Finite Group Databasebeta

← #6460469

Subgroups — #6460469

Order 512.

1804 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2#35629C211yesmaximal, characteristic
3#29062C214yesmaximal, complemented
4#29058C214yesmaximal, complemented
5#29053C214yesmaximal, complemented
6#29049C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23507C211yesmaximal, characteristic, complemented
9#23487C211yesmaximal, characteristic
10#23487C211yesmaximal, characteristic
11C2×#1960C211yesmaximal, characteristic, complemented
12#21154C211yesmaximal, characteristic, complemented
13#21088C211yesmaximal, characteristic, complemented
14#21066C211yesmaximal, characteristic, complemented
15#20974C211yesmaximal, characteristic
16#20974C211yesmaximal, characteristic
17#20880C211yesmaximal, characteristic, complemented
18#20417C211yesmaximal, characteristic, complemented
19#20384C211yesmaximal, characteristic
20C22×#131C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2546—22no—
39#2545—22no—
40#2514—22no—
41#2513—22no—
42#2510—24no—
43#2495C2214yes—
44#2494C2214yes—
45#2429C2214yes—
46#2429C2214yes—
47#2413—24no—
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2310C2211yescharacteristic
52#2297C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2274C2211yescharacteristic
57#2101C2214yes—
58#2101C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2036C2214yes—
62#2036C2214yes—
63#2035C2214yes—
64#2035C2214yes—
65#1960C2212yes—
66#1960C2212yes—
67#1960C2212yes—
68#1960C2212yes—
69#1944C2212yes—
70#1944C2212yes—
71#1943C2212yes—
72#1943C2212yes—
73#1935C2214yes—
74#1933C2214yes—
75#1928C2214yes—
76#1927C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1865C2214yes—
82#1863C2214yes—
83#1763C2214yes—
84C4×#103C2214yes—
85#1753C2212yes—
86#1753C2212yes—
87#1752C2212yes—
88#1752C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#131—24no—
101C2×#131—42no—
102C2×#131—42no—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2312yes—
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2312yes—
109#536—24no—
110#536—24no—
111#536—42no—
112#536—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—24no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—42no—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2312yes—
120C2×#124—21no—
121C2×#124—21no—
122C2×#123—22no—
123C2×#123—22no—
124C2×C22⋊Q8C2314yes—
125C2×C22⋊Q8C2314yes—
126#519—24no—
127#519—42no—
128C2×C4×Q8—22no—
129C2×C4×Q8C2314yes—
130C2×#117—22no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—22no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#427—24no—
141#427—42no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#419—24no—
147#419—42no—
148#419—42no—
149#419—42no—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166#408—24no—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2312yes—
169#405—42no—
170C2×#97—42no—
171C2×#97C2311yescharacteristic
172C2×#97C2311yescharacteristic
173C2×#97C2311yescharacteristic
174C2×#97C2311yescharacteristic
175C2×#97C2312yes—
176C2×C4×C8C2312yes—
177#401—24no—
178#400—24no—
179#398—24no—
180#398—24no—
181#397—24no—
182#397—24no—
183#396—24no—
184#396—24no—
185#391—21no—
186#391—21no—
187#391—22no—
188#391D412yes—
189#391C2314yes—
190#390—21no—
191#390—21no—
192#390D412yes—
193#389—22no—
194#389—24no—
195#389—24no—
196#389C2314yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#388—24no—
202#388—24no—
203#388—24no—
204#388—24no—
205#387—24no—
206#386—22no—
207#386—22no—
208#386D414yes—
209#385—22no—
210#385—22no—
211#383—24no—
212#382—22no—
213#382C2312yes—
214#382C2312yes—
215#381—22no—
216#380—22no—
217#380C2314yes—
218#380C2314yes—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2314yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4C2312yes—
225C4×C22⋊C4C2312yes—
226C2×#95—22no—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2312yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8C2×D411yescharacteristic
259C22×Q8#4114yes—
260C22×Q8C2412yes—
261C22×Q8C2412yes—
262C22×D4C2×D412yes—
263C22×D4C2×D411yescharacteristic
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×D412yes—
269C23×C4C2412yes—
270C23×C4C2411yescharacteristic
271#131—416no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—24no—
279C2×M16—42no—
280C2×M16—42no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—24no—
286C22×C8—42no—
287C22×C8—42no—
288C22×C8#4114yes—
289C22×C8#4114yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—42no—
299C4⋊Q8—42no—
300C4⋊Q8—48no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303#124—44no—
304#124—44no—
305#123—48no—
306#123—48no—
307C22⋊Q8—28no—
308C22⋊Q8—28no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311C4×Q8—28no—
312C4×Q8—44no—
313C4×Q8—48no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—48no—
324#117#4114yes—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—48no—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D414yes—
359C2×C22⋊C4C2×D414yes—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—24no—
368C2×C42C2×D412yes—
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#107—24no—
373#107—42no—
374#106—24no—
375#106—42no—
376#103—28no—
377#103—24no—
378#103—24no—
379#103—24no—
380#103—24no—
381#103—24no—
382#103—24no—
383#103—24no—
384#103—24no—
385#103—44no—
386#103—44no—
387#103—44no—
388C22⋊C8—24no—
389C22⋊C8—24no—
390#97—41no—
391#97—41no—
392#97—41no—
393#97—41no—
394#97—42no—
395C4×C8—42no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95C2×D414yes—
409#95C2×D414yes—
410C24C22×D412yes—
411C24C22×D411yescharacteristic
412#41—44no—
413#41—44no—
414#41—44no—
415#41—44no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8C2×#4114yes—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4C2511yescharacteristic
490M16—48no—
491M16—44no—
492M16—44no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—48no—
510C2×C8C2×#4112yes—
511C2×C8C2×#4112yes—
512C2×C8C2×#4112yes—
513C2×C8C2×#4112yes—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—84no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4C2×#4114yes—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—24no—
551C42—24no—
552C42—21no—
553C42—21no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—48no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—84no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23#53212yes—
571C23#54612yes—
572C23#54612yes—
573C23C23×D411yescharacteristic
574Q8—24no—
575Q8—24no—
576Q8—42no—
577Q8—42no—
578Q8—48no—
579Q8—44no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—84no—
585D4—48no—
586D4—44no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—84no—
628C2×C4—162no—
629C2×C4—162no—
630C2×C4#52914yes—
631C2×C4#53212yes—
632C2×C4#53814yes—
633C2×C4#54612yes—
634C2×C4#54612yes—
635C2×C4#55314yes—
636C2×C4#55314yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C8—22no—
644C8—22no—
645C8—22no—
646C8—22no—
647C8—42no—
648C8—42no—
649C22—24no—
650C22—22no—
651C22—22no—
652C22—24no—
653C22—42no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22#255914yes—
659C22C2×#53211yescharacteristic
660C22C2×#54611yescharacteristic
661C22C2×#54611yescharacteristic
662C22#358612yes—
663C22#358612yes—
664C4—42no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#255914yes—
677C4C2×#52911yescharacteristic
678C4C2×#53811yescharacteristic
679C4C2×#55311yescharacteristic
680C4C2×#55311yescharacteristic
681C2—24no—
682C2—82no—
683C2C2×#255911yescharacteristic
684C2#6072011yescharacteristic
685C2#6072011yescharacteristic
686C2#6117612yes—
687C2#6122412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29062C2semidirect
4. #29058C2semidirect
5. #29053C2semidirect
6. #29049C2semidirect
7. #23746C2semidirect
8. #23507C2semidirect
11. C2×#1960C2semidirect
12. #21154C2semidirect
13. #21088C2semidirect
14. #21066C2semidirect
17. #20880C2semidirect
18. #20417C2semidirect

Maximal subgroups

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

subgroupindex
#356672
#356292
#290622
#290582
#290532
#290492
#237462
#235072
#234872
C2×#19602
#211542
#210882
#210662
#209742
#208802
#204172
#203842