Finite Group Databasebeta

← #6615418

Subgroups — #6615418

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36210C211yesmaximal, characteristic
2#36151C211yesmaximal, characteristic
3#33290C211yesmaximal, characteristic, complemented
4#33209C211yesmaximal, characteristic, complemented
5#33207C211yesmaximal, characteristic, complemented
6#28848C214yesmaximal, complemented
7#28844C214yesmaximal, complemented
8#28688C211yesmaximal, characteristic, complemented
9#28515C211yesmaximal, characteristic, complemented
10#27877C211yesmaximal, characteristic, complemented
11#27876C211yesmaximal, characteristic, complemented
12#25674C211yesmaximal, characteristic, complemented
13#25672C211yesmaximal, characteristic, complemented
14#25343C214yesmaximal
15#22967C214yesmaximal, complemented
16C4×#487C211yesmaximal, characteristic, complemented
17#21400C211yesmaximal, characteristic, complemented
18#21400C211yesmaximal, characteristic, complemented
19C4×C8×D4C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#479C2214yes—
26C2×C8×D4C2212yes—
27C22×C4×C8C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2562C2212yes—
36#2536C2212yes—
37#2514—21no—
38#2509—21no—
39#2495C2212yes—
40#2494—21no—
41#2494C2212yes—
42#2489—21no—
43#2489—21no—
44#2485—21no—
45#2453—21no—
46#2453—21no—
47#2421—21no—
48#2421C2212yes—
49#2421C2212yes—
50#2415—21no—
51#2415—21no—
52#2415—21no—
53#2370C2212yes—
54#2362—21no—
55#2362—21no—
56#2362—21no—
57#2351C2212yes—
58#2347—21no—
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61#2285C2211yescharacteristic
62#2285C2211yescharacteristic
63C42×D4C2211yescharacteristic
64#2097C2214yes—
65#2085C2214yes—
66#2077C2214yes—
67#2075C2214yes—
68#2036C2214yes—
69#2035C2214yes—
70#1991C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1949C2214yes—
82#1942C2214yes—
83#1941C2214yes—
84#1933C2214yes—
85#1919C2214yes—
86#1918C2214yes—
87#1892C2214yes—
88#1892C2214yes—
89#1865C2214yes—
90#1863C2214yes—
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1835C2214yes—
94#1834C2211yescharacteristic
95#1808C2214yes—
96#1807C2214yes—
97C8×C4⋊C4C2212yes—
98C4×C4⋊C8C2211yescharacteristic
99C4×#103C2214yes—
100C4×#103C2214yes—
101C4×#103C2214yes—
102C8×C22⋊C4C2212yes—
103C8×C22⋊C4C2212yes—
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×C8C2312yes—
108C23×C8C2312yes—
109#533—24no—
110#533—24no—
111#533—42no—
112#533—42no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×#125—21no—
118C2×#125—21no—
119C2×#125C2312yes—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×D4D411yescharacteristic
135C2×C4×D4D411yescharacteristic
136C2×C4×D4C2311yescharacteristic
137C2×#117—21no—
138C2×#117—21no—
139C2×#117—21no—
140C2×#117—21no—
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#500—42no—
144#498—42no—
145#487—24no—
146#479—216no—
147C8×D4—28no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C4×C8—42no—
160C2×C4×C8—42no—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2312yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2312yes—
169#401—22no—
170#400—22no—
171#400—22no—
172#400—22no—
173#400—22no—
174#398—22no—
175#397—22no—
176#397—22no—
177#397—22no—
178#397—22no—
179#394—22no—
180#394—22no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391—21no—
189#391—21no—
190#391C2312yes—
191#389—22no—
192#389—22no—
193#389—22no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#388—22no—
215#388—22no—
216#388—22no—
217#388—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#383—22no—
228#382—21no—
229#382—21no—
230#382—22no—
231#382C2312yes—
232#382C2312yes—
233#382C2312yes—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380C2312yes—
241#380C2312yes—
242#380C2312yes—
243#380C2312yes—
244#380C2312yes—
245#380C2312yes—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—22no—
252C4×C4⋊C4—22no—
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2311yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4C2311yescharacteristic
264C4×C22⋊C4C2311yescharacteristic
265C43C2311yescharacteristic
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#340C2314yes—
274#336C2312yes—
275#336C2312yes—
276#336C2312yes—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—21no—
280C22×Q8C2412yes—
281C22×D4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C23×C4C2411yescharacteristic
288C22×C8—24no—
289C22×C8—24no—
290C22×C8—24no—
291C22×C8—24no—
292C22×C8—42no—
293C22×C8—42no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8#4114yes—
306#125—28no—
307#125—44no—
308#125—44no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—42no—
312C22⋊Q8—42no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C22⋊Q8—44no—
317C22⋊Q8—44no—
318C22⋊Q8—82no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—28no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8—82no—
327C4×D4—24no—
328C4×D4—24no—
329C4×D4—24no—
330#117—44no—
331#117—44no—
332#117—44no—
333#117—82no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
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⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C22⋊C4C2×D411yescharacteristic
381C2×C22⋊C4C2×D411yescharacteristic
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#107—28no—
407#107—44no—
408#106—28no—
409#106—44no—
410C4⋊C8#4118yes—
411#103—28no—
412#103—28no—
413#103—28no—
414#103—28no—
415C22⋊C8—24no—
416C22⋊C8—24no—
417C4×C8—24no—
418C4×C8—24no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—42no—
424C4×C8—42no—
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#95—22no—
435#95—22no—
436#95—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
453C24C22×D411yescharacteristic
454C24C22×D411yescharacteristic
455C2×Q8—22no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8C2×#4114yes—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4#13514yes—
498C22×C4#13514yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C2511yescharacteristic
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—48no—
522C2×C8—48no—
523C2×C8C2×#4114yes—
524C2×C8C2×#4114yes—
525C2×C82−1+414yes—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—21no—
587C42—21no—
588C42—21no—
589C42—21no—
590C42—22no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—82no—
596C42—82no—
597C42C2×#4114yes—
598C42C2×#4112yes—
599C42H(5,𝔽2)12yes—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23C2×#13512yes—
613C23C2×#13512yes—
614C23C23×D411yescharacteristic
615Q8—28no—
616Q8—44no—
617Q8—84no—
618Q8—84no—
619Q8—84no—
620D4—28no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—24no—
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—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
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—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4#53212yes—
680C2×C4#53212yes—
681C2×C4#54612yes—
682C2×C4#54612yes—
683C2×C4C2×#13512yes—
684C2×C4C2×#13512yes—
685C2×C4C2×#13512yes—
686C2×C4C2×#13512yes—
687C2×C4#57712yes—
688C2×C4#57812yes—
689C2×C4C22×#4111yescharacteristic
690C2×C4C22×#4112yes—
691C2×C4C2×H(5,𝔽2)11yescharacteristic
692C2×C4C2×2−1+412yes—
693C8—24no—
694C8—24no—
695C8—24no—
696C8—42no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22C2×#53211yescharacteristic
712C22C2×#54611yescharacteristic
713C22#349911yescharacteristic
714C22C22×#13511yescharacteristic
715C22C22×#13511yescharacteristic
716C22C2×#57711yescharacteristic
717C22C2×#57811yescharacteristic
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—44no—
727C4—41no—
728C4—41no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4#330014yes—
737C4#330114yes—
738C4#348212yes—
739C4#352012yes—
740C2—42no—
741C2—42no—
742C2#3640611yescharacteristic
743C2#6067311yescharacteristic
744C2#6072311yescharacteristic
745C2C2×#330011yescharacteristic
746C2C2×#330111yescharacteristic
747C2#6247011yescharacteristic
748C2#6247311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33290C2semidirect
4. #33209C2semidirect
5. #33207C2semidirect
6. #28848C2semidirect
7. #28844C2semidirect
8. #28688C2semidirect
9. #28515C2semidirect
10. #27877C2semidirect
11. #27876C2semidirect
12. #25674C2semidirect
13. #25672C2semidirect
15. #22967C2semidirect
16. C4×#487C2semidirect
17. #21400C2semidirect
18. #21400C2semidirect

Maximal subgroups

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

subgroupindex
#362102
#361512
#332902
#332092
#332072
#288482
#288442
#286882
#285152
#278772
#278762
#256742
#256722
#253432
#229672
C4×#4872
#214002
C4×C8×D42