Finite Group Databasebeta

← #160488

Subgroups — #160488

Order 512.

1750 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4#7586C214yesmaximal, complemented
5#7573C212yesmaximal, complemented
6#7572C212yesmaximal, complemented
7#7565C214yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C411yescharacteristic
11C23×C42C2211yescharacteristic
12C22×C4⋊C8—21no—
13C22×C22⋊C8—21no—
14C22×C22⋊C8—21no—
15C22×C4×C8—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121C412yes—
21C4×#121C412yes—
22C4×#121C2212yes—
23C4×#121C2212yes—
24C2×#390C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388—21no—
27C2×#386—21no—
28C2×#380C411yescharacteristic
29C2×C4×C4⋊C4C411yescharacteristic
30C2×C4×C22⋊C4—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yescomplemented
35C4×C22⋊C8—21no—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38C4×C22⋊C8C2212yescomplemented
39C2×#336C2212yescomplemented
40C2×#336C2212yescomplemented
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—24no—
54C2×#121C818yescomplemented
55C2×#121C2×C412yes—
56C2×#121C2×C412yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—44no—
64C2×C4×D4—44no—
65C2×C4×D4—44no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—21no—
78C22×C42—44no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—44no—
94C2×C22⋊C8—28no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—44no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—44no—
103#390—21no—
104#390—21no—
105#390C2×C412yes—
106#390C2312yes—
107#389—21no—
108#389—21no—
109#389C2×C412yes—
110#389C2312yes—
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#380—21no—
120#380—21no—
121#380C2×C412yes—
122#380C2×C412yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C412yes—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134#336—22no—
135#336—22no—
136#336C2314yes—
137#336C2314yes—
138C25—21no—
139C25C22⋊C411yescharacteristic
140C22×D4—28no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4C2×C814yescomplemented
150C22×D4C22×C412yes—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C2×C814yes—
176C23×C4M1614yes—
177C23×C4D814yes—
178C23×C4SD1614yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C22×C8—28no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—44no—
192C22×C8—44no—
193C22×C8—44no—
194C22×C8—44no—
195C22×C8—44no—
196#121—28no—
197#121—28no—
198#121—24no—
199#121—24no—
200#121C2×C818yescomplemented
201#121C2×C818yescomplemented
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—44no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×C814yescomplemented
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2411yescharacteristic
273C4⋊C8—44no—
274C4⋊C8—44no—
275C22⋊C8—28no—
276C22⋊C8—28no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279C4×C8—44no—
280C4×C8—44no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287C24—22no—
288C24—24no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24C22⋊C814yes—
300C24#10214yes—
301C24#10414yes—
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C411yescharacteristic
305C2×D4—28no—
306C2×D4—28no—
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—432no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4C22×C814yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
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—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—416no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—816no—
387C22×C4C22⋊C814yes—
388C22×C4C22⋊C814yes—
389C22×C4C22⋊C814yes—
390C22×C4#10214yes—
391C22×C4#10214yes—
392C22×C4#10214yes—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C22×C812yes—
403C22×C4C22×C812yes—
404C22×C4C22×C812yes—
405C22×C4C2×M1612yes—
406C22×C4C2×M1612yes—
407C22×C4C2×M1612yes—
408C22×C4C2×D812yes—
409C22×C4C2×D812yes—
410C22×C4C2×D812yes—
411C22×C4C2×SD1612yes—
412C22×C4C2×SD1612yes—
413C22×C4C2×SD1612yes—
414C22×C4C23×C411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C2×C8—28no—
418C2×C8—44no—
419C2×C8—416no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—44no—
424C2×C8—88no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4C22×C814yes—
435C22⋊C4—28no—
436C22⋊C4—28no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—44no—
462C42—44no—
463C42#9914yes—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—48no—
484C23—48no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—48no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23D4⋊C8116yes—
497C23C2×C22⋊C812yes—
498C23C2×C22⋊C812yes—
499C23C2×C22⋊C812yes—
500C23C2×#10212yes—
501C23C2×#10212yes—
502C23C2×#10212yes—
503C23C2×#10412yes—
504C23C2×#10412yes—
505C23C2×#10412yes—
506C23C22×C22⋊C411yescharacteristic
507D4—44no—
508D4—44no—
509D4—416no—
510D4—416no—
511D4—48no—
512C2×C4—28no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—88no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—88no—
572C2×C4C2×C22⋊C812yes—
573C2×C4C2×C22⋊C812yes—
574C2×C4C2×C22⋊C812yes—
575C2×C4C2×#9912yes—
576C2×C4C2×#9912yes—
577C2×C4C2×#9912yes—
578C2×C4#41312yes—
579C2×C4C2×#10212yes—
580C2×C4C2×#10212yes—
581C2×C4C2×#10212yes—
582C2×C4#42112yes—
583C2×C4C23×C811yescharacteristic
584C2×C4C22×M1611yescharacteristic
585C2×C4C22×D811yescharacteristic
586C2×C4C22×SD1611yescharacteristic
587C2×C4#57512yes—
588C8—48no—
589C8—84no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—416no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—816no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22C2×D4⋊C814yes—
618C22C2×D4⋊C814yes—
619C22C2×D4⋊C814yes—
620C22C22×C22⋊C811yescharacteristic
621C22C22×#9911yescharacteristic
622C22C2×#41311yescharacteristic
623C22C22×#10211yescharacteristic
624C22#289611yescharacteristic
625C22C22×#10411yescharacteristic
626C22C2×#42111yescharacteristic
627C4—22no—
628C4—22no—
629C4—21no—
630C4—22no—
631C4—21no—
632C4—22no—
633C4—21no—
634C4—21no—
635C4—22no—
636C4—22no—
637C4—44no—
638C4—41no—
639C4—41no—
640C4—82no—
641C4—82no—
642C4#145914yes—
643C4#151914yes—
644C2—24no—
645C2—24no—
646C2—24no—
647C2—84no—
648C2C2×#145911yescharacteristic
649C2C22×D4⋊C811yescharacteristic
650C2#1743711yescharacteristic
651C2#1754211yescharacteristic
652C2C2×#151911yescharacteristic
653C2#1765011yescharacteristic
654C2#1770411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1797C2semidirect
3. C2×C4×C22⋊C8C2semidirect
4. #7586C2semidirect
5. #7573C2semidirect
6. #7572C2semidirect
7. #7565C2semidirect
34. #1797C22semidirect
38. C4×C22⋊C8C22semidirect
39. C2×#336C22semidirect
40. C2×#336C22semidirect
41. #1271C22semidirect
42. #1270C22semidirect
54. C2×#121C8semidirect
149. C22×D4C2×C8semidirect
200. #121C2×C8semidirect
201. #121C2×C8semidirect
222. C2×C4⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#17972
C2×C4×C22⋊C82
#75862
#75732
#75722
#75652