Finite Group Databasebeta

← #6496150

Subgroups — #6496150

Order 512.

3068 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#41C211yesmaximal, characteristic
2C2×#3230C211yesmaximal, characteristic, complemented
3C22×#477C211yesmaximal, characteristic, complemented
4#35279C211yesmaximal, characteristic
5#32957C214yesmaximal, complemented
6#29691C214yesmaximal, complemented
7#25888C212yesmaximal, complemented
8#25841C212yesmaximal, complemented
9#25706C212yesmaximal, complemented
10#25651C212yesmaximal, complemented
11#24098C214yesmaximal, complemented
12#24042C214yesmaximal, complemented
13#21885C211yesmaximal, characteristic, complemented
14#21847C211yesmaximal, characteristic
15#20734C211yesmaximal, characteristic, complemented
16Q8×#41—28no—
17Q8×#41—28no—
18C2×Q82C2211yescharacteristic
19C2×#557—21no—
20C2×#557C2211yescharacteristic
21C2×#554—22no—
22C2×#554—21no—
23C2×#554C2212yes—
24C2×#554C2211yescharacteristic
25C2×D4×Q8C2211yescharacteristic
26C2×D4×Q8C2212yes—
27C2×#533—21no—
28C2×#533—22no—
29C2×#533C2212yes—
30C2×#533C2211yescharacteristic
31C22×C4⋊Q8C2211yescharacteristic
32C2×C4×#41—21no—
33C2×C4×#41C2211yescharacteristic
34C22×C4×Q8C2212yes—
35C22×C4×Q8C2211yescharacteristic
36#3230—28no—
37C2×#500C2212yes—
38C2×#487C2212yes—
39C2×#484C2214yes—
40C2×#477—28no—
41C2×#477C2212yes—
42C2×#477C2212yes—
43C2×#477C2212yes—
44C2×#477C2212yes—
45C2×#477C2214yes—
46C2×#431C2211yescharacteristic
47C22×#107C2212yes—
48C22×C4⋊C8C2211yescharacteristic
49C22×#103C2212yes—
50#2857C2214yes—
51#2628—22no—
52#2568—21no—
53#2567C2212yes—
54#2563C2212yes—
55#2542—21no—
56#2487C2214yes—
57#2409—22no—
58#2317—21no—
59#2311—21no—
60#2285C2211yescharacteristic
61#2057C2218yes—
62#2043C2218yes—
63#2036C2218yes—
64#2036C2218yes—
65#1987C2212yes—
66#1984C2212yes—
67#1981C2212yes—
68#1954C2212yes—
69#1892C2214yes—
70#1869C2214yes—
71#1863C2218yes—
72#1863C2218yes—
73#1852C2218yes—
74#1824C2212yes—
75#1803C2214yes—
76#1803C2212yes—
77#1803C2212yes—
78#1797C2214yes—
79#1787C2218yes—
80#1787C2212yes—
81C4×#103C2212yes—
82C4×#103C2212yes—
83C22×#41D411yescharacteristic
84C23×Q8—21no—
85C23×Q8C2312yes—
86C23×Q8C2311yescharacteristic
87C23×C8C2312yes—
88Q82—24no—
89Q82—24no—
90#557—216no—
91#557—24no—
92#557—24no—
93#554—216no—
94#554—232no—
95#554—24no—
96#554—28no—
97#554—24no—
98#554—28no—
99D4×Q8—28no—
100D4×Q8—24no—
101D4×Q8—28no—
102D4×Q8—24no—
103#533—232no—
104#533—216no—
105#533—28no—
106#533—28no—
107#533—24no—
108#533D418yes—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—22no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—24no—
116C2×C4⋊Q8—24no—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2311yescharacteristic
123C2×#125—21no—
124C2×#125—22no—
125C2×#125—21no—
126C2×#125—22no—
127C2×#125—22no—
128C2×#125—21no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—24no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—22no—
134C2×C22⋊Q8—22no—
135C2×C22⋊Q8—22no—
136C2×C22⋊Q8—22no—
137C2×C22⋊Q8—22no—
138C2×C22⋊Q8C2312yes—
139C2×C22⋊Q8C2312yes—
140C2×C22⋊Q8C2312yes—
141C4×#41—216no—
142C4×#41—24no—
143C4×#41—24no—
144C2×C4×Q8—21no—
145C2×C4×Q8—21no—
146C2×C4×Q8—28no—
147C2×C4×Q8—28no—
148C2×C4×Q8—22no—
149C2×C4×Q8—21no—
150C2×C4×Q8—22no—
151C2×C4×Q8—21no—
152C2×C4×Q8—24no—
153C2×C4×Q8—22no—
154C2×C4×Q8—21no—
155C2×C4×Q8D418yes—
156C2×C4×Q8D411yescharacteristic
157C2×C4×Q8C2312yes—
158C2×C4×Q8C2312yes—
159C2×C4×Q8C2311yescharacteristic
160C2×C4×Q8C2312yes—
161C2×C4×Q8C2311yescharacteristic
162C2×C4×Q8C2311yescharacteristic
163C2×C4×Q8C2311yescharacteristic
164C2×C4×Q8C2311yescharacteristic
165C2×C4×D4—22no—
166C2×C4×D4—21no—
167C2×C4×D4—22no—
168C2×C4×D4—21no—
169C2×C4×D4D412yes—
170C2×C4×D4D411yescharacteristic
171C2×C4×D4C2311yescharacteristic
172C2×C4×D4C2312yes—
173C2×#117—22no—
174C2×#117—21no—
175C2×#117—21no—
176C2×#117D411yescharacteristic
177C2×#117D412yes—
178C22×C4⋊C4—22no—
179C22×C4⋊C4—21no—
180C22×C4⋊C4—21no—
181C22×C4⋊C4C2312yes—
182C22×C4⋊C4C2312yes—
183C22×C4⋊C4C2311yescharacteristic
184C22×C42—22no—
185C22×C42—21no—
186C22×C42C2312yes—
187C22×C42C2311yescharacteristic
188#500—28no—
189#487—28no—
190#484—216no—
191#477—28no—
192#477—28no—
193#477—28no—
194#477—28no—
195#477—216no—
196#431—24no—
197C2×#107—28no—
198C2×#107C2314yes—
199C2×#107C2314yes—
200C2×#107C2318yes—
201C2×C4⋊C8—24no—
202C2×C4⋊C8C2312yes—
203C2×C4⋊C8C2312yes—
204C2×C4⋊C8C2312yes—
205C2×#103—28no—
206C2×#103C2314yes—
207C2×#103C2314yes—
208C2×#103C2318yes—
209C2×#103C2314yes—
210C2×#103C2318yes—
211C2×#103C2314yes—
212C2×C22⋊C8C2312yes—
213C2×C22⋊C8C2318yes—
214C2×C4×C8C2312yes—
215#400—24no—
216#399—24no—
217#395—24no—
218#393—24no—
219#391—22no—
220#391—24no—
221#390—22no—
222#389—24no—
223#389C2318yes—
224#386—24no—
225#385—24no—
226#384—22no—
227#384C2312yes—
228#382—22no—
229#382—21no—
230#380—24no—
231#380—21no—
232#380C2318yes—
233#380C2312yes—
234C4×C4⋊C4—21no—
235C4×C4⋊C4C2312yes—
236#340C2314yes—
237#340C2318yes—
238#340C2314yes—
239#336C2318yes—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41Q1618yes—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—21no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—24no—
253C22×Q8—28no—
254C22×Q8—42no—
255C22×Q8—44no—
256C22×Q8—44no—
257C22×Q8—84no—
258C22×Q8C2×D414yes—
259C22×Q8C2×D411yescharacteristic
260C22×Q8C2412yes—
261C22×Q8C2411yescharacteristic
262C22×Q8C2411yescharacteristic
263C22×D4C2×D412yes—
264C22×D4C2×D411yescharacteristic
265C23×C4—22no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4C2×D412yes—
271C23×C4C2×D412yes—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C22×C8—24no—
277C22×C8—44no—
278C22×C8C2414yes—
279C22×C8C2414yes—
280C4⋊Q8—216no—
281C4⋊Q8—28no—
282C4⋊Q8—216no—
283C4⋊Q8—28no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C4⋊Q8C2×D414yes—
297C4⋊Q8C2×D414yes—
298#125—216no—
299#125—216no—
300#125—28no—
301#125—28no—
302#125—216no—
303#125—28no—
304C22⋊Q8—216no—
305C22⋊Q8—216no—
306C22⋊Q8—216no—
307C22⋊Q8—216no—
308C22⋊Q8—216no—
309C22⋊Q8—216no—
310C22⋊Q8—216no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313C22⋊Q8—28no—
314C22⋊Q8—24no—
315C22⋊Q8—24no—
316C22⋊Q8—24no—
317C22⋊Q8—88no—
318C22⋊Q8C2×D418yes—
319C4×Q8—216no—
320C4×Q8—216no—
321C4×Q8—28no—
322C4×Q8—28no—
323C4×Q8—216no—
324C4×Q8—24no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×Q8—28no—
329C4×Q8—28no—
330C4×Q8—28no—
331C4×Q8—24no—
332C4×Q8—22no—
333C4×Q8—24no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8C2×D414yes—
340C4×Q8C2×D418yes—
341C4×Q8C2×D414yes—
342C4×Q8C2×D414yes—
343C4×Q8C2×D414yes—
344C4×D4—216no—
345C4×D4—28no—
346C4×D4—216no—
347C4×D4—28no—
348C4×D4—28no—
349C4×D4—24no—
350C4×D4—24no—
351C4×D4—24no—
352C4×D4—22no—
353C4×D4—22no—
354#117—216no—
355#117—28no—
356#117—24no—
357#117—28no—
358#117—44no—
359C2×C4⋊C4—216no—
360C2×C4⋊C4—28no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—28no—
390C2×C4⋊C4—44no—
391C2×C4⋊C4C2×D411yescharacteristic
392C2×C4⋊C4C2×D412yes—
393C2×C4⋊C4C2×D411yescharacteristic
394C2×C4⋊C4C2×D412yes—
395C2×C4⋊C4C2×D414yes—
396C2×C4⋊C4C2414yes—
397C2×C4⋊C4C2414yes—
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2412yes—
400C2×C4⋊C4C2411yescharacteristic
401C2×C22⋊C4—22no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—28no—
406C2×C22⋊C4C2×D412yes—
407C2×C22⋊C4C2×D412yes—
408C2×C22⋊C4C2×D412yes—
409C2×C42—216no—
410C2×C42—28no—
411C2×C42—22no—
412C2×C42—21no—
413C2×C42—22no—
414C2×C42—21no—
415C2×C42—22no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—24no—
419C2×C42—24no—
420C2×C42—22no—
421C2×C42C2×D412yes—
422C2×C42C2×D412yes—
423C2×C42C2×D411yescharacteristic
424C2×C42C2×D411yescharacteristic
425C2×C42C2×D414yes—
426C2×C42C2×D414yes—
427C2×C42C2412yes—
428C2×C42C2411yescharacteristic
429C2×C42C2411yescharacteristic
430#107—28no—
431#107—28no—
432#107—216no—
433C4⋊C8—24no—
434C4⋊C8—24no—
435C4⋊C8—24no—
436#103—28no—
437#103—28no—
438#103—216no—
439#103—28no—
440#103—44no—
441#103—48no—
442C22⋊C8—24no—
443C22⋊C8—48no—
444C4×C8—24no—
445#95—22no—
446#95—22no—
447#95—28no—
448#95—28no—
449C24C22×D412yes—
450C24C22×D411yescharacteristic
451#41—232no—
452C2×Q8—28no—
453C2×Q8—28no—
454C2×Q8—22no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8—22no—
462C2×Q8—28no—
463C2×Q8—22no—
464C2×Q8—24no—
465C2×Q8—22no—
466C2×Q8—24no—
467C2×Q8—44no—
468C2×Q8—48no—
469C2×Q8—44no—
470C2×Q8—48no—
471C2×Q8—48no—
472C2×Q8—416no—
473C2×Q8—48no—
474C2×Q8—48no—
475C2×Q8—44no—
476C2×Q8—44no—
477C2×Q8—82no—
478C2×Q8—84no—
479C2×Q8—84no—
480C2×Q8—816no—
481C2×Q8C2×Q1614yes—
482C2×Q8C22×D414yes—
483C2×Q8C22×D412yes—
484C2×Q8C22×D412yes—
485C2×Q82−1+414yes—
486C2×D4—24no—
487C2×D4—24no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4C2×Q1618yes—
491C2×D4C2×Q1614yes—
492C22×C4—21no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—28no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—28no—
505C22×C4—24no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—42no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—48no—
517C22×C4—44no—
518C22×C4—82no—
519C22×C4—84no—
520C22×C4—84no—
521C22×C4#12014yes—
522C22×C4#12014yes—
523C22×C4C2×Q1618yes—
524C22×C4C2×Q1614yes—
525C22×C4#13714yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D412yes—
532C22×C4C22×D412yes—
533C22×C4C22×D412yes—
534C22×C4C22×D411yescharacteristic
535C22×C4C22×D411yescharacteristic
536C22×C4C2511yescharacteristic
537C2×C8—24no—
538C2×C8—24no—
539C2×C8—24no—
540C2×C8—24no—
541C2×C8—28no—
542C2×C8—48no—
543C4⋊C4—28no—
544C4⋊C4—28no—
545C4⋊C4—28no—
546C4⋊C4—28no—
547C4⋊C4—28no—
548C4⋊C4—28no—
549C4⋊C4—28no—
550C4⋊C4—28no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—28no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—24no—
559C4⋊C4—24no—
560C4⋊C4—28no—
561C4⋊C4—24no—
562C4⋊C4—24no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—21no—
566C4⋊C4—22no—
567C4⋊C4—21no—
568C4⋊C4—24no—
569C4⋊C4—24no—
570C4⋊C4—24no—
571C4⋊C4—24no—
572C4⋊C4—44no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—88no—
578C4⋊C4C22×D412yes—
579C4⋊C4C22×D414yes—
580C4⋊C4C22×D412yes—
581C22⋊C4—28no—
582C22⋊C4—28no—
583C22⋊C4—28no—
584C22⋊C4—24no—
585C22⋊C4—24no—
586C22⋊C4—24no—
587C22⋊C4—44no—
588C22⋊C4—88no—
589C42—28no—
590C42—28no—
591C42—24no—
592C42—28no—
593C42—24no—
594C42—24no—
595C42—24no—
596C42—24no—
597C42—24no—
598C42—22no—
599C42—22no—
600C42—44no—
601C42#12014yes—
602C42#12014yes—
603C42C22×D412yes—
604C42C22×D412yes—
605C42C22×D412yes—
606C42C22×D412yes—
607C23—22no—
608C23—22no—
609C23—24no—
610C23—22no—
611C23—24no—
612C23—22no—
613C23C2×#12012yes—
614C23C22×Q1614yes—
615C23C22×Q1612yes—
616C23C2×#13712yes—
617C23C23×D411yescharacteristic
618Q8—22no—
619Q8—24no—
620Q8—28no—
621Q8—22no—
622Q8—22no—
623Q8—28no—
624Q8—48no—
625Q8—416no—
626Q8—42no—
627Q8—42no—
628Q8—42no—
629Q8—84no—
630Q8—88no—
631Q8—88no—
632Q8—88no—
633Q8#53614yes—
634D4—216no—
635D4—28no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—216no—
647C2×C4—24no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—24no—
656C2×C4—44no—
657C2×C4—48no—
658C2×C4—416no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—48no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4—88no—
675C2×C4—88no—
676C2×C4#451116yes—
677C2×C4C2×#12012yes—
678C2×C4C2×#12012yes—
679C2×C4C2×#12012yes—
680C2×C4C2×#12012yes—
681C2×C4C2×#12012yes—
682C2×C4C22×Q1614yes—
683C2×C4C22×Q1612yes—
684C2×C4C22×Q1612yes—
685C2×C4C2×#13712yes—
686C2×C4C2×#13712yes—
687C2×C4C23×D411yescharacteristic
688C2×C4C23×D411yescharacteristic
689C2×C4C2×2−1+412yes—
690C2×C4#58512yes—
691C8—44no—
692C8—44no—
693C22—24no—
694C22—24no—
695C22—22no—
696C22—22no—
697C22—28no—
698C22—24no—
699C22—44no—
700C22—42no—
701C22—42no—
702C22C2×#45114yes—
703C22C22×#12011yescharacteristic
704C22#348912yes—
705C22C23×Q1611yescharacteristic
706C22C22×#13711yescharacteristic
707C22#358512yes—
708C4—24no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—24no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—44no—
717C4—42no—
718C4—42no—
719C4—44no—
720C4—84no—
721C4—84no—
722C4C2×#45114yes—
723C4C2×#45114yes—
724C4C2×#53611yescharacteristic
725C4#344911yescharacteristic
726C4#348912yes—
727C2—24no—
728C2—44no—
729C2#3530511yescharacteristic
730C2C22×#45111yescharacteristic
731C2#6036911yescharacteristic
732C2#6210012yes—
733C2#6211412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3230C2semidirect
3. C22×#477C2semidirect
5. #32957C2semidirect
6. #29691C2semidirect
7. #25888C2semidirect
8. #25841C2semidirect
9. #25706C2semidirect
10. #25651C2semidirect
11. #24098C2semidirect
12. #24042C2semidirect
13. #21885C2semidirect
15. #20734C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#412
C2×#32302
C22×#4772
#352792
#329572
#296912
#258882
#258412
#257062
#256512
#240982
#240422
#218852
#218472
#207342