Finite Group Databasebeta

← #6763232

Subgroups — #6763232

Order 512.

1521 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C211yesmaximal, characteristic
2#36129C211yesmaximal, characteristic
3#33207C211yesmaximal, characteristic, complemented
4#32131C214yesmaximal, complemented
5#31429C214yesmaximal, complemented
6#30229C211yesmaximal, characteristic, complemented
7#29876C211yesmaximal, characteristic, complemented
8#29177C211yesmaximal, characteristic, complemented
9#28688C211yesmaximal, characteristic, complemented
10#27961C211yesmaximal, characteristic, complemented
11#27875C211yesmaximal, characteristic, complemented
12#26000C214yesmaximal, complemented
13#25813C211yesmaximal, characteristic, complemented
14#25812C211yesmaximal, characteristic, complemented
15#24943C214yesmaximal
16#22247C211yesmaximal, characteristic, complemented
17#21835C211yesmaximal, characteristic, complemented
18C2×#1990C211yesmaximal, characteristic, complemented
19#21101C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#500C2211yescharacteristic
24C2×#498C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2570—21no—
34#2550C2212yes—
35#2511—21no—
36#2507C2212yes—
37#2497C2212yes—
38#2492—21no—
39#2487C2212yes—
40#2451—21no—
41#2446—21no—
42#2442—21no—
43#2442C2212yes—
44#2441—21no—
45#2439C2212yes—
46#2429—21no—
47#2423—21no—
48#2417—21no—
49#2417—21no—
50#2409—21no—
51#2360—21no—
52#2360—21no—
53#2352—21no—
54#2352C2212yes—
55#2345C2212yes—
56Q8×C22⋊C4—21no—
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59#2285C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C42×D4C2211yescharacteristic
62#2101C2214yes—
63#2101C2214yes—
64#2100C2214yes—
65#2088C2214yes—
66#2086C2214yes—
67#2085C2214yes—
68#2077C2214yes—
69#2075C2214yes—
70#2059C2214yes—
71#2057C2214yes—
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2214yes—
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1990C2211yescharacteristic
83#1989C2214yes—
84#1949C2214yes—
85#1941C2214yes—
86#1923C2212yes—
87#1920C2214yes—
88#1919C2214yes—
89#1918C2214yes—
90#1918C2214yes—
91#1916C2212yes—
92#1869C2212yes—
93#1854C2214yes—
94#1852C2214yes—
95#1850C2212yes—
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1825C2214yes—
100#1824C2214yes—
101#1807C2214yes—
102C4×C4⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1744C2211yescharacteristic
106C2×#340C2214yes—
107C23×Q8—21no—
108C23×C8C2312yes—
109#533—24no—
110#533—42no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—42no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2311yescharacteristic
120C2×#125—21no—
121C2×#125C2312yes—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×D4D411yescharacteristic
133C2×C4×D4D411yescharacteristic
134C2×C4×D4C2311yescharacteristic
135C2×#117—21no—
136C2×#117C2312yes—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#500—24no—
145#500—42no—
146#498—42no—
147#431—42no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C4×C8—42no—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2312yes—
163#400—22no—
164#400—22no—
165#400—22no—
166#399—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#396—22no—
172#396—22no—
173#395—22no—
174#393—22no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391—22no—
186#391—22no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—22no—
196#389—22no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#388—22no—
206#388—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#386—22no—
214#386—22no—
215#385—22no—
216#385—22no—
217#384—21no—
218#384—21no—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2312yes—
224#382—21no—
225#382—21no—
226#382C2312yes—
227#382C2312yes—
228#380—21no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—22no—
234#380—22no—
235#380C2312yes—
236#380C2312yes—
237#380C2312yes—
238#380C2312yes—
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C22⋊C4—22no—
250C4×C22⋊C4—22no—
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—22no—
280C22×Q8—44no—
281C22×Q8—44no—
282C22×Q8—82no—
283C22×D4C2×D411yescharacteristic
284C23×C4—22no—
285C23×C4—22no—
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2411yescharacteristic
291C23×C4C2411yescharacteristic
292C22×C8—24no—
293C22×C8—24no—
294C22×C8—42no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—24no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—42no—
308C4⋊Q8—44no—
309C4⋊Q8—44no—
310C4⋊Q8#4114yes—
311#125—24no—
312#125—42no—
313#125—82no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C22⋊Q8—44no—
317C22⋊Q8—44no—
318C22⋊Q8—82no—
319C22⋊Q8—82no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8—82no—
323C4×Q8—82no—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—24no—
327#117—24no—
328#117—42no—
329#117—82no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—24no—
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—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
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—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4—82no—
362C2×C4⋊C4—82no—
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
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×C22⋊C4C2×D411yescharacteristic
381C2×C22⋊C4C2×D411yescharacteristic
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407#107—28no—
408#107—44no—
409#106—28no—
410#106—44no—
411C4⋊C8—24no—
412C4⋊C8—42no—
413C22⋊C8—24no—
414C22⋊C8—24no—
415C22⋊C8—42no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8—42no—
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#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447C24C22×D411yescharacteristic
448C24C22×D411yescharacteristic
449C2×Q8—24no—
450C2×Q8—44no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—82no—
464C2×Q8—816no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
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—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4#13514yes—
500C22×C4#13714yes—
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×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C42−1+414yes—
514C22×C4C2511yescharacteristic
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—48no—
527C2×C82−1+414yes—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
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—44no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4C2×#4114yes—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—21no—
591C42—21no—
592C42—21no—
593C42—21no—
594C42—22no—
595C42—22no—
596C42—44no—
597C42—82no—
598C42—82no—
599C42—82no—
600C42C2×#4112yes—
601C42H(5,𝔽2)12yes—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23#53612yes—
614C23C2×#13512yes—
615C23C2×#13712yes—
616C23C23×D411yescharacteristic
617Q8—48no—
618Q8—88no—
619Q8—84no—
620Q8—84no—
621Q8—84no—
622D4—44no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—24no—
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—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
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—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—88no—
683C2×C4—164no—
684C2×C4#53612yes—
685C2×C4#54612yes—
686C2×C4#54612yes—
687C2×C4C2×#13512yes—
688C2×C4C2×#13512yes—
689C2×C4C2×#13712yes—
690C2×C4C2×#13712yes—
691C2×C4#57612yes—
692C2×C4#57712yes—
693C2×C4C22×#4111yescharacteristic
694C2×C4C2×H(5,𝔽2)11yescharacteristic
695C2×C4C2×2−1+412yes—
696C2×C4C2×2−1+412yes—
697C8—24no—
698C8—42no—
699C8—42no—
700C8—42no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22#323114yes—
715C22C2×#53611yescharacteristic
716C22C2×#54611yescharacteristic
717C22#349911yescharacteristic
718C22C22×#13511yescharacteristic
719C22C22×#13711yescharacteristic
720C22C2×#57611yescharacteristic
721C22C2×#57711yescharacteristic
722C4—22no—
723C4—22no—
724C4—22no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—44no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4#329714yes—
741C4#331614yes—
742C4#352312yes—
743C4#353812yes—
744C2—24no—
745C2—81no—
746C2#3808111yescharacteristic
747C2C2×#323111yescharacteristic
748C2#6087711yescharacteristic
749C2C2×#329711yescharacteristic
750C2C2×#331611yescharacteristic
751C2#6247011yescharacteristic
752C2#6248011yescharacteristic
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. #33207C2semidirect
4. #32131C2semidirect
5. #31429C2semidirect
6. #30229C2semidirect
7. #29876C2semidirect
8. #29177C2semidirect
9. #28688C2semidirect
10. #27961C2semidirect
11. #27875C2semidirect
12. #26000C2semidirect
13. #25813C2semidirect
14. #25812C2semidirect
16. #22247C2semidirect
17. #21835C2semidirect
18. C2×#1990C2semidirect

Maximal subgroups

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

subgroupindex
#361892
#361292
#332072
#321312
#314292
#302292
#298762
#291772
#286882
#279612
#278752
#260002
#258132
#258122
#249432
#222472
#218352
C2×#19902
#211012