Finite Group Databasebeta

← #6344968

Subgroups — #6344968

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic
2#60248C211yesmaximal, characteristic, complemented
3#60236C211yesmaximal, characteristic
4#18030C218yesmaximal, complemented
5#18004C214yesmaximal, complemented
6#17997C214yesmaximal, complemented
7#17896C212yesmaximal, complemented
8#17879C212yesmaximal, complemented
9#17878C212yesmaximal, complemented
10#17855C212yesmaximal, complemented
11#17803C211yesmaximal, characteristic, complemented
12#17478C211yesmaximal, characteristic
13C2×#1479C211yesmaximal, characteristic, complemented
14#17326C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17#3103C414yes—
18#3103C412yes—
19#3103C412yes—
20#3103C2212yes—
21#3103C2212yes—
22#3103C2214yes—
23C2×#481C412yes—
24C2×#475C411yescharacteristic
25C2×#475C411yescharacteristic
26C2×#460C411yescharacteristic
27C2×#460C411yescharacteristic
28C2×#448C412yes—
29#2984—22no—
30#2984C2214yescomplemented
31#2983—22no—
32#2983C2212yes—
33#2982C2212yes—
34#2981—22no—
35#2979C2214yes—
36#2975C2212yes—
37#2974—22no—
38#2972C2212yes—
39Q8×M16—22no—
40Q8×M16C2214yescomplemented
41C2×C4×SD16C2212yes—
42C2×C4×SD16C2212yes—
43#2935C2214yes—
44#2931—22no—
45#2920—22no—
46#2920C2211yescharacteristic
47#2920C2211yescharacteristic
48C2×#431C2211yescharacteristic
49C2×#431C2211yescharacteristic
50C2×#431C2211yescharacteristic
51#2880—22no—
52#1592C2218yescomplemented
53#1590C2218yescomplemented
54#1588C2218yescomplemented
55#1585C2218yescomplemented
56#1583C2218yescomplemented
57#1582C2218yescomplemented
58SD16⋊C8C2218yescomplemented
59C8×SD16C2218yescomplemented
60#1572C2214yescomplemented
61#1571C2212yescomplemented
62#1570C2212yescomplemented
63#1564C2212yes—
64#1564C2212yes—
65#1496C2214yes—
66#1492C2214yes—
67#1479C2211yescharacteristic
68#1479C2211yescharacteristic
69#1479C2211yescharacteristic
70#1479C2211yescharacteristic
71#1479C2214yescomplemented
72#1478C2212yes—
73#1478C2212yes—
74C2×Q8⋊C8C2214yescomplemented
75#1454C2214yescomplemented
76#1452C2212yes—
77#1451C2212yescomplemented
78#1450C2212yes—
79C8×M16C2212yescomplemented
80C22×#41—21no—
81C23×Q8C2×C411yescharacteristic
82C22×SD16C2×C412yes—
83C22×SD16C2×C412yes—
84C2×#131—24no—
85#532—21no—
86#532—21no—
87#532—21no—
88#532—21no—
89#532C2×C412yes—
90#532C2×C412yes—
91#532C2312yes—
92#532C2312yes—
93C2×C4⋊Q8C2×C411yescharacteristic
94C2×#127C2×C411yescharacteristic
95C2×#124—21no—
96C2×#121—21no—
97C2×#121C2×C412yes—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2×C411yescharacteristic
105C2×C4×Q8C2×C411yescharacteristic
106C2×C4×Q8C2×C412yes—
107C2×C4×Q8C2×C412yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2×C411yescharacteristic
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118#481C2×C414yes—
119#481C2×C414yes—
120#481C2×C414yes—
121#481C2×C414yes—
122#475C2×C412yes—
123#475C2×C412yes—
124#475C2×C412yes—
125#475C2×C412yes—
126#475C2×C414yes—
127#475C2×C414yes—
128#460C2×C412yes—
129#460C2×C412yes—
130#460C2×C414yes—
131#460C2×C412yes—
132#460C2×C412yes—
133#460C2×C414yes—
134#448C2×C414yes—
135#448C2×C414yes—
136#448C2×C414yes—
137#448C2×C414yes—
138#446—28no—
139#446—24no—
140#446—28no—
141#446C2318yes—
142C8×Q8—24no—
143C8×Q8C2318yes—
144C4×SD16C2×C414yes—
145C4×SD16C2×C414yes—
146C4×SD16C2×C414yes—
147C4×SD16C2×C414yes—
148C4×SD16C2314yes—
149C4×SD16C2314yes—
150C4×SD16C2314yes—
151C4×SD16C2314yes—
152#436—24no—
153#436—24no—
154#436C2318yes—
155#435—28no—
156C8×D4—24no—
157C8×D4—24no—
158C8×D4C2318yes—
159#432—21no—
160#432—24no—
161#432—24no—
162#432D412yes—
163#432D412yes—
164#432C2312yes—
165#431—21no—
166#431—21no—
167#431—21no—
168#431—21no—
169#431—21no—
170#431—24no—
171#431—24no—
172#431C2×C412yes—
173#431C2×C411yescharacteristic
174#431C2×C411yescharacteristic
175#431C2311yescharacteristic
176#431C2311yescharacteristic
177#431C2312yes—
178#431C2312yes—
179#431C2311yescharacteristic
180#431C2311yescharacteristic
181#431C2312yes—
182#431C2311yescharacteristic
183#431C2311yescharacteristic
184#431C2312yes—
185#431C2312yes—
186#431C2312yes—
187C2×#106C2×C412yes—
188C2×#106C2×C412yes—
189#423—24no—
190#423D412yes—
191#423C2312yes—
192C2×C4⋊C8—21no—
193C2×C4⋊C8C2×C411yescharacteristic
194C2×C4⋊C8C2×C411yescharacteristic
195C2×C4⋊C8C2312yes—
196C2×#103C2×C412yes—
197C2×#103C2×C412yes—
198C2×#102C2×C412yes—
199C2×#102C2×C412yes—
200#408—24no—
201#408—28no—
202#407—21no—
203#407—24no—
204#407D412yes—
205C2×C22⋊C8—21no—
206C2×C22⋊C8—21no—
207C2×C22⋊C8—24no—
208C2×C22⋊C8C2×C412yes—
209C4×M16—24no—
210C4×M16—21no—
211C4×M16C2312yes—
212C2×C4×C8—21no—
213C2×C4×C8C2312yes—
214C2×C4×C8C2312yes—
215#335C2314yes—
216#335C2314yes—
217#334C2314yes—
218#334C2314yes—
219Q8⋊C8C2314yes—
220Q8⋊C8C2314yes—
221Q8⋊C8C2314yes—
222Q8⋊C8C2314yes—
223D4⋊C8C2318yes—
224D4⋊C8C2318yes—
225#322C2314yes—
226#322C2314yes—
227#322C2314yes—
228C82C2314yes—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—84no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—24no—
239C22×Q8—42no—
240C22×Q8C22×C411yescharacteristic
241C22×Q8C22×C412yes—
242C22×Q8C22×C412yes—
243C22×Q8C22×C411yescharacteristic
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C22×C412yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4C22×C411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4#4111yescharacteristic
254C2×SD16—28no—
255C2×SD16—44no—
256C2×SD16C22×C414yes—
257C2×SD16C22×C414yes—
258C2×SD16C22×C414yes—
259C2×SD16C22×C414yes—
260#131—88no—
261C2×M16—22no—
262C2×M16—28no—
263C2×M16C2×D414yes—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—28no—
267C22×C8C22×C412yes—
268C22×C8C22×C412yes—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8C22×C412yes—
272C4⋊Q8C22×C412yes—
273#127—21no—
274#127—21no—
275#127C22×C412yes—
276#127C22×C412yes—
277#124—22no—
278#124—22no—
279#124—22no—
280#124—22no—
281#121—22no—
282#121—22no—
283#121—22no—
284#121—22no—
285#121—22no—
286#121—22no—
287#121C22×C414yes—
288#121C22×C414yes—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—21no—
301C4×Q8—22no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—22no—
305C4×Q8C22×C412yes—
306C4×Q8C22×C412yes—
307C4×Q8C22×C412yes—
308C4×Q8C22×C412yes—
309C4×Q8C22×C412yes—
310C4×Q8C22×C412yes—
311C4×Q8C2412yes—
312C4×Q8C2412yes—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4C22×C414yes—
320C4×D4C2414yes—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4C22×C411yescharacteristic
334C2×C4⋊C4C22×C411yescharacteristic
335C2×C4⋊C4C22×C412yes—
336C2×C4⋊C4C22×C412yes—
337C2×C4⋊C4C22×C412yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42C22×C411yescharacteristic
349C2×C42C22×C411yescharacteristic
350C2×C42C22×C411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42#4111yescharacteristic
355C2×C42#4111yescharacteristic
356C2×C42#4111yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#106C22×C414yes—
361#106C22×C414yes—
362#106C22×C414yes—
363#106C22×C414yes—
364C4⋊C8—28no—
365C4⋊C8—28no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368C4⋊C8—21no—
369C4⋊C8—21no—
370C4⋊C8—22no—
371C4⋊C8C22×C411yescharacteristic
372C4⋊C8C22×C411yescharacteristic
373C4⋊C8C22×C411yescharacteristic
374C4⋊C8C22×C411yescharacteristic
375C4⋊C8C22×C412yes—
376C4⋊C8C22×C412yes—
377C4⋊C8C2×D414yes—
378C4⋊C8C2×D414yes—
379C4⋊C8C2414yes—
380C4⋊C8C2412yes—
381C4⋊C8C2412yes—
382#103C22×C414yes—
383#103C22×C414yes—
384#103C22×C414yes—
385#103C22×C414yes—
386#102C22×C414yes—
387#102C22×C414yes—
388#102C22×C414yes—
389#102C22×C414yes—
390C22⋊C8—21no—
391C22⋊C8—21no—
392C22⋊C8—22no—
393C22⋊C8—22no—
394C22⋊C8—22no—
395C22⋊C8—21no—
396C22⋊C8—21no—
397C22⋊C8—22no—
398C22⋊C8—28no—
399C22⋊C8—28no—
400C22⋊C8C22×C412yes—
401C22⋊C8C22×C412yes—
402C22⋊C8C22×C412yes—
403C22⋊C8C22×C412yes—
404C22⋊C8C2×D414yes—
405C22⋊C8C2×D414yes—
406#97—28no—
407#97—22no—
408#97C2×D414yes—
409C4×C8—28no—
410C4×C8—21no—
411C4×C8—22no—
412C4×C8—21no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8C22×C412yes—
416C4×C8C22×C412yes—
417C4×C8C2412yes—
418C4×C8C2412yes—
419C4×C8C2412yes—
420C4×C8C2412yes—
421C4×C8C2414yes—
422C24—22no—
423C24C4×D411yescharacteristic
424#41—48no—
425#41—84no—
426#41—84no—
427#41—84no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—82no—
445C2×Q8#13114yescomplemented
446C2×Q8C23×C412yes—
447C2×Q8C23×C412yes—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—82no—
458C2×D4—82no—
459C2×D4—84no—
460C2×D4C23×C414yes—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—44no—
476C22×C4—84no—
477C22×C4—81no—
478C22×C4—82no—
479C22×C4C4×D411yescharacteristic
480C22×C4C4×D411yescharacteristic
481C22×C4C4×D411yescharacteristic
482C22×C4C4×D411yescharacteristic
483C22×C4C4×D411yescharacteristic
484C22×C4C4×D411yescharacteristic
485C22×C4C4×D411yescharacteristic
486C22×C4C4×D411yescharacteristic
487C22×C4C4×D411yescharacteristic
488C22×C4C23×C411yescharacteristic
489C22×C4C23×C411yescharacteristic
490C22×C4C23×C411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C2×#4111yescharacteristic
495C22×C4C2×#4111yescharacteristic
496C22×C4C2×#4111yescharacteristic
497SD16—28no—
498SD16—28no—
499SD16—44no—
500SD16—44no—
501M16—28no—
502M16—44no—
503M16—88no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—44no—
512C2×C8—88no—
513C2×C8—82no—
514C2×C8C23×C412yes—
515C2×C8C23×C412yes—
516C2×C8C23×C412yes—
517C2×C8C23×C412yes—
518C2×C8C22×D418yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4C23×C412yes—
535C4⋊C4C23×C412yes—
536C4⋊C4C23×C414yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42C4×D411yescharacteristic
550C42C4×D411yescharacteristic
551C42C4×D411yescharacteristic
552C42C4×D411yescharacteristic
553C42C4×D411yescharacteristic
554C42C4×D411yescharacteristic
555C42C23×C411yescharacteristic
556C42C23×C411yescharacteristic
557C42C23×C411yescharacteristic
558C42C22×D411yescharacteristic
559C42C22×D411yescharacteristic
560C42C22×D411yescharacteristic
561C42C2×#4111yescharacteristic
562C42C2×#4111yescharacteristic
563C42C2×#4111yescharacteristic
564C42C2511yescharacteristic
565C23—22no—
566C23—22no—
567C23—24no—
568C23—41no—
569C23—44no—
570C23—82no—
571C23—82no—
572C23C2×C4×D411yescharacteristic
573C23C2×C4×D411yescharacteristic
574C23C2×C4×D411yescharacteristic
575Q8—24no—
576Q8—22no—
577Q8—22no—
578Q8—22no—
579Q8—44no—
580Q8—44no—
581Q8—42no—
582Q8—44no—
583Q8—82no—
584Q8C2×#13114yes—
585D4—24no—
586D4—24no—
587D4—44no—
588D4—48no—
589D4—44no—
590D4—44no—
591D4—82no—
592D4—82no—
593D4—84no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—48no—
623C2×C4—84no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4C2×C4×D411yescharacteristic
629C2×C4C2×C4×D411yescharacteristic
630C2×C4C2×C4×D411yescharacteristic
631C2×C4C2×C4×D411yescharacteristic
632C2×C4C2×C4×D411yescharacteristic
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×C4×D411yescharacteristic
637C2×C4C2×#13112yes—
638C2×C4#56812yes—
639C2×C4#57714yes—
640C2×C4C24×C411yescharacteristic
641C2×C4C23×D411yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C8—22no—
644C8—22no—
645C8—22no—
646C8—22no—
647C8—28no—
648C8—42no—
649C8—42no—
650C8—42no—
651C8—42no—
652C8—44no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—164no—
665C22#293312yes—
666C22#294912yes—
667C22C22×C4×D411yescharacteristic
668C4—21no—
669C4—21no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—24no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4#293312yes—
682C4#294912yes—
683C4C22×#13111yescharacteristic
684C4C2×#56811yescharacteristic
685C4C2×#57712yes—
686C2—22no—
687C2—22no—
688C2—41no—
689C2—84no—
690C2C2×#293311yescharacteristic
691C2C2×#294911yescharacteristic
692C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2920 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60248C2semidirect
4. #18030C2semidirect
5. #18004C2semidirect
6. #17997C2semidirect
7. #17896C2semidirect
8. #17879C2semidirect
9. #17878C2semidirect
10. #17855C2semidirect
11. #17803C2semidirect
13. C2×#1479C2semidirect
14. #17326C2semidirect
30. #2984C22semidirect
40. Q8×M16C22semidirect
52. #1592C22semidirect
53. #1590C22semidirect
54. #1588C22semidirect
55. #1585C22semidirect
56. #1583C22semidirect
57. #1582C22semidirect
58. SD16⋊C8C22semidirect
59. C8×SD16C22semidirect
60. #1572C22semidirect
61. #1571C22semidirect
62. #1570C22semidirect
71. #1479C22semidirect
74. C2×Q8⋊C8C22semidirect
75. #1454C22semidirect
77. #1451C22semidirect
79. C8×M16C22semidirect
445. C2×Q8#131semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
#602482
#602362
#180302
#180042
#179972
#178962
#178792
#178782
#178552
#178032
#174782
C2×#14792
#173262