Finite Group Databasebeta

← #6449907

Subgroups — #6449907

Order 512.

2058 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3080C211yesmaximal, characteristic, complemented
2C2×#2921C211yesmaximal, characteristic, complemented
3C2×#2629C211yesmaximal, characteristic, complemented
4#34065C211yesmaximal, characteristic
5#24719C214yesmaximal, complemented
6#24717C214yesmaximal, complemented
7#24715C214yesmaximal, complemented
8#24713C214yesmaximal, complemented
9#23642C212yesmaximal, complemented
10#23482C212yesmaximal
11#21314C211yesmaximal, characteristic, complemented
12#21171C211yesmaximal, characteristic, complemented
13#21159C212yesmaximal, complemented
14#21093C212yesmaximal, complemented
15#20648C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#3080—28no—
21C2×#479C2214yescomplemented
22C2×#478C2214yescomplemented
23#2921—28no—
24C2×#433C2214yes—
25C2×#433C2214yes—
26C2×#428C2212yes—
27C2×#428C2212yes—
28C2×#423C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#417C2212yes—
31C2×#417C2212yes—
32C2×#407C2212yes—
33C2×C4×M16C2212yes—
34#2629—21no—
35#2629—21no—
36#2629—21no—
37#2629—21no—
38#2629C2214yes—
39#2629C2214yes—
40C2×#398—22no—
41C2×#396—22no—
42#2324C2212yes—
43#2324C2212yes—
44#2323C2212yes—
45#2323C2212yes—
46#2322—22no—
47#2321—22no—
48#2297C2211yescharacteristic
49#2296C2211yescharacteristic
50#2293—21no—
51#2293—21no—
52C2×#389C2212yes—
53#2290—21no—
54#2290—21no—
55C2×#384C2211yescharacteristic
56#2280C2211yescharacteristic
57C2×#380C2212yes—
58C4×#117C2211yescharacteristic
59#2077C2214yes—
60#2077C2214yes—
61#2076C2214yes—
62#2075C2214yes—
63#2036C2214yes—
64#2035C2214yes—
65#2034C2214yes—
66#2033C2214yes—
67#1989C2214yescomplemented
68#1989C2214yescomplemented
69#1948C2214yescomplemented
70#1947C2214yescomplemented
71#1944C2214yescomplemented
72#1943C2214yescomplemented
73#1942C2214yescomplemented
74#1941C2214yescomplemented
75#1936C2214yescomplemented
76#1935C2214yescomplemented
77#1934C2214yescomplemented
78#1933C2214yescomplemented
79#1803C2214yescomplemented
80#1802C2214yescomplemented
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1736C2212yes—
85C22×#41C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C22×M16C2312yes—
88C22×M16C2312yes—
89C2×#125—21no—
90C2×#125—21no—
91C2×#125—24no—
92C2×#125—24no—
93C2×#125C2314yes—
94#525—24no—
95#525—24no—
96C2×#123—21no—
97C2×C22⋊Q8C2311yescharacteristic
98C2×C22⋊Q8C2311yescharacteristic
99C2×#121C2311yescharacteristic
100C2×#121C2311yescharacteristic
101C2×#120—21no—
102C4×#41—24no—
103C4×#41—24no—
104C2×C4×Q8C2312yes—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—22no—
108C2×C4×D4C2312yes—
109C2×#117—24no—
110C2×#117—21no—
111C2×#117—24no—
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C2×#117D411yescharacteristic
115C2×#117D411yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4D412yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4—22no—
128C22×C22⋊C4D412yes—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#479—216no—
133#478—216no—
134#433—216no—
135#433—216no—
136#428—28no—
137#428—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142#423—24no—
143#423D418yescomplemented
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146#417—28no—
147#417—28no—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152#407—28no—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C4×M16—28no—
156C2×#97C2314yes—
157C2×C4×C8C2314yes—
158#398—24no—
159#398—24no—
160#398—24no—
161#398—24no—
162#396—24no—
163#396—24no—
164#396—24no—
165#396—24no—
166#391C2312yes—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2314yes—
195#381—21no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2312yes—
200#380C2314yes—
201#380C2314yes—
202#379—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212#376—21no—
213C2×#95—22no—
214C2×#95—22no—
215C43—21no—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224C25C2×D411yescharacteristic
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41#4114yes—
231C22×Q8C2411yescharacteristic
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2411yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×M16—24no—
255C2×M16—24no—
256C2×M16—24no—
257C2×M16—24no—
258C2×M16—24no—
259C2×M16C2×D418yes—
260C22×C8C2414yes—
261C22×C8C2414yes—
262#125—28no—
263#125—28no—
264#125—44no—
265#125—44no—
266#123—28no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271#121—22no—
272#121—22no—
273#121—22no—
274#121—22no—
275#120—28no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—48no—
283#117—22no—
284#117—24no—
285#117—24no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—44no—
297#117#4114yes—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—84no—
318C2×C4⋊C4—84no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—84no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—44no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D414yes—
363C2×C42#4114yes—
364C2×C42C2412yes—
365C2×C42C2412yes—
366#107—28no—
367#107—28no—
368#106—28no—
369#106—28no—
370C4⋊C8—28no—
371C4⋊C8C2×D4116yes—
372#103—28no—
373#103—28no—
374#102—28no—
375#102—28no—
376C22⋊C8—28no—
377C22⋊C8—28no—
378#97—28no—
379C4×C8—28no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392C24—22no—
393C24—21no—
394C24—22no—
395C24—22no—
396C24—42no—
397C24—42no—
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C24C22×D411yescharacteristic
401#41—28no—
402#41—28no—
403#41—44no—
404#41—44no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8C2×#4112yes—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4C2×#4112yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4#12118yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488M16—216no—
489M16—48no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8C22×D418yes—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4C2×#4112yes—
524C4⋊C4C2×#4112yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C42—22no—
544C42—22no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42—82no—
552C42—82no—
553C42#12114yes—
554C42#12114yes—
555C42C22×D414yes—
556C42C2×#4114yes—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—21no—
569C23—21no—
570C23—48no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—44no—
575C23—41no—
576C23—84no—
577C23—82no—
578C23—82no—
579C23C2×#12112yes—
580C23#52512yes—
581C23#53612yes—
582C23C23×D411yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587D4—22no—
588D4—22no—
589D4—48no—
590D4—41no—
591D4—41no—
592D4—84no—
593D4—84no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—88no—
643C2×C4—84no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4C2×#12114yes—
647C2×C4C2×#12114yes—
648C2×C4C2×#12112yes—
649C2×C4#52512yes—
650C2×C4#53612yes—
651C2×C4#54714yes—
652C2×C4#54814yes—
653C2×C4#55214yes—
654C2×C4#55614yes—
655C2×C4C23×D411yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C2×2−1+411yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C8—28no—
662C8—44no—
663C22—24no—
664C22—22no—
665C22—22no—
666C22—24no—
667C22—42no—
668C22—42no—
669C22—41no—
670C22—42no—
671C22—41no—
672C22—41no—
673C22—42no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—88no—
681C22—82no—
682C22—82no—
683C22#241514yes—
684C22C22×#12111yescharacteristic
685C22C2×#52511yescharacteristic
686C22C2×#53611yescharacteristic
687C22#358512yes—
688C22#358612yes—
689C4—24no—
690C4—24no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—41no—
696C4—41no—
697C4—42no—
698C4—84no—
699C4—82no—
700C4—84no—
701C4—82no—
702C4#241514yes—
703C4C2×#54711yescharacteristic
704C4C2×#54811yescharacteristic
705C4C2×#55211yescharacteristic
706C4C2×#55611yescharacteristic
707C2—24no—
708C2—81no—
709C2—81no—
710C2—82no—
711C2C2×#241511yescharacteristic
712C2#6045612yes—
713C2#6056811yescharacteristic
714C2#6056911yescharacteristic
715C2#6086712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3080C2semidirect
2. C2×#2921C2semidirect
3. C2×#2629C2semidirect
5. #24719C2semidirect
6. #24717C2semidirect
7. #24715C2semidirect
8. #24713C2semidirect
9. #23642C2semidirect
11. #21314C2semidirect
12. #21171C2semidirect
13. #21159C2semidirect
14. #21093C2semidirect
15. #20648C2semidirect
21. C2×#479C22semidirect
22. C2×#478C22semidirect
67. #1989C22semidirect
68. #1989C22semidirect
69. #1948C22semidirect
70. #1947C22semidirect
71. #1944C22semidirect
72. #1943C22semidirect
73. #1942C22semidirect
74. #1941C22semidirect
75. #1936C22semidirect
76. #1935C22semidirect
77. #1934C22semidirect
78. #1933C22semidirect
79. #1803C22semidirect
80. #1802C22semidirect
143. #423D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30802
C2×#29212
C2×#26292
#340652
#247192
#247172
#247152
#247132
#236422
#234822
#213142
#211712
#211592
#210932
#206482