Finite Group Databasebeta

← #6854291

Subgroups — #6854291

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36574C211yesmaximal, characteristic, complemented
2Q8×#127C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#32364C214yesmaximal, complemented
5#32362C214yesmaximal, complemented
6#30305C211yesmaximal, characteristic, complemented
7#30304C211yesmaximal, characteristic, complemented
8#29218C211yesmaximal, characteristic, complemented
9#29208C211yesmaximal, characteristic, complemented
10#29206C211yesmaximal, characteristic, complemented
11#29192C211yesmaximal, characteristic
12#29190C211yesmaximal, characteristic
13#28514C211yesmaximal, characteristic
14#25844C214yesmaximal, complemented
15#25465C214yesmaximal, complemented
16#22220C211yesmaximal, characteristic
17#22219C211yesmaximal, characteristic
18#21646C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×D4×Q8—21no—
21C2×D4×Q8—21no—
22C2×D4×Q8C2212yes—
23C2×#501C2214yes—
24C2×#486C2211yescharacteristic
25C2×#478C2214yes—
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2584—21no—
33#2584C2212yes—
34#2581—21no—
35#2542—21no—
36#2542—21no—
37#2542—21no—
38#2542—21no—
39#2542—21no—
40#2542—21no—
41#2542C2212yes—
42#2542C2212yes—
43#2542C2212yes—
44#2509—21no—
45#2509C2212yes—
46#2508—21no—
47#2508C2212yes—
48#2453—21no—
49#2453—21no—
50#2370—21no—
51#2370C2212yes—
52#2351—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2100C2214yes—
61#2092C2214yes—
62#2084C2214yes—
63#2083C2214yes—
64#2067C2214yes—
65#2065C2214yes—
66#2058C2214yes—
67#2058C2214yes—
68#2057C2214yes—
69#2057C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1987C2214yes—
74#1985C2212yes—
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1948C2214yes—
79#1947C2214yes—
80#1940C2212yes—
81#1936C2214yes—
82#1926C2214yes—
83#1925C2214yes—
84#1917C2212yes—
85#1917C2212yes—
86#1864C2214yes—
87#1864C2214yes—
88#1862C2214yes—
89#1862C2214yes—
90#1847C2214yes—
91#1845C2211yescharacteristic
92#1845C2211yescharacteristic
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C23×Q8—22no—
108D4×Q8—48no—
109D4×Q8—48no—
110D4×Q8—48no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2312yes—
121C2×#127C2311yescharacteristic
122C2×#125—21no—
123C2×#125—21no—
124C2×#125C2312yes—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—22no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—22no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8D411yescharacteristic
138C2×C4×Q8D411yescharacteristic
139C2×C4×Q8C2311yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4—21no—
142C2×C4×D4—21no—
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4—21no—
146C2×C4×D4C2312yes—
147C2×C4×D4C2312yes—
148C2×C4×D4C2312yes—
149#501—48no—
150#486Q818yescomplemented
151#478—48no—
152C2×#107C2314yes—
153C2×#107C2314yes—
154C2×#107C2314yes—
155C2×#106C2314yes—
156C2×#106C2314yes—
157C2×#106C2314yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2312yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#97C2312yes—
166C2×#97C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171#400—22no—
172#400—22no—
173#400—22no—
174#399—22no—
175#399—22no—
176#399—22no—
177#399—22no—
178#399—22no—
179#399—22no—
180#398—22no—
181#398—22no—
182#395—22no—
183#391—21no—
184#391—21no—
185#391—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390C2312yes—
193#390C2312yes—
194#390C2312yes—
195#389—22no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#389C2312yes—
216#389C2312yes—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#384—21no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384—21no—
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2312yes—
234#384C2312yes—
235#384C2312yes—
236#382—21no—
237#382—21no—
238#382—22no—
239#382C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—22no—
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8—22no—
275C22×Q8—22no—
276C22×Q8—48no—
277C22×Q8—48no—
278C22×Q8—48no—
279C22×Q8C2×D411yescharacteristic
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4—22no—
285C23×C4—22no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—28no—
291C4⋊Q8—22no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C4⋊Q8—44no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8C2×Q814yescomplemented
301#127—22no—
302#127C2×Q814yes—
303#125—28no—
304#125—44no—
305#125—82no—
306C22⋊Q8—48no—
307C22⋊Q8—48no—
308C22⋊Q8—48no—
309C22⋊Q8—48no—
310C22⋊Q8—48no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C22⋊Q8—48no—
314C22⋊Q8—48no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—44no—
319C4×Q8—82no—
320C4×Q8—82no—
321C4×D4—28no—
322C4×D4—44no—
323C4×D4—44no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—44no—
328C4×D4—44no—
329C4×D4—44no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
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—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
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×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408#107—44no—
409#107—44no—
410#107—44no—
411#106—44no—
412#106—44no—
413#106—44no—
414C4⋊C8—24no—
415C4⋊C8—24no—
416C4⋊C8—24no—
417#102—28no—
418#102—44no—
419#102—44no—
420#102C2×Q8116yescomplemented
421#97—24no—
422#97—24no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8C2×Q814yescomplemented
427#95—22no—
428#95—22no—
429#95—22no—
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#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445C24—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—44no—
459C2×Q8—44no—
460C2×Q8—42no—
461C2×Q8—832no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4C22×Q814yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C2511yescharacteristic
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8C22×Q814yes—
514C2×C82−1+414yes—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4C22×Q814yes—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—44no—
588C42—82no—
589C42—82no—
590C42AGL(1,ℤ/8)14yes—
591C42AGL(1,ℤ/8)14yes—
592C42C22×Q812yes—
593C42H(5,𝔽2)14yes—
594C422−1+412yes—
595C23—24no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23C23×D411yescharacteristic
603Q8—28no—
604Q8—48no—
605Q8—88no—
606Q8—88no—
607Q8—88no—
608D4—28no—
609D4—48no—
610D4—48no—
611D4—44no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—24no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—88no—
664C2×C4—88no—
665C2×C4—88no—
666C2×C4#53512yes—
667C2×C4#53512yes—
668C2×C4D4×Q812yes—
669C2×C4D4×Q812yes—
670C2×C4C2×AGL(1,ℤ/8)12yes—
671C2×C4C2×AGL(1,ℤ/8)12yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C2×AGL(1,ℤ/8)12yes—
674C2×C4C2×AGL(1,ℤ/8)12yes—
675C2×C4C2×AGL(1,ℤ/8)12yes—
676C2×C4#57712yes—
677C2×C4#57812yes—
678C2×C4C23×Q811yescharacteristic
679C2×C4C2×H(5,𝔽2)12yes—
680C2×C4C2×2−1+412yes—
681C2×C4C2×2−1+411yescharacteristic
682C8—24no—
683C8—24no—
684C8—42no—
685C8—42no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22C2×#53511yescharacteristic
694C22C2×D4×Q811yescharacteristic
695C22#349911yescharacteristic
696C22C22×AGL(1,ℤ/8)11yescharacteristic
697C22C22×AGL(1,ℤ/8)11yescharacteristic
698C22C2×#57711yescharacteristic
699C22C2×#57811yescharacteristic
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—44no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4#338414yes—
720C4#338614yes—
721C4#350312yes—
722C4#355212yes—
723C2—84no—
724C2#3961611yescharacteristic
725C2#6079811yescharacteristic
726C2#6082411yescharacteristic
727C2C2×#338411yescharacteristic
728C2C2×#338611yescharacteristic
729C2#6247711yescharacteristic
730C2#6247911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36574C2semidirect
3. #33316C2semidirect
4. #32364C2semidirect
5. #32362C2semidirect
6. #30305C2semidirect
7. #30304C2semidirect
8. #29218C2semidirect
9. #29208C2semidirect
10. #29206C2semidirect
14. #25844C2semidirect
15. #25465C2semidirect
19. #21340C2semidirect
150. #486Q8semidirect
300. C4⋊Q8C2×Q8semidirect
420. #102C2×Q8semidirect
426. C4×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365742
Q8×#1272
#333162
#323642
#323622
#303052
#303042
#292182
#292082
#292062
#291922
#291902
#285142
#258442
#254652
#222202
#222192
#216462
#213402