Finite Group Databasebeta

← #7501893

Subgroups — #7501893

Order 512.

1962 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic, complemented
2C22×#482C211yesmaximal, characteristic
3#35584C211yesmaximal, characteristic
4#35151C211yesmaximal, characteristic, complemented
5#30802C214yesmaximal, complemented
6#30794C214yesmaximal, complemented
7#30786C214yesmaximal, complemented
8#30778C214yesmaximal, complemented
9#23735C211yesmaximal, characteristic, complemented
10#23638C211yesmaximal, characteristic, complemented
11#23614C211yesmaximal, characteristic, complemented
12#23556C211yesmaximal, characteristic, complemented
13#23480C211yesmaximal, characteristic
14#23456C211yesmaximal, characteristic, complemented
15#21174C211yesmaximal, characteristic, complemented
16#21091C211yesmaximal, characteristic, complemented
17#21050C211yesmaximal, characteristic, complemented
18#21037C211yesmaximal, characteristic, complemented
19#20845C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C22×#121C2211yescharacteristic
23C22×#117C2211yescharacteristic
24#3164—28no—
25C2×#501C2212yescomplemented
26C2×#501C2212yescomplemented
27C2×#500C2212yescomplemented
28C2×#499C2212yescomplemented
29C2×#482—28no—
30C2×#482C2214yes—
31C2×#482C2214yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C2×#407C2211yescharacteristic
44C22×C22⋊C8C2211yescharacteristic
45C2×#405C2211yescharacteristic
46#2855—22no—
47#2626—22no—
48#2561—22no—
49#2491—22no—
50#2491C2214yes—
51#2490—22no—
52#2490C2214yes—
53#2441—22no—
54#2441C2214yescomplemented
55#2439—22no—
56#2439C2214yescomplemented
57#2435—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2316C2211yescharacteristic
61#2309C2211yescharacteristic
62#2309C2211yescharacteristic
63#2280C2211yescharacteristic
64#2280C2211yescharacteristic
65#2280C2211yescharacteristic
66#2274C2211yescharacteristic
67#2094C2212yescomplemented
68#2093C2212yescomplemented
69#2092C2212yescomplemented
70#2092C2212yescomplemented
71#2077C2212yescomplemented
72#2077C2212yescomplemented
73#2076C2212yescomplemented
74#2075C2212yescomplemented
75#2067C2214yescomplemented
76#2065C2214yescomplemented
77#2059C2214yescomplemented
78#2058C2214yescomplemented
79#2034C2214yes—
80#2033C2214yes—
81#2025C2214yescomplemented
82#2023C2214yescomplemented
83#1949C2214yescomplemented
84#1948C2214yescomplemented
85#1936C2214yescomplemented
86#1934C2214yescomplemented
87#1924C2214yescomplemented
88#1923C2214yescomplemented
89#1921C2214yescomplemented
90#1918C2214yescomplemented
91#1854C2214yescomplemented
92#1854C2214yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C23×D4—21no—
102C23×D4C2311yescharacteristic
103C24×C4C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—24no—
110#544—42no—
111#544—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×#125—21no—
117C2×#125—21no—
118C2×#125C2312yes—
119C2×#125C2312yes—
120C2×#124—24no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×#121—22no—
124C2×#121—24no—
125C2×#121—42no—
126C2×#121C2314yes—
127C2×#117—22no—
128C2×#117—24no—
129C2×#117—42no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2314yes—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C42—21no—
143#501—28no—
144#501—28no—
145#500—28no—
146#499—28no—
147#482—216no—
148#482—216no—
149#428—24no—
150#428—24no—
151#428—42no—
152#427—24no—
153C2×#107D418yes—
154C2×#107C2312yes—
155C2×#107C2311yescharacteristic
156C2×#107C2311yescharacteristic
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#106—24no—
162C2×#106C2314yes—
163C2×#106C2312yes—
164C2×#106C2311yescharacteristic
165C2×#106C2311yescharacteristic
166C2×#106C2312yes—
167C2×#106C2312yes—
168#424—24no—
169C2×C4⋊C8C2314yes—
170#418—24no—
171#418—42no—
172C2×#102—24no—
173C2×#102—24no—
174C2×#102C2314yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×#102C2314yes—
178C2×#102C2312yes—
179C2×#102C2312yes—
180#407—24no—
181C2×C22⋊C8—24no—
182C2×C22⋊C8C2312yes—
183C2×C22⋊C8C2312yes—
184C2×C22⋊C8C2314yes—
185#405—24no—
186C2×#97C2312yes—
187C2×C4×C8C2312yes—
188#399—24no—
189#397—24no—
190#396—24no—
191#396—24no—
192#394—24no—
193#394—24no—
194#393—24no—
195#393—24no—
196#391—24no—
197#390—21no—
198#390—22no—
199#390—21no—
200#390—24no—
201#390C2312yes—
202#390C2312yes—
203#390C2314yes—
204#389—22no—
205#389—22no—
206#389—24no—
207#389C2314yes—
208#389C2314yes—
209#388—24no—
210#388—24no—
211#387—24no—
212#387—24no—
213#385—22no—
214#385—22no—
215#384—21no—
216#384C2312yes—
217#382—21no—
218#382C2312yes—
219#381—21no—
220#381—21no—
221#381D412yes—
222#380—22no—
223#380D412yes—
224#380D412yes—
225#380C2312yes—
226#380C2312yes—
227#380C2312yes—
228#380C2312yes—
229#379—21no—
230#379—21no—
231#379—22no—
232#379—22no—
233#379D412yes—
234C4×C4⋊C4—22no—
235C4×C4⋊C4C2314yes—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4C2314yes—
240#340C2312yes—
241#340C2314yes—
242#340C2312yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2312yes—
251#336C2312yes—
252C25—21no—
253C25C2×D411yescharacteristic
254C22×Q8—22no—
255C22×Q8—22no—
256C22×D4—24no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—44no—
265C22×D4—84no—
266C22×D4C2412yes—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—22no—
275C23×C4—21no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D412yes—
278C23×C4C2412yes—
279C23×C4C2411yescharacteristic
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16#4114yes—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8C2×D414yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C22×C8C2412yes—
297C4⋊Q8—24no—
298C4⋊Q8—24no—
299C4⋊Q8—42no—
300C4⋊Q8—42no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303#125—24no—
304#125—24no—
305#125—42no—
306#125—42no—
307#125—44no—
308#125—44no—
309#124—88no—
310C22⋊Q8—48no—
311C22⋊Q8—48no—
312#121—28no—
313#121—44no—
314#121—48no—
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—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—28no—
328#117—44no—
329#117—48no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—44no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C22⋊C4C2×D414yes—
379C2×C22⋊C4C2×D414yes—
380C2×C42—24no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—24no—
386C2×C42—44no—
387C2×C42C2412yes—
388C2×C42C2412yes—
389#107—22no—
390#107—22no—
391#107—24no—
392#107—24no—
393#107—42no—
394#107C2×D418yes—
395#107C2×D418yes—
396#106—22no—
397#106—22no—
398#106—28no—
399#106—24no—
400#106—24no—
401#106—42no—
402C4⋊C8—28no—
403#102—24no—
404#102—24no—
405#102—28no—
406#102—28no—
407#102—42no—
408#102—42no—
409C22⋊C8—24no—
410C22⋊C8—24no—
411C22⋊C8—28no—
412#97—24no—
413C4×C8—24no—
414#95—24no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95C2×D414yes—
423#95C2×D414yes—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24—42no—
432C24—42no—
433C24—82no—
434C24—82no—
435C24C22×D412yes—
436C24C22×D411yescharacteristic
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—84no—
440C2×Q8—84no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—48no—
448C2×D4—48no—
449C2×D4—44no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—84no—
457C2×D4—816no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4C22×D412yes—
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C22×C42−1+412yes—
499C22×C42−1+412yes—
500C22×C4C2511yescharacteristic
501M16—28no—
502M16—44no—
503C2×C8—28no—
504C2×C8—22no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—44no—
517C2×C8C22×D412yes—
518C2×C8C22×D412yes—
519C2×C8C2×#4114yes—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—84no—
549C4⋊C4—84no—
550C4⋊C4—84no—
551C4⋊C4C22×D414yes—
552C4⋊C4C22×D414yes—
553C22⋊C4—24no—
554C22⋊C4—24no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—44no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—84no—
565C22⋊C4—84no—
566C22⋊C4—84no—
567C22⋊C4—84no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—48no—
576C42—42no—
577C42—42no—
578C42—42no—
579C42—42no—
580C42—84no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—84no—
599C23—84no—
600C23—84no—
601C23—84no—
602C23—84no—
603C23—164no—
604C23—164no—
605C23#53612yes—
606C23D4212yes—
607C23#54712yes—
608C23C23×D411yescharacteristic
609Q8—164no—
610Q8—164no—
611D4—42no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—816no—
616D4—84no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4#53612yes—
657C2×C4#53614yes—
658C2×C4#54214yes—
659C2×C4D4212yes—
660C2×C4#54712yes—
661C2×C4#54814yes—
662C2×C4#56314yes—
663C2×C4C23×D411yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×2−1+411yescharacteristic
667C2×C4C2×2−1+411yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C8—22no—
670C8—22no—
671C8—24no—
672C8—42no—
673C8—42no—
674C22—24no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—88no—
686C22—84no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—84no—
691C22—168no—
692C22#267414yes—
693C22C2×#53611yescharacteristic
694C22C2×D4211yescharacteristic
695C22C2×#54711yescharacteristic
696C22#358412yes—
697C22#358512yes—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—44no—
703C4—44no—
704C4—82no—
705C4—81no—
706C4—81no—
707C4—82no—
708C4—81no—
709C4—81no—
710C4—82no—
711C4#267414yes—
712C4C2×#53611yescharacteristic
713C4C2×#54211yescharacteristic
714C4C2×#54811yescharacteristic
715C4C2×#56311yescharacteristic
716C2—24no—
717C2—82no—
718C2—162no—
719C2C2×#267411yescharacteristic
720C2#6087512yes—
721C2#6105411yescharacteristic
722C2#6105811yescharacteristic
723C2#6127412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3164C2semidirect
4. #35151C2semidirect
5. #30802C2semidirect
6. #30794C2semidirect
7. #30786C2semidirect
8. #30778C2semidirect
9. #23735C2semidirect
10. #23638C2semidirect
11. #23614C2semidirect
12. #23556C2semidirect
14. #23456C2semidirect
15. #21174C2semidirect
16. #21091C2semidirect
17. #21050C2semidirect
18. #21037C2semidirect
19. #20845C2semidirect
25. C2×#501C22semidirect
26. C2×#501C22semidirect
27. C2×#500C22semidirect
28. C2×#499C22semidirect
54. #2441C22semidirect
56. #2439C22semidirect
67. #2094C22semidirect
68. #2093C22semidirect
69. #2092C22semidirect
70. #2092C22semidirect
71. #2077C22semidirect
72. #2077C22semidirect
73. #2076C22semidirect
74. #2075C22semidirect
75. #2067C22semidirect
76. #2065C22semidirect
77. #2059C22semidirect
78. #2058C22semidirect
81. #2025C22semidirect
82. #2023C22semidirect
83. #1949C22semidirect
84. #1948C22semidirect
85. #1936C22semidirect
86. #1934C22semidirect
87. #1924C22semidirect
88. #1923C22semidirect
89. #1921C22semidirect
90. #1918C22semidirect
91. #1854C22semidirect
92. #1854C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C22×#4822
#355842
#351512
#308022
#307942
#307862
#307782
#237352
#236382
#236142
#235562
#234802
#234562
#211742
#210912
#210502
#210372
#208452