Finite Group Databasebeta

← #6444629

Subgroups — #6444629

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3084C211yesmaximal, characteristic
2C22×#479C211yesmaximal, characteristic, complemented
3#35640C211yesmaximal, characteristic, complemented
4#34266C211yesmaximal, characteristic
5#26477C214yesmaximal, complemented
6#26473C214yesmaximal, complemented
7#26469C214yesmaximal, complemented
8#26465C214yesmaximal, complemented
9#23613C211yesmaximal, characteristic, complemented
10#23609C211yesmaximal, characteristic
11#23457C211yesmaximal, characteristic, complemented
12#23453C211yesmaximal, characteristic
13#21262C211yesmaximal, characteristic
14#21205C211yesmaximal, characteristic
15#21173C211yesmaximal, characteristic, complemented
16#21147C211yesmaximal, characteristic, complemented
17#20881C211yesmaximal, characteristic
18#20874C211yesmaximal, characteristic, complemented
19#20454C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25#3084—28no—
26C2×#479—28no—
27C2×#479C2214yes—
28C2×#479C2214yes—
29C2×#479C2212yes—
30C2×#475C2212yes—
31C2×#478C2212yes—
32C2×#476C2212yes—
33C2×#428C2211yescharacteristic
34C2×#427C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C2×#424C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#417C2211yescharacteristic
42C2×#419C2211yescharacteristic
43C2×#419C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C2×#416C2211yescharacteristic
46C22×#103C2211yescharacteristic
47C22×#103C2211yescharacteristic
48C2×#405C2211yescharacteristic
49#2583—22no—
50#2552—22no—
51#2552C2214yes—
52#2550—22no—
53#2550C2214yes—
54#2502—22no—
55#2367C2214yes—
56#2365C2214yes—
57#2362—22no—
58#2360—22no—
59#2322—22no—
60#2317C2211yescharacteristic
61#2317C2211yescharacteristic
62#2310C2211yescharacteristic
63C4×#123—22no—
64#2297C2211yescharacteristic
65#2296C2211yescharacteristic
66#2296C2211yescharacteristic
67#2274C2211yescharacteristic
68#2274C2211yescharacteristic
69#2068C2212yes—
70#2068C2214yes—
71#2067C2212yes—
72#2066C2212yes—
73#2066C2214yes—
74#2065C2212yes—
75#2026C2214yes—
76#2026C2214yes—
77#2025C2214yes—
78#2024C2214yes—
79#1983C2214yes—
80#1981C2214yes—
81#1960C2212yes—
82#1959C2212yes—
83#1958C2212yes—
84#1956C2212yes—
85#1949C2214yes—
86#1947C2214yes—
87#1940C2214yes—
88#1939C2214yes—
89#1865C2214yes—
90#1863C2214yes—
91#1863C2214yes—
92#1862C2214yes—
93#1776C2214yes—
94C4×#106C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#544—24no—
108#544—42no—
109#536—24no—
110#536Q818yescomplemented
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×#125—21no—
115C2×#125—24no—
116C2×#125—22no—
117C2×#125—42no—
118C2×#125C2312yes—
119C2×#125C2314yes—
120C2×#124C2312yes—
121C2×#123—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124#520—24no—
125#520—42no—
126C2×C4×Q8—22no—
127C2×C4×Q8—24no—
128C2×C4×Q8—42no—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2314yes—
131C2×C4×D4—22no—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#479—216no—
149#479—216no—
150#479—28no—
151#478—28no—
152#476—28no—
153#475—28no—
154#428—24no—
155#427—24no—
156C2×#107—24no—
157C2×#107C2312yes—
158C2×#107C2314yes—
159C2×#107C2312yes—
160C2×#106D418yes—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165#424—24no—
166#424—24no—
167C2×C4⋊C8—24no—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2314yes—
173#419—24no—
174#419—42no—
175#417—24no—
176#417—42no—
177#416—24no—
178#416—24no—
179C2×#103—24no—
180C2×#103—24no—
181C2×#103C2311yescharacteristic
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#103C2312yes—
186C2×#103C2314yes—
187C2×#103C2312yes—
188C2×#103C2314yes—
189C2×#103C2312yes—
190C2×#103C2312yes—
191C2×#103C2312yes—
192C2×#102C2311yescharacteristic
193C2×#102C2312yes—
194C2×#102C2311yescharacteristic
195C2×#102C2312yes—
196C2×#102C2312yes—
197#405—24no—
198C2×#97C2312yes—
199C2×C4×C8C2312yes—
200#398—24no—
201#398—24no—
202#398—24no—
203#395—24no—
204#391—21no—
205#391—21no—
206#391—24no—
207#391C2314yes—
208#391C2312yes—
209#390C2312yes—
210#389—22no—
211#389—24no—
212#389—24no—
213#389—24no—
214#389D412yes—
215#389D412yes—
216#389C2314yes—
217#389C2312yes—
218#389C2312yes—
219#388—24no—
220#388—24no—
221#387—24no—
222#387—24no—
223#386—21no—
224#386—21no—
225#385—21no—
226#385—21no—
227#385—24no—
228#385—22no—
229#385—22no—
230#385D412yes—
231#385D412yes—
232#384—21no—
233#384—24no—
234#384C2312yes—
235#382—21no—
236#382C2312yes—
237#381—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—24no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4C2314yes—
243C4×C4⋊C4C2314yes—
244C4×C4⋊C4C2314yes—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248C4×C22⋊C4—24no—
249#340C2312yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#336C2314yes—
255#336C2314yes—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41C2×Q814yes—
262C22×Q8—21no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8C2×Q814yes—
270C22×Q8C2412yes—
271C22×Q8C2411yescharacteristic
272C22×D4—21no—
273C22×D4C2411yescharacteristic
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—22no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8C2×D414yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C4⋊Q8—24no—
301C4⋊Q8—24no—
302C4⋊Q8—42no—
303C4⋊Q8—44no—
304C4⋊Q8C2×Q818yes—
305#125—28no—
306#125—24no—
307#125—44no—
308#125—42no—
309#125—48no—
310#125—44no—
311#124—24no—
312#124C2×Q818yes—
313#123—216no—
314C22⋊Q8—216no—
315C22⋊Q8—48no—
316C4×Q8—28no—
317C4×Q8—24no—
318C4×Q8—44no—
319C4×Q8—42no—
320C4×Q8—48no—
321C4×D4—24no—
322C4×D4—42no—
323C4×D4—48no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—44no—
332#117C2×Q814yes—
333#117#4114yes—
334#117#4114yes—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—24no—
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⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—24no—
388C2×C42—24no—
389C2×C42—24no—
390C2×C42—44no—
391C2×C42C2×D412yes—
392C2×C42C2412yes—
393C2×C42C2412yes—
394C2×C42C2412yes—
395#107—28no—
396#107—24no—
397#107—24no—
398#106—24no—
399#106—24no—
400#106C2×D418yes—
401#106C2×D418yes—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—28no—
407#103—24no—
408#103—24no—
409#103—22no—
410#103—22no—
411#103—28no—
412#103—24no—
413#103—24no—
414#103—28no—
415#103—42no—
416#103—42no—
417#103—42no—
418#102—22no—
419#102—22no—
420#102—24no—
421#102—24no—
422#102—42no—
423#97—24no—
424C4×C8—24no—
425#95—22no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—22no—
430#95—24no—
431#95—22no—
432#95—22no—
433#95—24no—
434#95—24no—
435#95C2×D414yes—
436#95C2×D414yes—
437C24—21no—
438C24C22×D411yescharacteristic
439#41—28no—
440#41—28no—
441#41—44no—
442#41—44no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8—28no—
447C2×Q8—28no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—44no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—84no—
462C2×Q8C22×Q814yes—
463C2×Q8C22×Q812yes—
464C2×D4—24no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4C22×Q812yes—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—84no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D414yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D412yes—
507C22×C4C22×Q812yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4H(5,𝔽2)12yes—
511C22×C42−1+412yes—
512C22×C4C2511yescharacteristic
513M16—44no—
514M16—44no—
515C2×C8—28no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—44no—
530C2×C8C22×D412yes—
531C2×C8C22×D412yes—
532C4⋊C4—216no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—48no—
552C4⋊C4—48no—
553C4⋊C4—48no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—84no—
559C4⋊C4—82no—
560C4⋊C4—84no—
561C4⋊C4—82no—
562C4⋊C4C22⋊Q818yes—
563C4⋊C4C22×D414yes—
564C4⋊C4C22×D414yes—
565C4⋊C4C2×#4114yes—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—216no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—82no—
574C22⋊C4C22⋊Q818yes—
575C42—24no—
576C42—24no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—48no—
582C42—44no—
583C42—42no—
584C42—42no—
585C42—84no—
586C42—84no—
587C42C22×Q814yes—
588C42C2×#4114yes—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—42no—
593C23—42no—
594C23—41no—
595C23—41no—
596C23C2×C22⋊Q812yes—
597C23D4212yes—
598C23#54812yes—
599C23C23×D411yescharacteristic
600Q8—24no—
601Q8—22no—
602Q8—24no—
603Q8—22no—
604Q8—42no—
605Q8—41no—
606Q8—42no—
607Q8—41no—
608Q8—48no—
609Q8—164no—
610D4—22no—
611D4—22no—
612D4—48no—
613D4—41no—
614D4—41no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—48no—
643C2×C4—48no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—164no—
657C2×C4C2×C22⋊Q818yes—
658C2×C4C2×C22⋊Q812yes—
659C2×C4#53214yes—
660C2×C4#54214yes—
661C2×C4D4212yes—
662C2×C4#54812yes—
663C2×C4D4×Q814yes—
664C2×C4#55414yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C23×Q811yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C8—22no—
672C8—22no—
673C8—42no—
674C8—42no—
675C8—42no—
676C22—24no—
677C22—22no—
678C22—22no—
679C22—24no—
680C22—44no—
681C22—42no—
682C22—81no—
683C22—81no—
684C22—82no—
685C22#245214yes—
686C22C22×C22⋊Q811yescharacteristic
687C22C2×D4211yescharacteristic
688C22C2×#54811yescharacteristic
689C22#358412yes—
690C22#358612yes—
691C4—28no—
692C4—42no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—44no—
700C4—44no—
701C4—81no—
702C4—82no—
703C4—81no—
704C4—82no—
705C4#245214yes—
706C4C2×#53211yescharacteristic
707C4C2×#54211yescharacteristic
708C4C2×D4×Q811yescharacteristic
709C4C2×#55411yescharacteristic
710C2—24no—
711C2—82no—
712C2C2×#245211yescharacteristic
713C2#6049912yes—
714C2#6105211yescharacteristic
715C2#6106011yescharacteristic
716C2#6135012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
2. C22×#479C2semidirect
3. #35640C2semidirect
5. #26477C2semidirect
6. #26473C2semidirect
7. #26469C2semidirect
8. #26465C2semidirect
9. #23613C2semidirect
11. #23457C2semidirect
15. #21173C2semidirect
16. #21147C2semidirect
18. #20874C2semidirect
19. #20454C2semidirect
110. #536Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30842
C22×#4792
#356402
#342662
#264772
#264732
#264692
#264652
#236132
#236092
#234572
#234532
#212622
#212052
#211732
#211472
#208812
#208742
#204542