Finite Group Databasebeta

← #6434776

Subgroups — #6434776

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#34272C211yesmaximal, characteristic, complemented
3#30072C214yesmaximal, complemented
4#30068C214yesmaximal, complemented
5#30067C212yesmaximal, complemented
6#30066C212yesmaximal, complemented
7#30063C212yesmaximal, complemented
8#30059C212yesmaximal, complemented
9#23737C212yesmaximal, complemented
10#23552C211yesmaximal, characteristic, complemented
11C2×#2057C211yesmaximal, characteristic, complemented
12#23508C211yesmaximal, characteristic
13#23504C211yesmaximal, characteristic
14#23458C212yesmaximal, complemented
15#21067C212yesmaximal, complemented
16#21062C212yesmaximal, complemented
17#20774C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#428C2212yes—
22C22×#107C2212yes—
23C2×#417C2212yes—
24C2×#417C2212yes—
25C2×#416C2212yes—
26C2×#416C2212yes—
27#2862C2212yes—
28#2858C2212yes—
29#2640C2214yes—
30#2563—21no—
31#2563—21no—
32#2547—21no—
33#2547—21no—
34#2504—22no—
35#2436—22no—
36#2370—21no—
37#2370—21no—
38#2369—21no—
39#2368—22no—
40#2365—21no—
41#2360—22no—
42#2331C2212yes—
43#2330C2212yes—
44#2317C2211yescharacteristic
45#2311C2212yes—
46#2311C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#125C2212yes—
49#2296C2211yescharacteristic
50C2×#389C2211yescharacteristic
51#2274C2211yescharacteristic
52#2274C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2097C2214yes—
55#2095C2214yes—
56#2093C2214yes—
57#2092C2214yes—
58#2058C2212yes—
59#2058C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2057C2212yes—
63#2057C2212yes—
64#2057C2214yes—
65#2043C2211yescharacteristic
66#2043C2212yes—
67#2043C2211yescharacteristic
68#2042C2212yes—
69#2042C2214yes—
70#2042C2212yes—
71#2041C2211yescharacteristic
72#2041C2212yes—
73#2041C2211yescharacteristic
74#2026C2214yes—
75#2025C2214yes—
76#2024C2214yes—
77#2023C2214yes—
78#1928C2214yes—
79#1928C2214yes—
80#1928C2214yes—
81#1927C2214yes—
82#1926C2214yes—
83#1925C2214yes—
84#1925C2214yes—
85#1925C2214yes—
86#1840C2214yes—
87#1839C2212yes—
88#1839C2212yes—
89#1736C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2312yes—
96C23×Q8—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—24no—
100#544—24no—
101#532—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#127C2311yescharacteristic
111C2×#125—21no—
112C2×#125C2312yes—
113C2×#124—22no—
114C2×#124—22no—
115C2×#124C2312yes—
116C2×C22⋊Q8—22no—
117C2×#121—21no—
118C2×#121—21no—
119C2×C4×Q8—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#428—44no—
133C2×#107—44no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138#417—28no—
139#417—44no—
140#416—28no—
141#416—44no—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154#397—24no—
155#397—24no—
156#393—24no—
157#393—24no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391C2314yes—
164#391C2314yes—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2314yes—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2311yescharacteristic
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#386—24no—
189#386—24no—
190#385—21no—
191#385—21no—
192#385—22no—
193#384C2312yes—
194#382—22no—
195#382—22no—
196#382C2312yes—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222#376C2312yes—
223C2×#95—21no—
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—24no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—44no—
248C22×Q8—82no—
249C22×Q8—82no—
250C22×Q8C2412yes—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C2412yes—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C4⋊Q8—28no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8—42no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279#127—42no—
280#125—28no—
281#125—24no—
282#125—24no—
283#124—42no—
284#124—42no—
285#124—84no—
286#124—84no—
287C22⋊Q8—216no—
288#121—44no—
289#121—44no—
290C4×Q8—84no—
291C4×D4—44no—
292C4×D4—44no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—24no—
297#117—24no—
298#117#4114yes—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4#4114yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—44no—
363#107—44no—
364#107—44no—
365#106—44no—
366#103—24no—
367#103—24no—
368#103—28no—
369#103—42no—
370#103—42no—
371#103—44no—
372#102—24no—
373#102—24no—
374#102—28no—
375#102—42no—
376#102—42no—
377#102—44no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—22no—
399C24C22×D411yescharacteristic
400#41—48no—
401#41—48no—
402#41—48no—
403#41—48no—
404C2×Q8—28no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—44no—
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—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C42−1+414yes—
474C22×C4C2511yescharacteristic
475M16—48no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—48no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—82no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4C2×#4114yes—
517C4⋊C4C2×#4112yes—
518C4⋊C4C2×#4112yes—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—24no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—82no—
543C42—82no—
544C42—82no—
545C42—82no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23#52512yes—
558C23#53512yes—
559C23#53612yes—
560C23C23×D411yescharacteristic
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—88no—
566Q8—88no—
567Q8—88no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—84no—
573D4—84no—
574D4—88no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—84no—
624C2×C4—88no—
625C2×C4—88no—
626C2×C4—162no—
627C2×C4#52512yes—
628C2×C4#53512yes—
629C2×C4#53612yes—
630C2×C4#54218yes—
631C2×C4#54218yes—
632C2×C4#57612yes—
633C2×C4#57812yes—
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4C2×2−1+412yes—
638C8—42no—
639C8—42no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22#263214yes—
655C22C2×#52511yescharacteristic
656C22C2×#53511yescharacteristic
657C22C2×#53611yescharacteristic
658C22C2×#57611yescharacteristic
659C22C2×#57811yescharacteristic
660C22#358511yescharacteristic
661C22#358611yescharacteristic
662C4—22no—
663C4—22no—
664C4—41no—
665C4—41no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—82no—
674C4—81no—
675C4—84no—
676C4—82no—
677C4—81no—
678C4#263214yes—
679C4C2×#54212yes—
680C4C2×#54212yes—
681C2—24no—
682C2—162no—
683C2C2×#263211yescharacteristic
684C2#6059611yescharacteristic
685C2#6060011yescharacteristic
686C2#6081111yescharacteristic
687C2#6081411yescharacteristic
688C2#6088011yescharacteristic
689C2#6088611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #34272C2semidirect
3. #30072C2semidirect
4. #30068C2semidirect
5. #30067C2semidirect
6. #30066C2semidirect
7. #30063C2semidirect
8. #30059C2semidirect
9. #23737C2semidirect
10. #23552C2semidirect
11. C2×#2057C2semidirect
14. #23458C2semidirect
15. #21067C2semidirect
16. #21062C2semidirect
17. #20774C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#342722
#300722
#300682
#300672
#300662
#300632
#300592
#237372
#235522
C2×#20572
#235082
#235042
#234582
#210672
#210622
#207742