Finite Group Databasebeta

← #6492139

Subgroups — #6492139

Order 512.

1980 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35273C211yesmaximal, characteristic
2C4⋊C4×#41C211yesmaximal, characteristic, complemented
3#33147C212yesmaximal, complemented
4#33107C212yesmaximal, complemented
5#29937C212yesmaximal, complemented
6#29921C212yesmaximal, complemented
7#26202C214yesmaximal, complemented
8#26130C214yesmaximal, complemented
9#25058C212yesmaximal, complemented
10#25012C212yesmaximal, complemented
11#24992C212yesmaximal, complemented
12#24967C212yesmaximal, complemented
13#21981C211yesmaximal, characteristic, complemented
14#21884C211yesmaximal, characteristic, complemented
15#21847C211yesmaximal, characteristic, complemented
16#21386C211yesmaximal, characteristic, complemented
17C2×#1948C211yesmaximal, characteristic, complemented
18C2×#537—21no—
19C2×#532C2211yescharacteristic
20C2×C4×#41—21no—
21C2×C4×#41C2211yescharacteristic
22C2×#481C2214yes—
23C2×#431C2211yescharacteristic
24C22×#106C2212yes—
25C22×C4⋊C8C2211yescharacteristic
26#2857C2212yes—
27#2856C2212yes—
28#2620—22no—
29#2538—21no—
30#2538C2212yes—
31#2537—21no—
32#2537C2212yes—
33#2416—22no—
34#2416C2214yes—
35#2351—21no—
36Q8×C4⋊C4C2211yescharacteristic
37Q8×C4⋊C4C2211yescharacteristic
38#2349—22no—
39#2349—21no—
40D4×C4⋊C4C2212yes—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2211yescharacteristic
43D4×C4⋊C4C2211yescharacteristic
44#2311—21no—
45#2296—21no—
46#2296—22no—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yes—
53#2094C2214yes—
54#2084C2214yes—
55#2077C2214yes—
56#2077C2214yes—
57#2071C2214yes—
58#2034C2214yes—
59#2034C2214yes—
60#1988C2211yescharacteristic
61#1988C2212yes—
62#1988C2211yescharacteristic
63#1984C2212yes—
64#1982C2211yescharacteristic
65#1982C2212yes—
66#1982C2211yescharacteristic
67#1954C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1926C2214yes—
75#1926C2214yes—
76#1919C2214yes—
77#1919C2214yes—
78#1893C2214yes—
79#1874C2212yes—
80#1874C2214yes—
81#1874C2212yes—
82#1869C2214yes—
83#1853C2214yes—
84#1853C2214yes—
85#1844C2212yes—
86#1834C2212yes—
87#1824C2214yes—
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91#1797C2212yes—
92C4×#102C2212yes—
93C4×#102C2212yes—
94C2×#340C2212yes—
95C22×#41—21no—
96C22×#41D411yescharacteristic
97C23×C8C2312yes—
98#537—48no—
99#532—24no—
100#532—24no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×#121—24no—
107C2×#121—22no—
108C2×#121—22no—
109C2×#121—21no—
110C2×#121—22no—
111C2×#121C2312yes—
112C2×#121C2314yes—
113C4×#41—24no—
114C4×#41—24no—
115C4×#41—48no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4D412yes—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C2×#117—21no—
132C2×#117D411yescharacteristic
133C2×#117D412yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C42—22no—
142C22×C42—21no—
143C22×C42C2312yes—
144C22×C42C2311yescharacteristic
145#481—48no—
146#431—24no—
147C2×#106—44no—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×C4⋊C8—24no—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2312yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C4×C8C2312yes—
167#396—24no—
168#396—24no—
169#394—24no—
170#394—24no—
171#391—22no—
172#390—22no—
173#390—22no—
174#390C2314yes—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389C2314yes—
185#386—24no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#384C2312yes—
199#384C2312yes—
200#382—22no—
201#382—21no—
202#380—22no—
203#380—22no—
204#380—21no—
205#380—22no—
206#380—22no—
207#380C2314yes—
208#380C2312yes—
209#380C2312yes—
210#380C2312yes—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C2×#95—22no—
225C2×#95—21no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—44no—
241C2×#41—44no—
242C2×#41—44no—
243C2×#41SD1618yes—
244C22×Q8—21no—
245C22×Q8C2×D411yescharacteristic
246C22×D4—22no—
247C22×D4—24no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C22×C8—24no—
266C22×C8—42no—
267C22×C8—42no—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273#127—22no—
274#127—22no—
275#124—44no—
276C22⋊Q8—48no—
277C22⋊Q8—48no—
278#121—24no—
279#121—24no—
280#121—48no—
281#121—416no—
282#121—44no—
283#121—48no—
284#121—44no—
285#121—44no—
286#121—84no—
287C4×Q8—24no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×D4—28no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—48no—
301C4×D4—48no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—44no—
305#117—24no—
306#117—28no—
307#117—48no—
308#117—44no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—82no—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—48no—
374C2×C42—44no—
375C2×C42C2×D412yes—
376C2×C42C2×D412yes—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42#4114yes—
380C2×C42#4114yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#106—28no—
385#106—44no—
386#106—44no—
387#106—44no—
388C4⋊C8—24no—
389C4⋊C8—24no—
390C4⋊C8—22no—
391C4⋊C8—22no—
392#102—24no—
393#102—24no—
394#102—42no—
395#102—44no—
396#102—42no—
397#102—44no—
398C22⋊C8—24no—
399C22⋊C8—44no—
400C22⋊C8—44no—
401C4×C8—24no—
402#95—21no—
403#95—22no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—21no—
408#95—21no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414C24—22no—
415C24C22×D412yes—
416C24C22×D411yescharacteristic
417#41—232no—
418#41—816no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8C2×SD1614yes—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—48no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—84no—
446C2×D4—84no—
447C2×D4C2×SD1618yes—
448C2×D4C2×SD1614yes—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—81no—
487C22×C4—82no—
488C22×C4—81no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4#12314yes—
493C22×C4#12314yes—
494C22×C4C2×SD1618yes—
495C22×C4C2×SD1614yes—
496C22×C4AGL(1,ℤ/8)14yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4C2511yescharacteristic
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—22no—
512C2×C8—24no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—22no—
517C2×C8—48no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—48no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—81no—
550C4⋊C4—81no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4H(5,𝔽2)12yes—
556C4⋊C42−1+412yes—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—48no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—48no—
567C22⋊C4—84no—
568C22⋊C4—84no—
569C42—24no—
570C42—24no—
571C42—22no—
572C42—22no—
573C42—44no—
574C42—44no—
575C42—42no—
576C42—44no—
577C42—42no—
578C42—42no—
579C42—42no—
580C42#12314yes—
581C42#12314yes—
582C42C2×#4112yes—
583C42C2×#4112yes—
584C42C2×#4112yes—
585C42C2×#4112yes—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23C2×#12312yes—
604C23C22×SD1614yes—
605C23C22×SD1612yes—
606C23C2×AGL(1,ℤ/8)12yes—
607C23C23×D411yescharacteristic
608Q8—28no—
609Q8—84no—
610D4—216no—
611D4—28no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—84no—
617D4—88no—
618D4—84no—
619D4—168no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—216no—
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—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—88no—
672C2×C4—88no—
673C2×C4—88no—
674C2×C4—164no—
675C2×C4#481116yescomplemented
676C2×C4C2×#12312yes—
677C2×C4C2×#12312yes—
678C2×C4C2×#12312yes—
679C2×C4C2×#12312yes—
680C2×C4C2×#12312yes—
681C2×C4C22×SD1614yes—
682C2×C4C22×SD1612yes—
683C2×C4C22×SD1612yes—
684C2×C4C2×AGL(1,ℤ/8)12yes—
685C2×C4C2×AGL(1,ℤ/8)12yes—
686C2×C4C22×#4111yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×2−1+411yescharacteristic
690C2×C4#58511yescharacteristic
691C2×C4#58511yescharacteristic
692C8—44no—
693C8—42no—
694C8—42no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—24no—
702C22—24no—
703C22—24no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22C2×#48114yes—
716C22C22×#12311yescharacteristic
717C22#348811yescharacteristic
718C22#348911yescharacteristic
719C22C23×SD1611yescharacteristic
720C22C22×AGL(1,ℤ/8)11yescharacteristic
721C22#358511yescharacteristic
722C22#358611yescharacteristic
723C4—24no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—81no—
732C4—82no—
733C4—82no—
734C4—81no—
735C4—82no—
736C4—84no—
737C4C2×#48114yes—
738C4C2×#48114yes—
739C4#346011yescharacteristic
740C4#346111yescharacteristic
741C4#351111yescharacteristic
742C4#351211yescharacteristic
743C2—24no—
744C2—44no—
745C2—162no—
746C2#3525511yescharacteristic
747C2C22×#48111yescharacteristic
748C2#6052011yescharacteristic
749C2#6195111yescharacteristic
750C2#6196211yescharacteristic
751C2#6209911yescharacteristic
752C2#6211011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×#41C2semidirect
3. #33147C2semidirect
4. #33107C2semidirect
5. #29937C2semidirect
6. #29921C2semidirect
7. #26202C2semidirect
8. #26130C2semidirect
9. #25058C2semidirect
10. #25012C2semidirect
11. #24992C2semidirect
12. #24967C2semidirect
13. #21981C2semidirect
14. #21884C2semidirect
15. #21847C2semidirect
16. #21386C2semidirect
17. C2×#1948C2semidirect
675. C2×C4#481semidirect

Maximal subgroups

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

subgroupindex
#352732
C4⋊C4×#412
#331472
#331072
#299372
#299212
#262022
#261302
#250582
#250122
#249922
#249672
#219812
#218842
#218472
#213862
C2×#19482