Finite Group Databasebeta

← #6519578

Subgroups — #6519578

Order 512.

2076 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#464C211yesmaximal, characteristic, complemented
2C2×#2992C211yesmaximal, characteristic, complemented
3#37546C211yesmaximal, characteristic
4#35247C211yesmaximal, characteristic, complemented
5#28107C212yesmaximal, complemented
6#28098C212yesmaximal, complemented
7#25403C214yesmaximal, complemented
8#25311C214yesmaximal, complemented
9#24769C214yesmaximal, complemented
10#24669C214yesmaximal, complemented
11#24273C212yesmaximal, complemented
12#24264C212yesmaximal, complemented
13#21959C211yesmaximal, characteristic, complemented
14#21808C211yesmaximal, characteristic, complemented
15#20740C211yesmaximal, characteristic, complemented
16C23×Q16C2211yescharacteristic
17C23×SD16C2211yescharacteristic
18C2×#529C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#470C2214yescomplemented
22C2×#465C2214yes—
23C2×#464—28no—
24C2×#464C2214yes—
25C2×#464C2214yes—
26#2992—28no—
27C2×#446C2212yescomplemented
28C2×#435C2212yes—
29C2×#435C2212yescomplemented
30C2×C8×D4C2212yes—
31C2×#433C2212yescomplemented
32C2×#432C2211yescharacteristic
33C2×#424C2212yescomplemented
34C22×C4⋊C8C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37#2855—21no—
38#2854—21no—
39#2820—21no—
40#2819—21no—
41#2662—22no—
42#2637—22no—
43#2626—21no—
44#2620C2214yes—
45#2617—21no—
46#2592—21no—
47#2592C2212yes—
48#2584—21no—
49#2582C2212yescomplemented
50#2540C2212yes—
51#2538C2212yescomplemented
52#2486C2212yes—
53#2413—22no—
54#2411—22no—
55#2409C2212yes—
56#2409C2212yes—
57#2408C2212yes—
58#2323C2214yescomplemented
59#2285C2211yescharacteristic
60#2285C2211yescharacteristic
61#2066C2214yescomplemented
62#2065C2214yes—
63#2056C2214yescomplemented
64#2055C2214yes—
65#2035C2214yescomplemented
66#2034C2214yes—
67#2001C2214yescomplemented
68#2000C2214yes—
69#1983C2212yes—
70#1983C2212yes—
71#1971C2214yescomplemented
72#1968C2214yes—
73#1959C2214yes—
74#1948C2214yescomplemented
75#1948C2214yescomplemented
76#1936C2214yes—
77#1933C2214yescomplemented
78#1895C2214yes—
79#1892C2214yescomplemented
80#1823C2212yes—
81#1807C2212yes—
82#1807C2212yes—
83#1803C2212yescomplemented
84#1803C2212yescomplemented
85#1787C2212yes—
86#1787C2212yes—
87#1786C2212yes—
88#1763C2214yescomplemented
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C23×D4C2311yescharacteristic
92C22×Q16—24no—
93C22×Q16—44no—
94C22×Q16C2314yes—
95C22×SD16—24no—
96C22×SD16—44no—
97C22×SD16C2314yes—
98C22×M16C2312yes—
99C23×C8C2312yes—
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102#529—24no—
103#529—24no—
104C2×#126D412yes—
105C2×#125D411yescharacteristic
106C2×#124—21no—
107C2×#124—21no—
108C2×#124—24no—
109C2×#124—24no—
110C2×#124D411yescharacteristic
111C2×#124C2312yes—
112C2×#124C2312yes—
113C2×#123—22no—
114C2×#123D412yes—
115C2×C22⋊Q8D411yescharacteristic
116C2×C22⋊Q8C2314yes—
117C2×C22⋊Q8C2314yes—
118C2×#121D411yescharacteristic
119C2×#121C2314yes—
120C2×#120—22no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C2×#117—21no—
132C2×#117C2312yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#470—48no—
140#465—48no—
141#464—216no—
142#464—216no—
143#446—28no—
144#435—28no—
145#435—28no—
146C8×D4—28no—
147#433—28no—
148#432—24no—
149C2×#106C2314yes—
150C2×#106C2314yes—
151#424—28no—
152C2×C4⋊C8D418yescomplemented
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×#103—24no—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102—24no—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2312yes—
172C2×#97C2312yes—
173C2×#97C2312yes—
174C2×C4×C8C2312yes—
175#399—22no—
176#398—24no—
177#398—22no—
178#397—22no—
179#397—22no—
180#396—24no—
181#396—24no—
182#396—24no—
183#395—22no—
184#395—22no—
185#394—22no—
186#394—24no—
187#394—24no—
188#393—22no—
189#393—22no—
190#393—22no—
191#392—22no—
192#392—22no—
193#391—22no—
194#391C2314yes—
195#391C2314yes—
196#390—22no—
197#390—24no—
198#390C2314yes—
199#389—24no—
200#389—22no—
201#389—22no—
202#389C2314yes—
203#388—24no—
204#388—24no—
205#388—24no—
206#387—24no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#384C2312yes—
212#384C2312yes—
213#382—21no—
214#382—21no—
215#381—22no—
216#380—21no—
217#380—21no—
218#380C2312yes—
219#380C2312yes—
220#380C2314yes—
221#379—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230C25—21no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—24no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8—42no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2412yes—
244C22×D4—24no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×Q16—28no—
261C2×Q16—28no—
262C2×Q16—44no—
263C2×Q16—416no—
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—44no—
267C2×SD16—416no—
268C2×M16—24no—
269C2×M16—44no—
270C22×C8—24no—
271C22×C8—42no—
272C22×C8—42no—
273C22×C8—44no—
274C22×C8C2×D414yes—
275C22×C8C2×D414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280#126—28no—
281#125—24no—
282#124—24no—
283#124—24no—
284#124—24no—
285#124—24no—
286#124—24no—
287#124—44no—
288#124—44no—
289#123—28no—
290#123—84no—
291C22⋊Q8—24no—
292C22⋊Q8—28no—
293C22⋊Q8—44no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296#121—24no—
297#121—44no—
298#121—44no—
299#120—84no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—22no—
311C4×D4—22no—
312#117—24no—
313#117—24no—
314#117—82no—
315#117—82no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2414yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—44no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D414yes—
365C2×C42#4114yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#106—44no—
370#106—44no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373C4⋊C8—22no—
374C4⋊C8—22no—
375C4⋊C8C2×D418yes—
376C4⋊C8C2×D418yes—
377#103—28no—
378#103—28no—
379#103—44no—
380#103—44no—
381#103—44no—
382#102—28no—
383#102—44no—
384#102—44no—
385C22⋊C8—24no—
386C22⋊C8—24no—
387C22⋊C8—24no—
388#97—24no—
389#97—24no—
390C4×C8—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401C24—22no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—82no—
406C24—82no—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—48no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—82no—
429C2×Q8—84no—
430C2×Q8H(5,𝔽2)14yes—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—84no—
444C2×D4—82no—
445C2×D4—84no—
446C2×D4—84no—
447C2×D4—84no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—82no—
469C22×C4#12114yes—
470C22×C4#12114yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C2×#4112yes—
481C22×C4C2511yescharacteristic
482Q16—416no—
483SD16—416no—
484M16—84no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8—84no—
494C2×C8C22×D414yes—
495C2×C8C22×D414yes—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C42—22no—
536C42—22no—
537C42—44no—
538C42—44no—
539C42—82no—
540C42—82no—
541C42#12114yes—
542C42#12114yes—
543C42C22×D412yes—
544C42C22×D412yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—22no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—44no—
555C23—42no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—164no—
563C23—164no—
564C23C2×#12112yes—
565C23#57712yes—
566C23#57712yes—
567C23C23×D411yescharacteristic
568Q8—24no—
569Q8—24no—
570Q8—48no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—44no—
576Q8—82no—
577Q8—84no—
578D4—48no—
579D4—42no—
580D4—42no—
581D4—44no—
582D4—82no—
583D4—84no—
584D4—168no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4C2×#12112yes—
625C2×C4C2×#12112yes—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#12112yes—
629C2×C4C23×D411yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)12yes—
632C2×C4#58512yes—
633C8—24no—
634C8—24no—
635C8—42no—
636C8—42no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—22no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—82no—
647C22—82no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—168no—
652C22#306014yes—
653C22C22×#12111yescharacteristic
654C22#348812yes—
655C22C2×#57711yescharacteristic
656C22C2×#57711yescharacteristic
657C22#358612yes—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—82no—
668C4—84no—
669C4—84no—
670C4—82no—
671C4#348411yescharacteristic
672C4#348711yescharacteristic
673C4#349412yes—
674C2—24no—
675C2—82no—
676C2—162no—
677C2#3556311yescharacteristic
678C2C2×#306011yescharacteristic
679C2#6045511yescharacteristic
680C2#6201912yes—
681C2#6202112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37546C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#529 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#464C2semidirect
2. C2×#2992C2semidirect
4. #35247C2semidirect
5. #28107C2semidirect
6. #28098C2semidirect
7. #25403C2semidirect
8. #25311C2semidirect
9. #24769C2semidirect
10. #24669C2semidirect
11. #24273C2semidirect
12. #24264C2semidirect
13. #21959C2semidirect
14. #21808C2semidirect
15. #20740C2semidirect
21. C2×#470C22semidirect
27. C2×#446C22semidirect
29. C2×#435C22semidirect
31. C2×#433C22semidirect
33. C2×#424C22semidirect
49. #2582C22semidirect
51. #2538C22semidirect
58. #2323C22semidirect
61. #2066C22semidirect
63. #2056C22semidirect
65. #2035C22semidirect
67. #2001C22semidirect
71. #1971C22semidirect
74. #1948C22semidirect
75. #1948C22semidirect
77. #1933C22semidirect
79. #1892C22semidirect
83. #1803C22semidirect
84. #1803C22semidirect
88. #1763C22semidirect
152. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4642
C2×#29922
#375462
#352472
#281072
#280982
#254032
#253112
#247692
#246692
#242732
#242642
#219592
#218082
#207402