Finite Group Databasebeta

← #195875

Subgroups — #195875

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1490C212yesmaximal, complemented
5#9078C214yesmaximal, complemented
6#9072C214yesmaximal, complemented
7#7904C212yesmaximal, complemented
8C23×M16—21no—
9C22×#121C411yescharacteristic
10C22×C4×D4—21no—
11C22×C4×D4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#432C2212yes—
14C2×C4×M16—21no—
15C2×C4×M16—21no—
16C2×C4×M16—28no—
17C2×C4×M16C2212yes—
18C2×C4×M16C2212yescomplemented
19C22×#97—21no—
20C22×C4×C8C2212yescomplemented
21C4×#121—21no—
22C4×#121—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121C412yes—
26C4×#121C412yes—
27C4×#121C2212yes—
28C4×#121C2212yes—
29C2×#390C411yescharacteristic
30C2×#389C411yescharacteristic
31C2×#388—21no—
32C2×#386—21no—
33C2×#380C411yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36#1817—21no—
37#1797C2212yescomplemented
38#1797C2212yescomplemented
39#1779—21no—
40#1749—21no—
41#1749—21no—
42#1490—28no—
43C2×D4⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C23×D4—21no—
46C23×D4C2×C411yescharacteristic
47C24×C4—21no—
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C22×M16—28no—
52C22×M16—28no—
53C22×M16—28no—
54C22×M16—22no—
55C22×M16—22no—
56C23×C8—22no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—28no—
60C2×#121C2×C412yes—
61C2×#121C2×C412yes—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4—48no—
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2312yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—24no—
83C22×C42—24no—
84C22×C42—21no—
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#432—24no—
98#432—24no—
99C2×C4⋊C8C2314yes—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C4×M16—28no—
104C4×M16—28no—
105C4×M16—24no—
106C4×M16—28no—
107C4×M16D418yescomplemented
108C2×#97—28no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112C2×C4×C8—22no—
113C2×C4×C8—28no—
114C2×C4×C8C2314yes—
115#390—21no—
116#390—21no—
117#390C2×C412yes—
118#390C2×C412yes—
119#389—21no—
120#389—21no—
121#389C2×C412yes—
122#389C2×C412yes—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#380—21no—
132#380—21no—
133#380C2×C412yes—
134#380C2×C412yes—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×C412yes—
138C4×C4⋊C4C2312yes—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146#336—22no—
147#336—22no—
148D4⋊C8—216no—
149C25—21no—
150C25C22⋊C411yescharacteristic
151C22×D4—28no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—24no—
155C22×D4—21no—
156C22×D4—24no—
157C22×D4—42no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4C22×C412yes—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C2×M16—264no—
198C2×M16—28no—
199C2×M16—28no—
200C2×M16—28no—
201C2×M16—28no—
202C2×M16—28no—
203C2×M16—28no—
204C22×C8—28no—
205C22×C8—28no—
206C22×C8—28no—
207C22×C8—24no—
208C22×C8—24no—
209#121—28no—
210#121—28no—
211#121—24no—
212#121—24no—
213#121—24no—
214#121M1618yescomplemented
215C4×D4—28no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—48no—
219C4×D4—48no—
220C4×D4—48no—
221C4×D4—48no—
222C4×D4—42no—
223C4×D4—44no—
224C4×D4—42no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C22⋊C4—28no—
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—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
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×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D414yes—
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42#4114yes—
289C2×C42C2411yescharacteristic
290C4⋊C8—24no—
291C4⋊C8—24no—
292C22⋊C8—44no—
293C22⋊C8—44no—
294C22⋊C8—44no—
295#97—28no—
296#97—28no—
297#97—44no—
298C4×C8—28no—
299C4×C8—24no—
300C4×C8C2×D418yescomplemented
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—22no—
313C24—41no—
314C24—41no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24#10414yes—
322C24C2×C22⋊C411yescharacteristic
323C24C2×C22⋊C411yescharacteristic
324C24C2×C22⋊C411yescharacteristic
325C2×D4—24no—
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—416no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—432no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4C2×M1614yescomplemented
340C22×C4—216no—
341C22×C4—216no—
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—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
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—22no—
367C22×C4—22no—
368C22×C4—24no—
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—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
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—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—816no—
413C22×C4#10414yes—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
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×C4C4×D414yes—
424C22×C4#12012yes—
425C22×C4#12114yes—
426C22×C4#12312yes—
427C22×C4C2×M1614yes—
428C22×C4AGL(1,ℤ/8)14yes—
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432M16—232no—
433M16—232no—
434C2×C8—232no—
435C2×C8—28no—
436C2×C8—28no—
437C2×C8—28no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8—44no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4C2×M1614yescomplemented
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C42—22no—
462C42—22no—
463C42—21no—
464C42—21no—
465C42—22no—
466C42—22no—
467C42—21no—
468C42—21no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—44no—
476C42—44no—
477C42C4×D414yes—
478C42#12012yes—
479C42#12114yes—
480C42#12312yes—
481C42C22×D412yes—
482C42C2×#4112yes—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—28no—
497C23—28no—
498C23—21no—
499C23—24no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23—84no—
518C23#38612yes—
519C23#40712yes—
520C23#41812yes—
521C23C2×#10412yes—
522C23C2×#10412yes—
523C23C2×#10412yes—
524C23C2×#10412yes—
525C23C22×C22⋊C411yescharacteristic
526D4—28no—
527D4—48no—
528D4—416no—
529D4—416no—
530D4—44no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—88no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—88no—
588C2×C4#38612yes—
589C2×C4#38612yes—
590C2×C4#38612yes—
591C2×C4#40712yes—
592C2×C4#41812yes—
593C2×C4C2×#10412yes—
594C2×C4C2×#10412yes—
595C2×C4C2×C4×D412yes—
596C2×C4C2×#12011yescharacteristic
597C2×C4C2×#12112yes—
598C2×C4C2×#12311yescharacteristic
599C2×C4C22×M1612yes—
600C2×C4#56812yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4#57512yes—
603C8—216no—
604C8—84no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—416no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—816no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22#149014yescomplemented
634C22#189614yes—
635C22C2×#38611yescharacteristic
636C22C2×#40711yescharacteristic
637C22#287911yescharacteristic
638C22C2×#41811yescharacteristic
639C22#289611yescharacteristic
640C22C22×#10411yescharacteristic
641C22C22×#10411yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—21no—
647C4—21no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—44no—
652C4—41no—
653C4—41no—
654C4—82no—
655C4—82no—
656C4D4×M1614yes—
657C4#305514yes—
658C2—24no—
659C2—24no—
660C2—42no—
661C2—84no—
662C2C2×#149011yescharacteristic
663C2#1744711yescharacteristic
664C2#1750011yescharacteristic
665C2#1750211yescharacteristic
666C2#2089811yescharacteristic
667C2#2097311yescharacteristic
668C2C2×#189611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×M16C2semidirect
3. #20715C2semidirect
4. C2×#1490C2semidirect
5. #9078C2semidirect
6. #9072C2semidirect
7. #7904C2semidirect
18. C2×C4×M16C22semidirect
20. C22×C4×C8C22semidirect
37. #1797C22semidirect
38. #1797C22semidirect
43. C2×D4⋊C8C22semidirect
44. #1279C22semidirect
107. C4×M16D4semidirect
214. #121M16semidirect
300. C4×C8C2×D4semidirect
339. C2×D4C2×M16semidirect
449. C4⋊C4C2×M16semidirect
633. C22#1490semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
C2×C4×#1212
#207152
C2×#14902
#90782
#90722
#79042