Finite Group Databasebeta

← #209196

Subgroups — #209196

Order 512.

1057 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2982C211yesmaximal, characteristic, complemented
2#22133C211yesmaximal, characteristic
3C4×#448C211yesmaximal, characteristic
4C2×SD16⋊C8C211yesmaximal, characteristic, complemented
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#8948C211yesmaximal, characteristic, complemented
7#8948C211yesmaximal, characteristic, complemented
8#8940C211yesmaximal, characteristic
9#8846C211yesmaximal, characteristic, complemented
10#8844C211yesmaximal, characteristic, complemented
11#8831C211yesmaximal, characteristic, complemented
12#8311C211yesmaximal, characteristic, complemented
13#8201C211yesmaximal, characteristic, complemented
14C8×#103C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C22×C4×Q8C2211yescharacteristic
17C2×#448C411yescharacteristic
18#2982—24no—
19#2982—24no—
20C2×C8×Q8—21no—
21C2×C8×Q8—21no—
22C2×C8×Q8C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C4×SD16C2211yescharacteristic
27C2×C4×SD16C2211yescharacteristic
28C2×C8×D4C2211yescharacteristic
29C2×#431—21no—
30C2×#431—21no—
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C4×#121C2211yescharacteristic
34#1954—21no—
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1923—21no—
38#1880C411yescharacteristic
39#1880C411yescharacteristic
40#1874C411yescharacteristic
41#1869—21no—
42#1865C411yescharacteristic
43#1865C411yescharacteristic
44#1853C411yescharacteristic
45#1850—21no—
46#1797C2211yescharacteristic
47#1787C411yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#103C2211yescharacteristic
50C4×#103C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53SD16⋊C8—24no—
54C8×SD16—24no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C23×Q8C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×SD16C2×C411yescharacteristic
74C23×C8—21no—
75C2×#121C2×C411yescharacteristic
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×Q8C2311yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4C2311yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#448C818yescomplemented
93C8×Q8—28no—
94C8×Q8—28no—
95C8×Q8—22no—
96C8×Q8—22no—
97C8×Q8—22no—
98C8×Q8—22no—
99C8×Q8—22no—
100C8×Q8—22no—
101C8×Q8D414yescomplemented
102C8×Q8D414yescomplemented
103C4×SD16—22no—
104C4×SD16—22no—
105C4×SD16—22no—
106C4×SD16—22no—
107C8×D4—22no—
108C8×D4—22no—
109#431—28no—
110#431—28no—
111#431—22no—
112#431—22no—
113#431—22no—
114#431—22no—
115C2×#106C2×C411yescharacteristic
116C2×#106C2×C411yescharacteristic
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8D411yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8C2×C411yescharacteristic
143C2×C22⋊C8C2×C411yescharacteristic
144C2×C22⋊C8D411yescharacteristic
145C2×C22⋊C8D411yescharacteristic
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158#390C2×C411yescharacteristic
159#389C2×C411yescharacteristic
160#388—21no—
161#386—21no—
162#380C2×C411yescharacteristic
163C4×C4⋊C4C2311yescharacteristic
164C4×C22⋊C4—21no—
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#335—22no—
176#334—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183#322—22no—
184C82—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—21no—
190C22×Q8—22no—
191C22×Q8C2×C814yescomplemented
192C22×Q8C22×C411yescharacteristic
193C22×Q8C22×C411yescharacteristic
194C22×D4—21no—
195C22×D4C22×C411yescharacteristic
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4C22×C411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4#4111yescharacteristic
202C2×SD16—22no—
203C2×SD16—22no—
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16C2×C814yes—
207C2×SD16C2×C814yes—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—42no—
215C22×C8—81no—
216C22×C8—81no—
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225#121—22no—
226#121C2×C814yes—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8C2×D412yes—
240C4×Q8C2×D412yes—
241C4×Q8#4112yes—
242C4×Q8#4112yes—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—81no—
248C4×D4—81no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42C2411yescharacteristic
284#106—22no—
285#106—22no—
286C4⋊C8—24no—
287C4⋊C8—24no—
288C4⋊C8—24no—
289C4⋊C8—24no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—81no—
299C4⋊C8C2×D412yes—
300C4⋊C8C2×D412yes—
301#103—22no—
302#103—22no—
303#103—22no—
304#103—22no—
305#103C2×C814yescomplemented
306#103C2×C814yescomplemented
307#102—22no—
308#102—22no—
309C22⋊C8—24no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—42no—
314C22⋊C8—42no—
315C22⋊C8—81no—
316C22⋊C8—81no—
317C22⋊C8—81no—
318C22⋊C8C2×C814yescomplemented
319C4×C8—24no—
320C4×C8—24no—
321C4×C8—24no—
322C4×C8—24no—
323C4×C8—24no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—81no—
335C4×C8C2×D412yes—
336C4×C8C2×D412yes—
337#95—21no—
338#95—21no—
339#95—21no—
340C24—21no—
341C24C4×D411yescharacteristic
342C2×Q8—24no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—24no—
346C2×Q8—28no—
347C2×Q8—28no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—24no—
355C2×Q8C4×D412yes—
356C2×Q8C4×D412yes—
357C2×Q8C22×C812yes—
358C2×Q8C22×C812yes—
359C2×Q8#13112yes—
360C2×Q8#13112yes—
361C2×D4—24no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4C22×C812yes—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4C22×C812yes—
407C22×C4#13112yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411SD16—28no—
412SD16—28no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—84no—
452C2×C8#12012yes—
453C2×C8#12012yes—
454C2×C8C22×C812yes—
455C2×C8C22×C812yes—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—81no—
471C4⋊C4C4×D412yes—
472C4⋊C4C4×D412yes—
473C4⋊C4C22×C812yes—
474C22⋊C4—24no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C42—24no—
481C42—24no—
482C42—21no—
483C42—21no—
484C42—22no—
485C42—22no—
486C42—21no—
487C42—22no—
488C42—22no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—41no—
493C42—41no—
494C42C22×D411yescharacteristic
495C42C22×D411yescharacteristic
496C42C2×#4111yescharacteristic
497C42C2×#4111yescharacteristic
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C8×D412yes—
512C23C2×C4×D411yescharacteristic
513Q8—216no—
514Q8—22no—
515Q8—22no—
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—22no—
520Q8C8×D414yes—
521Q8C8×D414yes—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526D4—84no—
527C2×C4—216no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—84no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4C8×D412yes—
575C2×C4C2×C4×D411yescharacteristic
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C2×#12011yescharacteristic
578C2×C4#52511yescharacteristic
579C2×C4#52511yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4C23×C811yescharacteristic
582C2×C4C2×#13111yescharacteristic
583C2×C4C2×#13111yescharacteristic
584C2×C4C2×#13111yescharacteristic
585C2×C4#57711yescharacteristic
586C2×C4#57711yescharacteristic
587C2×C4#57711yescharacteristic
588C2×C4#57811yescharacteristic
589C8—22no—
590C8—22no—
591C8—22no—
592C8—22no—
593C8—42no—
594C8—41no—
595C8—41no—
596C8—41no—
597C8—42no—
598C8—41no—
599C8—42no—
600C8—81no—
601C8—81no—
602C8—81no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C4×#12011yescharacteristic
618C22C2×C8×D411yescharacteristic
619C22#292911yescharacteristic
620C22#294711yescharacteristic
621C22#294911yescharacteristic
622C22#295611yescharacteristic
623C22#295711yescharacteristic
624C4—24no—
625C4—24no—
626C4—22no—
627C4—21no—
628C4—21no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—81no—
636C4C2×C8×D411yescharacteristic
637C4C2×C8×D411yescharacteristic
638C4#292911yescharacteristic
639C4#292911yescharacteristic
640C4#301011yescharacteristic
641C4#301311yescharacteristic
642C4#312111yescharacteristic
643C4#312611yescharacteristic
644C2—42no—
645C2—82no—
646C2#1785311yescharacteristic
647C2#1785511yescharacteristic
648C2#1791911yescharacteristic
649C2#1792311yescharacteristic
650C2C8×#12011yescharacteristic
651C2#2146211yescharacteristic
652C2#2146811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2982C2semidirect
4. C2×SD16⋊C8C2semidirect
5. C2×C8×SD16C2semidirect
6. #8948C2semidirect
7. #8948C2semidirect
9. #8846C2semidirect
10. #8844C2semidirect
11. #8831C2semidirect
12. #8311C2semidirect
13. #8201C2semidirect
14. C8×#103C2semidirect
15. #8148C2semidirect
92. #448C8semidirect
101. C8×Q8D4semidirect
102. C8×Q8D4semidirect
191. C22×Q8C2×C8semidirect
305. #103C2×C8semidirect
306. #103C2×C8semidirect
318. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#29822
#221332
C4×#4482
C2×SD16⋊C82
C2×C8×SD162
#89482
#89402
#88462
#88442
#88312
#83112
#82012
C8×#1032
#81482