Finite Group Databasebeta

← #7505025

Subgroups — #7505025

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35668C211yesmaximal, characteristic, complemented
2#35515C211yesmaximal, characteristic
3#31531C214yesmaximal, complemented
4#31526C214yesmaximal, complemented
5#31522C214yesmaximal, complemented
6#31519C214yesmaximal, complemented
7#23731C211yesmaximal, characteristic
8#23661C211yesmaximal, characteristic, complemented
9#23615C211yesmaximal, characteristic, complemented
10#23556C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic
12#23455C211yesmaximal, characteristic, complemented
13#21318C211yesmaximal, characteristic, complemented
14C2×#1968C211yesmaximal, characteristic
15#21020C211yesmaximal, characteristic, complemented
16#20936C211yesmaximal, characteristic
17#20933C211yesmaximal, characteristic
18#20877C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic
20C23×SD16C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2580—22no—
37#2568C2212yescomplemented
38#2567C2212yescomplemented
39#2565C2212yescomplemented
40#2565C2212yescomplemented
41#2538—22no—
42#2454C2214yes—
43#2453—22no—
44#2452C2214yes—
45#2451—22no—
46#2438—24no—
47#2437—24no—
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50#2316C2211yescharacteristic
51#2310C2211yescharacteristic
52#2309C2211yescharacteristic
53#2296C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#390C2211yescharacteristic
56#2285C2211yescharacteristic
57#2094C2212yes—
58#2093C2212yes—
59#2092C2212yes—
60#2092C2212yes—
61#2085C2214yescomplemented
62#2084C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2059C2214yescomplemented
66#2058C2214yescomplemented
67#2026C2214yescomplemented
68#2025C2214yes—
69#2024C2214yescomplemented
70#2023C2214yes—
71#1991C2212yescomplemented
72#1991C2212yescomplemented
73#1990C2212yescomplemented
74#1990C2212yescomplemented
75#1968C2214yes—
76#1968C2214yes—
77#1916C2214yescomplemented
78#1916C2214yescomplemented
79#1880C2214yes—
80#1880C2214yes—
81#1877C2214yes—
82#1877C2214yes—
83#1865C2214yescomplemented
84#1863C2214yescomplemented
85#1804C2214yes—
86#1802C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×SD16—42no—
100C22×SD16—42no—
101C22×SD16Q818yescomplemented
102C22×SD16C2314yes—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#544—42no—
108#544—42no—
109#533—24no—
110#533—42no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—22no—
120#519—42no—
121#519—42no—
122#518—24no—
123#518—42no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—22no—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—22no—
130C2×C4×D4C2314yes—
131C2×#117—22no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C22⋊C4—22no—
141C22×C22⋊C4—22no—
142C22×C42C2311yescharacteristic
143#428—42no—
144#428—42no—
145C2×#107C2312yes—
146C2×#107C2312yes—
147C2×#106—42no—
148C2×#106C2312yes—
149C2×#106C2314yes—
150C2×#106C2312yes—
151#424—42no—
152C2×C4⋊C8C2314yes—
153#419—24no—
154#419—42no—
155#418—42no—
156#418—42no—
157C2×#103—42no—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102—42no—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×C4×C8—42no—
168C2×C4×C8C2312yes—
169C2×C4×C8C2312yes—
170#400—24no—
171#399—24no—
172#398—24no—
173#398—24no—
174#396—24no—
175#396—24no—
176#395—24no—
177#393—24no—
178#391—22no—
179#391—24no—
180#391—22no—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#390—21no—
186#390—21no—
187#390—22no—
188#390C2314yes—
189#390C2312yes—
190#390C2312yes—
191#389—24no—
192#389—22no—
193#389—22no—
194#389D412yes—
195#389D412yes—
196#389C2314yes—
197#389C2314yes—
198#387—24no—
199#387—24no—
200#385—24no—
201#385—22no—
202#385—22no—
203#385—24no—
204#385D414yes—
205#384—21no—
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2311yescharacteristic
209#382—21no—
210#382C2311yescharacteristic
211#382C2312yes—
212#382C2311yescharacteristic
213#381—22no—
214#380—22no—
215#380—22no—
216#380—24no—
217#379—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C2×#95—22no—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2×Q814yes—
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4C2×Q814yes—
255C22×D4C2412yes—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—24no—
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×SD16—416no—
263C2×SD16—44no—
264C2×SD16—44no—
265C2×SD16C2×Q8116yes—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8C2×Q814yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—24no—
283C4⋊Q8—42no—
284C4⋊Q8—42no—
285C4⋊Q8—42no—
286C4⋊Q8—44no—
287#125—24no—
288#125—42no—
289#125—42no—
290#125—42no—
291#125—44no—
292C22⋊Q8—48no—
293C22⋊Q8—48no—
294C22⋊Q8—88no—
295C4×Q8—48no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—42no—
299C4×Q8—42no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×D4—28no—
303C4×D4—44no—
304C4×D4—48no—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—48no—
317#117#4114yes—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4—84no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42C2×D412yes—
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—42no—
377#107—42no—
378#106—44no—
379#106—42no—
380#106—42no—
381C4⋊C8—44no—
382#103—28no—
383#103—44no—
384#103—44no—
385#102—44no—
386#102—42no—
387#102—42no—
388#102—42no—
389#102—42no—
390C4×C8—42no—
391C4×C8—42no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95C2×D414yes—
402#95C2×D414yes—
403C24—22no—
404C24—24no—
405C24—42no—
406C24—42no—
407C24—82no—
408C24—82no—
409C24C22×D411yescharacteristic
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—28no—
413C2×Q8—28no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—82no—
426C2×Q8—84no—
427C2×Q8C22×Q814yes—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—28no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—42no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4C22×Q814yes—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×Q812yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476SD16—416no—
477M16—44no—
478M16—44no—
479C2×C8—28no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—44no—
492C2×C8C22×Q814yes—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—28no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—82no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C2×#4114yes—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—48no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—48no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—24no—
541C42—24no—
542C42—21no—
543C42—21no—
544C42—22no—
545C42—21no—
546C42—21no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—44no—
551C42—44no—
552C42—42no—
553C42—42no—
554C42—42no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—48no—
563C23—88no—
564C23—162no—
565C23—162no—
566C23—164no—
567C23#53512yes—
568C23#54612yes—
569C23D4×Q812yes—
570C23C23×D411yescharacteristic
571Q8—24no—
572Q8—24no—
573Q8—42no—
574Q8—48no—
575Q8—42no—
576Q8—88no—
577Q8—164no—
578D4—24no—
579D4—24no—
580D4—42no—
581D4—48no—
582D4—42no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—168no—
619C2×C4—162no—
620C2×C4—162no—
621C2×C4#53512yes—
622C2×C4#54214yes—
623C2×C4#54314yes—
624C2×C4#54612yes—
625C2×C4D4×Q812yes—
626C2×C4#55414yes—
627C2×C4#56214yes—
628C2×C4#57714yes—
629C2×C4C23×Q811yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C8—24no—
636C8—42no—
637C8—42no—
638C8—42no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22—44no—
644C22—48no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22—168no—
649C22#269214yes—
650C22C2×#53511yescharacteristic
651C22C2×#54611yescharacteristic
652C22C2×D4×Q811yescharacteristic
653C22C2×#57712yes—
654C22#358412yes—
655C4—24no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—44no—
662C4—81no—
663C4—82no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—82no—
668C4—82no—
669C4#269214yes—
670C4C2×#54211yescharacteristic
671C4C2×#54311yescharacteristic
672C4C2×#55411yescharacteristic
673C4C2×#56211yescharacteristic
674C2—24no—
675C2—84no—
676C2C2×#269211yescharacteristic
677C2#6081911yescharacteristic
678C2#6082011yescharacteristic
679C2#6117212yes—
680C2#6137112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35668C2semidirect
3. #31531C2semidirect
4. #31526C2semidirect
5. #31522C2semidirect
6. #31519C2semidirect
8. #23661C2semidirect
9. #23615C2semidirect
10. #23556C2semidirect
12. #23455C2semidirect
13. #21318C2semidirect
15. #21020C2semidirect
18. #20877C2semidirect
37. #2568C22semidirect
38. #2567C22semidirect
39. #2565C22semidirect
40. #2565C22semidirect
61. #2085C22semidirect
62. #2084C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
65. #2059C22semidirect
66. #2058C22semidirect
67. #2026C22semidirect
69. #2024C22semidirect
71. #1991C22semidirect
72. #1991C22semidirect
73. #1990C22semidirect
74. #1990C22semidirect
77. #1916C22semidirect
78. #1916C22semidirect
83. #1865C22semidirect
84. #1863C22semidirect
101. C22×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
#356682
#355152
#315312
#315262
#315222
#315192
#237312
#236612
#236152
#235562
#234562
#234552
#213182
C2×#19682
#210202
#209362
#209332
#208772
#206502