Finite Group Databasebeta

← #7506747

Subgroups — #7506747

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35673C211yesmaximal, characteristic
2#35619C211yesmaximal, characteristic
3#31940C214yesmaximal, complemented
4#31939C214yesmaximal, complemented
5#31936C214yesmaximal, complemented
6#31935C214yesmaximal, complemented
7#23732C211yesmaximal, characteristic, complemented
8#23660C211yesmaximal, characteristic, complemented
9#23629C211yesmaximal, characteristic, complemented
10#23615C211yesmaximal, characteristic, complemented
11#23484C211yesmaximal, characteristic
12#23389C211yesmaximal, characteristic
13C2×#1969C211yesmaximal, characteristic, complemented
14#21046C211yesmaximal, characteristic
15#20935C211yesmaximal, characteristic, complemented
16#20913C211yesmaximal, characteristic
17#20847C211yesmaximal, characteristic
18#20760C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C23×Q16C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#408C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38#2503C2212yes—
39#2502C2212yes—
40#2499C2212yes—
41#2499C2212yes—
42#2498—24no—
43#2448—22no—
44#2447—22no—
45#2446—22no—
46#2446—22no—
47#2423C2214yes—
48#2421C2214yes—
49#2419—24no—
50#2317C2211yescharacteristic
51#2316C2211yescharacteristic
52#2311C2211yescharacteristic
53#2310C2211yescharacteristic
54#2297C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#391C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2097C2212yes—
60#2097C2212yes—
61#2096C2212yes—
62#2095C2212yes—
63#2085C2214yes—
64#2083C2214yes—
65#2074C2214yes—
66#2072C2214yes—
67#2068C2214yes—
68#2066C2214yes—
69#2036C2214yes—
70#2035C2214yes—
71#2001C2214yes—
72#2001C2214yes—
73#1969C2214yes—
74#1969C2214yes—
75#1923C2214yes—
76#1923C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1873C2214yes—
80#1872C2214yes—
81#1854C2214yes—
82#1852C2214yes—
83#1835C2212yes—
84#1835C2212yes—
85#1834C2212yes—
86#1834C2212yes—
87#1805C2214yes—
88#1803C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—24no—
102C22×Q16—42no—
103C22×Q16C2314yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#536—24no—
109#536—42no—
110#536—42no—
111#536—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#123—22no—
117C2×#123—22no—
118C2×C22⋊Q8C2314yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121#520—42no—
122#520—42no—
123#519—24no—
124#519—42no—
125C2×C4×Q8—21no—
126C2×C4×Q8—22no—
127C2×C4×Q8C2314yes—
128C2×C4×D4—22no—
129C2×C4×D4—21no—
130C2×#117—22no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2312yes—
136C2×#117C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C42C2311yescharacteristic
141#428—42no—
142#428—42no—
143C2×#107—42no—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#107C2314yes—
147C2×#106C2312yes—
148C2×#106C2312yes—
149#424—42no—
150C2×C4⋊C8C2314yes—
151#419—42no—
152#419—42no—
153#419—42no—
154#419—42no—
155C2×#103—42no—
156C2×#103—42no—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165#408—24no—
166#408—42no—
167C2×C22⋊C8C2314yes—
168C2×C22⋊C8C2314yes—
169C2×C4×C8—24no—
170C2×C4×C8C2312yes—
171C2×C4×C8C2312yes—
172#400—24no—
173#400—24no—
174#399—24no—
175#398—24no—
176#398—24no—
177#397—24no—
178#397—24no—
179#397—24no—
180#391—21no—
181#391—22no—
182#391—22no—
183#391—22no—
184#391D412yes—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#390—21no—
192#390D412yes—
193#389—22no—
194#389—22no—
195#389C2314yes—
196#388—24no—
197#388—24no—
198#387—24no—
199#387—24no—
200#387—24no—
201#387—24no—
202#386—22no—
203#386D414yes—
204#385—22no—
205#385—24no—
206#384C2312yes—
207#384C2312yes—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#382C2312yes—
211#382C2312yes—
212#382C2311yescharacteristic
213#382C2311yescharacteristic
214#381—22no—
215#381—24no—
216#380—22no—
217#380—22no—
218#380—22no—
219#380C2314yes—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C22⋊C4C2312yes—
223C4×C22⋊C4C2312yes—
224C2×#95—22no—
225C2×#95—22no—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
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—22no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272C2×Q16—28no—
273C2×Q16—28no—
274C2×Q16—44no—
275C2×Q16—416no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8#4114yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294#124—44no—
295#124—44no—
296#123—48no—
297#123—48no—
298C22⋊Q8—28no—
299C22⋊Q8—44no—
300C22⋊Q8—42no—
301C22⋊Q8—42no—
302C22⋊Q8—42no—
303C22⋊Q8—42no—
304C4×Q8—28no—
305C4×Q8—44no—
306C4×Q8—48no—
307C4×Q8—44no—
308C4×D4—48no—
309C4×D4—44no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—48no—
321#117—42no—
322#117—42no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—42no—
341C2×C4⋊C4—42no—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D414yes—
362C2×C22⋊C4C2×D414yes—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42C2×D412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—44no—
376#107—42no—
377#107—42no—
378#106—42no—
379#106—42no—
380C4⋊C8—44no—
381#103—44no—
382#103—44no—
383#103—42no—
384#103—42no—
385#103—44no—
386#103—44no—
387#103—42no—
388#103—42no—
389C22⋊C8—28no—
390C22⋊C8—44no—
391C4×C8—24no—
392C4×C8—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95C2×D414yes—
407#95C2×D414yes—
408C24C22×D412yes—
409C24C22×D411yescharacteristic
410#41—44no—
411#41—44no—
412#41—44no—
413#41—44no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—22no—
422C2×Q8—22no—
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—42no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8C2×#4114yes—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
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—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—84no—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C4C2511yescharacteristic
492Q16—416no—
493M16—44no—
494M16—44no—
495C2×C8—28no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—44no—
508C2×C8C2×#4114yes—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C22⋊C4—24no—
540C22⋊C4—28no—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—84no—
551C42—24no—
552C42—21no—
553C42—21no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—48no—
559C42—44no—
560C42—42no—
561C42—42no—
562C42—42no—
563C42—42no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23#53512yes—
572C23#54612yes—
573C23#54712yes—
574C23C23×D411yescharacteristic
575Q8—24no—
576Q8—24no—
577Q8—42no—
578Q8—48no—
579Q8—44no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—84no—
586D4—48no—
587D4—44no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
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—42no—
628C2×C4—42no—
629C2×C4—84no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4—164no—
633C2×C4#53512yes—
634C2×C4#53914yes—
635C2×C4#54014yes—
636C2×C4#54612yes—
637C2×C4#54712yes—
638C2×C4#55114yes—
639C2×C4#56214yes—
640C2×C4#57814yes—
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C8—24no—
648C8—42no—
649C8—42no—
650C8—42no—
651C22—24no—
652C22—24no—
653C22—22no—
654C22—22no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22#270614yes—
661C22C2×#53511yescharacteristic
662C22C2×#54611yescharacteristic
663C22C2×#54711yescharacteristic
664C22C2×#57812yes—
665C22#358412yes—
666C4—24no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—82no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—82no—
680C4#270614yes—
681C4C2×#53911yescharacteristic
682C4C2×#54011yescharacteristic
683C4C2×#55111yescharacteristic
684C4C2×#56211yescharacteristic
685C2—24no—
686C2—82no—
687C2C2×#270611yescharacteristic
688C2#6081911yescharacteristic
689C2#6082111yescharacteristic
690C2#6121412yes—
691C2#6126412yes—
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. #31940C2semidirect
4. #31939C2semidirect
5. #31936C2semidirect
6. #31935C2semidirect
7. #23732C2semidirect
8. #23660C2semidirect
9. #23629C2semidirect
10. #23615C2semidirect
13. C2×#1969C2semidirect
15. #20935C2semidirect
18. #20760C2semidirect
19. #20651C2semidirect

Maximal subgroups

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

subgroupindex
#356732
#356192
#319402
#319392
#319362
#319352
#237322
#236602
#236292
#236152
#234842
#233892
C2×#19692
#210462
#209352
#209132
#208472
#207602
#206512