Finite Group Databasebeta

← #6849169

Subgroups — #6849169

Order 512.

1979 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic
2#34929C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#32050C214yesmaximal, complemented
5#31836C214yesmaximal, complemented
6#29661C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28400C212yesmaximal, complemented
9#28252C211yesmaximal, characteristic, complemented
10#27813C212yesmaximal, complemented
11#27680C212yesmaximal, complemented
12#25571C212yesmaximal, complemented
13#24653C212yesmaximal, complemented
14#24120C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21860C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547—21no—
20C2×#547C2212yes—
21C2×#533C2212yes—
22C2×#487C2212yes—
23C2×#475C2212yes—
24C2×#460C2211yescharacteristic
25C2×#460C2212yes—
26C2×#461C2212yes—
27C2×C4×SD16C2212yes—
28C2×#431C2212yes—
29#2868C2211yescharacteristic
30#2856C2212yes—
31#2556—22no—
32#2551C2211yescharacteristic
33#2546C2211yescharacteristic
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2542C2211yescharacteristic
38#2538C2211yescharacteristic
39#2506—21no—
40#2501—22no—
41#2501—22no—
42#2501C2214yes—
43#2492C2212yes—
44#2452—21no—
45#2429C2212yes—
46#2396—22no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51#2293—21no—
52C2×#384C2212yes—
53C42×Q8C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2099C2214yes—
56#2096C2214yes—
57#2072C2214yes—
58#2055C2214yes—
59#2034C2214yes—
60#2023C2214yes—
61#2013C2214yes—
62#2000C2214yes—
63#2000C2214yes—
64#2000C2214yes—
65#1990C2212yes—
66#1990C2214yes—
67#1988C2212yes—
68#1982C2211yescharacteristic
69#1968C2214yes—
70#1956C2214yes—
71#1954C2212yes—
72#1948C2211yescharacteristic
73#1948C2212yes—
74#1948C2212yes—
75#1934C2214yes—
76#1926C2212yes—
77#1923C2214yes—
78#1880C2212yes—
79#1872C2214yes—
80#1872C2214yes—
81#1872C2214yes—
82#1853C2214yes—
83#1842C2211yescharacteristic
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×#41—22no—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98#547—44no—
99#547—44no—
100#547—48no—
101#547—48no—
102#533—28no—
103#533—44no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×#127C2311yescharacteristic
112C2×#125—22no—
113C2×#123—24no—
114C2×#123—24no—
115C2×#123—24no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×#121—22no—
124C2×#121—22no—
125C2×#121C2314yes—
126C2×C4×Q8—22no—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2311yescharacteristic
130C2×C4×Q8C2311yescharacteristic
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4D411yescharacteristic
134C2×C4×D4D411yescharacteristic
135C2×C4×D4C2312yes—
136C2×C4×D4C2311yescharacteristic
137C2×#117—22no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—22no—
140C22×C4⋊C4C2314yes—
141C22×C42C2312yes—
142#487—44no—
143#475—44no—
144#461—44no—
145#460—24no—
146#460—44no—
147C4×SD16—44no—
148#431—44no—
149C2×#106C2314yes—
150C2×#106C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2312yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2314yes—
169#400—22no—
170#399—22no—
171#399—24no—
172#399—24no—
173#399—24no—
174#398—22no—
175#398—22no—
176#396—22no—
177#396—22no—
178#395—22no—
179#393—22no—
180#391—22no—
181#391C2312yes—
182#391C2312yes—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—22no—
188#390—22no—
189#390C2312yes—
190#390C2314yes—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—22no—
198#389C2312yes—
199#388—22no—
200#388—22no—
201#387—22no—
202#385—22no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384—22no—
207#384—22no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2312yes—
211#384C2312yes—
212#384C2314yes—
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—22no—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4C2312yes—
230C43—21no—
231C43C2311yescharacteristic
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#340C2314yes—
238#340C2312yes—
239#336C2314yes—
240#336C2312yes—
241C2×#41—48no—
242C2×#41—48no—
243C2×#41—48no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4—24no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C2×SD16—24no—
259C2×SD16—42no—
260C2×SD16—42no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×SD16—44no—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—42no—
274C4⋊Q8—42no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#127—22no—
279#127—22no—
280#125—48no—
281#123—416no—
282#123—416no—
283#123—416no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—42no—
288C22⋊Q8—42no—
289C22⋊Q8—42no—
290C22⋊Q8—48no—
291C22⋊Q8—48no—
292C22⋊Q8—48no—
293C22⋊Q8—48no—
294#121—44no—
295#121—44no—
296#121—48no—
297#121—48no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—48no—
305C4×Q8—41no—
306C4×Q8#4114yescomplemented
307C4×D4—24no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4SD1618yescomplemented
315#117—48no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
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—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—48no—
344C2×C4⋊C4—48no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—21no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D412yes—
378C2×C42C2×D412yes—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#106—42no—
387#106—44no—
388C4⋊C8—22no—
389C4⋊C8—22no—
390C4⋊C8—24no—
391C4⋊C8—24no—
392C4⋊C8—42no—
393#103—42no—
394#103—44no—
395#103—44no—
396#103—44no—
397#103—44no—
398#102—24no—
399#102—24no—
400#102—42no—
401#102—44no—
402C22⋊C8—44no—
403C22⋊C8—44no—
404C4×C8—24no—
405C4×C8—44no—
406#95—22no—
407#95—22no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95C2×D412yes—
416#95C2×D412yes—
417#95C2×D412yes—
418C24—22no—
419C24C22×D412yes—
420#41—832no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—82no—
438C2×Q8C2×#4112yes—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—48no—
450C2×D4—48no—
451C2×D4—48no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4C2×SD1614yes—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—42no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4C2×SD1618yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C2511yescharacteristic
493SD16—48no—
494SD16—88no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—48no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—81no—
540C4⋊C4C2×SD1614yes—
541C4⋊C4AGL(1,ℤ/8)14yes—
542C4⋊C4C2×#4112yes—
543C4⋊C42−1+412yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—82no—
556C22⋊C4C2×SD1618yes—
557C42—24no—
558C42—24no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—21no—
565C42—21no—
566C42—41no—
567C42—42no—
568C42—42no—
569C42—44no—
570C42—42no—
571C42—42no—
572C42—81no—
573C42C2×SD1614yes—
574C42C2×#4112yes—
575C42H(5,𝔽2)12yes—
576C42H(5,𝔽2)12yes—
577C42H(5,𝔽2)12yes—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23C22×SD1614yes—
592C23C23×D411yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—41no—
596Q8—41no—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—88no—
602Q8—88no—
603D4—28no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—84no—
609D4—816no—
610D4—84no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—216no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—48no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—816no—
658C2×C4—88no—
659C2×C4#53512yes—
660C2×C4#53512yes—
661C2×C4#54612yes—
662C2×C4#54612yes—
663C2×C4C22×SD1612yes—
664C2×C4C22×SD1614yes—
665C2×C4C22×SD1612yes—
666C2×C4C22×SD1612yes—
667C2×C4C2×AGL(1,ℤ/8)14yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4#57812yes—
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×2−1+411yescharacteristic
675C2×C4#58511yescharacteristic
676C8—42no—
677C8—44no—
678C8—42no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22C2×#53511yescharacteristic
694C22C2×#54611yescharacteristic
695C22#348911yescharacteristic
696C22C23×SD1611yescharacteristic
697C22C22×AGL(1,ℤ/8)11yescharacteristic
698C22C2×#57811yescharacteristic
699C22#358611yescharacteristic
700C4—22no—
701C4—22no—
702C4—24no—
703C4—22no—
704C4—24no—
705C4—42no—
706C4—42no—
707C4—41no—
708C4—42no—
709C4—42no—
710C4—82no—
711C4—82no—
712C4—84no—
713C4—81no—
714C4—82no—
715C4#320814yes—
716C4#329814yes—
717C4#350611yescharacteristic
718C4#351111yescharacteristic
719C4#354611yescharacteristic
720C4#354811yescharacteristic
721C2—44no—
722C2—162no—
723C2#3947111yescharacteristic
724C2C2×#320811yescharacteristic
725C2#6083711yescharacteristic
726C2C2×#329811yescharacteristic
727C2#6116111yescharacteristic
728C2#6209911yescharacteristic
729C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34929C2semidirect
3. #33196C2semidirect
4. #32050C2semidirect
5. #31836C2semidirect
6. #29661C2semidirect
7. #28717C2semidirect
8. #28400C2semidirect
9. #28252C2semidirect
10. #27813C2semidirect
11. #27680C2semidirect
12. #25571C2semidirect
13. #24653C2semidirect
14. #24120C2semidirect
15. #22795C2semidirect
16. #21860C2semidirect
17. C4×#460C2semidirect
306. C4×Q8#41semidirect
314. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361792
#349292
#331962
#320502
#318362
#296612
#287172
#284002
#282522
#278132
#276802
#255712
#246532
#241202
#227952
#218602
C4×#4602