Finite Group Databasebeta

← #6804371

Subgroups — #6804371

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic, complemented
2#36573C211yesmaximal, characteristic, complemented
3#33277C211yesmaximal, characteristic, complemented
4#32597C214yesmaximal, complemented
5#31162C214yesmaximal, complemented
6#30219C211yesmaximal, characteristic, complemented
7#30206C211yesmaximal, characteristic, complemented
8#29193C211yesmaximal, characteristic, complemented
9#29177C211yesmaximal, characteristic, complemented
10#28615C211yesmaximal, characteristic
11#28613C211yesmaximal, characteristic, complemented
12#28514C211yesmaximal, characteristic, complemented
13#28443C211yesmaximal, characteristic
14#26900C214yesmaximal, complemented
15#26168C214yesmaximal, complemented
16#22269C211yesmaximal, characteristic
17#22220C211yesmaximal, characteristic, complemented
18C4×#492C211yesmaximal, characteristic, complemented
19#21339C211yesmaximal, characteristic, complemented
20C2×#493C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#478C2214yes—
24C2×#463C2214yes—
25C2×C4×SD16C2214yes—
26#2862C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2592—21no—
33#2592C2212yes—
34#2580—21no—
35#2580—21no—
36#2580—21no—
37#2580C2212yes—
38#2513—21no—
39#2513—21no—
40#2454—21no—
41#2454—21no—
42#2454—21no—
43#2454—21no—
44#2454C2212yes—
45#2454C2212yes—
46#2448—21no—
47#2448C2212yes—
48#2447—21no—
49#2447C2212yes—
50#2367—21no—
51#2367C2212yes—
52#2357—21no—
53#2349—21no—
54#2349—21no—
55#2349C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C42×Q8C2211yescharacteristic
63#2092C2214yes—
64#2088C2214yes—
65#2074C2214yes—
66#2068C2214yes—
67#2055C2214yes—
68#2054C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2026C2214yes—
72#2012C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1984C2212yes—
77#1968C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1939C2212yes—
86#1934C2214yes—
87#1934C2214yes—
88#1925C2214yes—
89#1919C2214yes—
90#1916C2212yes—
91#1916C2212yes—
92#1879C2214yes—
93#1879C2214yes—
94#1879C2214yes—
95#1877C2214yes—
96#1863C2214yes—
97#1862C2214yes—
98#1845C2211yescharacteristic
99#1844C2211yescharacteristic
100#1834C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108C22×D8C2314yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×#127C2311yescharacteristic
112C2×#127C2311yescharacteristic
113C2×#125—21no—
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#124—21no—
117C2×#124—21no—
118C2×#124C2312yes—
119C2×#124C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137#493—24no—
138#492—24no—
139#486—24no—
140#478—48no—
141#463—48no—
142C4×SD16—48no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2312yes—
162#399—22no—
163#399—22no—
164#399—22no—
165#399—22no—
166#397—22no—
167#397—22no—
168#397—22no—
169#397—22no—
170#396—22no—
171#396—22no—
172#393—22no—
173#393—22no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390C2312yes—
195#390C2312yes—
196#390C2312yes—
197#390C2312yes—
198#390C2312yes—
199#390C2312yes—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#389C2312yes—
216#388—22no—
217#388—22no—
218#388—22no—
219#388—22no—
220#388—22no—
221#388—22no—
222#385—22no—
223#385—22no—
224#385—22no—
225#385—22no—
226#385—22no—
227#385—22no—
228#385—22no—
229#385—22no—
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#382—21no—
235#382—21no—
236#382—21no—
237#382—21no—
238#382C2312yes—
239#381—22no—
240#380—22no—
241#380—22no—
242#380—22no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2311yescharacteristic
256C4×C22⋊C4—22no—
257C2×#95—22no—
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8C2×D411yescharacteristic
274C22×Q8C2412yes—
275C22×D4—21no—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4C2412yes—
280C22×D4C2412yes—
281C23×C4—22no—
282C23×C4—22no—
283C23×C4—22no—
284C23×C4—22no—
285C2×SD16—28no—
286C2×SD16—28no—
287C2×SD16—44no—
288C2×D8—28no—
289C2×D8—44no—
290C2×D8—44no—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299#127—22no—
300#127—22no—
301#127—22no—
302#127#4114yes—
303#125—44no—
304#125—82no—
305#125—82no—
306#124—44no—
307#124—44no—
308#124—44no—
309#124—82no—
310C4×Q8—24no—
311C4×Q8—42no—
312C4×Q8—42no—
313C4×Q8—44no—
314C4×Q8—82no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—82no—
323C4×D4—82no—
324C4×D4—82no—
325C4×D4—82no—
326C4×D4—82no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401#107—44no—
402#106—28no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8—42no—
406#103—44no—
407#103—44no—
408#103—44no—
409#102—28no—
410#102—28no—
411#102—44no—
412#102—44no—
413#102—44no—
414C4×C8—24no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8—22no—
419C4×C8—42no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440C24—22no—
441C24—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—41no—
448C2×Q8—41no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4C2×#4114yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C2511yescharacteristic
505SD16—416no—
506D8—88no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8H(5,𝔽2)14yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—82no—
591C42—82no—
592C42#13514yes—
593C42C2×#4112yes—
594C422−1+412yes—
595C23—24no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23C23×D411yescharacteristic
610Q8—44no—
611Q8—44no—
612Q8—44no—
613Q8—84no—
614D4—28no—
615D4—48no—
616D4—48no—
617D4—44no—
618D4—44no—
619D4—44no—
620D4—84no—
621D4—84no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
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—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—164no—
675C2×C4—164no—
676C2×C4—164no—
677C2×C4—164no—
678C2×C4#54812yes—
679C2×C4#54812yes—
680C2×C4C2×#13512yes—
681C2×C4C2×#13512yes—
682C2×C4C2×#13512yes—
683C2×C4#57512yes—
684C2×C4#57612yes—
685C2×C4#57712yes—
686C2×C4C22×#4111yescharacteristic
687C2×C4C2×H(5,𝔽2)12yes—
688C2×C4C2×2−1+411yescharacteristic
689C2×C4#58512yes—
690C8—24no—
691C8—42no—
692C8—42no—
693C8—42no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22#344911yescharacteristic
709C22C2×#54811yescharacteristic
710C22#349811yescharacteristic
711C22C22×#13511yescharacteristic
712C22C2×#57511yescharacteristic
713C22C2×#57611yescharacteristic
714C22C2×#57711yescharacteristic
715C4—22no—
716C4—22no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4#336114yes—
734C4#337614yes—
735C4#351312yes—
736C4#355412yes—
737C2—84no—
738C2—162no—
739C2#3916111yescharacteristic
740C2#6094411yescharacteristic
741C2#6100611yescharacteristic
742C2C2×#336111yescharacteristic
743C2C2×#337611yescharacteristic
744C2#6241411yescharacteristic
745C2#6242611yescharacteristic
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. #36609C2semidirect
2. #36573C2semidirect
3. #33277C2semidirect
4. #32597C2semidirect
5. #31162C2semidirect
6. #30219C2semidirect
7. #30206C2semidirect
8. #29193C2semidirect
9. #29177C2semidirect
11. #28613C2semidirect
12. #28514C2semidirect
14. #26900C2semidirect
15. #26168C2semidirect
17. #22220C2semidirect
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
#366092
#365732
#332772
#325972
#311622
#302192
#302062
#291932
#291772
#286152
#286132
#285142
#284432
#269002
#261682
#222692
#222202
C4×#4922
#213392