Finite Group Databasebeta

← #208326

Subgroups — #208326

Order 512.

1948 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
1#36624C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×C4×#95C211yesmaximal, characteristic, complemented
4#9909C212yesmaximal, complemented
5#9712C212yesmaximal, complemented
6#9147C214yesmaximal, complemented
7C4×#327C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525—21no—
10C23×C42—21no—
11C23×C42C2211yescharacteristic
12C2×#431—21no—
13#2857C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2856C411yescharacteristic, complemented
16#2856C2211yescharacteristic
17#2854—21no—
18#2854C412yescomplemented
19#2850—21no—
20#2850—21no—
21#2847—21no—
22#2847C2212yes—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121C412yescomplemented
26C4×#121C2212yes—
27#2297—21no—
28#2285C411yescharacteristic, complemented
29C22×#95—21no—
30C22×#95C2212yes—
31#1817—21no—
32#1797—21no—
33#1797C2212yes—
34#1779C2212yes—
35C4×C22⋊C8—21no—
36C4×C22⋊C8C2212yes—
37C2×#327C2218yes—
38C4×#95—21no—
39C4×#95—21no—
40C4×#95C2212yes—
41C4×#95C2212yes—
42#1294C2218yes—
43C22×#41—21no—
44C24×C4—22no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48#532—44no—
49C2×C4⋊Q8C2×C411yescharacteristic, complemented
50C2×#127C2×C411yescharacteristic, complemented
51C2×#124—21no—
52#525—44no—
53#525—44no—
54C2×#123—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×C412yes—
63C2×#121C2×C412yes—
64C2×#121C2312yes—
65C2×#120—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C412yescomplemented
70C2×#117—21no—
71C22×C42—28no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—44no—
75C22×C42—44no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
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
89#431—44no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2314yes—
94C2×C4×C8—22no—
95#391—21no—
96#391—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390C2×C412yescomplemented
102#390C2×C412yescomplemented
103#389—21no—
104#389—21no—
105#389C2×C412yescomplemented
106#389C2×C412yescomplemented
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#384C2×C411yescharacteristic, complemented
118#384C2×C411yescharacteristic, complemented
119#382—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380C2×C412yes—
124#380C2×C412yes—
125#380C2312yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C412yescomplemented
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—28no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—28no—
135C2×#95C2314yes—
136#336—22no—
137#336—22no—
138#327—216no—
139C25C2×D411yescharacteristic
140C2×#41—44no—
141C2×#41—44no—
142C2×#41—44no—
143C22×Q8—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22×C412yes—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—28no—
154C23×C4—28no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
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×C4D814yes—
176C23×C4SD1614yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C22×C8—24no—
188C4⋊Q8—42no—
189#127—42no—
190#124—44no—
191#123—48no—
192C22⋊Q8—44no—
193C22⋊Q8—42no—
194C22⋊Q8—42no—
195#121—28no—
196#121—44no—
197#121—44no—
198#121—44no—
199#121—44no—
200#121—44no—
201#121—44no—
202#121—42no—
203#121—42no—
204#120—44no—
205#120—44no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—48no—
209C4×D4—44no—
210C4×D4—44no—
211#117—44no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
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—22no—
232C2×C22⋊C4—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—28no—
236C2×C42—28no—
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—42no—
248C2×C42—432no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42C22⋊C411yescharacteristic
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×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C4⋊C8—44no—
270C22⋊C8—28no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273C4×C8—44no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—24no—
284#95#4118yescomplemented
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24C2×C22⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294C24C2×D812yes—
295C24C2×SD1612yes—
296C24C22×D411yescharacteristic
297#41—816no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×D4—24no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—216no—
323C22×C4—216no—
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—28no—
333C22×C4—28no—
334C22×C4—232no—
335C22×C4—22no—
336C22×C4—28no—
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—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—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—44no—
374C22×C4—416no—
375C22×C4—416no—
376C22×C4—416no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—416no—
380C22×C4—44no—
381C22×C4—416no—
382C22×C4—416no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4#9914yes—
389C22×C4#10214yes—
390C22×C4#10214yes—
391C22×C4#10214yes—
392C22×C4#10214yes—
393C22×C4#10414yes—
394C22×C4#10414yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
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×D812yes—
406C22×C4C2×D812yes—
407C22×C4C2×SD1612yes—
408C22×C4C2×SD1612yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4114yes—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—44no—
415C4⋊C4—24no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—416no—
454C42—416no—
455C42—44no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—21no—
477C23—21no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23C2×#9912yes—
486C23#41012yes—
487C23C2×#10212yes—
488C23C2×#10212yes—
489C23C2×#10412yes—
490C23C2×#10412yes—
491C23C22×C22⋊C411yescharacteristic
492C23#52512yes—
493C23C22×D811yescharacteristic
494C23C22×SD1611yescharacteristic
495C23#57512yes—
496Q8—84no—
497D4—84no—
498D4—84no—
499D4—84no—
500D4—84no—
501D4—88no—
502D4—88no—
503D4—84no—
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—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—216no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—44no—
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—48no—
547C2×C4—464no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—48no—
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—88no—
565C2×C4#327116yescomplemented
566C2×C4C2×#9912yes—
567C2×C4C2×#9912yes—
568C2×C4C2×#10212yes—
569C2×C4C2×#10212yes—
570C2×C4C2×#10212yes—
571C2×C4C2×#10212yes—
572C2×C4C2×#10412yes—
573C2×C4C2×#10412yes—
574C2×C4C2×#10412yes—
575C2×C4C2×#10412yes—
576C2×C4C4×#4112yes—
577C2×C4#53212yes—
578C8—84no—
579C22—216no—
580C22—216no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—21no—
585C22—21no—
586C22—22no—
587C22—22no—
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—22no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22C2×#32714yes—
609C22#150214yes—
610C22#229711yescharacteristic
611C22C22×#9911yescharacteristic
612C22C2×#41011yescharacteristic
613C22C22×#10211yescharacteristic
614C22#289611yescharacteristic
615C22C22×#10411yescharacteristic
616C22C22×#10411yescharacteristic
617C22#310012yes—
618C22#310312yes—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—416no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4C2×#32714yes—
634C4C2×#32714yes—
635C2—22no—
636C2—22no—
637C2—24no—
638C2—24no—
639C2—162no—
640C2C22×#32711yescharacteristic
641C2C2×#150211yescharacteristic
642C2#1754111yescharacteristic
643C2#1754211yescharacteristic
644C2#2138611yescharacteristic
645C2#2141511yescharacteristic
646C2#2142011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ 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×#95C2semidirect
4. #9909C2semidirect
5. #9712C2semidirect
6. #9147C2semidirect
7. C4×#327C2semidirect
8. C2×#532C4semidirect
13. #2857C4semidirect
15. #2856C4semidirect
18. #2854C4semidirect
25. C4×#121C4semidirect
28. #2285C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
50. C2×#127C2×C4semidirect
69. C2×C4×D4C2×C4semidirect
101. #390C2×C4semidirect
102. #390C2×C4semidirect
105. #389C2×C4semidirect
106. #389C2×C4semidirect
117. #384C2×C4semidirect
118. #384C2×C4semidirect
127. C4×C4⋊C4C2×C4semidirect
284. #95#41semidirect
565. C2×C4#327semidirect

Maximal subgroups

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

subgroupindex
#366242
#207142
C2×C4×#952
#99092
#97122
#91472
C4×#3272