Finite Group Databasebeta

← #160720

Subgroups — #160720

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#7620C214yesmaximal, complemented
5#7607C212yesmaximal, complemented
6#7606C212yesmaximal, complemented
7#7565C214yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#407—21no—
13C22×C22⋊C8—21no—
14C2×C4×M16—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121C412yes—
20C4×#121C412yes—
21C4×#121C2212yes—
22C4×#121C2212yes—
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388—21no—
26C2×#386—21no—
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic
29C2×C4×C22⋊C4—21no—
30#1817—21no—
31#1817—21no—
32#1817C2212yes—
33#1817C2212yescomplemented
34#1749—21no—
35#1749C2212yescomplemented
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yescomplemented
38#1728C2212yes—
39C2×#336—21no—
40C2×#336C2212yescomplemented
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—22no—
50C23×C8—22no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—28no—
54C2×#121C2×C412yes—
55C2×#121C2×C412yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—44no—
63C2×C4×D4—48no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C412yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42—21no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#407—28no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—44no—
95C4×M16—44no—
96C2×#97—22no—
97C2×C4×C8—22no—
98#390—21no—
99#390—21no—
100#390C2×C412yes—
101#390C2312yes—
102#389—21no—
103#389—21no—
104#389C2×C412yes—
105#389C2312yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#380—21no—
115#380—21no—
116#380C2×C412yes—
117#380C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336C2314yes—
134#336C2314yes—
135C25—21no—
136C25C22⋊C411yescharacteristic
137C22×D4—28no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—21no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4C22×C412yes—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
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—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
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×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C2×M16—28no—
183C2×M16—44no—
184C2×M16—44no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—44no—
188C22×C8—44no—
189C22×C8—44no—
190#121—28no—
191#121—28no—
192#121—24no—
193#121—24no—
194#121—24no—
195#121M1618yescomplemented
196C4×D4—44no—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—48no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2411yescharacteristic
262C22⋊C8—28no—
263C22⋊C8—28no—
264C22⋊C8—44no—
265C22⋊C8—44no—
266#97—44no—
267C4×C8—44no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—24no—
279C24—22no—
280C24—41no—
281C24—41no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24#10414yes—
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C2×D4—24no—
293C2×D4—28no—
294C2×D4—28no—
295C2×D4—22no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—416no—
300C2×D4—44no—
301C2×D4—432no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4C2×M1614yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
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—22no—
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—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
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—416no—
362C22×C4—416no—
363C22×C4—416no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—816no—
379C22×C4#9914yes—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×M1614yes—
390C22×C4#13514yes—
391C22×C4AGL(1,ℤ/8)14yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395M16—416no—
396C2×C8—28no—
397C2×C8—44no—
398C2×C8—44no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—88no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4C2×M1614yes—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—44no—
439C42—44no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—28no—
455C23—28no—
456C23—24no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—48no—
463C23—48no—
464C23—48no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—84no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23#40712yes—
476C23C2×#9912yes—
477C23#41612yes—
478C23#41812yes—
479C23C2×#10412yes—
480C23C2×#10412yes—
481C23C2×#10412yes—
482C23C22×C22⋊C411yescharacteristic
483D4—44no—
484D4—44no—
485D4—48no—
486D4—416no—
487D4—416no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—88no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—88no—
546C2×C4#40712yes—
547C2×C4C2×#9912yes—
548C2×C4C2×#9912yes—
549C2×C4#41312yes—
550C2×C4#41612yes—
551C2×C4#41812yes—
552C2×C4#42112yes—
553C2×C4C22×M1612yes—
554C2×C4C2×#13512yes—
555C2×C4C2×AGL(1,ℤ/8)12yes—
556C8—48no—
557C8—84no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—416no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—816no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22#149014yes—
587C22#150114yes—
588C22C2×#40711yescharacteristic
589C22C22×#9911yescharacteristic
590C22C2×#41311yescharacteristic
591C22C2×#41611yescharacteristic
592C22C2×#41811yescharacteristic
593C22C22×#10411yescharacteristic
594C22C2×#42111yescharacteristic
595C4—22no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—44no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—82no—
608C4—82no—
609C4#151014yes—
610C2—24no—
611C2—24no—
612C2—42no—
613C2—84no—
614C2#1736211yescharacteristic
615C2#1744311yescharacteristic
616C2C2×#149011yescharacteristic
617C2C2×#150111yescharacteristic
618C2C2×#151011yescharacteristic
619C2#1765011yescharacteristic
620C2#1771711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C22×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×#1817C2semidirect
3. C4×#407C2semidirect
4. #7620C2semidirect
5. #7607C2semidirect
6. #7606C2semidirect
7. #7565C2semidirect
33. #1817C22semidirect
35. #1749C22semidirect
37. C4×C22⋊C8C22semidirect
40. C2×#336C22semidirect
41. #1271C22semidirect
42. #1270C22semidirect
195. #121M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#18172
C4×#4072
#76202
#76072
#76062
#75652