Finite Group Databasebeta

← #6484432

Subgroups — #6484432

Order 512.

1668 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#35294C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic
3#32964C212yesmaximal, complemented
4#32962C212yesmaximal
5#29963C212yesmaximal, complemented
6#29962C212yesmaximal
7#29269C212yesmaximal, complemented
8#29268C212yesmaximal, complemented
9#29198C212yesmaximal, complemented
10#29197C212yesmaximal, complemented
11#29146C211yesmaximal, characteristic, complemented
12#29146C211yesmaximal, characteristic, complemented
13#28520C211yesmaximal, characteristic, complemented
14#28519C211yesmaximal, characteristic, complemented
15#28289C212yesmaximal, complemented
16#28288C212yesmaximal, complemented
17#21988C212yesmaximal, complemented
18#21808C212yesmaximal
19#20811C211yesmaximal, characteristic, complemented
20C22×#124C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×#477C2212yescomplemented
24C2×#475C2212yescomplemented
25C2×#476C2212yes—
26C2×#474C2212yes—
27C2×#432C2212yes—
28#2868C2211yescharacteristic, complemented
29#2859C2212yescomplemented
30#2859C2211yescharacteristic, complemented
31#2629C2212yes—
32#2628C2212yes—
33#2570—22no—
34#2550—22no—
35#2441—22no—
36#2439—22no—
37#2367—22no—
38#2367—22no—
39#2342—22no—
40#2342—22no—
41#2331C2212yescomplemented
42#2330C2212yescomplemented
43#2326C2211yescharacteristic
44#2326C2211yescharacteristic
45#2326C2211yescharacteristic
46#2326C2211yescharacteristic
47#2310C2211yescharacteristic
48#2309C2211yescharacteristic
49C4×#124C2211yescharacteristic
50C4×#124C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2211yescharacteristic
53#2285C2212yes—
54#2285C2211yescharacteristic
55C4×#117C2211yescharacteristic
56#2096C2212yescomplemented
57#2095C2212yescomplemented
58#2094C2212yes—
59#2093C2212yes—
60#2068C2214yescomplemented
61#2067C2214yes—
62#2066C2214yescomplemented
63#2065C2214yes—
64#2026C2212yescomplemented
65#2026C2212yescomplemented
66#2025C2212yes—
67#2025C2212yes—
68#1983C2214yes—
69#1983C2214yes—
70#1959C2212yes—
71#1959C2212yes—
72#1959C2212yes—
73#1959C2212yes—
74#1948C2214yescomplemented
75#1947C2214yescomplemented
76#1865C2214yescomplemented
77#1864C2214yes—
78#1863C2214yescomplemented
79#1862C2214yes—
80#1854C2214yes—
81#1854C2214yes—
82#1847C2212yescomplemented
83#1847C2212yescomplemented
84#1843C2212yescomplemented
85#1836C2212yescomplemented
86#1836C2212yescomplemented
87#1832C2212yescomplemented
88#1805C2212yescomplemented
89#1805C2212yescomplemented
90#1804C2212yes—
91#1804C2212yes—
92#1798C2212yes—
93#1798C2212yes—
94#1787C2214yes—
95#1786C2214yes—
96#1767C2212yescomplemented
97#1767C2211yescharacteristic, complemented
98#1767C2211yescharacteristic, complemented
99C23×Q8C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2312yes—
103C2×#125—22no—
104C2×#125C2312yes—
105C2×#124—22no—
106C2×#124—24no—
107C2×#124Q818yescomplemented
108C2×#124C2311yescharacteristic
109C2×#124C2311yescharacteristic
110C2×#124C2311yescharacteristic
111C2×#124C2311yescharacteristic
112#519—44no—
113#518—44no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C22⋊C4—22no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C42C2311yescharacteristic
137#477—44no—
138#476—44no—
139#475—44no—
140#474—44no—
141#432—28no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2314yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161#400—22no—
162#399—22no—
163#398—24no—
164#396—24no—
165#395—22no—
166#393—22no—
167#391—22no—
168#391—24no—
169#391C2311yescharacteristic
170#391C2311yescharacteristic
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#390—22no—
174#390C2311yescharacteristic
175#390C2311yescharacteristic
176#390C2311yescharacteristic
177#390C2311yescharacteristic
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—24no—
183#389C2314yes—
184#389C2314yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#387—24no—
194#387—24no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#384—22no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#384C2312yes—
205#382—22no—
206#382—22no—
207#382C2312yes—
208#382C2311yescharacteristic
209#382C2311yescharacteristic
210#381—22no—
211#380—22no—
212#380—22no—
213#380C2314yes—
214#379—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226#376C2311yescharacteristic
227#376C2311yescharacteristic
228C43C2311yescharacteristic
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8C2×Q814yes—
244C22×Q8C2411yescharacteristic
245C22×Q8C2411yescharacteristic
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4C2×Q814yes—
251C22×D4C2411yescharacteristic
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—84no—
265#125—44no—
266#125—84no—
267#124—216no—
268#124—22no—
269#124—22no—
270#124—22no—
271#124—22no—
272#124C2×Q814yes—
273#124C2×Q814yes—
274#124C2×Q814yes—
275#124C2×Q814yes—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—44no—
287C4×D4—44no—
288#117—24no—
289#117—42no—
290#117—44no—
291#117—44no—
292#117—42no—
293#117—42no—
294#117—84no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2414yes—
319C2×C22⋊C4—216no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4C2×D411yescharacteristic
338C2×C22⋊C4C2×D411yescharacteristic
339C2×C22⋊C4C2×D412yes—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×Q814yes—
353C2×C42#4114yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—44no—
361#107—44no—
362#106—44no—
363#106—44no—
364C4⋊C8—28no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#102—44no—
373#102—44no—
374#102—44no—
375#102—44no—
376C22⋊C8—28no—
377#97—24no—
378#97—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24—21no—
397C24—24no—
398C24—42no—
399C24—42no—
400C24—82no—
401C24—82no—
402C24C22×D411yescharacteristic
403C2×Q8—24no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—48no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—48no—
414C2×Q8—44no—
415C2×Q8—84no—
416C2×Q8C22×Q812yes—
417C2×Q8C22×Q812yes—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—48no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—48no—
429C2×D4C22×Q812yes—
430C2×D4C22×Q812yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C22⋊Q818yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×Q812yes—
465C22×C4C2×#4112yes—
466C22×C4C2511yescharacteristic
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—84no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—84no—
531C42C22⋊Q818yes—
532C42C22×Q812yes—
533C42C22×Q812yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C42H(5,𝔽2)14yes—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—48no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—88no—
553C23—164no—
554C23—162no—
555C23—162no—
556C23C2×C22⋊Q812yes—
557C23C23×D411yescharacteristic
558Q8—24no—
559Q8—24no—
560Q8—48no—
561Q8—42no—
562Q8—42no—
563Q8—164no—
564D4—24no—
565D4—24no—
566D4—48no—
567D4—42no—
568D4—42no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—162no—
612C2×C4—162no—
613C2×C4—162no—
614C2×C4—162no—
615C2×C4C2×C22⋊Q812yes—
616C2×C4C2×C22⋊Q814yes—
617C2×C4C2×C22⋊Q814yes—
618C2×C4#53512yes—
619C2×C4#53512yes—
620C2×C4C23×Q811yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4#58512yes—
624C8—44no—
625C8—44no—
626C22—24no—
627C22—22no—
628C22—42no—
629C22—41no—
630C22—41no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—82no—
636C22—82no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—168no—
642C22C22×C22⋊Q811yescharacteristic
643C22C2×#53511yescharacteristic
644C22#344911yescharacteristic
645C22#358512yes—
646C22#358612yes—
647C4—24no—
648C4—24no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—84no—
659C4—81no—
660C4—81no—
661C4—84no—
662C4—81no—
663C4—81no—
664C4#350112yes—
665C4#350612yes—
666C2—42no—
667C2—82no—
668C2—82no—
669C2#3517211yescharacteristic
670C2#6049312yes—
671C2#6082511yescharacteristic
672C2#6082611yescharacteristic
673C2#6101612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35294C2semidirect
3. #32964C2semidirect
5. #29963C2semidirect
7. #29269C2semidirect
8. #29268C2semidirect
9. #29198C2semidirect
10. #29197C2semidirect
11. #29146C2semidirect
12. #29146C2semidirect
13. #28520C2semidirect
14. #28519C2semidirect
15. #28289C2semidirect
16. #28288C2semidirect
17. #21988C2semidirect
19. #20811C2semidirect
23. C2×#477C22semidirect
24. C2×#475C22semidirect
28. #2868C22semidirect
29. #2859C22semidirect
30. #2859C22semidirect
41. #2331C22semidirect
42. #2330C22semidirect
56. #2096C22semidirect
57. #2095C22semidirect
60. #2068C22semidirect
62. #2066C22semidirect
64. #2026C22semidirect
65. #2026C22semidirect
74. #1948C22semidirect
75. #1947C22semidirect
76. #1865C22semidirect
78. #1863C22semidirect
82. #1847C22semidirect
83. #1847C22semidirect
84. #1843C22semidirect
85. #1836C22semidirect
86. #1836C22semidirect
87. #1832C22semidirect
88. #1805C22semidirect
89. #1805C22semidirect
96. #1767C22semidirect
97. #1767C22semidirect
98. #1767C22semidirect
107. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#342112
#329642
#329622
#299632
#299622
#292692
#292682
#291982
#291972
#291462
#285202
#285192
#282892
#282882
#219882
#218082
#208112