Finite Group Databasebeta

← #6434763

Subgroups — #6434763

Order 512.

1771 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35264C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30069C214yesmaximal, complemented
4#30068C214yesmaximal, complemented
5#30064C212yesmaximal, complemented
6#30062C212yesmaximal, complemented
7#30061C212yesmaximal, complemented
8#30059C212yesmaximal, complemented
9#23731C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic, complemented
11#23557C212yesmaximal, complemented
12#23511C212yesmaximal
13#23452C211yesmaximal, characteristic, complemented
14C2×#2023C211yesmaximal, characteristic, complemented
15#21065C212yesmaximal, complemented
16#21062C212yesmaximal, complemented
17#20774C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#535C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#417C2212yes—
24C2×#418C2212yes—
25C2×#416C2212yes—
26C22×#102C2212yes—
27#2861C2212yes—
28#2858C2212yes—
29#2641C2212yes—
30#2640C2212yes—
31#2491—21no—
32#2491—21no—
33#2490—21no—
34#2490—21no—
35#2472—22no—
36#2472—22no—
37#2369—21no—
38#2369—21no—
39#2365—21no—
40#2365—21no—
41#2345—22no—
42Q8×C22⋊C4—22no—
43#2331C2214yescomplemented
44#2316C2212yes—
45#2311C2211yescharacteristic
46#2309C2211yescharacteristic
47C4×C4⋊Q8C2212yescomplemented
48C4×C4⋊Q8C2212yescomplemented
49#2297C2211yescharacteristic
50C2×#390C2211yescharacteristic
51#2285C2211yescharacteristic
52#2274C2212yes—
53C4×#117C2211yescharacteristic
54#2097C2212yescomplemented
55#2096C2211yescharacteristic, complemented
56#2095C2211yescharacteristic, complemented
57#2094C2212yes—
58#2094C2211yescharacteristic, complemented
59#2093C2212yes—
60#2093C2211yescharacteristic, complemented
61#2092C2214yes—
62#2092C2212yescomplemented
63#2059C2214yescomplemented
64#2059C2214yescomplemented
65#2058C2214yescomplemented
66#2057C2214yescomplemented
67#2042C2214yes—
68#2042C2214yes—
69#2041C2214yes—
70#2041C2214yes—
71#2024C2212yescomplemented
72#2024C2212yescomplemented
73#2023C2212yes—
74#2023C2212yes—
75#2023C2214yes—
76#2023C2212yescomplemented
77#2023C2212yescomplemented
78#1928C2214yescomplemented
79#1927C2214yescomplemented
80#1926C2214yes—
81#1926C2214yes—
82#1926C2214yescomplemented
83#1925C2214yes—
84#1925C2214yes—
85#1925C2214yescomplemented
86#1840C2214yescomplemented
87#1839C2212yescomplemented
88#1839C2212yescomplemented
89#1736C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×D4C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#544—44no—
101#535—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—28no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#127C2312yes—
110C2×#125—21no—
111C2×#125C2312yes—
112C2×#124—22no—
113C2×#124—22no—
114C2×#124C2312yes—
115C2×C22⋊Q8—22no—
116C2×#121—21no—
117C2×#121—21no—
118C2×#120—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8—22no—
121C2×#117C2311yescharacteristic
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C42C2311yescharacteristic
131#428—44no—
132#427—44no—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139#418—44no—
140#417—28no—
141#416—44no—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#102—28no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153#397—22no—
154#397—22no—
155#394—22no—
156#394—22no—
157#394—22no—
158#394—22no—
159#393—22no—
160#393—22no—
161#391—22no—
162#391—22no—
163#391—24no—
164#391—24no—
165#391C2312yes—
166#391C2311yescharacteristic
167#391C2311yescharacteristic
168#390—21no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2311yescharacteristic
178#390C2312yes—
179#390C2312yes—
180#390C2311yescharacteristic
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2314yes—
186#389C2314yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—24no—
192#387—24no—
193#386—21no—
194#386—21no—
195#386—22no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#382C2312yes—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—24no—
204#379—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C22⋊C4—24no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222#376C2312yes—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2314yes—
229#336C2314yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C25—21no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—22no—
241C2×#41—22no—
242C22×Q8—28no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—82no—
247C22×Q8—82no—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—24no—
254C22×D4—24no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4C2412yes—
258C22×D4C2411yescharacteristic
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C4⋊Q8—28no—
275C4⋊Q8—28no—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280#127—44no—
281#125—44no—
282#125—44no—
283#124—44no—
284#124—84no—
285#124—84no—
286C22⋊Q8—48no—
287#121—44no—
288#121—44no—
289#120—48no—
290C4×Q8—84no—
291C4×Q8—84no—
292#117—24no—
293#117—22no—
294#117—22no—
295#117—24no—
296#117—24no—
297#117—24no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4#4118yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4—82no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—42no—
363#107—42no—
364#107—44no—
365#106—42no—
366#106—42no—
367#106—44no—
368#103—28no—
369#103—42no—
370#103—42no—
371#102—28no—
372#102—28no—
373#102—28no—
374#102—42no—
375#102—42no—
376#102—44no—
377#102—44no—
378#95—24no—
379#95—24no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24—21no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—81no—
400C24—81no—
401C24—81no—
402C24—81no—
403C24C22×D411yescharacteristic
404#41—48no—
405#41—48no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8—816no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C42−1+412yes—
476C22×C42−1+412yes—
477C22×C4C2511yescharacteristic
478M16—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—48no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C2×#4118yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—82no—
526C22⋊C4—81no—
527C22⋊C4—82no—
528C22⋊C4—81no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—42no—
545C42—42no—
546C42—44no—
547C42—42no—
548C42—84no—
549C42—84no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—162no—
572C23—162no—
573C23—162no—
574C23—162no—
575C23#52512yes—
576C23#53512yes—
577C23#53612yes—
578C23C23×D411yescharacteristic
579Q8—416no—
580Q8—42no—
581Q8—42no—
582Q8—88no—
583Q8—164no—
584D4—44no—
585D4—44no—
586D4—42no—
587D4—42no—
588D4—84no—
589D4—84no—
590D4—164no—
591D4—164no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—88no—
638C2×C4—84no—
639C2×C4—162no—
640C2×C4#52512yes—
641C2×C4#53512yes—
642C2×C4#53612yes—
643C2×C4#54218yes—
644C2×C4#54218yes—
645C2×C4#57612yes—
646C2×C4#57712yes—
647C2×C4C22×#4112yes—
648C2×C4C2×H(5,𝔽2)12yes—
649C2×C4C2×2−1+411yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C8—42no—
652C8—42no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—81no—
661C22—82no—
662C22—81no—
663C22—82no—
664C22—81no—
665C22—82no—
666C22—81no—
667C22—82no—
668C22—81no—
669C22—82no—
670C22—81no—
671C22—82no—
672C22—81no—
673C22—82no—
674C22—168no—
675C22#263214yes—
676C22C2×#52511yescharacteristic
677C22C2×#53511yescharacteristic
678C22C2×#53611yescharacteristic
679C22C2×#57611yescharacteristic
680C22C2×#57711yescharacteristic
681C22#358511yescharacteristic
682C22#358611yescharacteristic
683C4—22no—
684C4—22no—
685C4—42no—
686C4—42no—
687C4—41no—
688C4—48no—
689C4—41no—
690C4—42no—
691C4—42no—
692C4—81no—
693C4—82no—
694C4—82no—
695C4—81no—
696C4—82no—
697C4—81no—
698C4#263214yes—
699C4C2×#54212yes—
700C4C2×#54212yes—
701C2—24no—
702C2—161no—
703C2—162no—
704C2C2×#263211yescharacteristic
705C2#6057811yescharacteristic
706C2#6058211yescharacteristic
707C2#6082211yescharacteristic
708C2#6082911yescharacteristic
709C2#6088811yescharacteristic
710C2#6089211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
2. #34216C2semidirect
3. #30069C2semidirect
4. #30068C2semidirect
5. #30064C2semidirect
6. #30062C2semidirect
7. #30061C2semidirect
8. #30059C2semidirect
9. #23731C2semidirect
10. #23730C2semidirect
11. #23557C2semidirect
13. #23452C2semidirect
14. C2×#2023C2semidirect
15. #21065C2semidirect
16. #21062C2semidirect
17. #20774C2semidirect
43. #2331C22semidirect
47. C4×C4⋊Q8C22semidirect
48. C4×C4⋊Q8C22semidirect
54. #2097C22semidirect
55. #2096C22semidirect
56. #2095C22semidirect
58. #2094C22semidirect
60. #2093C22semidirect
62. #2092C22semidirect
63. #2059C22semidirect
64. #2059C22semidirect
65. #2058C22semidirect
66. #2057C22semidirect
71. #2024C22semidirect
72. #2024C22semidirect
76. #2023C22semidirect
77. #2023C22semidirect
78. #1928C22semidirect
79. #1927C22semidirect
82. #1926C22semidirect
85. #1925C22semidirect
86. #1840C22semidirect
87. #1839C22semidirect
88. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#352642
#342162
#300692
#300682
#300642
#300622
#300612
#300592
#237312
#237302
#235572
#235112
#234522
C2×#20232
#210652
#210622
#207742