Finite Group Databasebeta

← #188452

Subgroups — #188452

Order 512.

1856 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
2#20626C211yesmaximal, characteristic, complemented
3C2×#1507C212yesmaximal
4C4×#380C211yesmaximal, characteristic, complemented
5#8577C214yesmaximal, complemented
6#8575C214yesmaximal, complemented
7#8019C212yesmaximal, complemented
8C22×#121C2211yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C411yescharacteristic, complemented
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C2×#407C2212yes—
14C4×#121—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121C412yescomplemented
19C4×#121C412yescomplemented
20C4×#121C2212yes—
21C4×#121C2212yes—
22C2×#390C411yescharacteristic, complemented
23C2×#389C411yescharacteristic, complemented
24C2×#388—21no—
25C2×#386—21no—
26C2×#380—21no—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4C2212yes—
32#1797—21no—
33#1797—21no—
34#1797C2212yescomplemented
35#1797C2212yescomplemented
36#1728—21no—
37#1507—28no—
38C2×#327C2218yes—
39C4⋊#95C2212yescomplemented
40#1434C2212yescomplemented
41C22⋊#95—21no—
42C4×#95—21no—
43#1294C2218yescomplemented
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4—21no—
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121C2×C412yes—
56C2×#121D414yes—
57C2×#121Q814yes—
58C2×#121C2312yes—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—44no—
66C2×C4×D4—48no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C412yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C42—21no—
81C22×C42—22no—
82C22×C42—44no—
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#423—44no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98#407—24no—
99#407—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8C2314yes—
102#390—21no—
103#390—21no—
104#390C2×C412yes—
105#390C2×C412yes—
106#389—21no—
107#389—21no—
108#389C2×C412yes—
109#389C2×C412yes—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#380—28no—
119#380—21no—
120#380—21no—
121#380C2×C412yes—
122#380C2312yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—44no—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—24no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95C2314yes—
141#336—22no—
142#336—22no—
143#327—216no—
144C25—21no—
145C25C2×D411yescharacteristic
146C22×D4—22no—
147C22×D4—24no—
148C22×D4—24no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4C4⋊C414yescomplemented
157C22×D4C22×C412yes—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×Q812yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4C2411yescharacteristic
197C2×M16—44no—
198C2×M16—44no—
199C2×M16—44no—
200C22×C8—24no—
201#121—24no—
202#121—24no—
203#121—24no—
204#121—24no—
205#121—24no—
206#121—22no—
207#121—22no—
208#121C4⋊C418yescomplemented
209#121C2×D414yes—
210#121C2×Q814yes—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—48no—
215C4×D4—48no—
216C4×D4—48no—
217C4×D4—48no—
218C4×D4—44no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C4⋊C414yescomplemented
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
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—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—44no—
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×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C4⋊C8—44no—
289C4⋊C8—44no—
290C22⋊C8—24no—
291C22⋊C8—24no—
292C22⋊C8—44no—
293#95—28no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—44no—
303#95—44no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24—41no—
311C24—41no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24C2×C22⋊C411yescharacteristic
319C24C2×C22⋊C411yescharacteristic
320C24#12112yes—
321C24C22⋊Q812yes—
322C24C22×D411yescharacteristic
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—28no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—44no—
333C2×D4—416no—
334C2×D4—416no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4C2×C4⋊C414yes—
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—22no—
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—28no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
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—416no—
397C22×C4—416no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—816no—
419C22×C4#10414yes—
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C4⋊C412yes—
431C22×C4C2×C4⋊C412yes—
432C22×C4#11712yes—
433C22×C4#11712yes—
434C22×C4#12112yes—
435C22×C4#12112yes—
436C22×C4#12112yes—
437C22×C4C22⋊Q812yes—
438C22×C4C22⋊Q812yes—
439C22×C4C22⋊Q812yes—
440C22×C4AGL(1,ℤ/8)14yes—
441C22×C4C23×C411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×Q811yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447M16—88no—
448C2×C8—44no—
449C2×C8—44no—
450C2×C8—44no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—416no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4C2×C4⋊C414yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—28no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—416no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—44no—
489C42—44no—
490C42—88no—
491C42#9914yes—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—24no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23#38012yes—
531C23#38012yes—
532C23C2×#10412yes—
533C23#42112yes—
534C23C22×C22⋊C411yescharacteristic
535C23C2×#12111yescharacteristic
536C23C2×C22⋊Q811yescharacteristic
537C23C2×AGL(1,ℤ/8)12yes—
538C23#57512yes—
539D4—44no—
540D4—44no—
541D4—416no—
542D4—48no—
543D4—48no—
544D4—48no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—21no—
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—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4#38012yes—
617C2×C4#38012yes—
618C2×C4C2×#9912yes—
619C2×C4C2×#9912yes—
620C2×C4C2×#9912yes—
621C2×C4#41012yes—
622C2×C4#41812yes—
623C2×C4#41812yes—
624C2×C4C2×#10412yes—
625C2×C4C2×#10412yes—
626C2×C4C22×C4⋊C411yescharacteristic
627C2×C4C2×#11711yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4C2×C22⋊Q811yescharacteristic
630C8—84no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—21no—
636C22—22no—
637C22—21no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—48no—
649C22—48no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—816no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#180614yes—
663C22C2×#38011yescharacteristic
664C22C22×#9911yescharacteristic
665C22C2×#41011yescharacteristic
666C22C2×#41811yescharacteristic
667C22#289611yescharacteristic
668C22C22×#10411yescharacteristic
669C22C2×#42111yescharacteristic
670C22#303612yes—
671C22#307712yes—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—41no—
684C4—44no—
685C4—41no—
686C4—88no—
687C4—82no—
688C4—82no—
689C4#179914yes—
690C2—22no—
691C2—24no—
692C2—22no—
693C2—42no—
694C2—84no—
695C2#1754211yescharacteristic
696C2#1754411yescharacteristic
697C2#1757711yescharacteristic
698C2#1757911yescharacteristic
699C2C2×#179911yescharacteristic
700C2#2064611yescharacteristic
701C2C2×#180611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
4. C4×#380C2semidirect
5. #8577C2semidirect
6. #8575C2semidirect
7. #8019C2semidirect
10. C22×C4×D4C4semidirect
18. C4×#121C4semidirect
19. C4×#121C4semidirect
22. C2×#390C4semidirect
23. C2×#389C4semidirect
29. C2×C4×C4⋊C4C4semidirect
34. #1797C22semidirect
35. #1797C22semidirect
39. C4⋊#95C22semidirect
40. #1434C22semidirect
43. #1294C22semidirect
156. C22×D4C4⋊C4semidirect
208. #121C4⋊C4semidirect
231. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#206262
C2×#15072
C4×#3802
#85772
#85752
#80192