Finite Group Databasebeta

← #207669

Subgroups — #207669

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4#9879C212yesmaximal, complemented
5#9649C212yesmaximal, complemented
6#9079C214yesmaximal, complemented
7C4×#323C214yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4C412yes—
10C22×C4×D4C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431—21no—
13C22×C22⋊C8C2212yes—
14C22×C4×C8—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121—21no—
21C4×#121—21no—
22C4×#121—21no—
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388C412yes—
26C2×#386C2212yes—
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30#1817—21no—
31#1797—21no—
32#1797C2212yes—
33#1779C2212yes—
34C4×C22⋊C8—21no—
35C4×C22⋊C8—21no—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39C4×C22⋊C8C2212yes—
40C2×#336—21no—
41C2×#323C2218yes—
42#1280C2218yes—
43C23×D4C2×C412yes—
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—24no—
55C2×#121C818yescomplemented
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—21no—
63C2×C4×D4—21no—
64C2×C4×D4—44no—
65C2×C4×D4—48no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2312yes—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C412yes—
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#431—44no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—28no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8C2314yes—
96C2×C22⋊C8C2314yes—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—44no—
101#390—21no—
102#390—21no—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
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#380—21no—
120#380—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2312yes—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#323—216no—
137C25C22×C411yescharacteristic
138C25C22×C412yes—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—24no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4C2×C818yes—
150C22×D4C2×C814yescomplemented
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—28no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
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×C4C2×C814yes—
173C23×C4M1614yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C22×C8—28no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—44no—
191C22×C8—44no—
192#121—28no—
193#121—28no—
194#121—28no—
195#121—28no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—44no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C2×C814yescomplemented
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
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—22no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4C2×C818yescomplemented
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—44no—
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C4⋊C8—44no—
265C22⋊C8—28no—
266C22⋊C8—24no—
267C22⋊C8—28no—
268C22⋊C8—44no—
269C22⋊C8—44no—
270C22⋊C8#4118yes—
271C4×C8—44no—
272C4×C8—44no—
273C4×C8—44no—
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—22no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—41no—
289C24—41no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C24C22×C812yes—
297C24C22×C814yes—
298C24C2×M1612yes—
299C24C23×C411yescharacteristic
300C2×D4—28no—
301C2×D4—28no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—44no—
309C2×D4—416no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—416no—
314C2×D4—42no—
315C2×D4—42no—
316C22×C4—22no—
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—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—48no—
352C22×C4—48no—
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—41no—
369C22×C4—416no—
370C22×C4—416no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—816no—
384C22×C4C22⋊C814yes—
385C22×C4C22⋊C814yes—
386C22×C4C22⋊C814yes—
387C22×C4C22⋊C814yes—
388C22×C4#10014yes—
389C22×C4C2×C22⋊C411yescharacteristic
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×C4C22×C812yes—
400C22×C4C22×C814yes—
401C22×C4C22×C812yes—
402C22×C4C2×M1612yes—
403C22×C4C2×M1612yes—
404C22×C4#13114yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4114yes—
408C2×C8—28no—
409C2×C8—44no—
410C2×C8—416no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—44no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—44no—
449C42—44no—
450C42#9914yes—
451C23—21no—
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—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—21no—
469C23—24no—
470C23—28no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23C2×C22⋊C812yes—
493C23C2×C22⋊C812yes—
494C23#41012yes—
495C23C2×#10012yes—
496C23C22×C22⋊C411yescharacteristic
497C23C4×#4112yes—
498C23C23×C811yescharacteristic
499C23C22×M1611yescharacteristic
500C23C2×#13112yes—
501D4—48no—
502D4—416no—
503D4—416no—
504D4—48no—
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—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
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—42no—
549C2×C4—42no—
550C2×C4—44no—
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—88no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4C2×C22⋊C812yes—
563C2×C4C2×C22⋊C812yes—
564C2×C4C2×C22⋊C812yes—
565C2×C4C2×C22⋊C812yes—
566C2×C4#40812yes—
567C2×C4#40812yes—
568C2×C4C2×#9912yes—
569C2×C4C2×#9912yes—
570C2×C4C2×#9912yes—
571C2×C4C2×#10012yes—
572C2×C4C2×#10012yes—
573C2×C4#41312yes—
574C2×C4#52512yes—
575C2×C4#53212yes—
576C8—48no—
577C8—84no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—48no—
596C22—48no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—88no—
605C22—88no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22#146014yes—
611C22#229711yescharacteristic
612C22C22×C22⋊C811yescharacteristic
613C22C2×#40811yescharacteristic
614C22C22×#9911yescharacteristic
615C22C2×#41011yescharacteristic
616C22C22×#10011yescharacteristic
617C22C2×#41311yescharacteristic
618C22C8×#4112yes—
619C22#296512yes—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—41no—
629C4—44no—
630C4—41no—
631C4—82no—
632C4—82no—
633C4#145914yes—
634C4#147114yes—
635C2—22no—
636C2—22no—
637C2—24no—
638C2—42no—
639C2—84no—
640C2C2×#145911yescharacteristic
641C2C2×#146011yescharacteristic
642C2#1737611yescharacteristic
643C2C2×#147111yescharacteristic
644C2#2137311yescharacteristic
645C2#2142011yescharacteristic
646C2#2142411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. C2×C4×C22⋊C8C2semidirect
4. #9879C2semidirect
5. #9649C2semidirect
6. #9079C2semidirect
7. C4×#323C2semidirect
55. C2×#121C8semidirect
150. C22×D4C2×C8semidirect
216. C2×C4⋊C4C2×C8semidirect
232. C2×C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#207142
C2×C4×C22⋊C82
#98792
#96492
#90792
C4×#3232