Finite Group Databasebeta

← #6971862

Subgroups — #6971862

Order 512.

1742 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36276C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#32762C212yesmaximal, complemented
4#31840C214yesmaximal, complemented
5#30449C214yesmaximal, complemented
6#30262C211yesmaximal, characteristic, complemented
7#29715C212yesmaximal, complemented
8#29664C212yesmaximal, complemented
9#29347C211yesmaximal, characteristic, complemented
10#28440C211yesmaximal, characteristic, complemented
11#25760C212yesmaximal
12#25573C212yesmaximal
13#25370C212yesmaximal, complemented
14#25324C212yesmaximal, complemented
15#23934C212yesmaximal, complemented
16#23816C211yesmaximal, characteristic, complemented
17#22008C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C2×D4×Q8C2211yescharacteristic
20C2×#533C2212yes—
21C2×#494C2212yes—
22C2×#494C2212yes—
23C2×#487C2212yes—
24C2×#462C2212yes—
25C2×#462C2212yes—
26C2×#462C2214yes—
27C2×#451C2214yes—
28C2×#451C2214yes—
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2857C2212yes—
32#2857C2212yes—
33#2567C2211yescharacteristic
34#2552—21no—
35#2550—21no—
36#2550—21no—
37#2549—22no—
38#2549—22no—
39#2546C2211yescharacteristic
40#2542C2212yes—
41#2538—21no—
42#2502C2212yes—
43#2441—22no—
44#2434C2212yes—
45#2429C2214yes—
46#2423—22no—
47#2365C2212yes—
48#2354—21no—
49Q8×C4⋊C4—21no—
50#2349—21no—
51#2349—21no—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C2×#391C2212yes—
55C42×D4C2211yescharacteristic
56#2100C2212yes—
57#2072C2211yescharacteristic
58#2072C2211yescharacteristic
59#2072C2212yes—
60#2072C2214yes—
61#2056C2214yes—
62#2056C2214yes—
63#2043C2212yes—
64#2024C2211yescharacteristic
65#2024C2211yescharacteristic
66#2024C2212yes—
67#2016C2212yes—
68#2016C2212yes—
69#2016C2212yes—
70#2016C2214yes—
71#2016C2214yes—
72#2016C2212yes—
73#2001C2214yes—
74#2001C2214yes—
75#2001C2214yes—
76#2001C2214yes—
77#1987C2212yes—
78#1981C2212yes—
79#1956C2212yes—
80#1954C2212yes—
81#1939C2212yes—
82#1933C2214yes—
83#1923C2214yes—
84#1878C2214yes—
85#1878C2214yes—
86#1872C2214yes—
87#1872C2214yes—
88#1817C2212yes—
89#1803C2214yes—
90#1797C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#103C2212yes—
94#1744C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100#554—44no—
101D4×Q8—44no—
102#533—48no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#125—21no—
113C2×#125—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8C2314yes—
118C2×C22⋊Q8C2314yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2314yes—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×#117—22no—
132C22×C4⋊C4C2312yes—
133C22×C42C2312yes—
134#494—44no—
135#494—44no—
136#487—44no—
137#462—48no—
138#462—44no—
139#462—44no—
140#451—48no—
141#451—48no—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2312yes—
158#399—24no—
159#398—24no—
160#398—22no—
161#398—22no—
162#398—22no—
163#396—24no—
164#395—22no—
165#395—22no—
166#395—22no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—22no—
171#391—22no—
172#391C2311yescharacteristic
173#391C2311yescharacteristic
174#391C2312yes—
175#391C2314yes—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#390—22no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—24no—
193#389C2312yes—
194#388—24no—
195#387—24no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—22no—
201#384—21no—
202#384—21no—
203#384—21no—
204#384—21no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2312yes—
208#382—21no—
209#382—21no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C2×#95—22no—
226C2×#95—22no—
227C43C2311yescharacteristic
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2311yescharacteristic
237#336C2314yes—
238#336C2312yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—24no—
243C22×Q8—44no—
244C22×Q8—42no—
245C22×Q8—44no—
246C22×Q8C2411yescharacteristic
247C22×Q8C2414yes—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C23×C4—22no—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C2×Q16—28no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×Q16—44no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×Q16—44no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C4⋊Q8—24no—
272C4⋊Q8—22no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—41no—
278C4⋊Q8—44no—
279C4⋊Q8—42no—
280C4⋊Q8—44no—
281C4⋊Q8—42no—
282C4⋊Q8—44no—
283#125—48no—
284#125—44no—
285C22⋊Q8—28no—
286C22⋊Q8—48no—
287C22⋊Q8—48no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291C22⋊Q8—48no—
292C22⋊Q8—44no—
293C22⋊Q8—44no—
294C4×Q8—28no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—44no—
298C4×Q8—48no—
299C4×Q8—44no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4Q1618yescomplemented
305#117—48no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C4⋊C8—24no—
367C4⋊C8—44no—
368#103—24no—
369#103—24no—
370#103—28no—
371#103—24no—
372#103—42no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378C22⋊C8—44no—
379C22⋊C8—44no—
380C4×C8—24no—
381C4×C8—42no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24C22×D412yes—
402C2×Q8—24no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—82no—
426C2×Q8—81no—
427C2×Q8—81no—
428C2×Q8—88no—
429C2×Q8—88no—
430C2×Q8H(5,𝔽2)12yes—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4C2×Q1614yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4C2×Q1618yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469Q16—88no—
470Q16—88no—
471Q16—88no—
472Q16—88no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—84no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4C2×Q1614yes—
520C4⋊C4H(5,𝔽2)14yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—84no—
527C22⋊C4C2×Q1618yes—
528C42—21no—
529C42—24no—
530C42—22no—
531C42—21no—
532C42—42no—
533C42—42no—
534C42—41no—
535C42—41no—
536C42—48no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42C2×Q1614yes—
541C42#13714yes—
542C42H(5,𝔽2)14yes—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23C22×Q1614yes—
550C23C23×D411yescharacteristic
551Q8—24no—
552Q8—48no—
553Q8—42no—
554Q8—42no—
555Q8—44no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—42no—
561Q8—44no—
562Q8—88no—
563Q8—162no—
564D4—28no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4—164no—
616C2×C4#53512yes—
617C2×C4#53512yes—
618C2×C4C22×Q1612yes—
619C2×C4C22×Q1612yes—
620C2×C4C22×Q1614yes—
621C2×C4C22×Q1612yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13712yes—
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4#58511yescharacteristic
629C8—42no—
630C8—44no—
631C8—42no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22C2×#53511yescharacteristic
640C22#348411yescharacteristic
641C22#348811yescharacteristic
642C22C23×Q1611yescharacteristic
643C22C22×#13711yescharacteristic
644C22#358611yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—24no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—81no—
662C4—84no—
663C4#320914yes—
664C4#354312yes—
665C4#354612yes—
666C2—44no—
667C2#4248311yescharacteristic
668C2C2×#320911yescharacteristic
669C2#6082611yescharacteristic
670C2#6156311yescharacteristic
671C2#6158611yescharacteristic
672C2#6195211yescharacteristic
673C2#6196611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #32762C2semidirect
4. #31840C2semidirect
5. #30449C2semidirect
6. #30262C2semidirect
7. #29715C2semidirect
8. #29664C2semidirect
9. #29347C2semidirect
10. #28440C2semidirect
13. #25370C2semidirect
14. #25324C2semidirect
15. #23934C2semidirect
16. #23816C2semidirect
304. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#362762
#361342
#327622
#318402
#304492
#302622
#297152
#296642
#293472
#284402
#257602
#255732
#253702
#253242
#239342
#238162
#220082