Finite Group Databasebeta

← #6566100

Subgroups — #6566100

Order 512.

2155 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3354C212yesmaximal, complemented
2#36179C211yesmaximal, characteristic, complemented
3#34321C211yesmaximal, characteristic, complemented
4#29864C211yesmaximal, characteristic, complemented
5#29758C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28578C212yesmaximal, complemented
8#28202C211yesmaximal, characteristic, complemented
9#27645C212yesmaximal, complemented
10#25451C212yesmaximal, complemented
11#25221C214yesmaximal, complemented
12#24811C212yesmaximal, complemented
13#24592C214yesmaximal, complemented
14#23676C211yesmaximal, characteristic, complemented
15#23658C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22500C212yesmaximal, complemented
18C2×#555C2211yescharacteristic
19C2×#547—21no—
20C2×#547—21no—
21C2×#547C2212yes—
22#3354—216no—
23C2×#498C2212yes—
24C2×#481C2214yes—
25C2×#482C2214yes—
26C2×#476C2212yes—
27C2×#474C2212yes—
28C2×#469C2214yes—
29C2×C4×D8C2212yes—
30C2×#436C2212yes—
31C2×#428C2214yes—
32#2868C2211yescharacteristic
33#2856C2212yes—
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2501—22no—
38#2501—22no—
39#2501C2214yes—
40#2369C2211yescharacteristic
41#2367—22no—
42#2363C2212yes—
43#2361—21no—
44#2354C2211yescharacteristic
45D4×C4⋊C4C2211yescharacteristic
46#2347—21no—
47#2347C2212yes—
48#2337—22no—
49#2324—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#121C2212yes—
53#2296C2212yes—
54#2293C2211yescharacteristic
55C2×#384C2212yes—
56#2274—21no—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2212yes—
60#2084C2212yes—
61#2084C2214yes—
62#2083C2212yes—
63#2083C2214yes—
64#2083C2212yes—
65#2083C2211yescharacteristic
66#2071C2211yescharacteristic
67#2071C2212yes—
68#2071C2212yes—
69#2067C2212yes—
70#2067C2214yes—
71#2065C2214yes—
72#2054C2212yes—
73#2041C2212yes—
74#2041C2211yescharacteristic
75#2011C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1948C2214yes—
79#1948C2214yes—
80#1939C2212yes—
81#1926C2214yes—
82#1926C2214yes—
83#1895C2214yes—
84#1879C2214yes—
85#1875C2214yes—
86#1851C2214yes—
87#1831C2211yescharacteristic
88#1824C2214yes—
89#1802C2212yes—
90#1802C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×#41—22no—
97C22×D8C2312yes—
98C22×M16C2314yes—
99#555—24no—
100#555D418yescomplemented
101#547—28no—
102#547—28no—
103#547—48no—
104#547—48no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#124—22no—
111C2×#123—24no—
112C2×#123—24no—
113C2×#123—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×#121—21no—
118C2×#121—22no—
119C2×#121—22no—
120C2×#121C2314yes—
121C2×#121C2312yes—
122C2×C4×Q8—21no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2312yes—
131C2×C4×D4C2311yescharacteristic
132C2×#117—21no—
133C2×#117—21no—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2314yes—
140C22×C42C2312yes—
141#498—28no—
142#482—216no—
143#481—216no—
144#476—28no—
145#474—28no—
146#469—216no—
147C4×D8—28no—
148#436—28no—
149#428—216no—
150C2×#107C2314yes—
151C2×#107C2312yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×#106C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2314yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C4×C8C2312yes—
167#399—24no—
168#399—24no—
169#399—24no—
170#391—21no—
171#391—22no—
172#390—21no—
173#390—22no—
174#390—21no—
175#390—21no—
176#390—22no—
177#390—22no—
178#390C2312yes—
179#390C2311yescharacteristic
180#390C2314yes—
181#390C2311yescharacteristic
182#389—21no—
183#389—21no—
184#389C2312yes—
185#389C2312yes—
186#389C2311yescharacteristic
187#389C2312yes—
188#389C2311yescharacteristic
189#388—22no—
190#388—22no—
191#388—22no—
192#388—21no—
193#388—22no—
194#388—21no—
195#388—22no—
196#386—22no—
197#386—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#384—21no—
202#384—21no—
203#384—22no—
204#384—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2314yes—
208#384C2312yes—
209#384C2312yes—
210#383—22no—
211#380—21no—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—22no—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4C2312yes—
231C43C2311yescharacteristic
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#340C2314yes—
238#340C2314yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2314yes—
244C2×#41—48no—
245C2×#41—48no—
246C2×#41—48no—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×D4—22no—
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C22×D4C2411yescharacteristic
256C23×C4—21no—
257C23×C4—24no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C2×D8—24no—
264C2×D8—24no—
265C2×D8—42no—
266C2×M16—28no—
267C2×M16—28no—
268C2×M16—44no—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—22no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8C2×D414yes—
278#127—42no—
279#124—216no—
280#123—416no—
281#123—416no—
282#123—416no—
283C22⋊Q8—48no—
284C22⋊Q8—48no—
285C22⋊Q8—48no—
286#121—28no—
287#121—24no—
288#121—28no—
289#121—28no—
290#121—48no—
291#121—48no—
292#121C2×D418yes—
293C4×Q8—82no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—84no—
303C4×D4C2×D414yes—
304C4×D4C2×D414yes—
305C4×D4#4114yes—
306#117—24no—
307#117—28no—
308#117—44no—
309#117—44no—
310#117C2×D418yes—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—48no—
332C2×C4⋊C4—48no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
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—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2×D411yescharacteristic
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—24no—
380#107—28no—
381#106—28no—
382#106—28no—
383#106—28no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386C4⋊C8—44no—
387#102—24no—
388#102—28no—
389#102—24no—
390#102—24no—
391#102—28no—
392#102—42no—
393#102—44no—
394C22⋊C8—28no—
395C4×C8—24no—
396#95—22no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—21no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—21no—
409C24—22no—
410C24C22×D412yes—
411#41—832no—
412C2×Q8—24no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—81no—
417C2×Q8—81no—
418C2×D4—28no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—48no—
435C2×D4—48no—
436C2×D4—48no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4#12114yes—
442C2×D4C22×D412yes—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D414yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4114yes—
481C22×C4C2511yescharacteristic
482D8—48no—
483M16—416no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C82−1+414yes—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—82no—
520C4⋊C4—81no—
521C4⋊C4#12114yes—
522C4⋊C4AGL(1,ℤ/8)14yes—
523C4⋊C4C22×D414yes—
524C4⋊C4C22×D412yes—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—44no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—48no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C22⋊C4—42no—
539C22⋊C4—81no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4#12118yes—
544C42—28no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—24no—
549C42—22no—
550C42—42no—
551C42—42no—
552C42—41no—
553C42—42no—
554C42—82no—
555C42—82no—
556C42—82no—
557C42AGL(1,ℤ/8)14yes—
558C42C22×D412yes—
559C42C2×#4112yes—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23C2×#12114yes—
581C23C23×D411yescharacteristic
582Q8—84no—
583Q8—88no—
584D4—24no—
585D4—28no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—84no—
592D4—84no—
593D4—88no—
594D4—816no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—816no—
640C2×C4—88no—
641C2×C4—164no—
642C2×C4C2×#12114yes—
643C2×C4C2×#12112yes—
644C2×C4C2×#12112yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4C2×AGL(1,ℤ/8)12yes—
647C2×C4C2×AGL(1,ℤ/8)14yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4#57712yes—
651C2×C4C23×D411yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×2−1+412yes—
654C2×C4#58512yes—
655C8—24no—
656C8—42no—
657C8—44no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22C22×#12111yescharacteristic
680C22#348511yescharacteristic
681C22#349911yescharacteristic
682C22C22×AGL(1,ℤ/8)11yescharacteristic
683C22C22×AGL(1,ℤ/8)11yescharacteristic
684C22C2×#57711yescharacteristic
685C22#358611yescharacteristic
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—24no—
691C4—24no—
692C4—44no—
693C4—42no—
694C4—41no—
695C4—42no—
696C4—81no—
697C4—81no—
698C4—82no—
699C4—82no—
700C4—84no—
701C4#303814yes—
702C4#348912yes—
703C4#351512yes—
704C2—44no—
705C2—82no—
706C2—162no—
707C2#3602011yescharacteristic
708C2C2×#303811yescharacteristic
709C2#6042911yescharacteristic
710C2#6171711yescharacteristic
711C2#6172811yescharacteristic
712C2#6247711yescharacteristic
713C2#6250911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3354C2semidirect
2. #36179C2semidirect
3. #34321C2semidirect
4. #29864C2semidirect
5. #29758C2semidirect
6. #28687C2semidirect
7. #28578C2semidirect
8. #28202C2semidirect
9. #27645C2semidirect
10. #25451C2semidirect
11. #25221C2semidirect
12. #24811C2semidirect
13. #24592C2semidirect
14. #23676C2semidirect
15. #23658C2semidirect
16. #22700C2semidirect
17. #22500C2semidirect
100. #555D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33542
#361792
#343212
#298642
#297582
#286872
#285782
#282022
#276452
#254512
#252212
#248112
#245922
#236762
#236582
#227002
#225002