Finite Group Databasebeta

← #6439104

Subgroups — #6439104

Order 512.

2292 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic, complemented
2C2×#2946C211yesmaximal, characteristic
3#35639C211yesmaximal, characteristic, complemented
4#34248C211yesmaximal, characteristic, complemented
5#26398C214yesmaximal, complemented
6#26394C214yesmaximal, complemented
7#26382C214yesmaximal, complemented
8#26378C214yesmaximal, complemented
9#23738C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23609C211yesmaximal, characteristic, complemented
12#23553C211yesmaximal, characteristic, complemented
13#21155C211yesmaximal, characteristic, complemented
14#21024C211yesmaximal, characteristic, complemented
15#20929C211yesmaximal, characteristic
16#20925C211yesmaximal, characteristic
17#20881C211yesmaximal, characteristic, complemented
18#20879C211yesmaximal, characteristic, complemented
19#20876C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C2×#478—28no—
27C2×#478C2214yes—
28C2×#478C2214yes—
29#2946—28no—
30C2×#442C2212yes—
31C2×#440C2212yes—
32C2×C4×SD16C2212yes—
33C2×C4×D8C2212yes—
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#427C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C22×#106C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#418C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C2×#405C2211yescharacteristic
47#2579—22no—
48#2487—22no—
49#2454C2214yescomplemented
50#2453C2214yescomplemented
51#2442—22no—
52#2441—22no—
53#2349—22no—
54#2349C2214yes—
55D4×C4⋊C4—22no—
56D4×C4⋊C4C2214yes—
57#2317C2211yescharacteristic
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2316C2211yescharacteristic
61#2315—22no—
62#2311C2211yescharacteristic
63#2297C2211yescharacteristic
64#2296C2211yescharacteristic
65#2296C2211yescharacteristic
66C2×#385—22no—
67#2097C2212yescomplemented
68#2096C2212yescomplemented
69#2094C2212yescomplemented
70#2092C2212yescomplemented
71#2085C2214yescomplemented
72#2083C2214yescomplemented
73#2068C2212yescomplemented
74#2067C2212yescomplemented
75#2066C2212yescomplemented
76#2065C2212yescomplemented
77#2059C2214yescomplemented
78#2059C2214yescomplemented
79#1943C2214yescomplemented
80#1941C2214yescomplemented
81#1917C2214yescomplemented
82#1916C2214yescomplemented
83#1879C2214yes—
84#1879C2214yes—
85#1877C2214yes—
86#1877C2214yes—
87#1864C2214yes—
88#1864C2214yes—
89#1863C2214yescomplemented
90#1862C2214yes—
91#1862C2214yes—
92#1862C2214yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C2×AGL(1,ℤ/8)—24no—
105C2×AGL(1,ℤ/8)—24no—
106C2×AGL(1,ℤ/8)Q818yes—
107C22×SD16C2312yes—
108C22×D8C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C2×#125—24no—
114C2×#125—22no—
115C2×#125—42no—
116C2×#125C2314yes—
117#520—24no—
118#520—42no—
119#518—24no—
120#518—24no—
121#518—24no—
122#518—42no—
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—22no—
128C2×C4×D4—24no—
129C2×C4×D4—22no—
130C2×C4×D4—42no—
131C2×C4×D4C2314yes—
132C2×C4×D4C2314yes—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×#117—21no—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—24no—
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C22⋊C4—22no—
150C22×C22⋊C4—21no—
151C22×C22⋊C4—22no—
152C22×C42C2311yescharacteristic
153#478—216no—
154#478—216no—
155#442—28no—
156#440—28no—
157C4×SD16—28no—
158C4×D8—28no—
159#428—24no—
160#428—42no—
161#427—24no—
162C2×#107—24no—
163C2×#107C2314yes—
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#106—24no—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×C4⋊C8—24no—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176#418—24no—
177#417—42no—
178#416—24no—
179#416—24no—
180C2×#103C2311yescharacteristic
181C2×#103C2312yes—
182C2×#103C2311yescharacteristic
183C2×#103C2312yes—
184C2×#102—24no—
185C2×#102D418yes—
186C2×#102C2314yes—
187C2×#102C2311yescharacteristic
188C2×#102C2312yes—
189C2×#102C2311yescharacteristic
190C2×#102C2314yes—
191C2×#102C2312yes—
192C2×#102C2312yes—
193C2×#102C2312yes—
194#405—24no—
195C2×#97C2312yes—
196C2×C4×C8C2312yes—
197#400—24no—
198#399—24no—
199#398—24no—
200#396—24no—
201#391C2312yes—
202#391C2312yes—
203#390C2312yes—
204#390C2312yes—
205#389—22no—
206#389—22no—
207#389—24no—
208#389D412yes—
209#389D412yes—
210#389C2312yes—
211#389C2312yes—
212#389C2314yes—
213#389C2314yes—
214#389C2314yes—
215#387—24no—
216#386—21no—
217#386—21no—
218#386—22no—
219#385—21no—
220#385—21no—
221#385—24no—
222#385—24no—
223#385—24no—
224#385—22no—
225#385—22no—
226#385—22no—
227#385—24no—
228#385—24no—
229#385D412yes—
230#385D412yes—
231#384—21no—
232#384—21no—
233#384C2312yes—
234#382—21no—
235#382—21no—
236#382C2312yes—
237#380—22no—
238#380—24no—
239#380—22no—
240#380—24no—
241#380—24no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4C2314yes—
244C4×C22⋊C4—22no—
245C2×#95—24no—
246#340C2314yes—
247#340C2314yes—
248#340C2312yes—
249#340C2312yes—
250#340C2314yes—
251#336C2314yes—
252#336C2314yes—
253#336C2312yes—
254#336C2312yes—
255#336C2314yes—
256C25—21no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41C2×Q814yes—
263C22×Q8C2411yescharacteristic
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—22no—
267C22×D4—22no—
268C22×D4—22no—
269C22×D4C2×D414yes—
270C22×D4C2×Q814yes—
271C22×D4#4114yes—
272C22×D4C2411yescharacteristic
273C22×D4C2412yes—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—24no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—24no—
280C23×C4—22no—
281C23×C4—22no—
282C23×C4—22no—
283C23×C4—44no—
284C23×C4—84no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288AGL(1,ℤ/8)—232no—
289C2×SD16—24no—
290C2×SD16—24no—
291C2×SD16C2×Q818yes—
292C2×D8—24no—
293C2×D8—24no—
294C2×D8C2×Q818yes—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—22no—
300C2×M16C2×Q814yes—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C2×D414yes—
307C22×C8C2412yes—
308C22×C8C2412yes—
309C22×C8C2411yescharacteristic
310C22×C8C2411yescharacteristic
311C22×C8C2412yes—
312#125—28no—
313#125—44no—
314#125—48no—
315C4×Q8—24no—
316C4×Q8—42no—
317C4×D4—216no—
318C4×D4—28no—
319C4×D4—28no—
320C4×D4—28no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—44no—
325C4×D4—42no—
326C4×D4—42no—
327C4×D4—42no—
328C4×D4—48no—
329C4×D4—48no—
330C4×D4—44no—
331C4×D4—44no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—44no—
342#117#4114yes—
343C2×C4⋊C4—21no—
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—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4—44no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4—88no—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2×D414yes—
368C2×C4⋊C4C2×D414yes—
369C2×C4⋊C4C2×D414yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2411yescharacteristic
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C22⋊C4—216no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—48no—
390C2×C22⋊C4—44no—
391C2×C22⋊C4C2×D412yes—
392C2×C22⋊C4C2×D412yes—
393C2×C42—21no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42C2×D412yes—
400C2×C42C2412yes—
401C2×C42C2412yes—
402C2×C42C2412yes—
403#107—28no—
404#107—24no—
405#107—24no—
406#107—42no—
407#106—24no—
408#106—24no—
409#106—24no—
410#106—24no—
411#106—42no—
412C4⋊C8—24no—
413C4⋊C8—24no—
414#103—22no—
415#103—22no—
416#103—24no—
417#103—42no—
418#102—28no—
419#102—28no—
420#102—22no—
421#102—22no—
422#102—24no—
423#102—42no—
424#102C2×D418yes—
425#102C2×D418yes—
426#97—24no—
427C4×C8—24no—
428#95—22no—
429#95—24no—
430#95—24no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—24no—
435#95—24no—
436#95—24no—
437#95C2×D414yes—
438#95C2×D414yes—
439C24—22no—
440C24—24no—
441C24—24no—
442C24—24no—
443C24—21no—
444C24—82no—
445C24—82no—
446C24C22×D411yescharacteristic
447#41—28no—
448#41—28no—
449#41—44no—
450#41—44no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8C22×Q812yes—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—28no—
460C2×D4—24no—
461C2×D4—28no—
462C2×D4—24no—
463C2×D4—28no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4C22⋊Q8116yes—
475C2×D4C22×D414yes—
476C2×D4C22×Q812yes—
477C2×D4C22×Q814yes—
478C2×D4C2×#4114yes—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—216no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—84no—
504C22×C4—84no—
505C22×C4—84no—
506C22×C4—84no—
507C22×C4—84no—
508C22×C4—84no—
509C22×C4—82no—
510C22×C4—82no—
511C22×C4—82no—
512C22×C4—816no—
513C22×C4C22×D414yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C22×Q812yes—
521C22×C4C2×#4112yes—
522C22×C4C2×#4112yes—
523C22×C4H(5,𝔽2)12yes—
524C22×C42−1+412yes—
525C22×C4C2511yescharacteristic
526SD16—216no—
527D8—216no—
528M16—28no—
529M16—44no—
530C2×C8—28no—
531C2×C8—22no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—22no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—44no—
544C2×C8C22×D412yes—
545C2×C8C22×D412yes—
546C2×C8C22×Q814yes—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—48no—
567C4⋊C4—48no—
568C4⋊C4—44no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—82no—
572C4⋊C4—84no—
573C4⋊C4—84no—
574C4⋊C4—84no—
575C4⋊C4—84no—
576C4⋊C4C22×D414yes—
577C4⋊C4C2×#4114yes—
578C22⋊C4—216no—
579C22⋊C4—24no—
580C22⋊C4—24no—
581C22⋊C4—24no—
582C22⋊C4—44no—
583C22⋊C4—48no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C22⋊C4—42no—
587C22⋊C4—42no—
588C22⋊C4—84no—
589C22⋊C4—82no—
590C22⋊C4—82no—
591C42—24no—
592C42—24no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—44no—
600C42—42no—
601C42—42no—
602C23—22no—
603C23—22no—
604C23—24no—
605C23—24no—
606C23—24no—
607C23—216no—
608C23—22no—
609C23—44no—
610C23—44no—
611C23—42no—
612C23—42no—
613C23—41no—
614C23—41no—
615C23—84no—
616C23—84no—
617C23—88no—
618C23—88no—
619C23#52512yes—
620C23D4212yes—
621C23D4×Q812yes—
622C23C23×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—41no—
626Q8—41no—
627D4—22no—
628D4—24no—
629D4—24no—
630D4—24no—
631D4—22no—
632D4—216no—
633D4—48no—
634D4—41no—
635D4—41no—
636D4C2×C22⋊Q818yes—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—48no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—84no—
673C2×C4—816no—
674C2×C4—816no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—164no—
678C2×C4—164no—
679C2×C4—164no—
680C2×C4C2×C22⋊Q814yes—
681C2×C4#52512yes—
682C2×C4#53914yes—
683C2×C4D4212yes—
684C2×C4#54714yes—
685C2×C4D4×Q812yes—
686C2×C4#55414yes—
687C2×C4C23×D411yescharacteristic
688C2×C4C23×Q811yescharacteristic
689C2×C4C22×#4111yescharacteristic
690C2×C4C22×#4111yescharacteristic
691C2×C4C2×H(5,𝔽2)11yescharacteristic
692C2×C4C2×2−1+411yescharacteristic
693C8—24no—
694C8—22no—
695C8—22no—
696C8—42no—
697C8—42no—
698C22—22no—
699C22—22no—
700C22—24no—
701C22—24no—
702C22—28no—
703C22—44no—
704C22—44no—
705C22—48no—
706C22—48no—
707C22—42no—
708C22—44no—
709C22—816no—
710C22—81no—
711C22—81no—
712C22—82no—
713C22#244914yes—
714C22C2×#52511yescharacteristic
715C22C2×D4211yescharacteristic
716C22C2×D4×Q811yescharacteristic
717C22#358412yes—
718C22#358512yes—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—44no—
725C4—44no—
726C4—44no—
727C4—82no—
728C4—81no—
729C4—82no—
730C4—81no—
731C4—82no—
732C4#244914yes—
733C4C22×C22⋊Q811yescharacteristic
734C4C2×#53911yescharacteristic
735C4C2×#54711yescharacteristic
736C4C2×#55411yescharacteristic
737C2—24no—
738C2—48no—
739C2—82no—
740C2C2×#244911yescharacteristic
741C2#6060112yes—
742C2#6105211yescharacteristic
743C2#6105811yescharacteristic
744C2#6138512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#478C2semidirect
3. #35639C2semidirect
4. #34248C2semidirect
5. #26398C2semidirect
6. #26394C2semidirect
7. #26382C2semidirect
8. #26378C2semidirect
9. #23738C2semidirect
10. #23660C2semidirect
11. #23609C2semidirect
12. #23553C2semidirect
13. #21155C2semidirect
14. #21024C2semidirect
17. #20881C2semidirect
18. #20879C2semidirect
19. #20876C2semidirect
49. #2454C22semidirect
50. #2453C22semidirect
67. #2097C22semidirect
68. #2096C22semidirect
69. #2094C22semidirect
70. #2092C22semidirect
71. #2085C22semidirect
72. #2083C22semidirect
73. #2068C22semidirect
74. #2067C22semidirect
75. #2066C22semidirect
76. #2065C22semidirect
77. #2059C22semidirect
78. #2059C22semidirect
79. #1943C22semidirect
80. #1941C22semidirect
81. #1917C22semidirect
82. #1916C22semidirect
89. #1863C22semidirect
92. #1862C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#29462
#356392
#342482
#263982
#263942
#263822
#263782
#237382
#236602
#236092
#235532
#211552
#210242
#209292
#209252
#208812
#208792
#208762