Finite Group Databasebeta

← #7506482

Subgroups — #7506482

Order 512.

2038 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic, complemented
2#35531C211yesmaximal, characteristic
3#31853C214yesmaximal, complemented
4#31852C214yesmaximal, complemented
5#31837C214yesmaximal, complemented
6#31836C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23633C211yesmaximal, characteristic, complemented
9#23541C211yesmaximal, characteristic, complemented
10C2×#2013C211yesmaximal, characteristic, complemented
11#23388C211yesmaximal, characteristic
12#23388C211yesmaximal, characteristic
13#21202C211yesmaximal, characteristic, complemented
14#21095C211yesmaximal, characteristic, complemented
15#21048C211yesmaximal, characteristic, complemented
16#20936C211yesmaximal, characteristic, complemented
17#20913C211yesmaximal, characteristic, complemented
18#20913C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#536C2211yescharacteristic
24C2×#535C2211yescharacteristic
25C22×C4⋊Q8C2211yescharacteristic
26C2×#520C2211yescharacteristic
27C2×#519C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C2×#418C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C2×#408C2211yescharacteristic
38C2×#408C2211yescharacteristic
39C2×C4×M16C2211yescharacteristic
40#2546—22no—
41#2545—22no—
42#2504C2212yes—
43#2501C2212yes—
44#2500C2212yes—
45#2500C2212yes—
46#2448—22no—
47#2445—22no—
48#2429C2214yescomplemented
49#2429C2214yescomplemented
50#2418—24no—
51#2413—24no—
52#2317C2211yescharacteristic
53#2310C2211yescharacteristic
54#2309C2211yescharacteristic
55#2297C2211yescharacteristic
56#2296C2211yescharacteristic
57C2×#390C2211yescharacteristic
58#2285C2211yescharacteristic
59#2101C2212yescomplemented
60#2101C2212yescomplemented
61#2100C2212yescomplemented
62#2099C2212yescomplemented
63#2074C2214yescomplemented
64#2072C2214yescomplemented
65#2055C2214yes—
66#2055C2214yes—
67#2013C2214yes—
68#2013C2214yes—
69#2000C2214yes—
70#2000C2214yes—
71#2000C2214yes—
72#2000C2214yes—
73#1960C2212yescomplemented
74#1960C2212yescomplemented
75#1956C2212yescomplemented
76#1956C2212yescomplemented
77#1936C2214yes—
78#1934C2214yes—
79#1924C2214yescomplemented
80#1923C2214yescomplemented
81#1880C2214yes—
82#1880C2214yes—
83#1873C2214yescomplemented
84#1873C2214yescomplemented
85#1872C2214yescomplemented
86#1872C2214yescomplemented
87#1805C2214yescomplemented
88#1803C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C22×SD16—24no—
102C22×SD16—42no—
103C22×SD16—42no—
104C22×SD16—42no—
105C22×SD16—42no—
106C22×SD16—42no—
107C22×SD16C2314yes—
108C22×SD16C2314yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#536—24no—
114#536—24no—
115#536—42no—
116#536—42no—
117#535—42no—
118#535—42no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8—24no—
122C2×C4⋊Q8—22no—
123C2×C4⋊Q8—42no—
124C2×C4⋊Q8C2312yes—
125C2×C4⋊Q8C2312yes—
126C2×#127—21no—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124—21no—
130C2×#123—22no—
131C2×#123—22no—
132C2×#123—24no—
133C2×C22⋊Q8C2314yes—
134C2×C22⋊Q8C2314yes—
135C2×#121C2312yes—
136C2×#121C2312yes—
137C2×#120—22no—
138#520—42no—
139#520—42no—
140#519—24no—
141#519—42no—
142C2×C4×Q8—21no—
143C2×C4×Q8—22no—
144C2×C4×Q8C2314yes—
145C2×C4×D4—22no—
146C2×C4×D4—21no—
147C2×#117—22no—
148C2×#117C2311yescharacteristic
149C2×#117C2311yescharacteristic
150C2×#117C2311yescharacteristic
151C2×#117C2311yescharacteristic
152C2×#117C2312yes—
153C2×#117C2311yescharacteristic
154C22×C4⋊C4—22no—
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2312yes—
157C22×C22⋊C4—22no—
158C22×C42C2311yescharacteristic
159#424—42no—
160C2×C4⋊C8C2314yes—
161#419—42no—
162#419—42no—
163#419—42no—
164#419—42no—
165#418—24no—
166#418—42no—
167C2×#103—42no—
168C2×#103—42no—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2314yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2314yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#102C2314yes—
180C2×#102C2314yes—
181#408—42no—
182#408—42no—
183C2×C22⋊C8C2314yes—
184C2×C22⋊C8C2314yes—
185C4×M16—42no—
186C2×#97C2312yes—
187C2×C4×C8C2312yes—
188#399—24no—
189#398—24no—
190#398—24no—
191#397—24no—
192#396—24no—
193#396—24no—
194#394—24no—
195#394—24no—
196#391—22no—
197#391D412yes—
198#391C2314yes—
199#390—22no—
200#390—22no—
201#390D412yes—
202#390C2312yes—
203#390C2312yes—
204#390C2312yes—
205#390C2312yes—
206#389—22no—
207#389—24no—
208#389—24no—
209#389C2312yes—
210#389C2312yes—
211#388—24no—
212#388—24no—
213#388—24no—
214#388—24no—
215#386—24no—
216#386D414yes—
217#385—22no—
218#385—22no—
219#384C2312yes—
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#382C2312yes—
223#382C2311yescharacteristic
224#382C2311yescharacteristic
225#381—22no—
226#380—22no—
227#379—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C22⋊C4C2312yes—
231C4×C22⋊C4C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#340C2312yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244#336C2312yes—
245C25—21no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41—22no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8#4114yes—
267C22×Q8C2412yes—
268C22×Q8C2412yes—
269C22×D4—22no—
270C22×D4—22no—
271C22×D4—21no—
272C22×D4—22no—
273C22×D4—22no—
274C22×D4—22no—
275C22×D4—22no—
276C22×D4—22no—
277C22×D4—22no—
278C22×D4C2×D412yes—
279C22×D4C2×D411yescharacteristic
280C22×D4C2412yes—
281C23×C4—21no—
282C23×C4—21no—
283C23×C4—21no—
284C23×C4—22no—
285C23×C4C2×D412yes—
286C23×C4C2412yes—
287C23×C4C2411yescharacteristic
288C2×SD16—28no—
289C2×SD16—44no—
290C2×SD16—44no—
291C2×SD16—44no—
292C2×SD16—44no—
293C2×SD16—44no—
294C2×SD16—416no—
295C2×SD16—416no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—22no—
300C2×M16—22no—
301C2×M16—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8—22no—
307C22×C8—22no—
308C22×C8C2412yes—
309C22×C8C2412yes—
310C22×C8C2412yes—
311C22×C8C2412yes—
312C4⋊Q8—24no—
313C4⋊Q8—24no—
314C4⋊Q8—42no—
315C4⋊Q8—42no—
316C4⋊Q8—48no—
317C4⋊Q8—44no—
318C4⋊Q8—44no—
319#127—44no—
320#124—44no—
321#124—44no—
322#124—44no—
323#123—48no—
324#123—48no—
325#123—88no—
326C22⋊Q8—28no—
327C22⋊Q8—28no—
328C22⋊Q8—44no—
329C22⋊Q8—44no—
330#121—42no—
331#121—42no—
332#121—42no—
333#121—42no—
334#120—48no—
335C4×Q8—28no—
336C4×Q8—44no—
337C4×Q8—48no—
338C4×Q8—44no—
339C4×D4—48no—
340C4×D4—44no—
341#117—22no—
342#117—22no—
343#117—22no—
344#117—22no—
345#117—22no—
346#117—22no—
347#117—22no—
348#117—22no—
349#117—22no—
350#117—48no—
351#117—42no—
352#117—42no—
353#117#4114yes—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—42no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—48no—
371C2×C4⋊C4C2×D412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2411yescharacteristic
374C2×C4⋊C4C2411yescharacteristic
375C2×C4⋊C4C2411yescharacteristic
376C2×C4⋊C4C2411yescharacteristic
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2412yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—84no—
393C2×C22⋊C4C2×D412yes—
394C2×C22⋊C4C2×D414yes—
395C2×C22⋊C4C2×D414yes—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—24no—
400C2×C42—21no—
401C2×C42—22no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42C2×D412yes—
405C2×C42C2412yes—
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C4⋊C8—44no—
409#103—44no—
410#103—44no—
411#103—42no—
412#103—42no—
413#103—42no—
414#103—42no—
415#103—42no—
416#103—42no—
417#103—42no—
418#103—42no—
419#102—28no—
420#102—44no—
421C22⋊C8—44no—
422C22⋊C8—44no—
423#97—42no—
424C4×C8—42no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95C2×D414yes—
433#95C2×D414yes—
434C24—22no—
435C24—42no—
436C24—42no—
437C24—42no—
438C24—82no—
439C24—82no—
440C24C22×D412yes—
441C24C22×D411yescharacteristic
442#41—44no—
443#41—44no—
444#41—44no—
445#41—44no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—28no—
452C2×Q8—28no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—44no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—44no—
470C2×Q8—44no—
471C2×Q8—44no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8C2×#4114yes—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—24no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—44no—
488C2×D4—44no—
489C2×D4—42no—
490C2×D4—44no—
491C2×D4—44no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—44no—
495C2×D4—44no—
496C2×D4—84no—
497C2×D4—84no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—21no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—44no—
523C22×C4—42no—
524C22×C4—82no—
525C22×C4—84no—
526C22×C4C22×D414yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D411yescharacteristic
531C22×C4C2×#4112yes—
532C22×C4C2×#4112yes—
533C22×C4H(5,𝔽2)12yes—
534C22×C4H(5,𝔽2)12yes—
535C22×C4H(5,𝔽2)12yes—
536C22×C4H(5,𝔽2)12yes—
537C22×C4C2511yescharacteristic
538SD16—416no—
539SD16—416no—
540M16—44no—
541M16—44no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—22no—
554C2×C8—44no—
555C2×C8—44no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—48no—
571C4⋊C4—48no—
572C4⋊C4—48no—
573C4⋊C4—48no—
574C4⋊C4—48no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—42no—
578C4⋊C4—44no—
579C4⋊C4—44no—
580C4⋊C4—82no—
581C4⋊C4—84no—
582C4⋊C4C2×#4114yes—
583C22⋊C4—24no—
584C22⋊C4—24no—
585C22⋊C4—28no—
586C22⋊C4—44no—
587C22⋊C4—48no—
588C22⋊C4—48no—
589C22⋊C4—44no—
590C22⋊C4—44no—
591C22⋊C4—44no—
592C22⋊C4—42no—
593C22⋊C4—42no—
594C22⋊C4—42no—
595C22⋊C4—48no—
596C22⋊C4—84no—
597C22⋊C4—84no—
598C22⋊C4—84no—
599C42—24no—
600C42—24no—
601C42—21no—
602C42—21no—
603C42—22no—
604C42—21no—
605C42—21no—
606C42—22no—
607C42—48no—
608C42—44no—
609C42—42no—
610C42—42no—
611C42—42no—
612C23—22no—
613C23—22no—
614C23—24no—
615C23—22no—
616C23—22no—
617C23—22no—
618C23—24no—
619C23—44no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—44no—
624C23—88no—
625C23—84no—
626C23—164no—
627C23—162no—
628C23—162no—
629C23#53512yes—
630C23#54612yes—
631C23#54612yes—
632C23C23×D411yescharacteristic
633Q8—24no—
634Q8—24no—
635Q8—42no—
636Q8—48no—
637Q8—44no—
638Q8—42no—
639Q8—42no—
640Q8—42no—
641Q8—42no—
642Q8—42no—
643Q8—84no—
644D4—48no—
645D4—42no—
646D4—42no—
647D4—44no—
648D4—42no—
649D4—42no—
650D4—84no—
651D4—164no—
652D4—164no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—24no—
665C2×C4—22no—
666C2×C4—48no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—48no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—42no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—84no—
693C2×C4—162no—
694C2×C4—164no—
695C2×C4#53512yes—
696C2×C4#53814yes—
697C2×C4#54014yes—
698C2×C4#54612yes—
699C2×C4#54612yes—
700C2×C4#55314yes—
701C2×C4#56014yes—
702C2×C4#57714yes—
703C2×C4C22×#4111yescharacteristic
704C2×C4C22×#4111yescharacteristic
705C2×C4C2×H(5,𝔽2)11yescharacteristic
706C2×C4C2×H(5,𝔽2)11yescharacteristic
707C2×C4C2×H(5,𝔽2)11yescharacteristic
708C2×C4C2×H(5,𝔽2)11yescharacteristic
709C8—42no—
710C8—42no—
711C8—42no—
712C8—42no—
713C22—24no—
714C22—24no—
715C22—22no—
716C22—22no—
717C22—44no—
718C22—44no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—84no—
723C22—84no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—168no—
728C22#270314yes—
729C22C2×#53511yescharacteristic
730C22C2×#54611yescharacteristic
731C22C2×#54611yescharacteristic
732C22C2×#57712yes—
733C22#358612yes—
734C4—24no—
735C4—42no—
736C4—44no—
737C4—44no—
738C4—42no—
739C4—44no—
740C4—82no—
741C4—82no—
742C4—81no—
743C4—81no—
744C4—81no—
745C4—81no—
746C4—82no—
747C4#270314yes—
748C4C2×#53811yescharacteristic
749C4C2×#54011yescharacteristic
750C4C2×#55311yescharacteristic
751C4C2×#56011yescharacteristic
752C2—24no—
753C2—82no—
754C2—162no—
755C2C2×#270311yescharacteristic
756C2#6081811yescharacteristic
757C2#6082611yescharacteristic
758C2#6118512yes—
759C2#6120912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
1. #35667C2semidirect
3. #31853C2semidirect
4. #31852C2semidirect
5. #31837C2semidirect
6. #31836C2semidirect
7. #23744C2semidirect
8. #23633C2semidirect
9. #23541C2semidirect
10. C2×#2013C2semidirect
13. #21202C2semidirect
14. #21095C2semidirect
15. #21048C2semidirect
16. #20936C2semidirect
17. #20913C2semidirect
18. #20913C2semidirect
19. #20651C2semidirect
48. #2429C22semidirect
49. #2429C22semidirect
59. #2101C22semidirect
60. #2101C22semidirect
61. #2100C22semidirect
62. #2099C22semidirect
63. #2074C22semidirect
64. #2072C22semidirect
73. #1960C22semidirect
74. #1960C22semidirect
75. #1956C22semidirect
76. #1956C22semidirect
79. #1924C22semidirect
80. #1923C22semidirect
83. #1873C22semidirect
84. #1873C22semidirect
85. #1872C22semidirect
86. #1872C22semidirect
87. #1805C22semidirect
88. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#356672
#355312
#318532
#318522
#318372
#318362
#237442
#236332
#235412
C2×#20132
#233882
#212022
#210952
#210482
#209362
#209132
#206512