Finite Group Databasebeta

← #7511140

Subgroups — #7511140

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35515C211yesmaximal, characteristic, complemented
2#35280C211yesmaximal, characteristic, complemented
3#32665C214yesmaximal, complemented
4#32659C214yesmaximal, complemented
5#32657C214yesmaximal, complemented
6#32648C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic, complemented
8C2×#2093C211yesmaximal, characteristic, complemented
9#23660C211yesmaximal, characteristic, complemented
10#23658C211yesmaximal, characteristic, complemented
11#23614C211yesmaximal, characteristic, complemented
12#23611C211yesmaximal, characteristic, complemented
13#21304C211yesmaximal, characteristic, complemented
14#21235C211yesmaximal, characteristic
15#21158C211yesmaximal, characteristic, complemented
16#20929C211yesmaximal, characteristic
17#20928C211yesmaximal, characteristic
18#20879C211yesmaximal, characteristic, complemented
19#20872C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2857—22no—
38#2629—22no—
39#2582C2214yescomplemented
40#2580C2214yes—
41#2538C2214yes—
42#2538C2214yescomplemented
43#2454—22no—
44#2452—22no—
45#2437—24no—
46#2417—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52#2311C2211yescharacteristic
53#2309C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2097C2212yescomplemented
58#2095C2212yescomplemented
59#2093C2214yes—
60#2093C2214yes—
61#2093C2212yescomplemented
62#2092C2212yescomplemented
63#2085C2214yescomplemented
64#2084C2214yescomplemented
65#2083C2214yescomplemented
66#2083C2214yescomplemented
67#2068C2214yescomplemented
68#2067C2214yes—
69#2067C2214yescomplemented
70#2065C2214yes—
71#1985C2214yescomplemented
72#1984C2214yescomplemented
73#1973C2212yes—
74#1972C2212yes—
75#1970C2212yes—
76#1968C2212yes—
77#1943C2214yescomplemented
78#1942C2214yescomplemented
79#1880C2214yes—
80#1879C2214yes—
81#1879C2214yes—
82#1877C2214yes—
83#1865C2212yescomplemented
84#1864C2214yes—
85#1864C2212yescomplemented
86#1863C2212yescomplemented
87#1862C2214yes—
88#1862C2212yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C2×AGL(1,ℤ/8)—42no—
100C2×AGL(1,ℤ/8)—42no—
101C2×AGL(1,ℤ/8)Q818yescomplemented
102C22×SD16C2312yes—
103C22×D8C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#533—42no—
109#533—42no—
110C2×C4⋊Q8D412yes—
111C2×#125D412yes—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8D414yes—
115#518—42no—
116#518—42no—
117C4×#41—42no—
118C4×#41—42no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2314yes—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4C2314yes—
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#428—42no—
140#428—42no—
141#427—42no—
142C2×#107—24no—
143C2×#107—42no—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#107C2312yes—
147C2×#107C2314yes—
148C2×#107C2312yes—
149C2×#107C2312yes—
150C2×#106C2312yes—
151C2×#106C2312yes—
152C2×#106C2312yes—
153#424—42no—
154C2×C4⋊C8C2314yes—
155#418—42no—
156#417—42no—
157#416—24no—
158#416—42no—
159C2×#103C2311yescharacteristic
160C2×#103C2311yescharacteristic
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#102—24no—
164C2×#102—42no—
165C2×#102C2311yescharacteristic
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2311yescharacteristic
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173#405—42no—
174C2×#97C2312yes—
175C2×C4×C8C2312yes—
176#399—24no—
177#398—24no—
178#398—24no—
179#396—24no—
180#396—24no—
181#396—24no—
182#396—24no—
183#393—24no—
184#391—24no—
185#391C2312yes—
186#391C2312yes—
187#390—22no—
188#390—21no—
189#390—21no—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#390C2312yes—
194#390C2314yes—
195#389—22no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—24no—
200#389C2314yes—
201#389C2314yes—
202#389C2314yes—
203#389C2312yes—
204#389C2312yes—
205#386—22no—
206#386—22no—
207#385—22no—
208#385—24no—
209#385—24no—
210#385—22no—
211#384—21no—
212#384—21no—
213#384C2312yes—
214#384C2314yes—
215#382C2312yes—
216#380—22no—
217#380—22no—
218#380—24no—
219#380—24no—
220#380—22no—
221#380—22no—
222#380—22no—
223#379—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25—21no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41C2×Q814yes—
244C22×Q8C2×D414yes—
245C22×Q8C2411yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4C2×Q814yes—
254C22×D4C2411yescharacteristic
255C22×D4C2412yes—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—24no—
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264AGL(1,ℤ/8)—416no—
265C2×SD16—42no—
266C2×SD16—42no—
267C2×SD16C2×Q818yes—
268C2×D8—42no—
269C2×D8—42no—
270C2×D8C2×Q818yes—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16C2×Q814yes—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8#4114yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2411yescharacteristic
287C22×C8C2411yescharacteristic
288C4⋊Q8—44no—
289#125—44no—
290C22⋊Q8—48no—
291C22⋊Q8—88no—
292C22⋊Q8—88no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8—42no—
296C4×Q8—42no—
297C4×D4—42no—
298C4×D4—42no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—48no—
302C4×D4—48no—
303#117—22no—
304#117—22no—
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—44no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—84no—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2×D414yes—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—48no—
357C2×C22⋊C4—84no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2412yes—
371C2×C42C2412yes—
372#107—24no—
373#107—24no—
374#107—44no—
375#107—42no—
376#107—42no—
377#107—42no—
378#106—42no—
379#106—42no—
380#106—42no—
381C4⋊C8—44no—
382#103—22no—
383#103—22no—
384#103—42no—
385#103—42no—
386#102—24no—
387#102—24no—
388#102—22no—
389#102—22no—
390#102—44no—
391#102—42no—
392#102—42no—
393#102—44no—
394#97—42no—
395C4×C8—42no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404C24—21no—
405C24—22no—
406C24—24no—
407C24—42no—
408C24—42no—
409C24—82no—
410C24—82no—
411C24C22×D411yescharacteristic
412#41—28no—
413#41—28no—
414#41—44no—
415#41—44no—
416C2×Q8—28no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8C22×Q812yes—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—24no—
428C2×D4—28no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4C22×Q812yes—
446C2×D4C22×Q814yes—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×Q812yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C42−1+412yes—
484C22×C4C2511yescharacteristic
485SD16—48no—
486D8—48no—
487M16—28no—
488M16—44no—
489C2×C8—28no—
490C2×C8—21no—
491C2×C8—22no—
492C2×C8—21no—
493C2×C8—22no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—44no—
503C2×C8C22×Q814yes—
504C2×C8C2×#4112yes—
505C2×C8C2×#4112yes—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—82no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—48no—
540C22⋊C4—44no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—48no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—42no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—44no—
567C23—42no—
568C23—41no—
569C23—42no—
570C23—41no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—48no—
575C23—88no—
576C23—164no—
577C23—162no—
578C23—162no—
579C23#53412yes—
580C23D4×Q812yes—
581C23#55012yes—
582C23C23×D411yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587Q8—88no—
588D4—22no—
589D4—24no—
590D4—24no—
591D4—22no—
592D4—48no—
593D4—42no—
594D4—41no—
595D4—48no—
596D4—42no—
597D4—41no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—88no—
631C2×C4—168no—
632C2×C4—162no—
633C2×C4—162no—
634C2×C4—162no—
635C2×C4#53412yes—
636C2×C4#53914yes—
637C2×C4#54314yes—
638C2×C4D4×Q812yes—
639C2×C4#55012yes—
640C2×C4#55414yes—
641C2×C4#56214yes—
642C2×C4#57614yes—
643C2×C4C23×Q811yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C8—22no—
650C8—22no—
651C8—24no—
652C8—42no—
653C8—42no—
654C22—24no—
655C22—24no—
656C22—22no—
657C22—22no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—82no—
663C22—84no—
664C22—81no—
665C22—82no—
666C22—81no—
667C22—82no—
668C22—168no—
669C22#273214yes—
670C22C2×#53411yescharacteristic
671C22C2×D4×Q811yescharacteristic
672C22C2×#55011yescharacteristic
673C22C2×#57612yes—
674C22#358612yes—
675C4—24no—
676C4—42no—
677C4—44no—
678C4—44no—
679C4—42no—
680C4—82no—
681C4—82no—
682C4—81no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—81no—
687C4#273214yes—
688C4C2×#53911yescharacteristic
689C4C2×#54311yescharacteristic
690C4C2×#55411yescharacteristic
691C4C2×#56211yescharacteristic
692C2—24no—
693C2—82no—
694C2—84no—
695C2C2×#273211yescharacteristic
696C2#6076212yes—
697C2#6138112yes—
698C2#6140911yescharacteristic
699C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35515C2semidirect
2. #35280C2semidirect
3. #32665C2semidirect
4. #32659C2semidirect
5. #32657C2semidirect
6. #32648C2semidirect
7. #23737C2semidirect
8. C2×#2093C2semidirect
9. #23660C2semidirect
10. #23658C2semidirect
11. #23614C2semidirect
12. #23611C2semidirect
13. #21304C2semidirect
15. #21158C2semidirect
18. #20879C2semidirect
19. #20872C2semidirect
39. #2582C22semidirect
42. #2538C22semidirect
57. #2097C22semidirect
58. #2095C22semidirect
61. #2093C22semidirect
62. #2092C22semidirect
63. #2085C22semidirect
64. #2084C22semidirect
65. #2083C22semidirect
66. #2083C22semidirect
67. #2068C22semidirect
69. #2067C22semidirect
71. #1985C22semidirect
72. #1984C22semidirect
77. #1943C22semidirect
78. #1942C22semidirect
83. #1865C22semidirect
85. #1864C22semidirect
86. #1863C22semidirect
88. #1862C22semidirect
101. C2×AGL(1,ℤ/8)Q8semidirect

Maximal subgroups

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

subgroupindex
#355152
#352802
#326652
#326592
#326572
#326482
#237372
C2×#20932
#236602
#236582
#236142
#236112
#213042
#212352
#211582
#209292
#209282
#208792
#208722