Finite Group Databasebeta

← #214150

Subgroups — #214150

Order 512.

1732 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#9078C214yesmaximal, complemented
5#8939C214yesmaximal, complemented
6#8707C212yesmaximal, complemented
7#8310C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×C4⋊C8C2212yescomplemented
13C2×#407—21no—
14C22×C22⋊C8—21no—
15C2×C4×M16C2212yescomplemented
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121C412yes—
21C4×#121C412yes—
22C4×#121C2212yes—
23C4×#121C2212yes—
24C2×#390C411yescharacteristic
25C2×#389C411yescharacteristic
26C2×#388—21no—
27C2×#386—21no—
28C2×#380C411yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yescomplemented
35#1749—21no—
36#1749—21no—
37C4×C22⋊C8C2212yes—
38C4×C22⋊C8C2212yescomplemented
39#1728—21no—
40C2×#336—21no—
41C2×D4⋊C8C2218yescomplemented
42#1279C2218yescomplemented
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×D4C2312yes—
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
89C2×C4⋊C8—22no—
90C2×C4⋊C8—44no—
91C2×C4⋊C8C2314yes—
92#407—28no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—44no—
98C4×M16—44no—
99C2×#97—22no—
100C2×C4×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#380—21no—
118#380—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2312yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136D4⋊C8—48no—
137C25—21no—
138C25C22⋊C411yescharacteristic
139C22×D4—28no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—21no—
144C22×D4—24no—
145C22×D4—42no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4C22×C412yes—
149C23×C4—24no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4—42no—
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×C4C22⋊C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C2×M16—28no—
187C2×M16—44no—
188C2×M16—44no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—44no—
192C22×C8—44no—
193C22×C8—44no—
194#121—28no—
195#121—28no—
196#121—24no—
197#121—24no—
198#121—24no—
199#121M1618yescomplemented
200C4×D4—28no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—48no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—48no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C22⋊C4—28no—
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—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—44no—
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2411yescharacteristic
266C4⋊C8—28no—
267C4⋊C8—44no—
268C22⋊C8—28no—
269C22⋊C8—28no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272#97—44no—
273C4×C8—28no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—22no—
286C24—41no—
287C24—41no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24#10414yes—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298C2×D4—24no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—24no—
304C2×D4—416no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—432no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4C2×M1614yes—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—24no—
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—21no—
343C22×C4—22no—
344C22×C4—21no—
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—22no—
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—48no—
370C22×C4—48no—
371C22×C4—416no—
372C22×C4—416no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
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×M1614yes—
400C22×C4#13114yes—
401C22×C4AGL(1,ℤ/8)12yes—
402C22×C4AGL(1,ℤ/8)14yes—
403C22×C4#13712yes—
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407M16—416no—
408C2×C8—28no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—88no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4C2×M1614yes—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—44no—
450C42—44no—
451C42H(5,𝔽2)14yes—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—28no—
467C23—28no—
468C23—24no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—48no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23#40712yes—
488C23#40812yes—
489C23#41712yes—
490C23#41812yes—
491C23C2×#10412yes—
492C23C2×#10412yes—
493C23C2×#10412yes—
494C23C22×C22⋊C411yescharacteristic
495D4—28no—
496D4—48no—
497D4—416no—
498D4—416no—
499D4—44no—
500C2×C4—28no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—88no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4#40712yes—
561C2×C4#40812yes—
562C2×C4#41712yes—
563C2×C4#41812yes—
564C2×C4#42112yes—
565C2×C4#51812yes—
566C2×C4#53412yes—
567C2×C4#53412yes—
568C2×C4C22×M1612yes—
569C2×C4C2×#13112yes—
570C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
571C2×C4C2×AGL(1,ℤ/8)12yes—
572C2×C4C2×#13711yescharacteristic
573C8—48no—
574C8—84no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—416no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—816no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22#148314yes—
604C22#149014yes—
605C22#230811yescharacteristic
606C22C2×#40711yescharacteristic
607C22C2×#40811yescharacteristic
608C22C2×#41811yescharacteristic
609C22C2×#41711yescharacteristic
610C22C22×#10411yescharacteristic
611C22C2×#42111yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—44no—
622C4—41no—
623C4—82no—
624C4—82no—
625C4#297414yes—
626C4#319212yes—
627C4#319312yes—
628C2—24no—
629C2—24no—
630C2—42no—
631C2—84no—
632C2C2×#149011yescharacteristic
633C2#1745111yescharacteristic
634C2C2×#148311yescharacteristic
635C2#1746311yescharacteristic
636C2#2174011yescharacteristic
637C2#2176411yescharacteristic
638C2#2177311yescharacteristic
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
2. C2×#1797C2semidirect
3. C4×#407C2semidirect
4. #9078C2semidirect
5. #8939C2semidirect
6. #8707C2semidirect
7. #8310C2semidirect
12. C22×C4⋊C8C22semidirect
15. C2×C4×M16C22semidirect
34. #1797C22semidirect
38. C4×C22⋊C8C22semidirect
41. C2×D4⋊C8C22semidirect
42. #1279C22semidirect
199. #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×#17972
C4×#4072
#90782
#89392
#87072
#83102