Finite Group Databasebeta

← #6487675

Subgroups — #6487675

Order 512.

2084 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3117C212yesmaximal, complemented
2#35103C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#32909C212yesmaximal, complemented
5#32906C211yesmaximal, characteristic, complemented
6#32903C211yesmaximal, characteristic
7#29761C212yesmaximal, complemented
8#29757C211yesmaximal, characteristic, complemented
9#29755C211yesmaximal, characteristic
10#24826C214yesmaximal, complemented
11#24824C214yesmaximal, complemented
12#24741C214yesmaximal, complemented
13#24739C214yesmaximal, complemented
14#21955C211yesmaximal, characteristic, complemented
15#21867C212yesmaximal, complemented
16C2×#529C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#3117—216no—
21C2×#484C2212yes—
22C2×#481C2212yes—
23C2×#479C2214yes—
24C2×#464C2214yes—
25C2×#464C2214yes—
26C2×#462C2212yes—
27C2×#460C2212yes—
28C2×#451C2212yes—
29C2×#448C2212yes—
30C2×#441C2214yes—
31C2×#440C2214yes—
32C2×#432C2212yes—
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2850C2212yes—
36#2618—21no—
37#2618—21no—
38#2601—22no—
39#2601—22no—
40#2566C2212yes—
41#2565C2212yes—
42#2545C2212yes—
43#2544C2212yes—
44Q8×C4⋊C4—22no—
45Q8×C4⋊C4—22no—
46Q8×C4⋊C4C2214yes—
47Q8×C4⋊C4C2214yes—
48#2325—21no—
49#2325—21no—
50C2×#389—21no—
51C2×#389—21no—
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#384—21no—
55C2×#384C2211yescharacteristic
56C2×C4×C4⋊C4—21no—
57C2×C4×C4⋊C4C2211yescharacteristic
58#2085C2214yes—
59#2084C2212yes—
60#2083C2212yes—
61#2068C2214yes—
62#2067C2214yes—
63#2066C2214yes—
64#2065C2214yes—
65#2043C2212yes—
66#2042C2214yes—
67#2041C2212yes—
68#1992C2214yes—
69#1959C2214yes—
70#1940C2214yes—
71#1940C2214yes—
72#1880C2214yes—
73#1880C2214yes—
74#1878C2214yes—
75#1878C2214yes—
76#1874C2212yes—
77#1874C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1805C2214yes—
81#1803C2214yes—
82#1803C2212yes—
83#1803C2212yes—
84#1802C2214yes—
85#1797C2211yescharacteristic
86#1797C2211yescharacteristic
87#1797C2212yes—
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90C22×SD16C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93#529—28no—
94#529—28no—
95C2×#126—22no—
96C2×#125C2312yes—
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×C22⋊Q8C2312yes—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—24no—
109C2×C4×Q8—44no—
110C2×C4×Q8—44no—
111C2×C4×Q8D418yescomplemented
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2314yes—
117C2×C4×D4—22no—
118C2×C4×D4C2312yes—
119C2×#117—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C42—21no—
133C22×C42—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#484—28no—
139#481—28no—
140#479—216no—
141#464—216no—
142#464—48no—
143#462—44no—
144#460—44no—
145#451—28no—
146#448—28no—
147#441—216no—
148#440—216no—
149#432—28no—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2312yes—
166C2×#97C2314yes—
167#398—22no—
168#398—22no—
169#396—22no—
170#396—22no—
171#396—22no—
172#396—22no—
173#394—22no—
174#394—22no—
175#391C2312yes—
176#391C2312yes—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—21no—
185#389—21no—
186#389—24no—
187#389—24no—
188#389—24no—
189#389—24no—
190#389C2312yes—
191#389C2314yes—
192#389C2312yes—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#387—22no—
198#387—22no—
199#386—22no—
200#386—22no—
201#386—22no—
202#386—22no—
203#385—22no—
204#385—22no—
205#384—24no—
206#384—24no—
207#384—22no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2314yes—
211#382C2312yes—
212#380—21no—
213#380—21no—
214C4×C4⋊C4—24no—
215C4×C4⋊C4—24no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95D411yescharacteristic
223C2×#95D411yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—24no—
232C22×Q8—24no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2×D414yes—
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×Q16—28no—
255C2×Q16—28no—
256C2×Q16—28no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—28no—
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—42no—
263#127—42no—
264#126—48no—
265#125—24no—
266#125—24no—
267#124—24no—
268#124—24no—
269#124—44no—
270#123—48no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273#121—24no—
274#121—24no—
275#121—44no—
276#121—44no—
277C4×Q8—216no—
278C4×Q8—28no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—28no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—44no—
285C4×Q8—42no—
286C4×Q8—42no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—48no—
290C4×Q8C2×D4116yes—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—48no—
294#117—48no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C42—28no—
351C2×C42—24no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—42no—
357C2×C42—42no—
358C2×C42—44no—
359C2×C42—44no—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D414yes—
365C2×C42#4114yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370#107—28no—
371#106—28no—
372C4⋊C8—28no—
373C4⋊C8—42no—
374C4⋊C8—42no—
375C4⋊C8—44no—
376#103—28no—
377#103—28no—
378#103—28no—
379#103—44no—
380#103—44no—
381#102—28no—
382#102—44no—
383#102—44no—
384C22⋊C8—24no—
385C22⋊C8—24no—
386#97—28no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—22no—
401C24C22×D411yescharacteristic
402C2×Q8—24no—
403C2×Q8—216no—
404C2×Q8—24no—
405C2×Q8—216no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8C22×D414yes—
415C2×Q8H(5,𝔽2)14yes—
416C2×Q82−1+414yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—81no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4#12114yes—
470C22×C4#12114yes—
471C22×C4#13714yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484Q16—416no—
485SD16—416no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C4⋊C4—216no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—81no—
525C4⋊C4—81no—
526C4⋊C4C22×D414yes—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—216no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—41no—
543C42—41no—
544C42—42no—
545C42—44no—
546C42—44no—
547C42—48no—
548C42—44no—
549C42#12118yes—
550C42C22×D414yes—
551C42C2×#4114yes—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C2×#12112yes—
564C23C2×#13712yes—
565C23#57512yes—
566C23C23×D411yescharacteristic
567Q8—216no—
568Q8—24no—
569Q8—24no—
570Q8—24no—
571Q8—44no—
572Q8—44no—
573Q8#53718yes—
574D4—44no—
575D4—44no—
576D4—88no—
577D4—162no—
578D4—162no—
579C2×C4—216no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—88no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4C2×#12114yes—
636C2×C4C2×#12114yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#13712yes—
639C2×C4C2×#13712yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)12yes—
643C2×C4C2×2−1+412yes—
644C8—44no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22#303714yes—
660C22C22×#12111yescharacteristic
661C22#348411yescharacteristic
662C22#348611yescharacteristic
663C22C22×#13711yescharacteristic
664C22C2×#57511yescharacteristic
665C22#358511yescharacteristic
666C22#358611yescharacteristic
667C4—24no—
668C4—22no—
669C4—22no—
670C4—28no—
671C4—22no—
672C4—24no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—41no—
677C4—41no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4C2×#53712yes—
684C4#349212yes—
685C2—24no—
686C2—162no—
687C2#3522311yescharacteristic
688C2C2×#303711yescharacteristic
689C2#6042511yescharacteristic
690C2#6157411yescharacteristic
691C2#6157711yescharacteristic
692C2#6183011yescharacteristic
693C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3117C2semidirect
3. C2×Q8×C4⋊C4C2semidirect
4. #32909C2semidirect
5. #32906C2semidirect
7. #29761C2semidirect
8. #29757C2semidirect
10. #24826C2semidirect
11. #24824C2semidirect
12. #24741C2semidirect
13. #24739C2semidirect
14. #21955C2semidirect
15. #21867C2semidirect
111. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31172
#351032
C2×Q8×C4⋊C42
#329092
#329062
#329032
#297612
#297572
#297552
#248262
#248242
#247412
#247392
#219552
#218672