Finite Group Databasebeta

← #6432488

Subgroups — #6432488

Order 512.

2018 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2324C211yesmaximal, characteristic
2#34048C211yesmaximal, characteristic, complemented
3#23771C214yesmaximal, complemented
4#23770C214yesmaximal, complemented
5#23767C214yesmaximal, complemented
6#23766C214yesmaximal, complemented
7#23747C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic, complemented
9#23657C211yesmaximal, characteristic, complemented
10#23642C212yesmaximal, complemented
11#23628C211yesmaximal, characteristic, complemented
12#23480C212yesmaximal
13C4×#428C212yesmaximal, complemented
14C4×#418C212yesmaximal, complemented
15#20377C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#418C2212yes—
22C2×#418C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2324—22no—
26#2324—22no—
27#2324C2214yes—
28#2324C2214yes—
29#2314C2212yescomplemented
30#2314C2212yescomplemented
31#2314C2212yescomplemented
32#2314C2212yescomplemented
33#2313—21no—
34#2313—21no—
35#2312—21no—
36#2312—21no—
37#2298—22no—
38#2298—22no—
39C2×#390—21no—
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#386—21no—
44C2×#385—21no—
45#2285C2211yescharacteristic
46#2285C2211yescharacteristic
47#2280C2212yes—
48C2×#380C2211yescharacteristic
49C4×#117C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yescomplemented
52#2100C2212yescomplemented
53#2099C2212yescomplemented
54#2094C2212yescomplemented
55#2093C2212yescomplemented
56#2092C2214yescomplemented
57#2085C2214yescomplemented
58#2084C2212yescomplemented
59#2083C2212yescomplemented
60#2077C2214yescomplemented
61#2077C2214yescomplemented
62#2076C2214yescomplemented
63#2075C2214yescomplemented
64#2073C2214yescomplemented
65#2071C2212yescomplemented
66#2071C2212yescomplemented
67#2034C2214yes—
68#2034C2214yes—
69#2033C2214yes—
70#2033C2214yes—
71#1776C2214yescomplemented
72#1776C2214yescomplemented
73C4×#107C2214yescomplemented
74C4×#106C2214yescomplemented
75#1762C2214yescomplemented
76#1762C2214yescomplemented
77C4×#102C2214yescomplemented
78C4×#102C2214yescomplemented
79#1750C2214yescomplemented
80#1750C2214yescomplemented
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×M16C2312yes—
92C22×M16C2312yes—
93C2×#121—24no—
94C2×#121—22no—
95C2×#121—24no—
96C2×#121C2314yes—
97C2×C4×D4—22no—
98C2×C4×D4—28no—
99C2×C4×D4C2314yes—
100C2×#117—28no—
101C2×#117D412yes—
102C2×#117D412yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#428—28no—
124#428—28no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129#418—28no—
130#418—28no—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135#407—28no—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C4×M16—28no—
139C2×#97C2314yes—
140C2×C4×C8C2314yes—
141#390—24no—
142#390—24no—
143#390—21no—
144#390—21no—
145#390C2312yes—
146#390C2312yes—
147#389—21no—
148#389—21no—
149#389C2312yes—
150#389C2312yes—
151#388—24no—
152#388—24no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#386—24no—
170#386—24no—
171#385—24no—
172#385—24no—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#382C2312yes—
178#382C2312yes—
179#381—22no—
180#380—21no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—22no—
185#380C2312yes—
186#380C2312yes—
187#380C2312yes—
188#380C2312yes—
189#380C2314yes—
190#379—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2314yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205#376—22no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C43—22no—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2312yes—
221#336C2312yes—
222C25—21no—
223C25C2×D411yescharacteristic
224C22×D4—28no—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—82no—
231C22×D4—82no—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—24no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—44no—
243C23×C4—82no—
244C23×C4—82no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—24no—
256C2×M16—24no—
257C2×M16—24no—
258C2×M16#4118yes—
259C22×C8C2414yes—
260C22×C8C2414yes—
261#121—216no—
262#121—28no—
263#121—28no—
264C4×D4—28no—
265C4×D4—28no—
266C4×D4—48no—
267#117—28no—
268#117—28no—
269#117—24no—
270#117—24no—
271#117—24no—
272#117—24no—
273#117—24no—
274#117—24no—
275#117—24no—
276#117—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4#4114yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—82no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42#4118yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348#107—28no—
349#107—28no—
350#106—28no—
351#106—28no—
352#102—28no—
353#102—28no—
354#102—28no—
355#102—28no—
356C22⋊C8—28no—
357C22⋊C8—28no—
358#97—28no—
359C4×C8—28no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393C2×D4—216no—
394C2×D4—24no—
395C2×D4—216no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—84no—
403C2×D4—84no—
404C2×D4—816no—
405C2×D4C2×#4114yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—816no—
450C22×C4#12314yes—
451C22×C4#12314yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C4C2511yescharacteristic
465M16—216no—
466M16—48no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8C2×#4118yes—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4C2×#4114yes—
502C22⋊C4—216no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—44no—
524C42—44no—
525C42—82no—
526C42—82no—
527C42—82no—
528C42—82no—
529C42—84no—
530C42#12318yes—
531C42C2×#4114yes—
532C42C2×#4114yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—22no—
545C23—44no—
546C23—48no—
547C23—44no—
548C23—48no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—48no—
555C23—42no—
556C23—42no—
557C23—82no—
558C23—82no—
559C23—82no—
560C23—82no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23C2×#12312yes—
566C23#52512yes—
567C23#53212yes—
568C23C23×D411yescharacteristic
569D4—28no—
570D4—48no—
571D4—44no—
572D4—816no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—816no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4C2×#12314yes—
628C2×C4C2×#12314yes—
629C2×C4C2×#12312yes—
630C2×C4#52512yes—
631C2×C4#52918yes—
632C2×C4#52918yes—
633C2×C4#53212yes—
634C2×C4C22×#4112yes—
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4112yes—
638C8—28no—
639C8—44no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22—44no—
656C22—88no—
657C22—816no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#239514yes—
663C22C22×#12311yescharacteristic
664C22C2×#52511yescharacteristic
665C22C2×#53211yescharacteristic
666C22#358511yescharacteristic
667C22#358511yescharacteristic
668C22#358511yescharacteristic
669C22#358611yescharacteristic
670C4—24no—
671C4—24no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—44no—
677C4—42no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#239514yes—
685C4C2×#52912yes—
686C4C2×#52912yes—
687C2—24no—
688C2—82no—
689C2—84no—
690C2C2×#239511yescharacteristic
691C2#6051811yescharacteristic
692C2#6052011yescharacteristic
693C2#6056411yescharacteristic
694C2#6056511yescharacteristic
695C2#6070311yescharacteristic
696C2#6070411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34048C2semidirect
3. #23771C2semidirect
4. #23770C2semidirect
5. #23767C2semidirect
6. #23766C2semidirect
7. #23747C2semidirect
8. #23731C2semidirect
9. #23657C2semidirect
10. #23642C2semidirect
11. #23628C2semidirect
13. C4×#428C2semidirect
14. C4×#418C2semidirect
15. #20377C2semidirect
29. #2314C22semidirect
30. #2314C22semidirect
31. #2314C22semidirect
32. #2314C22semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
54. #2094C22semidirect
55. #2093C22semidirect
56. #2092C22semidirect
57. #2085C22semidirect
58. #2084C22semidirect
59. #2083C22semidirect
60. #2077C22semidirect
61. #2077C22semidirect
62. #2076C22semidirect
63. #2075C22semidirect
64. #2073C22semidirect
65. #2071C22semidirect
66. #2071C22semidirect
71. #1776C22semidirect
72. #1776C22semidirect
73. C4×#107C22semidirect
74. C4×#106C22semidirect
75. #1762C22semidirect
76. #1762C22semidirect
77. C4×#102C22semidirect
78. C4×#102C22semidirect
79. #1750C22semidirect
80. #1750C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23242
#340482
#237712
#237702
#237672
#237662
#237472
#237312
#236572
#236422
#236282
#234802
C4×#4282
C4×#4182
#203772