Finite Group Databasebeta

← #188470

Subgroups — #188470

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 699 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×#1510C212yesmaximal
4#14685C211yesmaximal, characteristic, complemented
5#8577C214yesmaximal, complemented
6#8573C214yesmaximal, complemented
7#8015C212yesmaximal, 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×#381C2212yes—
27C2×#380—21no—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30C2×C4×C22⋊C4—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yescomplemented
34#1797C2212yescomplemented
35#1728—21no—
36#1510—28no—
37C2×#327C2218yes—
38C4⋊#95—21no—
39C4⋊#95C2212yescomplemented
40#1434—21no—
41#1434C2212yescomplemented
42C22⋊#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—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C42—21no—
81C22×C42—44no—
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×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#423—44no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97#407—24no—
98#407—24no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2314yes—
101#390—21no—
102#390—21no—
103#390C2×C412yes—
104#390C2×C412yes—
105#389—21no—
106#389—21no—
107#389C2×C412yes—
108#389C2×C412yes—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#381—24no—
118#381—24no—
119#380—21no—
120#380—21no—
121#380—44no—
122#380C2×C412yes—
123#380C2312yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95C2314yes—
140#336—22no—
141#336—22no—
142#327—216no—
143C25—21no—
144C25C2×D411yescharacteristic
145C22×D4—22no—
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4C4⋊C414yescomplemented
156C22×D4C22×C412yes—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
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×C4—44no—
174C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×Q812yes—
193C23×C4#4112yes—
194C23×C4#4112yes—
195C23×C4C2411yescharacteristic
196C2×M16—44no—
197C2×M16—44no—
198C2×M16—44no—
199C22×C8—24no—
200#121—24no—
201#121—24no—
202#121—24no—
203#121—24no—
204#121—24no—
205#121—22no—
206#121—22no—
207#121C4⋊C418yescomplemented
208#121C2×D414yes—
209#121C2×Q814yes—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—48no—
215C4×D4—48no—
216C4×D4—48no—
217C4×D4—48no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C4⋊C414yescomplemented
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
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—22no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
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—42no—
267C2×C42—44no—
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C4⋊C8—44no—
285C4⋊C8—44no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—44no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—41no—
308C24—41no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24#12112yes—
318C24C22⋊Q812yes—
319C24C22×D411yescharacteristic
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—28no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—42no—
326C2×D4—48no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—48no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—416no—
333C2×D4—416no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4C2×C4⋊C414yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
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—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
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—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—41no—
392C22×C4—416no—
393C22×C4—416no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
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—816no—
416C22×C4#9914yes—
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
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×C4⋊C412yes—
428C22×C4C2×C4⋊C412yes—
429C22×C4#11712yes—
430C22×C4#11712yes—
431C22×C4#12112yes—
432C22×C4#12112yes—
433C22×C4#12112yes—
434C22×C4C22⋊Q812yes—
435C22×C4C22⋊Q812yes—
436C22×C4C22⋊Q812yes—
437C22×C4AGL(1,ℤ/8)14yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×Q811yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444M16—88no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—88no—
457C4⋊C4C2×C4⋊C414yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—28no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—88no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—44no—
487C42—44no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—24no—
506C23—24no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23#38012yes—
527C23#38012yes—
528C23C2×#9912yes—
529C23#41012yes—
530C23C22×C22⋊C411yescharacteristic
531C23C2×#12111yescharacteristic
532C23C2×C22⋊Q811yescharacteristic
533C23C2×AGL(1,ℤ/8)12yes—
534C23#57512yes—
535D4—44no—
536D4—44no—
537D4—48no—
538D4—416no—
539D4—48no—
540D4—48no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
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—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
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—48no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4#38012yes—
615C2×C4#38012yes—
616C2×C4C2×#9912yes—
617C2×C4C2×#9912yes—
618C2×C4#41812yes—
619C2×C4#41812yes—
620C2×C4#42112yes—
621C2×C4#42112yes—
622C2×C4C22×C4⋊C411yescharacteristic
623C2×C4C2×#11711yescharacteristic
624C2×C4C2×#12111yescharacteristic
625C2×C4C2×C22⋊Q811yescharacteristic
626C8—84no—
627C22—22no—
628C22—22no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—21no—
637C22—22no—
638C22—21no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—48no—
645C22—48no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—816no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22#179914yes—
659C22C2×#38011yescharacteristic
660C22C22×#9911yescharacteristic
661C22C2×#41011yescharacteristic
662C22C2×#41811yescharacteristic
663C22#289611yescharacteristic
664C22C2×#42111yescharacteristic
665C22C2×#42111yescharacteristic
666C22#303612yes—
667C22#307712yes—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—21no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—44no—
681C4—41no—
682C4—41no—
683C4—82no—
684C4—82no—
685C4—164no—
686C2—22no—
687C2—24no—
688C2—22no—
689C2—42no—
690C2—84no—
691C2#1755411yescharacteristic
692C2#1756111yescharacteristic
693C2#1757711yescharacteristic
694C2#1758711yescharacteristic
695C2C2×#179911yescharacteristic
696C2#2064611yescharacteristic
697C2#2065411yescharacteristic
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. #14685C2semidirect
5. #8577C2semidirect
6. #8573C2semidirect
7. #8015C2semidirect
10. C22×C4×D4C4semidirect
18. C4×#121C4semidirect
19. C4×#121C4semidirect
22. C2×#390C4semidirect
23. C2×#389C4semidirect
29. C2×C4×C4⋊C4C4semidirect
33. #1797C22semidirect
34. #1797C22semidirect
39. C4⋊#95C22semidirect
41. #1434C22semidirect
43. #1294C22semidirect
155. C22×D4C4⋊C4semidirect
207. #121C4⋊C4semidirect
230. 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×#15102
#146852
#85772
#85732
#80152