Finite Group Databasebeta

← #196385

Subgroups — #196385

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C2×#2854C211yesmaximal, characteristic
3C2×#1507C212yesmaximal
4C4×#381C211yesmaximal, characteristic, complemented
5#9118C214yesmaximal, complemented
6#9114C214yesmaximal, complemented
7#8019C212yesmaximal, complemented
8C22×#123—21no—
9C22×#121C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432C2212yescomplemented
14C2×#431—21no—
15C2×#431C2212yescomplemented
16C2×#423—21no—
17C2×#407C2212yes—
18C2×C4×M16—21no—
19#2854—21no—
20#2854—21no—
21#2854—21no—
22#2854—21no—
23#2854C412yescomplemented
24#2854C412yescomplemented
25#2854C2212yes—
26#2854C2212yes—
27C2×#390C411yescharacteristic, complemented
28C2×#390C411yescharacteristic, complemented
29C2×#389C411yescharacteristic, complemented
30C2×#389C411yescharacteristic, complemented
31C2×#388—21no—
32C2×#388—21no—
33C2×#386—21no—
34C2×#381—21no—
35C2×#380C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37#1507—28no—
38C2×#327C2218yes—
39#1434C2212yescomplemented
40#1434C2212yescomplemented
41C22⋊#95—21no—
42C4×#95—21no—
43C4×#95—21no—
44#1294C2218yescomplemented
45C23×D4—21no—
46C23×D4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C2×#123—22no—
52C2×#123—22no—
53C2×#123—22no—
54C2×#123—22no—
55C2×#123—48no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—24no—
59C2×#121—24no—
60C2×#121C2×C412yes—
61C2×#121C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—24no—
75C22×C42—24no—
76C22×C42—22no—
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#432—28no—
90#432—28no—
91#431—28no—
92#431—28no—
93#423—28no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96#407—24no—
97#407D418yes—
98C2×C22⋊C8—22no—
99C2×C22⋊C8C2314yes—
100C4×M16—28no—
101C2×#97—22no—
102C2×C4×C8—22no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390C2×C412yes—
108#390C2×C412yes—
109#390C2×C412yes—
110#390C2×C412yes—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389C2×C412yes—
116#389C2×C412yes—
117#389C2×C412yes—
118#389C2×C412yes—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#381—28no—
132#380—21no—
133#380—21no—
134#380C2×C412yes—
135#380C2312yes—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—24no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—21no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95C2314yes—
146#327—216no—
147C25—21no—
148C25C2×D411yescharacteristic
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4C22×C412yes—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
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×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D414yes—
189C23×C4#4114yes—
190C23×C4C2411yescharacteristic
191C2×M16—28no—
192C2×M16—44no—
193C2×M16—44no—
194C22×C8—24no—
195#123—48no—
196#123—48no—
197#123—48no—
198#123—48no—
199#121—28no—
200#121—24no—
201#121—24no—
202#121—42no—
203#121—44no—
204#121—42no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
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—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C4⋊C8—28no—
280C4⋊C8—28no—
281C22⋊C8—28no—
282C22⋊C8—24no—
283C22⋊C8C2×D418yes—
284#97—28no—
285C4×C8—28no—
286#95—28no—
287#95—28no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—44no—
297#95—44no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24#12114yes—
315C24C22×D411yescharacteristic
316C2×D4—24no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—88no—
329C2×D4—816no—
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—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—22no—
361C22×C4—22no—
362C22×C4—28no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—48no—
367C22×C4—48no—
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—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—416no—
383C22×C4—416no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4#10414yes—
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×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C4×D414yes—
411C22×C4C4×D414yes—
412C22×C4#12012yes—
413C22×C4#12012yes—
414C22×C4#12114yes—
415C22×C4#12312yes—
416C22×C4#12312yes—
417C22×C4AGL(1,ℤ/8)14yes—
418C22×C4C23×C411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C2×#4112yes—
422M16—416no—
423C2×C8—28no—
424C2×C8—44no—
425C2×C8—44no—
426C4⋊C4—24no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—416no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—88no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—28no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—28no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23#38612yes—
507C23#38612yes—
508C23C2×#10412yes—
509C23#42112yes—
510C23C22×C22⋊C411yescharacteristic
511C23C2×#12112yes—
512C23C2×AGL(1,ℤ/8)12yes—
513C23#57512yes—
514D4—84no—
515D4—88no—
516D4—88no—
517D4—84no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—28no—
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—28no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4#38612yes—
585C2×C4#38612yes—
586C2×C4#41012yes—
587C2×C4#41012yes—
588C2×C4#41812yes—
589C2×C4#41812yes—
590C2×C4C2×#10412yes—
591C2×C4C2×#10412yes—
592C2×C4C2×C4×D412yes—
593C2×C4C2×#12011yescharacteristic
594C2×C4C2×#12311yescharacteristic
595C8—48no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—416no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—168no—
624C22#187114yes—
625C22C2×#38611yescharacteristic
626C22C2×#41011yescharacteristic
627C22C2×#41011yescharacteristic
628C22C2×#41811yescharacteristic
629C22#289611yescharacteristic
630C22C22×#10411yescharacteristic
631C22C2×#42111yescharacteristic
632C22#305514yes—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—21no—
638C4—22no—
639C4—21no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—88no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4#189914yes—
650C2—24no—
651C2—24no—
652C2—42no—
653C2—162no—
654C2#1753711yescharacteristic
655C2#1754411yescharacteristic
656C2#1757911yescharacteristic
657C2#1758311yescharacteristic
658C2C2×#187111yescharacteristic
659C2#2097311yescharacteristic
660C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
4. C4×#381C2semidirect
5. #9118C2semidirect
6. #9114C2semidirect
7. #8019C2semidirect
13. C2×#432C22semidirect
15. C2×#431C22semidirect
23. #2854C4semidirect
24. #2854C4semidirect
27. C2×#390C4semidirect
28. C2×#390C4semidirect
29. C2×#389C4semidirect
30. C2×#389C4semidirect
39. #1434C22semidirect
40. #1434C22semidirect
44. #1294C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×#28542
C2×#15072
C4×#3812
#91182
#91142
#80192