Finite Group Databasebeta

← #204374

Subgroups — #204374

Order 512.

1083 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3102C211yesmaximal, characteristic
2#21380C211yesmaximal, characteristic, complemented
3#21378C211yesmaximal, characteristic, complemented
4C2×#1572C211yesmaximal, characteristic, complemented
5C2×#1494C211yesmaximal, characteristic, complemented
6C2×#1476C211yesmaximal, characteristic, complemented
7C2×C8×M16C211yesmaximal, characteristic, complemented
8#9663C211yesmaximal, characteristic, complemented
9#9662C211yesmaximal, characteristic, complemented
10#9661C211yesmaximal, characteristic, complemented
11#9661C211yesmaximal, characteristic, complemented
12#9657C211yesmaximal, characteristic, complemented
13#9656C211yesmaximal, characteristic, complemented
14#9655C211yesmaximal, characteristic, complemented
15#9655C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18#3102—24no—
19#3102—24no—
20C2×#485C411yescharacteristic
21C2×#482C411yescharacteristic
22C2×#479C411yescharacteristic
23C2×#478C411yescharacteristic
24C2×#464C411yescharacteristic
25C2×#463C411yescharacteristic
26C2×#452C411yescharacteristic
27C2×#449C411yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29C2×C4×SD16C2211yescharacteristic
30C2×C4×SD16C2211yescharacteristic
31C2×C4×D8C2211yescharacteristic
32C2×#432C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#1984C2211yescharacteristic
38#1984C2211yescharacteristic
39#1984C2211yescharacteristic
40#1984C2211yescharacteristic
41#1954C2211yescharacteristic
42#1954C2211yescharacteristic
43#1954C2211yescharacteristic
44#1954C2211yescharacteristic
45#1870—22no—
46#1870—22no—
47#1750—22no—
48#1749—22no—
49#1572—24no—
50C2×#335C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1494—24no—
55#1476—24no—
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C8×M16—24no—
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67C2×C82C2211yescharacteristic
68C2×C82C2211yescharacteristic
69C22×#41C2×C411yescharacteristic
70C22×Q16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×D8C2×C411yescharacteristic
74C22×M16D412yes—
75C23×C8—21no—
76#529—22no—
77#529—22no—
78#529—22no—
79#529—22no—
80C2×#126—21no—
81C2×#125C2×C411yescharacteristic
82C2×#124C2×C411yescharacteristic
83C2×#123—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C4×#41—22no—
87C4×#41—22no—
88C4×#41—22no—
89C4×#41—22no—
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2311yescharacteristic
92C2×C4×Q8C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2×C411yescharacteristic
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2311yescharacteristic
99C2×#117—21no—
100C2×#117—21no—
101C2×#117C2×C411yescharacteristic
102C22×C42—21no—
103C22×C42C2311yescharacteristic
104#485—24no—
105#482—24no—
106#479—24no—
107#478—24no—
108#464—24no—
109#463—24no—
110#452—24no—
111#449—24no—
112C4×Q16—22no—
113C4×Q16—22no—
114C4×SD16—22no—
115C4×SD16—22no—
116C4×SD16—22no—
117C4×SD16—22no—
118C4×D8—22no—
119C4×D8—22no—
120#432—22no—
121#432—22no—
122#431—22no—
123#431—22no—
124#431D414yes—
125#431D414yes—
126C2×#107C2×C411yescharacteristic
127C2×#107C2×C411yescharacteristic
128C2×#106C2×C411yescharacteristic
129C2×#106C2×C411yescharacteristic
130C2×C4⋊C8C2×C411yescharacteristic
131C2×C4⋊C8C2×C411yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#103C2×C411yescharacteristic
137C2×#103C2×C411yescharacteristic
138C2×#103C2×C411yescharacteristic
139C2×#103C2×C411yescharacteristic
140C2×#102C2×C411yescharacteristic
141C2×#102C2×C411yescharacteristic
142C2×#102C2×C411yescharacteristic
143C2×#102C2×C411yescharacteristic
144C2×C22⋊C8—21no—
145C2×C22⋊C8—21no—
146C2×C22⋊C8—22no—
147C2×C22⋊C8—22no—
148C2×C22⋊C8—22no—
149C2×C22⋊C8—22no—
150C2×C22⋊C8C2×C411yescharacteristic
151C2×C22⋊C8C2×C411yescharacteristic
152C4×M16—22no—
153C4×M16—22no—
154C2×#97—21no—
155C2×#97D412yes—
156C2×C4×C8—22no—
157C2×C4×C8—21no—
158C2×C4×C8—21no—
159C2×C4×C8D414yes—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166#336—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170#335—22no—
171#335—22no—
172#334—22no—
173#334—22no—
174Q8⋊C8—22no—
175Q8⋊C8—22no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178D4⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182#322—22no—
183#322—22no—
184C82—22no—
185C82—22no—
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—41no—
191C2×#41—41no—
192C22×Q8C22×C411yescharacteristic
193C22×Q8C22×C411yescharacteristic
194C22×D4—21no—
195C22×D4—21no—
196C22×D4C22×C411yescharacteristic
197C22×D4C22×C411yescharacteristic
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4C22×C411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4#4111yescharacteristic
205C2×Q16—22no—
206C2×Q16—22no—
207C2×Q16—41no—
208C2×SD16—22no—
209C2×SD16—22no—
210C2×SD16—22no—
211C2×SD16—22no—
212C2×SD16—41no—
213C2×SD16—41no—
214C2×D8—22no—
215C2×D8—22no—
216C2×D8—41no—
217C2×M16—24no—
218C2×M16—24no—
219C2×M16—24no—
220C22×C8—21no—
221C22×C8—24no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—44no—
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C2×D412yes—
233C22×C8C2×D412yes—
234#126—42no—
235#126—42no—
236#125—22no—
237#125—22no—
238#124—22no—
239#124—22no—
240#123—42no—
241#123—42no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244#121—22no—
245#121—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—42no—
269C4×D4—42no—
270C4×D4—42no—
271C4×D4—42no—
272#117—22no—
273#117—22no—
274#117—42no—
275#117—42no—
276#117—42no—
277#117—42no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42C22×C411yescharacteristic
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D411yescharacteristic
307C2×C42#4112yes—
308C2×C42#4111yescharacteristic
309C2×C42C2411yescharacteristic
310#107—22no—
311#107—22no—
312#106—22no—
313#106—22no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8C22⋊C414yes—
323C4⋊C8C22⋊C414yes—
324#103—22no—
325#103—22no—
326#103—22no—
327#103—22no—
328#102—22no—
329#102—22no—
330#102—22no—
331#102—22no—
332C22⋊C8—42no—
333C22⋊C8—42no—
334C22⋊C8—44no—
335C22⋊C8—44no—
336C22⋊C8—44no—
337C22⋊C8—44no—
338C22⋊C8C22⋊C414yes—
339C22⋊C8C22⋊C414yes—
340#97—24no—
341#97—42no—
342C4×C8—24no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—42no—
350C4×C8C2×D412yes—
351C4×C8C2×D412yes—
352C4×C8C2×D412yes—
353C4×C8C2×D412yes—
354C4×C8C2×D412yes—
355C4×C8C2×D412yes—
356C24—21no—
357C24—21no—
358C24C4×D411yescharacteristic
359#41—28no—
360#41—44no—
361#41—44no—
362#41—44no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8#13112yes—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—82no—
391C2×D4#13112yes—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—82no—
418C22×C4C2×C22⋊C412yes—
419C22×C4C4×D412yes—
420C22×C4C4×D411yescharacteristic
421C22×C4C4×D411yescharacteristic
422C22×C4C4×D411yescharacteristic
423C22×C4#12112yes—
424C22×C4#12112yes—
425C22×C4#12412yes—
426C22×C4#12712yes—
427C22×C4C23×C411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4111yescharacteristic
430Q16—44no—
431SD16—44no—
432SD16—44no—
433D8—44no—
434M16—216no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—48no—
451C2×C8—42no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8C2×C22⋊C412yes—
461C2×C8C2×C22⋊C412yes—
462C2×C8C2×C22⋊C412yes—
463C2×C8C2×C22⋊C412yes—
464C2×C8#13514yes—
465C2×C8#13514yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—42no—
497C42—42no—
498C42C2×C22⋊C412yes—
499C42C4×D412yes—
500C42#12112yes—
501C42#12112yes—
502C42#12412yes—
503C42#12712yes—
504C42C22×D411yescharacteristic
505C42C22×D411yescharacteristic
506C42C22×D411yescharacteristic
507C42C2×#4111yescharacteristic
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23#39012yes—
523C23C2×C4×D411yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532D4—22no—
533D4—22no—
534D4—48no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—84no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—48no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—84no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4#39012yes—
579C2×C4#39012yes—
580C2×C4#39012yes—
581C2×C4C22×C22⋊C411yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×#12111yescharacteristic
584C2×C4C2×#12111yescharacteristic
585C2×C4C2×#12411yescharacteristic
586C2×C4C2×#12711yescharacteristic
587C2×C4C2×#13111yescharacteristic
588C2×C4C2×#13111yescharacteristic
589C2×C4#56811yescharacteristic
590C2×C4#56811yescharacteristic
591C2×C4C2×#13512yes—
592C2×C4C2×#13512yes—
593C8—24no—
594C8—22no—
595C8—22no—
596C8—22no—
597C8—22no—
598C8—24no—
599C8—24no—
600C8—42no—
601C8—81no—
602C8—81no—
603C8—81no—
604C8—81no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—82no—
613C22—81no—
614C22—81no—
615C22—82no—
616C22—82no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22C2×#39011yescharacteristic
621C22#293311yescharacteristic
622C22#293311yescharacteristic
623C22C4×#13512yes—
624C22#295612yes—
625C4—22no—
626C4—22no—
627C4—21no—
628C4—22no—
629C4—21no—
630C4—42no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4#288011yescharacteristic
637C4#288011yescharacteristic
638C4#292911yescharacteristic
639C4#293011yescharacteristic
640C4#305012yes—
641C4#314711yescharacteristic
642C4#314711yescharacteristic
643C2—42no—
644C2—82no—
645C2—161no—
646C2#1799612yes—
647C2#1799912yes—
648C2#2119111yescharacteristic
649C2#2122411yescharacteristic
650C2#2124111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#449C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21380C2semidirect
3. #21378C2semidirect
4. C2×#1572C2semidirect
5. C2×#1494C2semidirect
6. C2×#1476C2semidirect
7. C2×C8×M16C2semidirect
8. #9663C2semidirect
9. #9662C2semidirect
10. #9661C2semidirect
11. #9661C2semidirect
12. #9657C2semidirect
13. #9656C2semidirect
14. #9655C2semidirect
15. #9655C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31022
#213802
#213782
C2×#15722
C2×#14942
C2×#14762
C2×C8×M162
#96632
#96622
#96612
#96572
#96562
#96552