Finite Group Databasebeta

← #7505052

Subgroups — #7505052

Order 512.

1804 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic, complemented
2#34745C211yesmaximal, characteristic
3#31548C214yesmaximal, complemented
4#31546C214yesmaximal, complemented
5#31528C214yesmaximal, complemented
6#31526C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic
8#23658C211yesmaximal, characteristic, complemented
9#23619C211yesmaximal, characteristic, complemented
10#23553C211yesmaximal, characteristic, complemented
11#23455C211yesmaximal, characteristic, complemented
12#23453C211yesmaximal, characteristic
13#21323C211yesmaximal, characteristic, complemented
14#21237C211yesmaximal, characteristic
15#21023C211yesmaximal, characteristic, complemented
16#20927C211yesmaximal, characteristic
17#20926C211yesmaximal, characteristic
18#20874C211yesmaximal, characteristic, complemented
19#20648C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C2×#428C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#419C2211yescharacteristic
34C2×#419C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36#2565—22no—
37#2565—22no—
38#2565C2214yes—
39#2565C2214yes—
40#2454C2212yes—
41#2453C2212yes—
42#2452C2212yes—
43#2451C2212yes—
44#2450—22no—
45#2449—22no—
46#2396—22no—
47C2×#398—22no—
48C4×C22⋊Q8—22no—
49#2297C2211yescharacteristic
50#2297C2211yescharacteristic
51#2296C2211yescharacteristic
52#2296C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#387—22no—
56C2×#384C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2097C2212yes—
60#2097C2212yes—
61#2092C2212yes—
62#2092C2212yes—
63#2084C2214yes—
64#2083C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#2059C2214yes—
68#2059C2214yes—
69#2026C2214yes—
70#2026C2214yes—
71#2025C2214yes—
72#2024C2214yes—
73#1993C2214yes—
74#1991C2214yes—
75#1973C2212yes—
76#1971C2212yes—
77#1969C2212yes—
78#1968C2212yes—
79#1917C2214yes—
80#1916C2214yes—
81#1880C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1877C2214yes—
85#1865C2214yes—
86#1863C2214yes—
87#1803C2214yes—
88#1802C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1728C2211yescharacteristic
96#1728C2211yescharacteristic
97#1728C2211yescharacteristic
98#1728C2211yescharacteristic
99C22×#41C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101C2×#137—42no—
102C2×#137—42no—
103C2×#137Q818yescomplemented
104C22×Q16C2312yes—
105C22×SD16C2312yes—
106C22×M16C2311yescharacteristic
107C22×M16C2311yescharacteristic
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110#533—24no—
111#533—42no—
112C2×C4⋊Q8C2312yes—
113C2×#125—22no—
114C2×#125—42no—
115C2×#125—42no—
116C2×#125C2312yes—
117C2×#125C2314yes—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C4×#41—24no—
121C4×#41—42no—
122C2×C4×Q8—22no—
123C2×C4×Q8—22no—
124C2×C4×Q8—42no—
125C2×C4×Q8—42no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2314yes—
128C2×C4×D4—22no—
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4D412yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C42—21no—
143C22×C42—21no—
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#428—42no—
148#428—42no—
149#428—42no—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#106C2312yes—
154C2×#106C2312yes—
155C2×#106C2312yes—
156#423—42no—
157C2×C4⋊C8C2314yes—
158#419—24no—
159#419—42no—
160#419—42no—
161#417—42no—
162#417—42no—
163#417—42no—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C4×M16—42no—
175C2×#97C2312yes—
176C2×C4×C8C2312yes—
177#400—22no—
178#399—22no—
179#398—24no—
180#398—24no—
181#398—24no—
182#398—24no—
183#398—22no—
184#398—22no—
185#396—22no—
186#396—22no—
187#395—22no—
188#393—22no—
189#391—21no—
190#391—22no—
191#391—21no—
192#391—22no—
193#391C2314yes—
194#391C2312yes—
195#391C2312yes—
196#391C2311yescharacteristic
197#391C2311yescharacteristic
198#391C2312yes—
199#390C2311yescharacteristic
200#390C2311yescharacteristic
201#390C2312yes—
202#389—22no—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#387—22no—
208#387—24no—
209#387—24no—
210#387—24no—
211#387—24no—
212#387—22no—
213#386—21no—
214#386—22no—
215#386—21no—
216#385—21no—
217#385—22no—
218#385—21no—
219#385—22no—
220#385—22no—
221#384—21no—
222#384—21no—
223#384C2314yes—
224#382—22no—
225#382C2312yes—
226#382C2312yes—
227#380—22no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4D412yes—
232C4×C4⋊C4D412yes—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2312yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C2×#95—22no—
238C2×#95—22no—
239C2×#95—22no—
240C2×#95D412yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#336C2314yes—
248#336C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41C2×Q814yes—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8C2×Q814yes—
266C22×Q8C2412yes—
267C22×Q8C2411yescharacteristic
268C22×D4—21no—
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—22no—
276C23×C4C2×D412yes—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D412yes—
280C23×C4C2411yescharacteristic
281#137—416no—
282C2×Q16—42no—
283C2×Q16—42no—
284C2×Q16C2×Q818yes—
285C2×SD16—42no—
286C2×SD16—42no—
287C2×SD16C2×Q818yes—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16C2×Q814yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C4⋊Q8—24no—
305C4⋊Q8—42no—
306#125—24no—
307#125—44no—
308#125—44no—
309#125—42no—
310#125—48no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C4×Q8—24no—
314C4×Q8—48no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×Q8—42no—
318C4×Q8—48no—
319C4×D4—24no—
320C4×D4—42no—
321C4×D4—48no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—44no—
328#117#4114yes—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4—48no—
353C2×C4⋊C4—44no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—24no—
386C2×C42—44no—
387C2×C42—44no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D412yes—
390C2×C42C2×D412yes—
391C2×C42C2×D411yescharacteristic
392C2×C42C2412yes—
393C2×C42C2412yes—
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396#107—42no—
397#107—42no—
398#107—42no—
399#106—42no—
400#106—42no—
401#106—42no—
402C4⋊C8—44no—
403#103—28no—
404#103—42no—
405#103—42no—
406#103—42no—
407#103—42no—
408#103—42no—
409#103—44no—
410#102—42no—
411#102—42no—
412#102—42no—
413#97—42no—
414C4×C8—42no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95C2×D412yes—
430#95C2×D412yes—
431#95C2×D412yes—
432#95C2×D412yes—
433C24—21no—
434C24C22×D411yescharacteristic
435#41—28no—
436#41—28no—
437#41—44no—
438#41—44no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—28no—
443C2×Q8—28no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—21no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—42no—
455C2×Q8—44no—
456C2×Q8—42no—
457C2×Q8—44no—
458C2×Q8—82no—
459C2×Q8C22×Q814yes—
460C2×Q8C22×Q812yes—
461C2×D4—24no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4C22×Q812yes—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—44no—
503C22×C4—82no—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×Q812yes—
512C22×C4C2×#4112yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C42−1+412yes—
517C22×C4C2511yescharacteristic
518Q16—48no—
519SD16—48no—
520M16—28no—
521M16—44no—
522M16—44no—
523M16—44no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8C22×Q814yes—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—48no—
558C4⋊C4—48no—
559C4⋊C4—48no—
560C4⋊C4—48no—
561C4⋊C4—44no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—48no—
566C4⋊C4—44no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4C2×#4114yes—
570C22⋊C4—24no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C42—24no—
579C42—22no—
580C42—22no—
581C42—24no—
582C42—24no—
583C42—22no—
584C42—21no—
585C42—21no—
586C42—21no—
587C42—21no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—44no—
592C42—48no—
593C42—42no—
594C42—42no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—42no—
599C23—41no—
600C23—41no—
601C23—42no—
602C23#53512yes—
603C23#54612yes—
604C23D4×Q812yes—
605C23C23×D411yescharacteristic
606Q8—24no—
607Q8—22no—
608Q8—24no—
609Q8—22no—
610Q8—42no—
611Q8—41no—
612Q8—48no—
613Q8—42no—
614Q8—41no—
615Q8—88no—
616D4—22no—
617D4—22no—
618D4—48no—
619D4—41no—
620D4—41no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—48no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—48no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—48no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—88no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4#53512yes—
671C2×C4#54214yes—
672C2×C4#54314yes—
673C2×C4#54612yes—
674C2×C4D4×Q812yes—
675C2×C4#55414yes—
676C2×C4#56214yes—
677C2×C4#57514yes—
678C2×C4C23×Q811yescharacteristic
679C2×C4C22×#4111yescharacteristic
680C2×C4C2×H(5,𝔽2)11yescharacteristic
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4C2×H(5,𝔽2)11yescharacteristic
683C2×C4C2×2−1+411yescharacteristic
684C8—24no—
685C8—42no—
686C8—42no—
687C8—42no—
688C22—22no—
689C22—22no—
690C22—24no—
691C22—24no—
692C22—42no—
693C22—44no—
694C22—81no—
695C22—81no—
696C22—82no—
697C22#269214yes—
698C22C2×#53511yescharacteristic
699C22C2×#54611yescharacteristic
700C22C2×D4×Q811yescharacteristic
701C22C2×#57512yes—
702C22#358612yes—
703C4—24no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—44no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—44no—
712C4—81no—
713C4—82no—
714C4—82no—
715C4—81no—
716C4—82no—
717C4—82no—
718C4#269214yes—
719C4C2×#54211yescharacteristic
720C4C2×#54311yescharacteristic
721C4C2×#55411yescharacteristic
722C4C2×#56211yescharacteristic
723C2—24no—
724C2—82no—
725C2C2×#269211yescharacteristic
726C2#6080211yescharacteristic
727C2#6080711yescharacteristic
728C2#6116312yes—
729C2#6136912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2565C2semidirect
3. #31548C2semidirect
4. #31546C2semidirect
5. #31528C2semidirect
6. #31526C2semidirect
8. #23658C2semidirect
9. #23619C2semidirect
10. #23553C2semidirect
11. #23455C2semidirect
13. #21323C2semidirect
15. #21023C2semidirect
18. #20874C2semidirect
103. C2×#137Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25652
#347452
#315482
#315462
#315282
#315262
#237332
#236582
#236192
#235532
#234552
#234532
#213232
#212372
#210232
#209272
#209262
#208742
#206482