Finite Group Databasebeta

← #7505055

Subgroups — #7505055

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2#34745C211yesmaximal, characteristic
3#31545C214yesmaximal, complemented
4#31543C214yesmaximal, complemented
5#31533C214yesmaximal, complemented
6#31531C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic
8#23659C211yesmaximal, characteristic, complemented
9#23619C211yesmaximal, characteristic, complemented
10#23552C211yesmaximal, characteristic, complemented
11#23455C211yesmaximal, characteristic, complemented
12#23452C211yesmaximal, characteristic
13#21323C211yesmaximal, characteristic, complemented
14#21237C211yesmaximal, characteristic
15#21023C211yesmaximal, characteristic, complemented
16#20927C211yesmaximal, characteristic
17#20926C211yesmaximal, characteristic
18#20874C211yesmaximal, characteristic, complemented
19#20649C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
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#2568—22no—
37#2568—22no—
38#2568C2214yes—
39#2568C2214yes—
40#2454C2212yes—
41#2453C2212yes—
42#2452C2212yes—
43#2451C2212yes—
44#2450—22no—
45#2449—22no—
46#2396—22no—
47C2×#400—22no—
48C4×C22⋊Q8—22no—
49#2297C2211yescharacteristic
50#2297C2211yescharacteristic
51#2296C2211yescharacteristic
52#2296C2211yescharacteristic
53C2×#391—22no—
54C2×#391C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#382C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2097C2212yes—
60#2097C2212yes—
61#2092C2212yes—
62#2092C2212yes—
63#2085C2214yes—
64#2085C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#2058C2214yes—
68#2057C2214yes—
69#2026C2214yes—
70#2024C2214yes—
71#2024C2214yes—
72#2023C2214yes—
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#1805C2214yes—
88#1804C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1728C2211yescharacteristic
96#1728C2211yescharacteristic
97#1728C2211yescharacteristic
98#1728C2211yescharacteristic
99C22×#41C2311yescharacteristic
100C23×Q8—21no—
101C23×Q8C2311yescharacteristic
102C2×#137—42no—
103C2×#137—42no—
104C2×#137Q818yescomplemented
105C22×Q16C2312yes—
106C22×SD16C2312yes—
107C22×M16C2311yescharacteristic
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111#533—24no—
112#533—42no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8—42no—
115C2×C4⋊Q8—42no—
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2314yes—
118C2×#125C2312yes—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C4×#41—24no—
122C4×#41—42no—
123C2×C4×Q8—22no—
124C2×C4×Q8—22no—
125C2×C4×Q8—42no—
126C2×C4×Q8—42no—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2314yes—
129C2×C4×D4—22no—
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4D412yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C42—21no—
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146#428—42no—
147#428—42no—
148#428—42no—
149C2×#107C2312yes—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#106C2312yes—
153C2×#106C2312yes—
154C2×#106C2312yes—
155#423—42no—
156C2×C4⋊C8C2314yes—
157#419—24no—
158#419—42no—
159#419—42no—
160#417—42no—
161#417—42no—
162#417—42no—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C4×M16—42no—
174C2×#97C2312yes—
175C2×C4×C8C2312yes—
176#400—24no—
177#400—24no—
178#400—24no—
179#400—24no—
180#400—22no—
181#399—22no—
182#398—22no—
183#398—22no—
184#396—22no—
185#396—22no—
186#395—22no—
187#393—22no—
188#391—24no—
189#391—24no—
190#391—24no—
191#391—24no—
192#391—21no—
193#391—22no—
194#391—21no—
195#391—22no—
196#391C2314yes—
197#391C2312yes—
198#391C2312yes—
199#391C2311yescharacteristic
200#391C2311yescharacteristic
201#391C2312yes—
202#390C2311yescharacteristic
203#390C2311yescharacteristic
204#390C2312yes—
205#389—22no—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#387—22no—
211#387—22no—
212#386—21no—
213#386—22no—
214#386—21no—
215#385—21no—
216#385—22no—
217#385—21no—
218#385—22no—
219#385—22no—
220#382—21no—
221#382—22no—
222#382—21no—
223#382C2314yes—
224#382C2312yes—
225#382C2312yes—
226#380—22no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4D412yes—
231C4×C4⋊C4D412yes—
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2312yes—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C2×#95—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—24no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—44no—
267C22×Q8—48no—
268C22×Q8C2×Q814yes—
269C22×Q8C2412yes—
270C22×Q8C2411yescharacteristic
271C22×D4—21no—
272C22×D4C2411yescharacteristic
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—22no—
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D412yes—
282C23×C4C2×D412yes—
283C23×C4C2411yescharacteristic
284#137—416no—
285C2×Q16—42no—
286C2×Q16—42no—
287C2×Q16C2×Q818yescomplemented
288C2×SD16—42no—
289C2×SD16—42no—
290C2×SD16C2×Q818yescomplemented
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×M16—22no—
300C2×M16—22no—
301C2×M16—22no—
302C2×M16C2×Q814yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C22×C8C2412yes—
307C4⋊Q8—24no—
308C4⋊Q8—44no—
309C4⋊Q8—44no—
310C4⋊Q8—42no—
311C4⋊Q8—48no—
312#125—24no—
313#125—42no—
314C22⋊Q8—48no—
315C22⋊Q8—48no—
316C4×Q8—24no—
317C4×Q8—48no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8—42no—
321C4×Q8—48no—
322C4×D4—24no—
323C4×D4—42no—
324C4×D4—48no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—44no—
331#117#4114yes—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
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—44no—
353C2×C4⋊C4—44no—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—24no—
384C2×C42—44no—
385C2×C42—44no—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D412yes—
388C2×C42C2×D412yes—
389C2×C42C2×D411yescharacteristic
390C2×C42C2412yes—
391C2×C42C2412yes—
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394#107—42no—
395#107—42no—
396#107—42no—
397#106—42no—
398#106—42no—
399#106—42no—
400C4⋊C8—44no—
401#103—28no—
402#103—42no—
403#103—42no—
404#103—42no—
405#103—42no—
406#103—42no—
407#103—44no—
408#102—42no—
409#102—42no—
410#102—42no—
411#97—42no—
412C4×C8—42no—
413#95—24no—
414#95—24no—
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—48no—
455C2×Q8—48no—
456C2×Q8—42no—
457C2×Q8—44no—
458C2×Q8—42no—
459C2×Q8—44no—
460C2×Q8—48no—
461C2×Q8—82no—
462C2×Q8—816no—
463C2×Q8C22×Q814yes—
464C2×Q8C22×Q812yes—
465C2×D4—24no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4C22×Q812yes—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—82no—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C22×Q812yes—
516C22×C4C2×#4112yes—
517C22×C4H(5,𝔽2)12yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C42−1+412yes—
521C22×C4C2511yescharacteristic
522Q16—48no—
523SD16—48no—
524M16—28no—
525M16—44no—
526M16—44no—
527M16—44no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8C22×Q814yes—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—48no—
562C4⋊C4—48no—
563C4⋊C4—44no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—44no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4C2×#4114yes—
571C22⋊C4—24no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—44no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C42—24no—
580C42—22no—
581C42—22no—
582C42—24no—
583C42—24no—
584C42—22no—
585C42—21no—
586C42—21no—
587C42—21no—
588C42—21no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—44no—
593C42—48no—
594C42—42no—
595C42—42no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—42no—
600C23—41no—
601C23—41no—
602C23—42no—
603C23#53512yes—
604C23#54612yes—
605C23D4×Q812yes—
606C23C23×D411yescharacteristic
607Q8—24no—
608Q8—22no—
609Q8—24no—
610Q8—22no—
611Q8—42no—
612Q8—41no—
613Q8—48no—
614Q8—42no—
615Q8—41no—
616Q8—816no—
617Q8—88no—
618D4—22no—
619D4—22no—
620D4—48no—
621D4—41no—
622D4—41no—
623C2×C4—28no—
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—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—48no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—48no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4—88no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4#53512yes—
673C2×C4#54214yes—
674C2×C4#54314yes—
675C2×C4#54612yes—
676C2×C4D4×Q812yes—
677C2×C4#55414yes—
678C2×C4#56214yes—
679C2×C4#57514yes—
680C2×C4C23×Q811yescharacteristic
681C2×C4C22×#4111yescharacteristic
682C2×C4C2×H(5,𝔽2)11yescharacteristic
683C2×C4C2×H(5,𝔽2)11yescharacteristic
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C2×C4C2×2−1+411yescharacteristic
686C8—24no—
687C8—42no—
688C8—42no—
689C8—42no—
690C22—22no—
691C22—22no—
692C22—24no—
693C22—24no—
694C22—42no—
695C22—44no—
696C22—81no—
697C22—81no—
698C22—82no—
699C22#269214yes—
700C22C2×#53511yescharacteristic
701C22C2×#54611yescharacteristic
702C22C2×D4×Q811yescharacteristic
703C22C2×#57512yes—
704C22#358612yes—
705C4—24no—
706C4—42no—
707C4—42no—
708C4—44no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—44no—
714C4—81no—
715C4—82no—
716C4—82no—
717C4—81no—
718C4—82no—
719C4—82no—
720C4#269214yes—
721C4C2×#54211yescharacteristic
722C4C2×#54311yescharacteristic
723C4C2×#55411yescharacteristic
724C4C2×#56211yescharacteristic
725C2—24no—
726C2—82no—
727C2C2×#269211yescharacteristic
728C2#6079911yescharacteristic
729C2#6080711yescharacteristic
730C2#6116512yes—
731C2#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×#2568C2semidirect
3. #31545C2semidirect
4. #31543C2semidirect
5. #31533C2semidirect
6. #31531C2semidirect
8. #23659C2semidirect
9. #23619C2semidirect
10. #23552C2semidirect
11. #23455C2semidirect
13. #21323C2semidirect
15. #21023C2semidirect
18. #20874C2semidirect
104. C2×#137Q8semidirect
287. C2×Q16C2×Q8semidirect
290. C2×SD16C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
#347452
#315452
#315432
#315332
#315312
#237332
#236592
#236192
#235522
#234552
#234522
#213232
#212372
#210232
#209272
#209262
#208742
#206492