Finite Group Databasebeta

← #6496149

Subgroups — #6496149

Order 512.

3144 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#41C211yesmaximal, characteristic
2C2×#3229C211yesmaximal, characteristic, complemented
3C22×#475C211yesmaximal, characteristic, complemented
4#35316C211yesmaximal, characteristic
5#32954C214yesmaximal, complemented
6#29688C214yesmaximal, complemented
7#25884C212yesmaximal, complemented
8#25837C212yesmaximal, complemented
9#25702C212yesmaximal, complemented
10#25647C212yesmaximal, complemented
11#24097C214yesmaximal, complemented
12#24041C214yesmaximal, 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×#536—21no—
28C2×#533—21no—
29C2×#533—22no—
30C2×#533C2212yes—
31C2×#533C2211yescharacteristic
32C22×C4⋊Q8C2211yescharacteristic
33C2×C4×#41—21no—
34C2×C4×#41C2211yescharacteristic
35C22×C4×Q8C2212yes—
36C22×C4×Q8C2211yescharacteristic
37#3229—28no—
38C2×#498C2212yes—
39C2×#487C2212yes—
40C2×#483C2214yes—
41C2×#475—28no—
42C2×#475C2212yes—
43C2×#475C2212yes—
44C2×#475C2212yes—
45C2×#475C2212yes—
46C2×#475C2214yes—
47C2×#431C2211yescharacteristic
48C22×#106C2212yes—
49C22×C4⋊C8C2211yescharacteristic
50C22×#103C2212yes—
51#2857C2214yes—
52#2621—22no—
53#2547—21no—
54#2547C2212yes—
55#2546—21no—
56#2546C2212yes—
57#2486—22no—
58#2486C2214yes—
59#2310—21no—
60#2285C2211yescharacteristic
61#2058C2218yes—
62#2043C2218yes—
63#2035C2218yes—
64#2035C2218yes—
65#1987C2212yes—
66#1984C2212yes—
67#1981C2212yes—
68#1954C2212yes—
69#1892C2214yes—
70#1869C2214yes—
71#1865C2218yes—
72#1865C2218yes—
73#1853C2218yes—
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#536—48no—
104#533—232no—
105#533—216no—
106#533—28no—
107#533—28no—
108#533—24no—
109#533D418yes—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—22no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—24no—
118C2×C4⋊Q8—24no—
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2311yescharacteristic
125C2×#125—21no—
126C2×#125—22no—
127C2×#125—21no—
128C2×#125—22no—
129C2×#125—22no—
130C2×#125—21no—
131C2×#124—22no—
132C2×#123—24no—
133C2×C22⋊Q8—22no—
134C2×C22⋊Q8—21no—
135C2×C22⋊Q8—24no—
136C2×C22⋊Q8—22no—
137C2×C22⋊Q8—22no—
138C2×C22⋊Q8—22no—
139C2×C22⋊Q8—22no—
140C2×C22⋊Q8—22no—
141C2×C22⋊Q8—22no—
142C2×C22⋊Q8C2312yes—
143C2×C22⋊Q8C2312yes—
144C2×C22⋊Q8C2312yes—
145C4×#41—216no—
146C4×#41—24no—
147C4×#41—24no—
148C2×C4×Q8—21no—
149C2×C4×Q8—21no—
150C2×C4×Q8—28no—
151C2×C4×Q8—28no—
152C2×C4×Q8—22no—
153C2×C4×Q8—21no—
154C2×C4×Q8—22no—
155C2×C4×Q8—21no—
156C2×C4×Q8—24no—
157C2×C4×Q8—22no—
158C2×C4×Q8—21no—
159C2×C4×Q8D418yes—
160C2×C4×Q8D411yescharacteristic
161C2×C4×Q8C2312yes—
162C2×C4×Q8C2312yes—
163C2×C4×Q8C2311yescharacteristic
164C2×C4×Q8C2312yes—
165C2×C4×Q8C2311yescharacteristic
166C2×C4×Q8C2311yescharacteristic
167C2×C4×Q8C2311yescharacteristic
168C2×C4×Q8C2311yescharacteristic
169C2×C4×D4—22no—
170C2×C4×D4—21no—
171C2×C4×D4—22no—
172C2×C4×D4—21no—
173C2×C4×D4D412yes—
174C2×C4×D4D411yescharacteristic
175C2×C4×D4C2312yes—
176C2×C4×D4C2311yescharacteristic
177C2×#117—22no—
178C2×#117—21no—
179C2×#117—21no—
180C2×#117D411yescharacteristic
181C2×#117D412yes—
182C22×C4⋊C4—22no—
183C22×C4⋊C4—21no—
184C22×C4⋊C4—21no—
185C22×C4⋊C4C2312yes—
186C22×C4⋊C4C2312yes—
187C22×C4⋊C4C2311yescharacteristic
188C22×C42—22no—
189C22×C42—21no—
190C22×C42C2312yes—
191C22×C42C2311yescharacteristic
192#498—28no—
193#487—28no—
194#483—216no—
195#475—28no—
196#475—28no—
197#475—28no—
198#475—28no—
199#475—216no—
200#431—24no—
201C2×#106—28no—
202C2×#106C2314yes—
203C2×#106C2314yes—
204C2×#106C2318yes—
205C2×C4⋊C8—24no—
206C2×C4⋊C8C2312yes—
207C2×C4⋊C8C2312yes—
208C2×C4⋊C8C2312yes—
209C2×#103—28no—
210C2×#103C2314yes—
211C2×#103C2314yes—
212C2×#103C2318yes—
213C2×#103C2314yes—
214C2×#103C2318yes—
215C2×#103C2314yes—
216C2×C22⋊C8C2312yes—
217C2×C22⋊C8C2318yes—
218C2×C4×C8C2312yes—
219#398—24no—
220#398—24no—
221#397—24no—
222#397—24no—
223#391—22no—
224#391—22no—
225#391—24no—
226#389—24no—
227#389C2318yes—
228#384C2312yes—
229#382—21no—
230#381—24no—
231#380—24no—
232#380—21no—
233#380C2312yes—
234#380C2318yes—
235C4×C4⋊C4—21no—
236C4×C4⋊C4C2312yes—
237#340C2314yes—
238#340C2318yes—
239#340C2314yes—
240#336C2318yes—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41SD1618yes—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—22no—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—21no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—24no—
254C22×Q8—28no—
255C22×Q8—42no—
256C22×Q8—44no—
257C22×Q8—44no—
258C22×Q8—44no—
259C22×Q8—44no—
260C22×Q8C2×D414yes—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2412yes—
263C22×Q8C2411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×D4C2×D412yes—
266C22×D4C2×D411yescharacteristic
267C23×C4—22no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—21no—
271C23×C4—22no—
272C23×C4C2×D412yes—
273C23×C4C2×D412yes—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2412yes—
277C23×C4C2411yescharacteristic
278C22×C8—24no—
279C22×C8—44no—
280C22×C8C2414yes—
281C22×C8C2414yes—
282C4⋊Q8—216no—
283C4⋊Q8—28no—
284C4⋊Q8—216no—
285C4⋊Q8—28no—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—24no—
295C4⋊Q8—24no—
296C4⋊Q8—48no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299C4⋊Q8C2×D414yes—
300C4⋊Q8C2×D414yes—
301#125—216no—
302#125—216no—
303#125—28no—
304#125—28no—
305#125—216no—
306#125—28no—
307#124—48no—
308#123—416no—
309C22⋊Q8—216no—
310C22⋊Q8—216no—
311C22⋊Q8—216no—
312C22⋊Q8—216no—
313C22⋊Q8—216no—
314C22⋊Q8—216no—
315C22⋊Q8—216no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—28no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8—24no—
322C22⋊Q8—416no—
323C22⋊Q8C2×D418yes—
324C4×Q8—216no—
325C4×Q8—216no—
326C4×Q8—28no—
327C4×Q8—28no—
328C4×Q8—216no—
329C4×Q8—24no—
330C4×Q8—24no—
331C4×Q8—24no—
332C4×Q8—24no—
333C4×Q8—28no—
334C4×Q8—28no—
335C4×Q8—28no—
336C4×Q8—24no—
337C4×Q8—22no—
338C4×Q8—24no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8C2×D414yes—
345C4×Q8C2×D418yes—
346C4×Q8C2×D414yes—
347C4×Q8C2×D414yes—
348C4×Q8C2×D414yes—
349C4×D4—216no—
350C4×D4—28no—
351C4×D4—216no—
352C4×D4—28no—
353C4×D4—28no—
354C4×D4—24no—
355C4×D4—24no—
356C4×D4—24no—
357C4×D4—22no—
358C4×D4—22no—
359#117—216no—
360#117—28no—
361#117—24no—
362#117—28no—
363#117—44no—
364C2×C4⋊C4—216no—
365C2×C4⋊C4—28no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—28no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—44no—
396C2×C4⋊C4C2×D411yescharacteristic
397C2×C4⋊C4C2×D412yes—
398C2×C4⋊C4C2×D411yescharacteristic
399C2×C4⋊C4C2×D412yes—
400C2×C4⋊C4C2×D414yes—
401C2×C4⋊C4C2414yes—
402C2×C4⋊C4C2414yes—
403C2×C4⋊C4C2411yescharacteristic
404C2×C4⋊C4C2412yes—
405C2×C4⋊C4C2411yescharacteristic
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—28no—
411C2×C22⋊C4C2×D412yes—
412C2×C22⋊C4C2×D412yes—
413C2×C22⋊C4C2×D412yes—
414C2×C42—216no—
415C2×C42—28no—
416C2×C42—22no—
417C2×C42—21no—
418C2×C42—22no—
419C2×C42—21no—
420C2×C42—22no—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42—24no—
424C2×C42—24no—
425C2×C42—22no—
426C2×C42C2×D412yes—
427C2×C42C2×D412yes—
428C2×C42C2×D411yescharacteristic
429C2×C42C2×D411yescharacteristic
430C2×C42C2×D414yes—
431C2×C42C2×D414yes—
432C2×C42C2412yes—
433C2×C42C2411yescharacteristic
434C2×C42C2411yescharacteristic
435#106—28no—
436#106—28no—
437#106—216no—
438C4⋊C8—24no—
439C4⋊C8—24no—
440C4⋊C8—24no—
441#103—28no—
442#103—28no—
443#103—216no—
444#103—28no—
445#103—44no—
446#103—48no—
447C22⋊C8—24no—
448C22⋊C8—48no—
449C4×C8—24no—
450#95—22no—
451#95—22no—
452#95—28no—
453#95—28no—
454C24C22×D412yes—
455C24C22×D411yescharacteristic
456#41—232no—
457C2×Q8—28no—
458C2×Q8—28no—
459C2×Q8—22no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—22no—
463C2×Q8—21no—
464C2×Q8—22no—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8—28no—
468C2×Q8—22no—
469C2×Q8—24no—
470C2×Q8—22no—
471C2×Q8—24no—
472C2×Q8—44no—
473C2×Q8—48no—
474C2×Q8—44no—
475C2×Q8—48no—
476C2×Q8—416no—
477C2×Q8—48no—
478C2×Q8—48no—
479C2×Q8—44no—
480C2×Q8—44no—
481C2×Q8—44no—
482C2×Q8—44no—
483C2×Q8—44no—
484C2×Q8—44no—
485C2×Q8—82no—
486C2×Q8—84no—
487C2×Q8—816no—
488C2×Q8C2×SD1614yes—
489C2×Q8C22×D414yes—
490C2×Q8C22×D412yes—
491C2×Q8C22×D412yes—
492C2×Q82−1+414yes—
493C2×D4—24no—
494C2×D4—24no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4C2×SD1618yes—
498C2×D4C2×SD1614yes—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—28no—
505C22×C4—24no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—28no—
512C22×C4—24no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—24no—
516C22×C4—24no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—24no—
520C22×C4—42no—
521C22×C4—44no—
522C22×C4—44no—
523C22×C4—48no—
524C22×C4—44no—
525C22×C4—44no—
526C22×C4—44no—
527C22×C4—82no—
528C22×C4—84no—
529C22×C4#12014yes—
530C22×C4#12014yes—
531C22×C4C2×SD1618yes—
532C22×C4C2×SD1614yes—
533C22×C4#13714yes—
534C22×C4C22×D412yes—
535C22×C4C22×D412yes—
536C22×C4C22×D411yescharacteristic
537C22×C4C22×D411yescharacteristic
538C22×C4C22×D411yescharacteristic
539C22×C4C22×D412yes—
540C22×C4C22×D412yes—
541C22×C4C22×D412yes—
542C22×C4C22×D411yescharacteristic
543C22×C4C22×D411yescharacteristic
544C22×C4C2511yescharacteristic
545C2×C8—24no—
546C2×C8—24no—
547C2×C8—24no—
548C2×C8—24no—
549C2×C8—28no—
550C2×C8—48no—
551C4⋊C4—28no—
552C4⋊C4—28no—
553C4⋊C4—28no—
554C4⋊C4—28no—
555C4⋊C4—28no—
556C4⋊C4—28no—
557C4⋊C4—28no—
558C4⋊C4—28no—
559C4⋊C4—24no—
560C4⋊C4—24no—
561C4⋊C4—28no—
562C4⋊C4—24no—
563C4⋊C4—24no—
564C4⋊C4—24no—
565C4⋊C4—24no—
566C4⋊C4—24no—
567C4⋊C4—24no—
568C4⋊C4—28no—
569C4⋊C4—24no—
570C4⋊C4—24no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4—21no—
574C4⋊C4—22no—
575C4⋊C4—21no—
576C4⋊C4—24no—
577C4⋊C4—24no—
578C4⋊C4—24no—
579C4⋊C4—24no—
580C4⋊C4—44no—
581C4⋊C4—44no—
582C4⋊C4—44no—
583C4⋊C4—44no—
584C4⋊C4—44no—
585C4⋊C4—416no—
586C4⋊C4C22×D412yes—
587C4⋊C4C22×D414yes—
588C4⋊C4C22×D412yes—
589C22⋊C4—28no—
590C22⋊C4—28no—
591C22⋊C4—28no—
592C22⋊C4—24no—
593C22⋊C4—24no—
594C22⋊C4—24no—
595C22⋊C4—44no—
596C22⋊C4—416no—
597C42—28no—
598C42—28no—
599C42—24no—
600C42—28no—
601C42—24no—
602C42—24no—
603C42—24no—
604C42—24no—
605C42—24no—
606C42—22no—
607C42—22no—
608C42—44no—
609C42#12014yes—
610C42#12014yes—
611C42C22×D412yes—
612C42C22×D412yes—
613C42C22×D412yes—
614C42C22×D412yes—
615C23—22no—
616C23—22no—
617C23—24no—
618C23—22no—
619C23—24no—
620C23—22no—
621C23C2×#12012yes—
622C23C22×SD1614yes—
623C23C22×SD1612yes—
624C23C2×#13712yes—
625C23C23×D411yescharacteristic
626Q8—22no—
627Q8—24no—
628Q8—28no—
629Q8—22no—
630Q8—22no—
631Q8—28no—
632Q8—48no—
633Q8—416no—
634Q8—42no—
635Q8—42no—
636Q8—42no—
637Q8—84no—
638Q8—88no—
639Q8—88no—
640Q8—88no—
641Q8#53614yescomplemented
642D4—216no—
643D4—28no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—216no—
655C2×C4—24no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—24no—
662C2×C4—22no—
663C2×C4—24no—
664C2×C4—44no—
665C2×C4—48no—
666C2×C4—416no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—88no—
684C2×C4—88no—
685C2×C4#448116yes—
686C2×C4C2×#12012yes—
687C2×C4C2×#12012yes—
688C2×C4C2×#12012yes—
689C2×C4C2×#12012yes—
690C2×C4C2×#12012yes—
691C2×C4C22×SD1614yes—
692C2×C4C22×SD1612yes—
693C2×C4C22×SD1612yes—
694C2×C4C2×#13712yes—
695C2×C4C2×#13712yes—
696C2×C4C23×D411yescharacteristic
697C2×C4C23×D411yescharacteristic
698C2×C4C2×2−1+412yes—
699C2×C4#58512yes—
700C8—44no—
701C8—44no—
702C22—24no—
703C22—24no—
704C22—22no—
705C22—22no—
706C22—28no—
707C22—24no—
708C22—44no—
709C22—42no—
710C22—42no—
711C22C2×#44814yes—
712C22C22×#12011yescharacteristic
713C22#348912yes—
714C22C23×SD1611yescharacteristic
715C22C22×#13711yescharacteristic
716C22#358612yes—
717C4—24no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—24no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—44no—
726C4—42no—
727C4—42no—
728C4—44no—
729C4—84no—
730C4—84no—
731C4C2×#44814yes—
732C4C2×#44814yes—
733C4C2×#53611yescharacteristic
734C4#344911yescharacteristic
735C4#348912yes—
736C2—24no—
737C2—44no—
738C2#3530511yescharacteristic
739C2C22×#44811yescharacteristic
740C2#6037011yescharacteristic
741C2#6209912yes—
742C2#6211312yes—
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×#3229C2semidirect
3. C22×#475C2semidirect
5. #32954C2semidirect
6. #29688C2semidirect
7. #25884C2semidirect
8. #25837C2semidirect
9. #25702C2semidirect
10. #25647C2semidirect
11. #24097C2semidirect
12. #24041C2semidirect
13. #21885C2semidirect
15. #20734C2semidirect
641. Q8#536semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#412
C2×#32292
C22×#4752
#353162
#329542
#296882
#258842
#258372
#257022
#256472
#240972
#240412
#218852
#218472
#207342