Finite Group Databasebeta

← #7563370

Subgroups — #7563370

Order 512.

2221 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D8C212yesmaximal, complemented
2#59817C211yesmaximal, characteristic, complemented
3D4×C4⋊Q8C211yesmaximal, characteristic, complemented
4#30946C214yesmaximal, complemented
5#28813C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28508C211yesmaximal, characteristic, complemented
8#26460C212yesmaximal, complemented
9#26368C214yesmaximal, complemented
10#26040C212yesmaximal, complemented
11#25838C212yesmaximal, complemented
12#25469C212yesmaximal, complemented
13#24326C212yesmaximal, complemented
14#23149C211yesmaximal, characteristic, complemented
15#22995C212yesmaximal
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×D4×Q8—21no—
19C2×D4×Q8C2212yes—
20C2×D4×Q8C2212yes—
21C22×C4⋊Q8—21no—
22Q8×D8—216no—
23C2×#500C2212yes—
24C2×#500C2214yes—
25C2×#474C2214yes—
26C2×#474C2214yes—
27C2×#474C2212yes—
28C2×#474C2212yes—
29C2×#474C2211yescharacteristic
30C2×#459C2211yescharacteristic
31C2×C8×Q8C2212yes—
32C2×C4×D8C2214yes—
33C2×C4×D8C2212yes—
34#2868C2211yescharacteristic
35#2857—21no—
36#2683—22no—
37#2551C2212yes—
38#2551C2212yes—
39#2551C2212yes—
40#2546C2212yes—
41#2538—21no—
42#2538C2212yes—
43#2452—22no—
44#2452C2214yes—
45#2440—22no—
46#2440—22no—
47#2434—22no—
48#2409—22no—
49#2366C2212yes—
50D4×C4⋊C4C2212yes—
51D4×C4⋊C4C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2093C2214yes—
57#2086C2214yes—
58#2083C2214yes—
59#2067C2214yes—
60#2057C2214yes—
61#2057C2214yes—
62#2057C2214yes—
63#2023C2214yes—
64#1990C2212yes—
65#1988C2211yescharacteristic
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1957C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1947C2211yescharacteristic
72#1947C2211yescharacteristic
73#1942C2214yes—
74#1939C2212yes—
75#1934C2214yes—
76#1925C2212yes—
77#1918C2212yes—
78#1916C2214yes—
79#1879C2212yes—
80#1879C2214yes—
81#1862C2212yes—
82#1862C2214yes—
83#1862C2214yes—
84#1862C2214yes—
85#1845C2212yes—
86#1802C2211yescharacteristic
87#1802C2211yescharacteristic
88C4×#107C2212yes—
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93C4×#102C2212yes—
94C23×Q8—22no—
95C23×Q8—22no—
96C22×D8C2312yes—
97D4×Q8—28no—
98D4×Q8—28no—
99D4×Q8—44no—
100D4×Q8—44no—
101D4×Q8—48no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—48no—
109C2×C4⋊Q8D411yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2314yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×#127C2311yescharacteristic
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—24no—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2314yes—
129C2×C4×D4C2314yes—
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C42—21no—
135#500—28no—
136#500—216no—
137#474—216no—
138#474—28no—
139#474—216no—
140#474—44no—
141#474D418yescomplemented
142#459Q818yescomplemented
143C8×Q8—28no—
144C4×D8—28no—
145C4×D8—216no—
146C2×#107C2312yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#107C2312yes—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#107C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2311yescharacteristic
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2314yes—
168C2×C4×C8C2312yes—
169#398—24no—
170#396—24no—
171#396—24no—
172#395—24no—
173#395—24no—
174#395—24no—
175#393—24no—
176#391—22no—
177#391—21no—
178#391D412yes—
179#390—22no—
180#390—21no—
181#390—21no—
182#390C2312yes—
183#390C2314yes—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—21no—
188#389D412yes—
189#389D412yes—
190#389C2312yes—
191#389C2314yes—
192#389C2314yes—
193#389C2314yes—
194#386—24no—
195#385—24no—
196#385—24no—
197#385—24no—
198#384—21no—
199#384—21no—
200#384D411yescharacteristic
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2312yes—
205#380—22no—
206#380—22no—
207#380—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2312yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—24no—
234C22×Q8—24no—
235C22×Q8—48no—
236C22×Q8—48no—
237C22×Q8—48no—
238C22×Q8—48no—
239C22×Q8—84no—
240C22×Q8—84no—
241C22×Q8C2×D412yes—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2412yes—
245C22×D4C2411yescharacteristic
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—24no—
251C23×C4—24no—
252C2×D8—24no—
253C2×D8—24no—
254C2×D8C2×Q818yescomplemented
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268C4⋊Q8—48no—
269C4⋊Q8—48no—
270C4⋊Q8—44no—
271C4⋊Q8—82no—
272C4⋊Q8D818yescomplemented
273C4⋊Q8C2×D414yes—
274#127—22no—
275#127C2×Q814yescomplemented
276C22⋊Q8—416no—
277C22⋊Q8—416no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—416no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—44no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—24no—
292C4×D4—22no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—48no—
296C4×D4—44no—
297C4×D4—82no—
298C4×D4C2×D414yes—
299C4×D4C2×Q814yescomplemented
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2414yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—28no—
355#107—24no—
356#107—28no—
357#107—28no—
358#107—24no—
359#107—44no—
360#107C2×D418yes—
361C4⋊C8—28no—
362C4⋊C8—22no—
363C4⋊C8—22no—
364C4⋊C8C2×D414yes—
365C4⋊C8C2×Q814yescomplemented
366#102—24no—
367#102—24no—
368#102—28no—
369#102—24no—
370#102—24no—
371#102—44no—
372#102C2×D418yes—
373#102C2×Q818yescomplemented
374C4×C8—28no—
375C4×C8—24no—
376C4×C8—24no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95C2×D414yes—
386#95C2×D414yes—
387C24—21no—
388C24—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×Q8—832no—
404C2×Q8—832no—
405C2×Q8C2×D818yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4C22×D412yes—
421C2×D4C22×Q814yes—
422C2×D4C22×Q812yes—
423C2×D42−1+412yes—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—48no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D414yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2511yescharacteristic
458D8—216no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8C22×D414yes—
470C2×C8C22×Q814yes—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4C2×D818yes—
502C4⋊C4C2×D818yes—
503C4⋊C4C22×D412yes—
504C4⋊C4C22×D414yes—
505C4⋊C4C22×Q812yes—
506C4⋊C42−1+412yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—84no—
516C22⋊C4—82no—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—24no—
521C42—21no—
522C42—21no—
523C42—41no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42C2×D814yes—
529C42C22×D412yes—
530C42C22×Q812yes—
531C42H(5,𝔽2)12yes—
532C422−1+412yes—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23C23×D411yescharacteristic
548Q8—216no—
549Q8—48no—
550Q8—816no—
551Q8—816no—
552Q8—88no—
553D4—24no—
554D4—22no—
555D4—22no—
556D4—24no—
557D4—22no—
558D4—48no—
559D4—42no—
560D4—42no—
561D4—84no—
562D4D4×Q814yescomplemented
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—48no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—816no—
602C2×C4—816no—
603C2×C4—88no—
604C2×C4D4212yes—
605C2×C4D4212yes—
606C2×C4D4×Q812yes—
607C2×C4D4×Q812yes—
608C2×C4C22×D812yes—
609C2×C4C22×D818yes—
610C2×C4C22×D812yes—
611C2×C4C22×D812yes—
612C2×C4#57612yes—
613C2×C4#57812yes—
614C2×C4C23×D411yescharacteristic
615C2×C4C23×Q811yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C8—24no—
621C8—24no—
622C8—42no—
623C8—42no—
624C22—24no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22C2×D4211yescharacteristic
639C22C2×D4×Q811yescharacteristic
640C22#348611yescharacteristic
641C22C23×D811yescharacteristic
642C22C2×#57611yescharacteristic
643C22C2×#57811yescharacteristic
644C22#358611yescharacteristic
645C4—28no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—41no—
655C4—82no—
656C4—84no—
657C4—82no—
658C4D4×D814yes—
659C4Q8×D814yescomplemented
660C4C2×D4×Q811yescharacteristic
661C4#349911yescharacteristic
662C4#350311yescharacteristic
663C4#355211yescharacteristic
664C2—44no—
665C2—84no—
666C2#5912711yescharacteristic
667C2C2×D4×D811yescharacteristic
668C2#6109711yescharacteristic
669C2C2×Q8×D811yescharacteristic
670C2#6138111yescharacteristic
671C2#6181711yescharacteristic
672C2#6187011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D8C2semidirect
2. #59817C2semidirect
3. D4×C4⋊Q8C2semidirect
4. #30946C2semidirect
5. #28813C2semidirect
6. #28689C2semidirect
7. #28508C2semidirect
8. #26460C2semidirect
9. #26368C2semidirect
10. #26040C2semidirect
11. #25838C2semidirect
12. #25469C2semidirect
13. #24326C2semidirect
14. #23149C2semidirect
16. C4×#474C2semidirect
141. #474D4semidirect
142. #459Q8semidirect
254. C2×D8C2×Q8semidirect
272. C4⋊Q8D8semidirect
275. #127C2×Q8semidirect
299. C4×D4C2×Q8semidirect
365. C4⋊C8C2×Q8semidirect
373. #102C2×Q8semidirect
562. D4D4×Q8semidirect
659. C4Q8×D8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D82
#598172
D4×C4⋊Q82
#309462
#288132
#286892
#285082
#264602
#263682
#260402
#258382
#254692
#243262
#231492
#229952
C4×#4742
C4×#4592