Finite Group Databasebeta

← #214210

Subgroups — #214210

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#9078C214yesmaximal, complemented
5#8944C214yesmaximal, complemented
6#8707C212yesmaximal, complemented
7#8310C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423C2212yescomplemented
13C2×#407—21no—
14C2×#407—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—
40#1728—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—
50C22×M16—22no—
51C2×#121—24no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—24no—
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—48no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2312yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—21no—
77C22×C42—44no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#423—44no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2314yes—
93#407—28no—
94#407—44no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C4×M16—44no—
100C2×#97—22no—
101C2×C4×C8C2314yes—
102#390—21no—
103#390—21no—
104#390C2×C412yes—
105#390C2×C412yes—
106#389—21no—
107#389—21no—
108#389C2×C412yes—
109#389C2×C412yes—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#380—21no—
119#380—21no—
120#380C2×C412yes—
121#380C2×C412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2312yes—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137D4⋊C8—48no—
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×D4M1614yescomplemented
150C22×D4C22×C412yes—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—42no—
170C23×C4—42no—
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×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C2×M16—28no—
190C2×M16—44no—
191C2×M16—44no—
192C2×M16—44no—
193C2×M16—44no—
194C2×M16—44no—
195C22×C8—24no—
196C22×C8—24no—
197#121—28no—
198#121—28no—
199#121—24no—
200#121—24no—
201#121—24no—
202#121—24no—
203C4×D4—28no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—48no—
208C4×D4—48no—
209C4×D4—48no—
210C4×D4—48no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4M1614yescomplemented
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C22⋊C4—28no—
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—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—44no—
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2411yescharacteristic
270C4⋊C8—28no—
271C4⋊C8—44no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274C22⋊C8—44no—
275C22⋊C8—44no—
276#97—44no—
277C4×C8—28no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—22no—
290C24—22no—
291C24—41no—
292C24—41no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24C2×C22⋊C411yescharacteristic
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—48no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—48no—
312C2×D4—44no—
313C2×D4—432no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4C2×M1614yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
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—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
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—22no—
358C22×C4—22no—
359C22×C4—22no—
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—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—416no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4#10414yes—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×M1612yes—
409C22×C4C2×M1612yes—
410C22×C4AGL(1,ℤ/8)12yes—
411C22×C4AGL(1,ℤ/8)14yes—
412C22×C4#13712yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416M16—416no—
417M16—88no—
418C2×C8—28no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—44no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4C2×M1614yes—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—44no—
459C42—44no—
460C42H(5,𝔽2)14yes—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—28no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—24no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—48no—
485C23—48no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—48no—
494C23—84no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23#40712yes—
499C23#41712yes—
500C23#41812yes—
501C23C2×#10412yes—
502C23#42112yes—
503C23C22×C22⋊C411yescharacteristic
504D4—28no—
505D4—48no—
506D4—416no—
507D4—416no—
508D4—44no—
509C2×C4—28no—
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—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—88no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4#40712yes—
573C2×C4#41712yes—
574C2×C4#41812yes—
575C2×C4C2×#10412yes—
576C2×C4C2×#10412yes—
577C2×C4#51812yes—
578C2×C4#53412yes—
579C2×C4#53412yes—
580C2×C4C22×M1611yescharacteristic
581C2×C4C22×M1611yescharacteristic
582C2×C4#56812yes—
583C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
584C2×C4C2×AGL(1,ℤ/8)12yes—
585C2×C4C2×#13711yescharacteristic
586C8—48no—
587C8—84no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—416no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—816no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22#148914yes—
618C22#230811yescharacteristic
619C22C2×#40711yescharacteristic
620C22#287911yescharacteristic
621C22C2×#41811yescharacteristic
622C22C2×#41711yescharacteristic
623C22C22×#10411yescharacteristic
624C22C2×#42111yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—44no—
634C4—41no—
635C4—41no—
636C4—82no—
637C4—82no—
638C4#297312yes—
639C4#297512yes—
640C4#319212yes—
641C4#319312yes—
642C2—24no—
643C2—42no—
644C2—42no—
645C2—84no—
646C2C2×#148911yescharacteristic
647C2#1744811yescharacteristic
648C2#1749211yescharacteristic
649C2#1750011yescharacteristic
650C2#2174211yescharacteristic
651C2#2176411yescharacteristic
652C2#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. #20626C2semidirect
3. C4×#407C2semidirect
4. #9078C2semidirect
5. #8944C2semidirect
6. #8707C2semidirect
7. #8310C2semidirect
12. C2×#423C22semidirect
15. C2×C4×M16C22semidirect
34. #1797C22semidirect
38. C4×C22⋊C8C22semidirect
41. C2×D4⋊C8C22semidirect
42. #1279C22semidirect
149. C22×D4M16semidirect
220. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#206262
C4×#4072
#90782
#89442
#87072
#83102