Finite Group Databasebeta

← #6433319

Subgroups — #6433319

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35253C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#29978C214yesmaximal, complemented
4#29975C214yesmaximal, complemented
5#29973C212yesmaximal, complemented
6#29973C212yesmaximal, complemented
7#29969C212yesmaximal, complemented
8#29967C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23671C212yesmaximal, complemented
11#23669C212yesmaximal, complemented
12#23557C212yesmaximal, complemented
13#20846C211yesmaximal, characteristic
14#20843C211yesmaximal, characteristic
15#20700C211yesmaximal, characteristic
16C2×#1824C211yesmaximal, characteristic
17#20628C211yesmaximal, characteristic
18C2×#544C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#424C2212yes—
25C2×#407C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27#2857C2212yes—
28#2851C2212yes—
29#2629C2212yes—
30#2628C2212yes—
31#2442—21no—
32#2441—21no—
33#2440—21no—
34#2439—21no—
35#2438—22no—
36#2437—22no—
37#2317C2211yescharacteristic
38#2317C2211yescharacteristic
39#2316—22no—
40#2316—21no—
41#2316—21no—
42#2316C2212yes—
43#2316C2214yes—
44#2316C2212yes—
45#2316C2211yescharacteristic
46C2×#389—21no—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#389C2212yes—
50#2280C2211yescharacteristic
51C2×#380—22no—
52C2×#380C2211yescharacteristic
53#2101C2214yescomplemented
54#2101C2214yescomplemented
55#2100C2214yescomplemented
56#2099C2214yescomplemented
57#2088C2214yescomplemented
58#2088C2214yescomplemented
59#2088C2214yescomplemented
60#2088C2214yescomplemented
61#2087C2214yescomplemented
62#2087C2214yescomplemented
63#2086C2214yescomplemented
64#2086C2214yescomplemented
65#2059C2214yescomplemented
66#2059C2214yescomplemented
67#2058C2214yescomplemented
68#2057C2214yescomplemented
69#1854C2212yes—
70#1854C2214yes—
71#1854C2212yes—
72#1853C2211yescharacteristic
73#1853C2212yes—
74#1853C2211yescharacteristic
75#1852C2211yescharacteristic
76#1852C2212yes—
77#1852C2211yescharacteristic
78#1824C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1824C2212yes—
82#1824C2214yes—
83#1823C2212yes—
84#1823C2212yes—
85#1798C2214yes—
86#1797C2212yes—
87#1797C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C24×C4—21no—
96C24×C4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×#125—21no—
103C2×#125C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117—44no—
109C2×#117—44no—
110C2×#117D412yes—
111C2×#117D411yescharacteristic
112C2×#117D411yescharacteristic
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4Q818yescomplemented
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4D412yes—
139C22×C42—21no—
140C22×C42C2312yes—
141#428—44no—
142#427—44no—
143C2×#107C2312yes—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×#106C2312yes—
147C2×#106C2314yes—
148C2×#106C2312yes—
149#424—44no—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152#407—24no—
153C2×C22⋊C8—24no—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159#400—22no—
160#399—22no—
161#398—22no—
162#398—22no—
163#396—22no—
164#396—22no—
165#395—22no—
166#393—22no—
167#391—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—24no—
173#389—24no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#389C2314yes—
186#389C2314yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#387—22no—
192#387—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#384—21no—
198#384C2312yes—
199#382—21no—
200#382C2312yes—
201#381—21no—
202#380—22no—
203#380—21no—
204#380—21no—
205#380—24no—
206#380—24no—
207#380—24no—
208#380—24no—
209#380—22no—
210#380C2311yescharacteristic
211#380C2311yescharacteristic
212#380C2311yescharacteristic
213#380C2312yes—
214#380C2311yescharacteristic
215#380C2312yes—
216#379—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C2×#95—21no—
220C2×#95—22no—
221C2×#95—22no—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2314yes—
238C25C2×D411yescharacteristic
239C22×Q8—22no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—28no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—82no—
251C23×C4—82no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2×Q814yes—
258C23×C4#4114yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273#125—44no—
274#125—44no—
275C22⋊Q8—48no—
276C22⋊Q8—84no—
277#117—24no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—24no—
283#117—44no—
284#117—44no—
285#117—42no—
286#117—42no—
287#117—42no—
288#117—42no—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4—84no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×Q818yes—
332C2×C4⋊C4C2×Q818yes—
333C2×C4⋊C4C2×Q818yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2414yes—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—82no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—44no—
365C2×C42—44no—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2412yes—
369C2×C42C2412yes—
370#107—42no—
371#107—42no—
372#107—44no—
373#106—42no—
374#106—42no—
375#106—44no—
376C4⋊C8—44no—
377C4⋊C8—44no—
378C22⋊C8—22no—
379C22⋊C8—22no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382C22⋊C8—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—41no—
404C24—41no—
405C24C22⋊Q814yes—
406C24C22×D411yescharacteristic
407C24C22×D411yescharacteristic
408C24C22×D411yescharacteristic
409C2×Q8—44no—
410C2×Q8—82no—
411C2×Q8—82no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—816no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4C22⋊Q814yes—
462C22×C4C22⋊Q814yes—
463C22×C4C22⋊Q814yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×Q812yes—
472C22×C4C22×Q812yes—
473C22×C4C22×Q812yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C42−1+414yes—
479C22×C4C2511yescharacteristic
480M16—48no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—48no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4C22×Q818yes—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C22⋊C4—82no—
543C42—24no—
544C42—24no—
545C42—42no—
546C42—42no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—42no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—28no—
562C23—28no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23C2×C22⋊Q812yes—
573C23C2×C22⋊Q812yes—
574C23C2×C22⋊Q812yes—
575C23#53412yes—
576C23#53612yes—
577C23#57712yes—
578C23#57812yes—
579C23C23×D411yescharacteristic
580Q8—164no—
581C2×C4—28no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—416no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4—88no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—162no—
639C2×C4—162no—
640C2×C4C2×C22⋊Q812yes—
641C2×C4C2×C22⋊Q812yes—
642C2×C4C2×C22⋊Q812yes—
643C2×C4#53412yes—
644C2×C4#53612yes—
645C2×C4#53918yes—
646C2×C4#54418yes—
647C2×C4C23×Q811yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×H(5,𝔽2)12yes—
650C2×C4C2×2−1+412yes—
651C8—42no—
652C8—42no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—416no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22#262714yes—
675C22#308214yes—
676C22C22×C22⋊Q811yescharacteristic
677C22C2×#53411yescharacteristic
678C22C2×#53611yescharacteristic
679C22C2×#57711yescharacteristic
680C22C2×#57811yescharacteristic
681C22#358511yescharacteristic
682C22#358511yescharacteristic
683C4—22no—
684C4—22no—
685C4—48no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—81no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—81no—
696C4—82no—
697C4—81no—
698C4#262714yes—
699C4C2×#53912yes—
700C4C2×#54412yes—
701C2—24no—
702C2—24no—
703C2—42no—
704C2C2×#262711yescharacteristic
705C2C2×#308211yescharacteristic
706C2#6049011yescharacteristic
707C2#6076411yescharacteristic
708C2#6076511yescharacteristic
709C2#6087311yescharacteristic
710C2#6087411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #29978C2semidirect
4. #29975C2semidirect
5. #29973C2semidirect
6. #29973C2semidirect
7. #29969C2semidirect
8. #29967C2semidirect
9. #23749C2semidirect
10. #23671C2semidirect
11. #23669C2semidirect
12. #23557C2semidirect
53. #2101C22semidirect
54. #2101C22semidirect
55. #2100C22semidirect
56. #2099C22semidirect
57. #2088C22semidirect
58. #2088C22semidirect
59. #2088C22semidirect
60. #2088C22semidirect
61. #2087C22semidirect
62. #2087C22semidirect
63. #2086C22semidirect
64. #2086C22semidirect
65. #2059C22semidirect
66. #2059C22semidirect
67. #2058C22semidirect
68. #2057C22semidirect
128. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#352532
C2×#23162
#299782
#299752
#299732
#299692
#299672
#237492
#236712
#236692
#235572
#208462
#208432
#207002
C2×#18242
#206282