Finite Group Databasebeta

← #6560054

Subgroups — #6560054

Order 512.

1882 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#30205C211yesmaximal, characteristic, complemented
4#30199C212yesmaximal, complemented
5#29355C211yesmaximal, characteristic, complemented
6#29148C211yesmaximal, characteristic, complemented
7#29107C212yesmaximal, complemented
8#27813C212yesmaximal, complemented
9#27769C212yesmaximal, complemented
10#27722C214yesmaximal, complemented
11#27274C214yesmaximal, complemented
12#25902C212yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14#23452C212yesmaximal, complemented
15#22504C212yesmaximal, complemented
16#22431C212yesmaximal
17#20567C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×D4×Q8—21no—
20C2×#547C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C2×#487C2214yes—
23C2×#436C2212yes—
24C2×#417C2214yes—
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2857C2212yes—
28#2857C2212yes—
29#2553—22no—
30#2553—22no—
31#2551—21no—
32#2551C2212yes—
33#2546—21no—
34#2546C2211yescharacteristic
35#2542C2211yescharacteristic
36#2499C2212yes—
37#2440—22no—
38#2440C2214yes—
39#2434—22no—
40#2432C2212yes—
41#2427—21no—
42#2427—21no—
43#2360—21no—
44#2360—21no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#124C2211yescharacteristic
49#2297C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2212yes—
52#2099C2214yes—
53#2072C2211yescharacteristic
54#2072C2214yes—
55#2072C2212yes—
56#2071C2211yescharacteristic
57#2043C2214yes—
58#2042C2212yes—
59#2024C2211yescharacteristic
60#2024C2212yes—
61#2024C2212yes—
62#2024C2212yes—
63#2024C2214yes—
64#2024C2212yes—
65#2023C2211yescharacteristic
66#2023C2214yes—
67#2023C2212yes—
68#1990C2214yes—
69#1990C2212yes—
70#1988C2212yes—
71#1981C2212yes—
72#1959C2212yes—
73#1956C2212yes—
74#1935C2214yes—
75#1934C2214yes—
76#1933C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1872C2214yes—
80#1872C2214yes—
81#1836C2212yes—
82#1825C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1753C2214yes—
91C4×C22⋊C8C2212yes—
92#1744C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41C2312yes—
96C23×Q8—22no—
97C22×M16C2314yes—
98#556—28no—
99D4×Q8—48no—
100#547—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—48no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2312yes—
113C2×#126—22no—
114C2×#124—21no—
115C2×#124C2311yescharacteristic
116C2×#123—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×#121—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—22no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117C2312yes—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2314yes—
136C22×C42C2312yes—
137#487—48no—
138#436—28no—
139C2×C4⋊C8C2312yes—
140#417—48no—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97C2312yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2314yes—
155#399—22no—
156#398—24no—
157#398—22no—
158#397—22no—
159#397—22no—
160#395—24no—
161#395—24no—
162#394—22no—
163#394—22no—
164#391—21no—
165#391—22no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2311yescharacteristic
170#391C2312yes—
171#391C2312yes—
172#390C2311yescharacteristic
173#390C2312yes—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2314yes—
184#388—21no—
185#388—21no—
186#387—22no—
187#387—22no—
188#386—22no—
189#386—22no—
190#385—24no—
191#385—24no—
192#385—22no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#384C2312yes—
200#382—21no—
201#382—21no—
202#382C2311yescharacteristic
203#380—22no—
204#380—22no—
205#380—22no—
206#380C2314yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2314yes—
235C2×#41—24no—
236C2×#41—28no—
237C2×#41—42no—
238C2×#41—44no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—48no—
244C22×Q8—48no—
245C22×Q8—84no—
246C22×Q8C2411yescharacteristic
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4C2×D411yescharacteristic
250C22×D4C2411yescharacteristic
251C23×C4—24no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C2×M16—28no—
258C2×M16—44no—
259C2×M16—44no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—28no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272C4⋊Q8—48no—
273C4⋊Q8—48no—
274C4⋊Q8—82no—
275#126—48no—
276#124—22no—
277#124—41no—
278#124—44no—
279#123—48no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—416no—
283C22⋊Q8—48no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286#121—44no—
287C4×Q8—24no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—48no—
294C4×D4#4118yes—
295#117—28no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C4⋊C8—24no—
360#103—28no—
361#103—24no—
362#103—44no—
363#103—42no—
364#103—44no—
365#103—44no—
366#103—44no—
367#102—24no—
368#102—24no—
369#102—44no—
370C22⋊C8—28no—
371#97—42no—
372C4×C8—24no—
373C4×C8—44no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24C22×D412yes—
393#41—416no—
394#41—88no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—48no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—48no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—84no—
415C2×Q8—832no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—82no—
424C2×D4#12414yes—
425C2×D42−1+412yes—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4—84no—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4118yes—
462C22×C4C2511yescharacteristic
463M16—88no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4#12414yes—
504C4⋊C4AGL(1,ℤ/8)14yes—
505C4⋊C4#13714yes—
506C4⋊C4H(5,𝔽2)14yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4#12418yes—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—42no—
533C42—44no—
534C42C2×#4114yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×#12414yes—
549C23C23×D411yescharacteristic
550Q8—48no—
551Q8—42no—
552Q8—44no—
553Q8—42no—
554Q8—44no—
555Q8—816no—
556Q8—88no—
557Q8—84no—
558D4—28no—
559D4—24no—
560D4—42no—
561D4—88no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—88no—
609C2×C4—816no—
610C2×C4—84no—
611C2×C4C2×#12414yes—
612C2×C4C2×#12412yes—
613C2×C4C2×#12412yes—
614C2×C4C2×AGL(1,ℤ/8)14yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4C2×#13714yes—
617C2×C4C2×#13712yes—
618C2×C4#57712yes—
619C2×C4#57812yes—
620C2×C4C22×#4112yes—
621C2×C4C2×H(5,𝔽2)12yes—
622C2×C4C2×2−1+411yescharacteristic
623C2×C4#58511yescharacteristic
624C8—44no—
625C8—42no—
626C8—42no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22C22×#12411yescharacteristic
642C22#348411yescharacteristic
643C22#348911yescharacteristic
644C22C22×AGL(1,ℤ/8)11yescharacteristic
645C22C22×#13711yescharacteristic
646C22C2×#57711yescharacteristic
647C22C2×#57811yescharacteristic
648C4—24no—
649C4—22no—
650C4—24no—
651C4—22no—
652C4—24no—
653C4—22no—
654C4—44no—
655C4—41no—
656C4—41no—
657C4—82no—
658C4—81no—
659C4—84no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4#314114yes—
665C4#351412yes—
666C4#352012yes—
667C2—44no—
668C2—82no—
669C2#3593911yescharacteristic
670C2C2×#314111yescharacteristic
671C2#6062311yescharacteristic
672C2#6160011yescharacteristic
673C2#6160211yescharacteristic
674C2#6214711yescharacteristic
675C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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
2. D4×C4⋊Q8C2semidirect
3. #30205C2semidirect
4. #30199C2semidirect
5. #29355C2semidirect
6. #29148C2semidirect
7. #29107C2semidirect
8. #27813C2semidirect
9. #27769C2semidirect
10. #27722C2semidirect
11. #27274C2semidirect
12. #25902C2semidirect
13. #23817C2semidirect
14. #23452C2semidirect
15. #22504C2semidirect
17. #20567C2semidirect

Maximal subgroups

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

subgroupindex
#364062
D4×C4⋊Q82
#302052
#301992
#293552
#291482
#291072
#278132
#277692
#277222
#272742
#259022
#238172
#234522
#225042
#224312
#205672