Finite Group Databasebeta

← #6432494

Subgroups — #6432494

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34268C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#23773C214yesmaximal, complemented
4#23769C214yesmaximal, complemented
5#23763C212yesmaximal, complemented
6#23762C212yesmaximal, complemented
7#23757C212yesmaximal, complemented
8#23756C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic, complemented
10#23732C211yesmaximal, characteristic, complemented
11#23658C211yesmaximal, characteristic, complemented
12#23643C212yesmaximal
13C2×#2072C211yesmaximal, characteristic, complemented
14#23484C212yesmaximal
15#20453C212yesmaximal, complemented
16#20417C212yesmaximal, complemented
17#20378C211yesmaximal, characteristic
18C2×#536C2211yescharacteristic
19C2×#519C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#419C2212yes—
24C22×#103C2212yes—
25C2×#408C2212yes—
26C2×#405C2212yes—
27#2367—21no—
28#2367—21no—
29#2365—21no—
30#2365—21no—
31#2359—22no—
32#2359—22no—
33#2355—21no—
34#2355—21no—
35#2354—21no—
36#2354—21no—
37#2352—22no—
38#2352—22no—
39#2323C2212yes—
40#2323C2212yes—
41#2314C2214yes—
42#2311C2211yescharacteristic
43#2311C2212yes—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2212yes—
46#2296C2211yescharacteristic
47C2×#391C2211yescharacteristic
48#2285C2212yes—
49#2285C2212yes—
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2212yes—
53#2274C2211yescharacteristic
54#2101C2212yes—
55#2101C2212yes—
56#2100C2211yescharacteristic
57#2100C2211yescharacteristic
58#2099C2211yescharacteristic
59#2099C2211yescharacteristic
60#2097C2214yes—
61#2096C2212yes—
62#2095C2212yes—
63#2084C2212yes—
64#2084C2212yes—
65#2083C2212yes—
66#2083C2212yes—
67#2077C2214yes—
68#2077C2214yes—
69#2076C2214yes—
70#2075C2214yes—
71#2072C2212yes—
72#2072C2212yes—
73#2072C2214yes—
74#2036C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2035C2214yes—
78#1776C2214yes—
79#1776C2214yes—
80C4×#107C2214yes—
81C4×#106C2214yes—
82#1763C2214yes—
83#1763C2214yes—
84C4×#103C2214yes—
85C4×#103C2214yes—
86#1750C2214yes—
87#1749C2212yes—
88C4×C22⋊C8C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#536—24no—
100#536—24no—
101C2×C4⋊Q8—21no—
102C2×#124—21no—
103C2×#123—22no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106#519—28no—
107C4×#41—28no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4D412yes—
115C2×C4×D4D412yes—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×#117—22no—
119C2×#117—22no—
120C2×#117—22no—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#428—28no—
133#428—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138#419—28no—
139C2×#103—28no—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144#408—28no—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147#405—28no—
148C2×#97C2314yes—
149C2×C4×C8C2314yes—
150#391—22no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391C2312yes—
155#391C2312yes—
156#390—22no—
157#390—21no—
158#389—22no—
159#389—22no—
160#389C2311yescharacteristic
161#389C2311yescharacteristic
162#389C2311yescharacteristic
163#389C2311yescharacteristic
164#388—24no—
165#388—24no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#387—24no—
171#387—24no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#386—24no—
177#386—22no—
178#385—21no—
179#385—21no—
180#385—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2312yes—
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#382—22no—
189#382—22no—
190#382C2312yes—
191#380—21no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380C2314yes—
196#380C2314yes—
197#380C2312yes—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—24no—
218C4×C22⋊C4—22no—
219C2×#95—21no—
220C2×#95—21no—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—24no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8#4114yes—
248C22×Q8C2412yes—
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×M16—24no—
261C2×M16—24no—
262C2×M16#4118yes—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C4⋊Q8—28no—
270#124—28no—
271#123—216no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—28no—
279C4×Q8—28no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—28no—
285C4×D4—28no—
286#117—22no—
287#117—28no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—48no—
292#117—84no—
293#117—84no—
294#117#4114yes—
295#117#4118yes—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42—28no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361#107—28no—
362#107—28no—
363#106—28no—
364#106—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—28no—
369C22⋊C8—28no—
370C22⋊C8—28no—
371#97—28no—
372C4×C8—28no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393#41—48no—
394#41—48no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—216no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—42no—
404C2×Q8—48no—
405C2×Q8—42no—
406C2×Q8C2×#4114yes—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4114yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4114yes—
459C22×C4C2511yescharacteristic
460M16—216no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—48no—
470C2×C8C2×#4118yes—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4#12314yes—
503C4⋊C4#12314yes—
504C4⋊C4C2×#4112yes—
505C4⋊C4C2×#4112yes—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4#12318yes—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—42no—
526C42—44no—
527C42—44no—
528C42—44no—
529C42—82no—
530C42—82no—
531C42—84no—
532C42—84no—
533C42C2×#4118yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23C2×#12312yes—
546C23#52512yes—
547C23#53212yes—
548C23C23×D411yescharacteristic
549Q8—28no—
550Q8—48no—
551Q8—44no—
552Q8—44no—
553D4—48no—
554D4—44no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—162no—
607C2×C4—164no—
608C2×C4C2×#12314yes—
609C2×C4C2×#12314yes—
610C2×C4C2×#12312yes—
611C2×C4#52512yes—
612C2×C4#52918yes—
613C2×C4#52918yes—
614C2×C4#53212yes—
615C2×C4#57712yes—
616C2×C4#57812yes—
617C2×C4C22×#4112yes—
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4112yes—
621C8—28no—
622C8—42no—
623C8—42no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22#239514yes—
638C22C22×#12311yescharacteristic
639C22C2×#52511yescharacteristic
640C22C2×#53211yescharacteristic
641C22C2×#57711yescharacteristic
642C22C2×#57811yescharacteristic
643C22#358511yescharacteristic
644C22#358611yescharacteristic
645C4—22no—
646C4—24no—
647C4—22no—
648C4—24no—
649C4—41no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4—81no—
660C4—84no—
661C4#239514yes—
662C4C2×#52912yes—
663C4C2×#52912yes—
664C2—24no—
665C2—82no—
666C2C2×#239511yescharacteristic
667C2#6052211yescharacteristic
668C2#6052611yescharacteristic
669C2#6059311yescharacteristic
670C2#6059411yescharacteristic
671C2#6072311yescharacteristic
672C2#6072711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23773C2semidirect
4. #23769C2semidirect
5. #23763C2semidirect
6. #23762C2semidirect
7. #23757C2semidirect
8. #23756C2semidirect
9. #23744C2semidirect
10. #23732C2semidirect
11. #23658C2semidirect
13. C2×#2072C2semidirect
15. #20453C2semidirect
16. #20417C2semidirect

Maximal subgroups

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

subgroupindex
#342682
#341882
#237732
#237692
#237632
#237622
#237572
#237562
#237442
#237322
#236582
#236432
C2×#20722
#234842
#204532
#204172
#203782