Finite Group Databasebeta

← #214168

Subgroups — #214168

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4#9072C214yesmaximal, complemented
5#8944C214yesmaximal, complemented
6#8706C212yesmaximal, complemented
7#8310C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423C2212yescomplemented
13C2×#407—21no—
14C22×C22⋊C8—21no—
15C22×C4×C8C2212yescomplemented
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
35C4×C22⋊C8—21no—
36C4×C22⋊C8—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—44no—
64C2×C4×D4—44no—
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—44no—
94C2×C22⋊C8—28no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—44no—
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—24no—
144C22×D4—21no—
145C22×D4—24no—
146C22×D4—42no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4C22×C412yes—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4—42no—
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×C4D814yes—
177C23×C4SD1614yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C2×M16—44no—
188C2×M16—44no—
189C2×M16—44no—
190C22×C8—28no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—44no—
194C22×C8—44no—
195#121—28no—
196#121—28no—
197#121—24no—
198#121—24no—
199#121—24no—
200#121M1618yescomplemented
201C4×D4—28no—
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—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—28no—
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—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—42no—
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—42no—
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—
276C4×C8—28no—
277C4×C8—44no—
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—41no—
291C24—41no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24#10214yes—
299C24C2×C22⋊C411yescharacteristic
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302C2×D4—24no—
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—24no—
308C2×D4—416no—
309C2×D4—42no—
310C2×D4—432no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4C2×M1614yes—
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—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—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—22no—
350C22×C4—48no—
351C22×C4—48no—
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—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—416no—
369C22×C4—416no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4#10214yes—
388C22×C4#10214yes—
389C22×C4#10214yes—
390C22×C4#10414yes—
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×C22⋊C411yescharacteristic
400C22×C4C2×M1614yes—
401C22×C4#13114yes—
402C22×C4C2×D812yes—
403C22×C4C2×D812yes—
404C22×C4C2×D812yes—
405C22×C4C2×SD1612yes—
406C22×C4C2×SD1612yes—
407C22×C4C2×SD1612yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411M16—88no—
412C2×C8—28no—
413C2×C8—416no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8—44no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4C2×M1614yes—
428C22⋊C4—28no—
429C22⋊C4—28no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—42no—
454C42—42no—
455C42—44no—
456C42H(5,𝔽2)14yes—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—28no—
472C23—28no—
473C23—24no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23#40712yes—
493C23#40812yes—
494C23C2×#10212yes—
495C23C2×#10212yes—
496C23C2×#10212yes—
497C23C2×#10412yes—
498C23#42112yes—
499C23C22×C22⋊C411yescharacteristic
500D4—28no—
501D4—48no—
502D4—416no—
503D4—416no—
504D4—44no—
505C2×C4—28no—
506C2×C4—21no—
507C2×C4—21no—
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—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—28no—
529C2×C4—28no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—88no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4#40712yes—
564C2×C4#40812yes—
565C2×C4C2×#10212yes—
566C2×C4C2×#10212yes—
567C2×C4C2×#10212yes—
568C2×C4C2×#10412yes—
569C2×C4C2×#10412yes—
570C2×C4#51812yes—
571C2×C4#53412yes—
572C2×C4#53412yes—
573C2×C4C22×M1612yes—
574C2×C4C2×#13112yes—
575C2×C4C22×D811yescharacteristic
576C2×C4C22×SD1611yescharacteristic
577C2×C4#57512yes—
578C8—48no—
579C8—84no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—416no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—816no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22#148614yes—
609C22#149214yes—
610C22#230811yescharacteristic
611C22C2×#40711yescharacteristic
612C22C2×#40811yescharacteristic
613C22C22×#10211yescharacteristic
614C22#289611yescharacteristic
615C22C22×#10411yescharacteristic
616C22C2×#42111yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—44no—
627C4—41no—
628C4—82no—
629C4—82no—
630C4#297414yes—
631C4#318612yes—
632C4#318712yes—
633C2—24no—
634C2—24no—
635C2—42no—
636C2—84no—
637C2C2×#148611yescharacteristic
638C2#1744711yescharacteristic
639C2C2×#149211yescharacteristic
640C2#1748011yescharacteristic
641C2#2174011yescharacteristic
642C2#2175811yescharacteristic
643C2#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. C2×C4×C22⋊C8C2semidirect
4. #9072C2semidirect
5. #8944C2semidirect
6. #8706C2semidirect
7. #8310C2semidirect
12. C2×#423C22semidirect
15. C22×C4×C8C22semidirect
34. #1797C22semidirect
38. C4×C22⋊C8C22semidirect
41. C2×D4⋊C8C22semidirect
42. #1279C22semidirect
200. #121M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#206262
C2×C4×C22⋊C82
#90722
#89442
#87062
#83102