complemented
10C22×C4×D4C2212yes—
11C23×C42C2211yescharacteristic
12C22×#99C2218yes—
13C4×#121—21no—
14C4×#121—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121—21no—
21C2×#390C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#388C412yescomplemented
24C2×#386C2212yes—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C411yescharacteristic, complemented
29C2×C4×C22⋊C4C412yescomplemented
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2212yes—
32C2×#376—21no—
33C4×#99—216no—
34C2×#342C2218yes—
35C4⋊#95—21no—
36#1434—21no—
37#1434—21no—
38#1434—21no—
39C22⋊#95C2212yes—
40C22⋊#95C2212yes—
41C22⋊#95C2212yes—
42C4×#95—21no—
43C4×#95—21no—
44C23×D4C2×C411yescharacteristic
45C23×D4C2312yes—
46C24×C4C2×C412yes—
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C2×#121—28no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—21no—
55C2×C4×D4—28no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—28no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—44no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—44no—
81C22×C42C2×C412yes—
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C2×#99—216no—
95C2×#99—216no—
96C2×#99—216no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—28no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—44no—
123C4×C22⋊C4—24no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—28no—
129C4×C22⋊C4D418yescomplemented
130#376—44no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143#342—216no—
144C25C22×C412yes—
145C25C2411yescharacteristic
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—28no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4#4118yes—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—28no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C412yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D414yes—
195C23×C4#4114yes—
196#121—28no—
197#121—28no—
198#121—28no—
199#121—28no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—48no—
204C4×D4—48no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4—44no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×D418yes—
249C2×C22⋊C4#4118yes—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—28no—
256C2×C42—28no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287#99—264no—
288#95—28no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—28no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24#11714yes—
316C24C23×C411yescharacteristic
317C24C22×D411yescharacteristic
318C24C22×D411yescharacteristic
319C24C22×D412yes—
320C24C2×#4114yes—
321C24C2×#4112yes—
322C24C2×#4114yes—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—216no—
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—216no—
329C2×D4—42no—
330C2×D4—416no—
331C2×D4—42no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—48no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—42no—
340C2×D4—42no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
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—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
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—28no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—816no—
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4#11714yes—
439C22×C4C4×D414yes—
440C22×C4C4×D414yes—
441C22×C4#12114yes—
442C22×C4#12114yes—
443C22×C4#12412yes—
444C22×C4#12412yes—
445C22×C4#12712yes—
446C22×C4#12712yes—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—416no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—216no—
459C22⋊C4—216no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—42no—
465C22⋊C4—48no—
466C22⋊C4—42no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—416no—
487C42—44no—
488C42—44no—
489C42—88no—
490C42#9914yescomplemented
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—24no—
504C23—28no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—44no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23—84no—
528C23C22×C22⋊C411yescharacteristic
529C23C2×#11712yes—
530C23C2×C4×D412yes—
531C23C4×#4112yes—
532C23C2×#12112yes—
533C23#52512yes—
534C23#52512yes—
535C23#52512yes—
536C23#52518yes—
537C23C2×#12411yescharacteristic
538C23#52918yes—
539C23C2×#12711yescharacteristic
540C23#53212yes—
541D4—48no—
542D4—48no—
543D4—416no—
544D4—48no—
545D4—48no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
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—21no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—88no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4#39012yes—
622C2×C4#39012yes—
623C2×C4#39012yes—
624C2×C4#39012yes—
625C2×C4C2×#9912yes—
626C2×C4C2×#9912yes—
627C2×C4C2×#9912yes—
628C2×C4#41012yes—
629C2×C4#41012yes—
630C2×C4#41012yes—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—21no—
639C22—22no—
640C22—21no—
641C22—22no—
642C22—21no—
643C22—22no—
644C22—21no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—48no—
652C22—48no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—88no—
657C22—88no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#228011yescharacteristic
663C22#229014yes—
664C22C2×#39011yescharacteristic
665C22#229711yescharacteristic
666C22#239214yes—
667C22#239712yes—
668C22#239812yes—
669C22C22×#9911yescharacteristic
670C22C2×#41011yescharacteristic
671C22C2×#41011yescharacteristic
672C22C2×#41011yescharacteristic
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—44no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—88no—
686C4—88no—
687C4—82no—
688C4—82no—
689C4#196414yes—
690C2—22no—
691C2—24no—
692C2—24no—
693C2—22no—
694C2—84no—
695C2#1384711yescharacteristic
696C2#2072211yescharacteristic
697C2#2072311yescharacteristic
698C2C2×#196411yescharacteristic
699C2#2121211yescharacteristic
700C2#2142011yescharacteristic
701C2#2142111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
socleC23C22×C22⋊C4
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#99C2semidirect
3. C4×#380C2semidirect
4. #13753C2semidirect
5. #11353C2semidirect
6. #11043C2semidirect
9. C22×C4×D4C4semidirect
21. C2×#390C4semidirect
22. C2×#389C4semidirect
23. C2×#388C4semidirect
28. C2×C4×C4⋊C4C4semidirect
29. C2×C4×C22⋊C4C4semidirect
129. C4×C22⋊C4D4semidirect
490. C42#99semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×C4×#992
C4×#3802
#137532
#113532
#110432
#109802