Finite Group Databasebeta

← #6804538

Subgroups — #6804538

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic, complemented
3#33277C211yesmaximal, characteristic
4#32577C214yesmaximal, complemented
5#32487C214yesmaximal, complemented
6#30229C211yesmaximal, characteristic, complemented
7#30203C211yesmaximal, characteristic
8#29206C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic
10#29177C211yesmaximal, characteristic, complemented
11#28688C211yesmaximal, characteristic, complemented
12#28655C211yesmaximal, characteristic
13#28654C211yesmaximal, characteristic
14#27367C214yesmaximal, complemented
15#27129C214yesmaximal, complemented
16#22248C211yesmaximal, characteristic, complemented
17#22219C211yesmaximal, characteristic
18C4×#498C211yesmaximal, characteristic, complemented
19#21339C211yesmaximal, characteristic, complemented
20C2×#557—21no—
21C2×#500C2211yescharacteristic
22C2×#498C2211yescharacteristic
23C2×#486C2211yescharacteristic
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2858C2211yescharacteristic
29#2858C2211yescharacteristic
30#2592—21no—
31#2592—21no—
32#2580C2212yes—
33#2568—21no—
34#2565—21no—
35#2565—21no—
36#2546—21no—
37#2546C2212yes—
38#2513C2212yes—
39#2513C2212yes—
40#2506—21no—
41#2506—21no—
42#2506C2212yes—
43#2506C2212yes—
44#2448—21no—
45#2448—21no—
46#2447—21no—
47#2447—21no—
48#2367—21no—
49#2367—21no—
50#2365—21no—
51#2365C2212yes—
52#2357C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2214yes—
61#2101C2214yes—
62#2088C2214yes—
63#2085C2214yes—
64#2071C2214yes—
65#2067C2214yes—
66#2065C2214yes—
67#2065C2214yes—
68#2058C2214yes—
69#2023C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1984C2212yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1949C2214yes—
82#1948C2214yes—
83#1941C2214yes—
84#1939C2212yes—
85#1934C2214yes—
86#1934C2214yes—
87#1926C2214yes—
88#1926C2214yes—
89#1925C2214yes—
90#1921C2214yes—
91#1921C2214yes—
92#1919C2214yes—
93#1916C2212yes—
94#1916C2212yes—
95#1864C2214yes—
96#1864C2214yes—
97#1844C2211yescharacteristic
98#1844C2211yescharacteristic
99#1834C2211yescharacteristic
100C4×#106C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#557—48no—
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#125—22no—
117C2×#125—22no—
118C2×#125—21no—
119C2×#125—21no—
120C2×#124—21no—
121C2×#124—21no—
122C2×#124—21no—
123C2×#124—21no—
124C2×C4×Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8D411yescharacteristic
131C2×C4×Q8D411yescharacteristic
132C2×C4×Q8C2311yescharacteristic
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136#500—24no—
137#498—24no—
138#486—24no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2312yes—
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156#400—22no—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#397—22no—
164#397—22no—
165#397—22no—
166#397—22no—
167#393—22no—
168#391—22no—
169#391—22no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390C2312yes—
186#390C2312yes—
187#390C2312yes—
188#390C2312yes—
189#390C2312yes—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#385—22no—
219#385—22no—
220#384—21no—
221#384—21no—
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2312yes—
227#382—21no—
228#382—21no—
229#382C2312yes—
230#382C2312yes—
231#381—22no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4D411yescharacteristic
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2311yescharacteristic
250C4×C4⋊C4C2311yescharacteristic
251C4×C22⋊C4—22no—
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8—21no—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8C2×D411yescharacteristic
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C2412yes—
278C23×C4—22no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—48no—
291C4⋊Q8—44no—
292C4⋊Q8#4114yes—
293#127—22no—
294#127—22no—
295#125—44no—
296#125—48no—
297#125—48no—
298#125—44no—
299#124—44no—
300#124—44no—
301#124—82no—
302#124—82no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—42no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8—48no—
309C4×Q8—44no—
310C4×Q8—82no—
311C4×Q8—82no—
312C4×D4—44no—
313C4×D4—82no—
314C4×D4—82no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—28no—
390#106—28no—
391#106—44no—
392#106—44no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—42no—
396#102—28no—
397#102—28no—
398#102—44no—
399#102—44no—
400C4×C8—24no—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8—22no—
405C4×C8—42no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430C24—22no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—41no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2511yescharacteristic
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C82−1+414yes—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4C2×#4114yes—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—44no—
563C42—44no—
564C42—44no—
565C42—82no—
566C42—82no—
567C42#13514yes—
568C42AGL(1,ℤ/8)14yes—
569C42C2×#4112yes—
570C422−1+412yes—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23C23×D411yescharacteristic
579Q8—44no—
580Q8—48no—
581Q8—88no—
582Q8—84no—
583Q8—84no—
584D4—44no—
585D4—44no—
586D4—84no—
587D4—84no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
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—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—164no—
642C2×C4#54812yes—
643C2×C4#54812yes—
644C2×C4C2×#13512yes—
645C2×C4C2×#13512yes—
646C2×C4C2×#13512yes—
647C2×C4C2×AGL(1,ℤ/8)12yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4#57712yes—
651C2×C4#57712yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×2−1+412yes—
654C2×C4C2×2−1+411yescharacteristic
655C2×C4#58512yes—
656C8—24no—
657C8—42no—
658C8—42no—
659C8—42no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22#344911yescharacteristic
668C22C2×#54811yescharacteristic
669C22#349911yescharacteristic
670C22C22×#13511yescharacteristic
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C2×#57711yescharacteristic
673C22C2×#57711yescharacteristic
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—44no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4#336514yes—
694C4#337314yes—
695C4#351612yes—
696C4#355512yes—
697C2—162no—
698C2#3916211yescharacteristic
699C2#6094611yescharacteristic
700C2#6101211yescharacteristic
701C2C2×#336511yescharacteristic
702C2C2×#337311yescharacteristic
703C2#6247011yescharacteristic
704C2#6247711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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
2. #36560C2semidirect
4. #32577C2semidirect
5. #32487C2semidirect
6. #30229C2semidirect
8. #29206C2semidirect
10. #29177C2semidirect
11. #28688C2semidirect
14. #27367C2semidirect
15. #27129C2semidirect
16. #22248C2semidirect
18. C4×#498C2semidirect
19. #21339C2semidirect

Maximal subgroups

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

subgroupindex
#366092
#365602
#332772
#325772
#324872
#302292
#302032
#292062
#291902
#291772
#286882
#286552
#286542
#273672
#271292
#222482
#222192
C4×#4982
#213392