Finite Group Databasebeta

← #6513420

Subgroups — #6513420

Order 512.

2160 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3213C211yesmaximal, characteristic
2C22×#464C211yesmaximal, characteristic, complemented
3#35249C211yesmaximal, characteristic, complemented
4#34377C211yesmaximal, characteristic, complemented
5#28262C214yesmaximal, complemented
6#28172C214yesmaximal, complemented
7#25383C214yesmaximal, complemented
8#25336C214yesmaximal, complemented
9#24528C212yesmaximal, complemented
10#24526C212yesmaximal, complemented
11#24361C212yesmaximal, complemented
12#24359C212yesmaximal, complemented
13#21977C211yesmaximal, characteristic, complemented
14#21884C211yesmaximal, characteristic, complemented
15#20740C211yesmaximal, characteristic, complemented
16C23×Q16C2211yescharacteristic
17C23×SD16C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3213—28no—
22C2×#496C2212yes—
23C2×#489C2212yes—
24C2×#464—28no—
25C2×#464C2214yes—
26C2×#464C2214yes—
27C2×#464C2214yescomplemented
28C2×#463C2212yes—
29C2×#463C2212yes—
30C2×#463C2214yes—
31C2×#448C2212yes—
32C2×#447C2212yes—
33C2×#432C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37#2854—21no—
38#2854—21no—
39#2632—22no—
40#2580—22no—
41#2567C2212yescomplemented
42#2548C2212yescomplemented
43#2501C2212yes—
44#2436C2212yes—
45#2406—22no—
46#2379—22no—
47#2367C2214yes—
48#2342—22no—
49Q8×C22⋊C4C2212yes—
50D4×C22⋊C4C2212yes—
51#2337—22no—
52#2326C2214yes—
53#2326C2214yescomplemented
54#2315—21no—
55#2312—21no—
56#2285C2211yescharacteristic
57#2283C2211yescharacteristic
58#2072C2214yescomplemented
59#2071C2214yes—
60#2056C2214yescomplemented
61#2055C2214yes—
62#2055C2214yescomplemented
63#2054C2214yes—
64#2015C2214yescomplemented
65#2012C2214yes—
66#1988C2212yescomplemented
67#1982C2212yescomplemented
68#1940C2214yescomplemented
69#1893C2212yes—
70#1893C2212yes—
71#1880C2214yescomplemented
72#1879C2214yes—
73#1878C2214yescomplemented
74#1877C2214yes—
75#1850C2212yes—
76#1850C2212yes—
77#1843C2212yescomplemented
78#1832C2212yescomplemented
79#1823C2212yes—
80#1803C2212yescomplemented
81#1803C2212yescomplemented
82#1803C2214yescomplemented
83#1802C2214yes—
84#1802C2214yescomplemented
85#1787C2212yes—
86#1763C2214yescomplemented
87C22×#41—21no—
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C22×Q16—24no—
92C22×Q16—44no—
93C22×Q16C2314yes—
94C22×SD16—24no—
95C22×SD16—44no—
96C22×SD16C2314yes—
97C22×SD16C2312yes—
98C22×SD16C2312yes—
99C22×D8C2312yes—
100C22×D8C2312yes—
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#532—24no—
104#532—24no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#124—21no—
108C2×#124—21no—
109C2×#124—21no—
110C2×#124—24no—
111C2×#124D418yes—
112C2×#124C2314yes—
113C2×#124C2312yes—
114C2×#124C2312yes—
115C2×#123—22no—
116C2×#123—22no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—24no—
122C2×C4×Q8—24no—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2314yes—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2314yes—
129C2×#117—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#496—28no—
138#489—28no—
139#464—216no—
140#464—216no—
141#464—48no—
142#463—28no—
143#463—28no—
144#463—48no—
145#448—28no—
146#447—28no—
147#432—24no—
148C2×C4⋊C8—24no—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155C2×#103—24no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#102—24no—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C22⋊C8C2312yes—
168C2×#97C2312yes—
169#399—24no—
170#395—24no—
171#394—24no—
172#393—24no—
173#391—24no—
174#391C2314yes—
175#391C2314yes—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#389—22no—
181#389—22no—
182#388—24no—
183#388—24no—
184#388—24no—
185#388—24no—
186#388—24no—
187#388—24no—
188#386—22no—
189#386—22no—
190#386—24no—
191#385—24no—
192#385—24no—
193#385—24no—
194#385—24no—
195#385—24no—
196#384C2314yes—
197#384C2312yes—
198#383D412yes—
199#382—21no—
200#382D412yes—
201#381D412yes—
202#380—21no—
203#380—22no—
204#380D411yescharacteristic
205#380D411yescharacteristic
206#380C2312yes—
207#379—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—24no—
213C4×C22⋊C4C2312yes—
214#376C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25—21no—
222C2×#41—44no—
223C2×#41—82no—
224C2×#41—82no—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×Q8—22no—
229C22×Q8—24no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—42no—
233C22×Q8—42no—
234C22×Q8—42no—
235C22×Q8C2×D414yes—
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—42no—
244C22×D4—42no—
245C22×D4C2×D414yes—
246C22×D4C2412yes—
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×Q16—28no—
257C2×Q16—28no—
258C2×Q16—44no—
259C2×Q16—416no—
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—44no—
265C2×SD16—44no—
266C2×SD16—44no—
267C2×SD16—416no—
268C2×D8—24no—
269C2×D8—24no—
270C2×D8—44no—
271C2×D8—44no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—42no—
275C22×C8—42no—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282#127—22no—
283#127—22no—
284#124—28no—
285#124—28no—
286#124—28no—
287#124—24no—
288#124—24no—
289#124—44no—
290#124—44no—
291#124C2×D418yes—
292#124C2×D418yes—
293#123—84no—
294#123—84no—
295#121—24no—
296#121—24no—
297#121—44no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—44no—
309C4×D4—44no—
310C4×D4—44no—
311#117—82no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—84no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D414yes—
356C2×C42#4114yes—
357C2×C42C2414yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—22no—
363C4⋊C8—22no—
364C4⋊C8—44no—
365C4⋊C8—44no—
366#103—28no—
367#103—24no—
368#103—24no—
369#103—28no—
370#103—44no—
371#102—28no—
372#102—24no—
373#102—24no—
374#102—44no—
375#102—44no—
376C22⋊C8—24no—
377#97—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D414yes—
387#95C2×D414yes—
388#95C2×D414yes—
389C24—22no—
390C24—24no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—82no—
395C24—82no—
396C24C22×D411yescharacteristic
397#41—816no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—44no—
409C2×Q8—48no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—82no—
416C2×Q8—81no—
417C2×Q8—81no—
418C2×Q8—84no—
419C2×Q8C22×D414yes—
420C2×Q8H(5,𝔽2)14yes—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—84no—
438C2×D4—82no—
439C2×D4—81no—
440C2×D4—81no—
441C2×D4C22×D414yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4—82no—
465C22×C4#12114yes—
466C22×C4#12114yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C2×#4112yes—
476C22×C4C2511yescharacteristic
477Q16—416no—
478SD16—416no—
479SD16—84no—
480SD16—84no—
481D8—84no—
482D8—84no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—44no—
492C2×C8—44no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—28no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—24no—
527C42—24no—
528C42—44no—
529C42—44no—
530C42—82no—
531C42—84no—
532C42#12114yes—
533C42#12114yes—
534C42C22×D412yes—
535C42C22×D412yes—
536C42C2×#4112yes—
537C42C2×#4112yes—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—48no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—84no—
552C23—84no—
553C23—164no—
554C23—164no—
555C23C2×#12112yes—
556C23#57612yes—
557C23#57712yes—
558C23C23×D411yescharacteristic
559Q8—24no—
560Q8—24no—
561Q8—24no—
562Q8—24no—
563Q8—48no—
564Q8—42no—
565Q8—42no—
566Q8—44no—
567Q8—84no—
568Q8—84no—
569D4—24no—
570D4—24no—
571D4—42no—
572D4—42no—
573D4—44no—
574D4—44no—
575D4—84no—
576D4—84no—
577D4—84no—
578D4—88no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—88no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—168no—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C23×D411yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4#58512yes—
625C8—42no—
626C8—42no—
627C8—42no—
628C8—42no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—22no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—84no—
637C22—84no—
638C22—82no—
639C22—82no—
640C22—84no—
641C22—84no—
642C22—82no—
643C22—168no—
644C22#304514yes—
645C22C22×#12111yescharacteristic
646C22#348812yes—
647C22C2×#57611yescharacteristic
648C22C2×#57711yescharacteristic
649C22#358612yes—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4#348411yescharacteristic
663C4#348511yescharacteristic
664C4#349412yes—
665C2—24no—
666C2—84no—
667C2—162no—
668C2#3551411yescharacteristic
669C2C2×#304511yescharacteristic
670C2#6042711yescharacteristic
671C2#6202112yes—
672C2#6202212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3213C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#464C2semidirect
3. #35249C2semidirect
4. #34377C2semidirect
5. #28262C2semidirect
6. #28172C2semidirect
7. #25383C2semidirect
8. #25336C2semidirect
9. #24528C2semidirect
10. #24526C2semidirect
11. #24361C2semidirect
12. #24359C2semidirect
13. #21977C2semidirect
14. #21884C2semidirect
15. #20740C2semidirect
27. C2×#464C22semidirect
41. #2567C22semidirect
42. #2548C22semidirect
53. #2326C22semidirect
58. #2072C22semidirect
60. #2056C22semidirect
62. #2055C22semidirect
64. #2015C22semidirect
66. #1988C22semidirect
67. #1982C22semidirect
68. #1940C22semidirect
71. #1880C22semidirect
73. #1878C22semidirect
77. #1843C22semidirect
78. #1832C22semidirect
80. #1803C22semidirect
81. #1803C22semidirect
82. #1803C22semidirect
84. #1802C22semidirect
86. #1763C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32132
C22×#4642
#352492
#343772
#282622
#281722
#253832
#253362
#245282
#245262
#243612
#243592
#219772
#218842
#207402