Finite Group Databasebeta

← #6479719

Subgroups — #6479719

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33209C211yesmaximal, characteristic, complemented
4#33170C211yesmaximal, characteristic
5#33043C211yesmaximal, characteristic, complemented
6#33042C211yesmaximal, characteristic
7#29775C214yesmaximal, complemented
8#29435C214yesmaximal, complemented
9#29433C214yesmaximal, complemented
10#29193C211yesmaximal, characteristic, complemented
11#29192C211yesmaximal, characteristic, complemented
12#29177C211yesmaximal, characteristic, complemented
13#29176C211yesmaximal, characteristic, complemented
14#28656C211yesmaximal, characteristic, complemented
15#28655C211yesmaximal, characteristic, complemented
16#28614C211yesmaximal, characteristic, complemented
17#28613C211yesmaximal, characteristic, complemented
18#22185C214yesmaximal, complemented
19C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
20C2×#529—21no—
21C2×#500C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#465C2214yes—
26C22×C4⋊C8C2212yes—
27C22×C4×C8C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2864—21no—
30#2863—21no—
31#2863C2212yes—
32#2862C2211yescharacteristic
33#2861—21no—
34#2860—21no—
35#2860C2212yes—
36#2859C2211yescharacteristic
37#2859C2211yescharacteristic
38#2858C2211yescharacteristic
39#2857C2211yescharacteristic
40#2857C2211yescharacteristic
41#2857C2211yescharacteristic
42#2856C2211yescharacteristic
43#2855—21no—
44#2855C2212yes—
45#2854—21no—
46#2854C2212yes—
47#2851—21no—
48#2850—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#127C2211yescharacteristic
53C4×#125—21no—
54C4×#124—21no—
55C4×C22⋊Q8—21no—
56C4×C22⋊Q8C2212yes—
57C4×#121—21no—
58C4×#121C2212yes—
59#2295—21no—
60#2295C2212yes—
61#2293—21no—
62#2293C2212yes—
63#2283—21no—
64C2×C43C2211yescharacteristic
65#2088C2214yes—
66#2088C2214yes—
67#2068C2214yes—
68#2068C2214yes—
69#2067C2214yes—
70#2067C2214yes—
71#2055C2214yes—
72#2055C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1968C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1942C2214yes—
83#1934C2214yes—
84#1933C2214yes—
85#1928C2214yes—
86#1925C2214yes—
87#1920C2214yes—
88#1919C2214yes—
89#1893C2214yes—
90#1892C2214yes—
91#1880C2214yes—
92#1877C2214yes—
93#1863C2214yes—
94#1862C2214yes—
95#1845C2211yescharacteristic
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1808C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2212yes—
105C4×C4⋊C8C2212yes—
106C2×#336C2212yes—
107C22×SD16C2314yes—
108C23×C8C2312yes—
109C23×C8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114#529—48no—
115C2×#126—22no—
116C2×#125—21no—
117C2×#125—21no—
118C2×#124—21no—
119C2×#124—21no—
120C2×#123—22no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8C2312yes—
124C2×#121—21no—
125C2×#121—21no—
126C2×#121C2312yes—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×C4×D4—22no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×#117—22no—
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#500—42no—
143#492—42no—
144#487—42no—
145#486—42no—
146#465—216no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8—28no—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C4×C8—42no—
162C2×C4×C8—42no—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2312yes—
168C2×C4×C8C2312yes—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390C2312yes—
186#390C2312yes—
187#390C2312yes—
188#390C2312yes—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#388—22no—
215#388—22no—
216#388—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#387—22no—
225#386—22no—
226#386—22no—
227#385—22no—
228#385—22no—
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#383—22no—
234#382—21no—
235#382—21no—
236#382—21no—
237#382—21no—
238#381—22no—
239#380—21no—
240#380—21no—
241#380—21no—
242#380—21no—
243#380C2312yes—
244#380C2312yes—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—22no—
248C4×C4⋊C4—22no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C22⋊C4—22no—
254C4×C22⋊C4—22no—
255C4×C22⋊C4—22no—
256#376—22no—
257C43D411yescharacteristic
258C43D411yescharacteristic
259C43D411yescharacteristic
260C43D411yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263C43C2311yescharacteristic
264C43C2311yescharacteristic
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8C2412yes—
278C22×D4—22no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C2412yes—
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C2×SD16—28no—
291C2×SD16—44no—
292C2×SD16—44no—
293C22×C8—24no—
294C22×C8—42no—
295C22×C8—42no—
296C22×C8—42no—
297C22×C8—42no—
298C22×C8#4118yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309#127—22no—
310#127—22no—
311#126—48no—
312#125—44no—
313#125—44no—
314#124—44no—
315#124—44no—
316#123—48no—
317C22⋊Q8—28no—
318C22⋊Q8—44no—
319C22⋊Q8—44no—
320#121—28no—
321#121—44no—
322#121—44no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×D4—44no—
327C4×D4—44no—
328C4×D4—48no—
329C4×D4—44no—
330#117—48no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C42—24no—
373C2×C42—24no—
374C2×C42—24no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—42no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411#107—44no—
412#107—44no—
413#106—28no—
414C4⋊C8—24no—
415C4⋊C8—24no—
416C4⋊C8—24no—
417C4⋊C8—24no—
418#103—28no—
419#103—44no—
420#103—44no—
421#102—28no—
422#102—44no—
423#102—44no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—22no—
428C4×C8—42no—
429C4×C8—42no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95—22no—
450C24—22no—
451C24C22×D411yescharacteristic
452C2×Q8—24no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—24no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4AGL(1,ℤ/8)14yes—
505C22×C4#13714yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C2×#4114yes—
520C22×C4C2511yescharacteristic
521SD16—88no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—48no—
533C2×C8—48no—
534C2×C8C2×#4114yes—
535C2×C8C2×#4114yes—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—44no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—44no—
565C4⋊C4—42no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—21no—
600C42—21no—
601C42—22no—
602C42—22no—
603C42—41no—
604C42—41no—
605C42—44no—
606C42—44no—
607C42—44no—
608C42#13514yes—
609C42#13514yes—
610C42H(5,𝔽2)12yes—
611C422−1+412yes—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23#53212yes—
624C23C2×AGL(1,ℤ/8)12yes—
625C23C2×#13712yes—
626C23C23×D411yescharacteristic
627Q8—44no—
628Q8—44no—
629Q8—84no—
630Q8—84no—
631D4—44no—
632D4—44no—
633D4—88no—
634D4—84no—
635D4—84no—
636C2×C4—28no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—88no—
692C2×C4#53212yes—
693C2×C4#53712yes—
694C2×C4#53712yes—
695C2×C4C2×#13512yes—
696C2×C4C2×#13512yes—
697C2×C4C2×#13512yes—
698C2×C4C2×#13512yes—
699C2×C4C2×#13512yes—
700C2×C4C2×#13512yes—
701C2×C4C2×AGL(1,ℤ/8)12yes—
702C2×C4C2×AGL(1,ℤ/8)12yes—
703C2×C4C2×#13712yes—
704C2×C4C2×#13712yes—
705C2×C4C22×#4112yes—
706C2×C4C2×H(5,𝔽2)11yescharacteristic
707C2×C4C2×2−1+411yescharacteristic
708C2×C4#58512yes—
709C8—24no—
710C8—24no—
711C8—42no—
712C8—42no—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—22no—
718C22—22no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22#318214yes—
727C22C2×#53211yescharacteristic
728C22C2×#53711yescharacteristic
729C22#344911yescharacteristic
730C22C22×#13511yescharacteristic
731C22C22×#13511yescharacteristic
732C22C22×AGL(1,ℤ/8)11yescharacteristic
733C22C22×#13711yescharacteristic
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—22no—
740C4—22no—
741C4—41no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4#324714yes—
753C4#324814yes—
754C4#350512yes—
755C4#350712yes—
756C2—24no—
757C2—162no—
758C2#3510311yescharacteristic
759C2#6067311yescharacteristic
760C2C2×#318211yescharacteristic
761C2C2×#324711yescharacteristic
762C2C2×#324811yescharacteristic
763C2#6094611yescharacteristic
764C2#6094711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #33209C2semidirect
5. #33043C2semidirect
7. #29775C2semidirect
8. #29435C2semidirect
9. #29433C2semidirect
10. #29193C2semidirect
11. #29192C2semidirect
12. #29177C2semidirect
13. #29176C2semidirect
14. #28656C2semidirect
15. #28655C2semidirect
16. #28614C2semidirect
17. #28613C2semidirect
18. #22185C2semidirect
19. C2×C4×C4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#332092
#331702
#330432
#330422
#297752
#294352
#294332
#291932
#291922
#291772
#291762
#286562
#286552
#286142
#286132
#221852
C2×C4×C4⋊C82