Finite Group Databasebeta

← #6483196

Subgroups — #6483196

Order 512.

2164 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3218C212yesmaximal, complemented
2#36631C211yesmaximal, characteristic
3C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4#32906C212yesmaximal, complemented
5#32903C212yesmaximal
6#29719C214yesmaximal, complemented
7#28725C212yesmaximal, complemented
8#28619C214yesmaximal, complemented
9#28451C212yesmaximal, complemented
10#28255C212yesmaximal, complemented
11#28252C212yesmaximal, complemented
12#28159C214yesmaximal, complemented
13#21867C212yesmaximal, complemented
14#20810C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529C2212yes—
17C22×C4×Q8C2212yes—
18#3218—216no—
19C2×#497C2214yes—
20C2×#491C2214yes—
21C2×#462C2214yes—
22C2×#462C2214yes—
23C2×#460C2214yes—
24C2×#461C2214yes—
25C2×#451C2214yes—
26C2×#448C2214yes—
27C2×#432C2212yes—
28#2861C2212yes—
29#2860—21no—
30#2860—21no—
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2854—21no—
36#2854—21no—
37#2850C2212yes—
38C4×C4⋊Q8—21no—
39C4×C4⋊Q8—21no—
40C4×C4⋊Q8—21no—
41C4×C4⋊Q8—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#127C2211yescharacteristic
48C4×#124C2212yes—
49C4×#121—21no—
50C4×#121—21no—
51C2×#391C2212yes—
52#2293—21no—
53#2293—21no—
54C2×#389—21no—
55C2×#389—21no—
56C2×#384C2211yescharacteristic
57C2×C4×C4⋊C4—21no—
58C2×C4×C4⋊C4—21no—
59C2×C43C2211yescharacteristic
60#2074C2218yes—
61#2056C2218yes—
62#2055C2218yes—
63#2015C2218yes—
64#1992C2214yes—
65#1959C2214yes—
66#1880C2218yes—
67#1878C2218yes—
68#1874C2214yes—
69#1872C2214yes—
70#1842C2212yes—
71#1831C2212yes—
72#1803C2214yes—
73#1803C2214yes—
74#1803C2214yes—
75#1802C2214yes—
76#1797C2212yes—
77#1797C2212yes—
78#1767C2212yes—
79#1767C2212yes—
80C23×Q8C2312yes—
81C22×Q16C2318yes—
82C22×SD16C2318yes—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8D418yescomplemented
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91#529—28no—
92#529—28no—
93C2×#126—22no—
94C2×#125C2312yes—
95C2×#124C2312yes—
96C2×#124C2312yes—
97C2×#123—22no—
98C2×C22⋊Q8C2312yes—
99C2×#121—21no—
100C2×#121—21no—
101C2×#121C2312yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—48no—
105C2×C4×Q8C2314yes—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2314yes—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C42—22no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#497—216no—
128#491—216no—
129#462—216no—
130#462—48no—
131#461—216no—
132#460—48no—
133#451—216no—
134#448—216no—
135#432—28no—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2318yes—
140C2×#103C2318yes—
141C2×#103C2318yes—
142C2×#102C2318yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2314yes—
145#391—22no—
146#391—22no—
147#391C2314yes—
148#391C2314yes—
149#391C2312yes—
150#391C2312yes—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390C2312yes—
160#390C2312yes—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#386—22no—
186#386—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#382C2312yes—
194#380—21no—
195#380—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C2×#95—22no—
209C2×#95—22no—
210C43D411yescharacteristic
211C43D411yescharacteristic
212C43D411yescharacteristic
213C43D411yescharacteristic
214C43C2311yescharacteristic
215C43C2311yescharacteristic
216C43C2311yescharacteristic
217C43C2311yescharacteristic
218#340C2318yes—
219#340C2318yes—
220#336C2314yes—
221#336C2314yes—
222C22×Q8—22no—
223C22×Q8—24no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8C2×D418yes—
227C22×Q8C2412yes—
228C22×Q8C2414yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2412yes—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×Q16—216no—
244C2×Q16—48no—
245C2×Q16—48no—
246C2×SD16—216no—
247C2×SD16—48no—
248C2×SD16—48no—
249C22×C8C2418yes—
250C4⋊Q8—28no—
251C4⋊Q8—28no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8C2×D418yes—
256C4⋊Q8C2×D418yes—
257#127—24no—
258#126—48no—
259#125—24no—
260#125—24no—
261#124—24no—
262#124—24no—
263#124—44no—
264#123—48no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267#121—24no—
268#121—24no—
269#121—44no—
270#121—44no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—28no—
274C4×Q8—28no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×Q8—48no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4—48no—
283C4×D4—44no—
284#117—48no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—48no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D414yes—
358C2×C42C2×D411yescharacteristic
359C2×C42#4114yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C4⋊C8—28no—
367C4⋊C8—28no—
368C4⋊C8—44no—
369#103—216no—
370#103—216no—
371#103—48no—
372#102—48no—
373C22⋊C8—28no—
374#97—28no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385C24—22no—
386C24C22×D411yescharacteristic
387C2×Q8—28no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—416no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—416no—
401C2×Q8C22×D418yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4#12114yes—
447C22×C4#12114yes—
448C22×C4#13714yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4112yes—
464C22×C4C2511yescharacteristic
465Q16—816no—
466SD16—816no—
467C2×C8—28no—
468C2×C8—28no—
469C2×C8—28no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—48no—
531C42—44no—
532C42—48no—
533C42#12114yes—
534C42#12114yes—
535C42#13714yes—
536C42C22×D412yes—
537C42C22×D412yes—
538C42C2×#4112yes—
539C42C2×#4112yes—
540C42H(5,𝔽2)14yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23C2×#12112yes—
553C23C2×#13712yes—
554C23#57512yes—
555C23C23×D411yescharacteristic
556Q8—28no—
557Q8—28no—
558Q8—416no—
559Q8—44no—
560Q8—44no—
561Q8—44no—
562Q8—44no—
563D4—44no—
564D4—44no—
565D4—84no—
566D4—88no—
567D4—84no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—28no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
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—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—416no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—88no—
619C2×C4—88no—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C2×#12112yes—
624C2×C4C2×#12112yes—
625C2×C4#53512yes—
626C2×C4#53512yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4C2×#13712yes—
630C2×C4C2×#13712yes—
631C2×C4C2×#13712yes—
632C2×C4#57512yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)12yes—
636C2×C4#58512yes—
637C8—48no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22#303714yes—
652C22C22×#12111yescharacteristic
653C22C2×#53511yescharacteristic
654C22#344911yescharacteristic
655C22C22×#13711yescharacteristic
656C22C22×#13711yescharacteristic
657C22C2×#57511yescharacteristic
658C22C2×#57511yescharacteristic
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—48no—
670C4—41no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4#303714yes—
676C4#321814yescomplemented
677C4#349612yes—
678C4#350612yes—
679C2—24no—
680C2—162no—
681C2#3516511yescharacteristic
682C2C2×#303711yescharacteristic
683C2C2×#303711yescharacteristic
684C2C2×#321811yescharacteristic
685C2#6080511yescharacteristic
686C2#6095211yescharacteristic
687C2#6096311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3218C2semidirect
3. C2×C4×C4⋊Q8C2semidirect
4. #32906C2semidirect
6. #29719C2semidirect
7. #28725C2semidirect
8. #28619C2semidirect
9. #28451C2semidirect
10. #28255C2semidirect
11. #28252C2semidirect
12. #28159C2semidirect
13. #21867C2semidirect
14. #20810C2semidirect
86. C2×C4⋊Q8D4semidirect
676. C4#3218semidirect

Maximal subgroups

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

subgroupindex
C2×#32182
#366312
C2×C4×C4⋊Q82
#329062
#329032
#297192
#287252
#286192
#284512
#282552
#282522
#281592
#218672
#208102