Finite Group Databasebeta

← #6805857

Subgroups — #6805857

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#36573C211yesmaximal, characteristic, complemented
3#33226C211yesmaximal, characteristic, complemented
4#30886C214yesmaximal, complemented
5#30883C214yesmaximal, complemented
6#30220C211yesmaximal, characteristic, complemented
7#30219C211yesmaximal, characteristic, complemented
8#29191C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic, complemented
10#29144C211yesmaximal, characteristic, complemented
11#28443C211yesmaximal, characteristic
12#28443C211yesmaximal, characteristic
13#28438C211yesmaximal, characteristic, complemented
14#24737C214yesmaximal, complemented
15#24193C214yesmaximal, complemented
16#22269C211yesmaximal, characteristic
17#22268C211yesmaximal, characteristic, complemented
18C4×#492C211yesmaximal, characteristic, complemented
19#21339C211yesmaximal, characteristic, complemented
20C2×#556C2212yes—
21C2×#493C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#492C2211yescharacteristic
24C2×#488C2214yes—
25C2×#463C2214yes—
26C2×#463C2214yes—
27C2×#461C2214yes—
28C2×C4×SD16C2214yes—
29#2868C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2858C2211yescharacteristic
35#2580—21no—
36#2580—21no—
37#2580C2212yes—
38#2547C2212yes—
39#2545C2212yes—
40#2545C2212yes—
41#2538—21no—
42#2538—21no—
43#2512—21no—
44#2512—21no—
45#2512—21no—
46#2512—21no—
47#2454—21no—
48#2454—21no—
49#2454—21no—
50#2454—21no—
51#2454C2212yes—
52#2454C2212yes—
53#2354—21no—
54#2354—21no—
55#2349—21no—
56#2349—21no—
57#2349C2212yes—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C4×#127C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C42×Q8C2211yescharacteristic
65#2066C2214yes—
66#2065C2214yes—
67#2043C2214yes—
68#2043C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2026C2214yes—
72#2013C2214yes—
73#2012C2214yes—
74#2012C2214yes—
75#1984C2212yes—
76#1968C2214yes—
77#1958C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1956C2211yescharacteristic
88#1939C2212yes—
89#1916C2212yes—
90#1916C2212yes—
91#1880C2214yes—
92#1879C2214yes—
93#1879C2214yes—
94#1877C2214yes—
95#1865C2214yes—
96#1864C2214yes—
97#1844C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1805C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108C22×SD16C2314yes—
109C22×D8C2314yes—
110#556—48no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#127C2311yescharacteristic
117C2×#126—22no—
118C2×#126—22no—
119C2×#125—21no—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#124C2312yes—
123C2×#124C2312yes—
124C2×#124C2312yes—
125C2×#124C2312yes—
126C2×C22⋊Q8—22no—
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8D411yescharacteristic
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4—21no—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×#117—22no—
145#493—24no—
146#492—24no—
147#492—24no—
148#488—48no—
149#463—48no—
150#463—48no—
151#461—48no—
152C4×SD16—48no—
153C2×#106C2314yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2312yes—
171C2×C4×C8C2312yes—
172C2×C4×C8C2311yescharacteristic
173#399—22no—
174#399—22no—
175#399—22no—
176#397—22no—
177#397—22no—
178#396—22no—
179#396—22no—
180#396—22no—
181#396—22no—
182#396—22no—
183#396—22no—
184#393—22no—
185#391C2312yes—
186#391C2312yes—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390—21no—
203#390C2312yes—
204#390C2312yes—
205#390C2312yes—
206#390C2312yes—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389C2312yes—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#388—22no—
224#388—22no—
225#388—22no—
226#388—22no—
227#385—22no—
228#385—22no—
229#385—22no—
230#385—22no—
231#385—22no—
232#385—22no—
233#384—21no—
234#384—21no—
235#384—21no—
236#384—21no—
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2311yescharacteristic
240#384C2311yescharacteristic
241#382—21no—
242#382—21no—
243#382C2312yes—
244#380—22no—
245#380—22no—
246#380—22no—
247#380—22no—
248#380—22no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4D411yescharacteristic
259C4×C4⋊C4C2311yescharacteristic
260C4×C4⋊C4C2311yescharacteristic
261C4×C4⋊C4C2311yescharacteristic
262C2×#95—22no—
263C43C2311yescharacteristic
264C43C2311yescharacteristic
265C43C2311yescharacteristic
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8C2×D411yescharacteristic
276C22×Q8C2412yes—
277C22×Q8C2412yes—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C22×D4C2412yes—
284C23×C4—22no—
285C23×C4—22no—
286C23×C4—22no—
287C23×C4—22no—
288C2×SD16—28no—
289C2×SD16—28no—
290C2×SD16—28no—
291C2×SD16—44no—
292C2×SD16—44no—
293C2×SD16—44no—
294C2×D8—28no—
295C2×D8—44no—
296C2×D8—44no—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—44no—
306C4⋊Q8—44no—
307#127—22no—
308#127—22no—
309#127—22no—
310#127#4114yes—
311#126—48no—
312#126—48no—
313#125—44no—
314#125—82no—
315#125—82no—
316#124—28no—
317#124—44no—
318#124—44no—
319#124—44no—
320C22⋊Q8—48no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—28no—
324C4×Q8—42no—
325C4×Q8—44no—
326C4×D4—28no—
327C4×D4—44no—
328C4×D4—44no—
329C4×D4—44no—
330C4×D4—44no—
331C4×D4—44no—
332C4×D4—44no—
333C4×D4—44no—
334C4×D4—82no—
335C4×D4—82no—
336C4×D4—82no—
337C4×D4—82no—
338#117—48no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416C2×C42C2411yescharacteristic
417#106—44no—
418C4⋊C8—24no—
419C4⋊C8—24no—
420C4⋊C8—24no—
421#103—28no—
422#103—44no—
423#103—44no—
424#103—44no—
425#103—44no—
426#103—44no—
427#102—28no—
428#102—44no—
429#102—44no—
430#102—44no—
431C4×C8—24no—
432C4×C8—22no—
433C4×C8—22no—
434C4×C8—22no—
435C4×C8—24no—
436C4×C8—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453C24—22no—
454C24—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—24no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—41no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—22no—
469C2×D4—24no—
470C2×D4—24no—
471C2×D4—24no—
472C2×D4—24no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4C2×#4114yes—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C2511yescharacteristic
519SD16—416no—
520SD16—88no—
521D8—88no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8H(5,𝔽2)14yes—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C22⋊C4—82no—
587C22⋊C4—82no—
588C22⋊C4—82no—
589C22⋊C4—82no—
590C22⋊C4—82no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42—44no—
608C42—44no—
609C42#13514yes—
610C42C2×#4112yes—
611C42H(5,𝔽2)14yes—
612C422−1+412yes—
613C23—24no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23C23×D411yescharacteristic
628Q8—44no—
629Q8—44no—
630Q8—44no—
631Q8—44no—
632Q8—44no—
633D4—28no—
634D4—48no—
635D4—48no—
636D4—44no—
637D4—44no—
638D4—44no—
639D4—84no—
640D4—84no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—88no—
692C2×C4—164no—
693C2×C4—164no—
694C2×C4—164no—
695C2×C4#53512yes—
696C2×C4#53512yes—
697C2×C4#54812yes—
698C2×C4#54812yes—
699C2×C4C2×#13512yes—
700C2×C4C2×#13512yes—
701C2×C4C2×#13512yes—
702C2×C4#57512yes—
703C2×C4#57712yes—
704C2×C4#57712yes—
705C2×C4C22×#4111yescharacteristic
706C2×C4C2×H(5,𝔽2)12yes—
707C2×C4C2×H(5,𝔽2)12yes—
708C2×C4C2×2−1+411yescharacteristic
709C8—24no—
710C8—42no—
711C8—42no—
712C8—42no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22C2×#53511yescharacteristic
728C22C2×#54811yescharacteristic
729C22#349811yescharacteristic
730C22C22×#13511yescharacteristic
731C22C2×#57511yescharacteristic
732C22C2×#57711yescharacteristic
733C22C2×#57711yescharacteristic
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—41no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4#336014yes—
753C4#337614yes—
754C4#351912yes—
755C4#354812yes—
756C2—84no—
757C2—162no—
758C2#3916911yescharacteristic
759C2#6078211yescharacteristic
760C2#6082011yescharacteristic
761C2C2×#336011yescharacteristic
762C2C2×#337611yescharacteristic
763C2#6241411yescharacteristic
764C2#6242711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#493C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. #36573C2semidirect
3. #33226C2semidirect
4. #30886C2semidirect
5. #30883C2semidirect
6. #30220C2semidirect
7. #30219C2semidirect
8. #29191C2semidirect
9. #29190C2semidirect
10. #29144C2semidirect
13. #28438C2semidirect
14. #24737C2semidirect
15. #24193C2semidirect
17. #22268C2semidirect
18. C4×#492C2semidirect
19. #21339C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#365732
#332262
#308862
#308832
#302202
#302192
#291912
#291902
#291442
#284432
#284382
#247372
#241932
#222692
#222682
C4×#4922
#213392