Finite Group Databasebeta

← #6805898

Subgroups — #6805898

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic, complemented
3#33230C211yesmaximal, characteristic
4#30904C214yesmaximal, complemented
5#30901C214yesmaximal, complemented
6#30220C211yesmaximal, characteristic, complemented
7#30219C211yesmaximal, characteristic
8#29191C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic
10#29141C211yesmaximal, characteristic, complemented
11#28444C211yesmaximal, characteristic
12#28444C211yesmaximal, characteristic
13#28440C211yesmaximal, characteristic, complemented
14#24741C214yesmaximal, complemented
15#24207C214yesmaximal, complemented
16#22270C211yesmaximal, characteristic, complemented
17#22268C211yesmaximal, characteristic
18#21691C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×#556C2212yes—
21C2×#494C2211yescharacteristic
22C2×#494C2211yescharacteristic
23C2×#492C2211yescharacteristic
24C2×#489C2214yes—
25C2×#464C2214yes—
26C2×#464C2214yes—
27C2×#462C2214yes—
28C2×#441C2214yes—
29#2868C2211yescharacteristic
30#2868C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2858C2211yescharacteristic
35#2582C2212yes—
36#2567—21no—
37#2552—21no—
38#2550—21no—
39#2550—21no—
40#2547C2212yes—
41#2545C2212yes—
42#2545C2212yes—
43#2538—21no—
44#2538—21no—
45#2512—21no—
46#2512—21no—
47#2512—21no—
48#2512—21no—
49#2507—21no—
50#2507—21no—
51#2451C2212yes—
52#2451C2212yes—
53#2369—21no—
54#2355—21no—
55#2354—21no—
56#2354—21no—
57Q8×C4⋊C4C2212yes—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C42×Q8C2211yescharacteristic
65#2066C2214yes—
66#2065C2214yes—
67#2043C2214yes—
68#2043C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2024C2214yes—
72#2016C2214yes—
73#2015C2214yes—
74#2015C2214yes—
75#1985C2212yes—
76#1971C2214yes—
77#1959C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1956C2211yescharacteristic
88#1940C2212yes—
89#1917C2212yes—
90#1917C2212yes—
91#1880C2214yes—
92#1878C2214yes—
93#1878C2214yes—
94#1877C2214yes—
95#1865C2214yes—
96#1864C2214yes—
97#1844C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1803C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×Q16C2314yes—
108C22×Q16C2314yes—
109C22×SD16C2314yes—
110#556—48no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2312yes—
119C2×#127C2311yescharacteristic
120C2×#126—22no—
121C2×#126—22no—
122C2×#125—21no—
123C2×#125—21no—
124C2×#124C2312yes—
125C2×#124C2312yes—
126C2×#124C2312yes—
127C2×#124C2312yes—
128C2×C22⋊Q8—22no—
129C2×C4×Q8D411yescharacteristic
130C2×C4×Q8D411yescharacteristic
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4C2312yes—
140C2×#117—22no—
141#494—24no—
142#494—24no—
143#492—24no—
144#489—48no—
145#464—48no—
146#464—48no—
147#462—48no—
148#441—48no—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#398—22no—
170#398—22no—
171#398—22no—
172#398—22no—
173#397—22no—
174#397—22no—
175#396—22no—
176#396—22no—
177#396—22no—
178#396—22no—
179#395—22no—
180#395—22no—
181#391—22no—
182#391—22no—
183#391—22no—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390C2312yes—
200#389—22no—
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#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—22no—
216#389—22no—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#389C2312yes—
221#388—22no—
222#388—22no—
223#388—22no—
224#388—22no—
225#387—22no—
226#387—22no—
227#384—22no—
228#384—21no—
229#384—21no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#384C2312yes—
239#382—21no—
240#382—21no—
241#380—22no—
242#380—22no—
243C4×C4⋊C4—22no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4D411yescharacteristic
258C4×C4⋊C4D411yescharacteristic
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4C2311yescharacteristic
261C4×C4⋊C4C2311yescharacteristic
262C4×C4⋊C4C2311yescharacteristic
263C43C2311yescharacteristic
264C43C2311yescharacteristic
265C43C2311yescharacteristic
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—22no—
276C22×Q8C2×D411yescharacteristic
277C22×Q8C2412yes—
278C22×Q8C2412yes—
279C22×Q8C2412yes—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C2×Q16—28no—
285C2×Q16—28no—
286C2×Q16—28no—
287C2×Q16—44no—
288C2×Q16—44no—
289C2×Q16—44no—
290C2×SD16—28no—
291C2×SD16—44no—
292C2×SD16—44no—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—44no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306C4⋊Q8—82no—
307C4⋊Q8#4114yes—
308#127—22no—
309#127—22no—
310#126—48no—
311#126—48no—
312#125—82no—
313#125—82no—
314#124—28no—
315#124—44no—
316#124—44no—
317#124—44no—
318C22⋊Q8—48no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—28no—
322C4×Q8—28no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×D4—44no—
328C4×D4—44no—
329C4×D4—44no—
330#117—48no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
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×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
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×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
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×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409#106—44no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413#103—28no—
414#103—44no—
415#103—44no—
416#103—44no—
417#103—44no—
418#103—44no—
419#103—44no—
420#102—28no—
421#102—44no—
422#102—44no—
423#97—24no—
424#97—24no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—22no—
428C4×C8—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#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449C24—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—41no—
464C2×Q8—44no—
465C2×Q8—82no—
466C2×Q8—82no—
467C2×Q8C2×#4114yes—
468C2×D4—22no—
469C2×D4—24no—
470C2×D4—24no—
471C2×D4—22no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
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
503Q16—416no—
504Q16—88no—
505SD16—88no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8H(5,𝔽2)14yes—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
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—22no—
588C42—44no—
589C42—44no—
590C42—44no—
591C42—82no—
592C42—82no—
593C42#13714yes—
594C42C2×#4112yes—
595C42H(5,𝔽2)14yes—
596C422−1+412yes—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23C23×D411yescharacteristic
605Q8—28no—
606Q8—44no—
607Q8—44no—
608Q8—44no—
609Q8—44no—
610Q8—44no—
611Q8—44no—
612Q8—164no—
613D4—44no—
614D4—44no—
615D4—84no—
616D4—84no—
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—21no—
627C2×C4—21no—
628C2×C4—24no—
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—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
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—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—88no—
671C2×C4#53512yes—
672C2×C4#53512yes—
673C2×C4#54812yes—
674C2×C4#54812yes—
675C2×C4C2×#13712yes—
676C2×C4C2×#13712yes—
677C2×C4C2×#13712yes—
678C2×C4#57512yes—
679C2×C4#57712yes—
680C2×C4#57812yes—
681C2×C4C22×#4111yescharacteristic
682C2×C4C2×H(5,𝔽2)12yes—
683C2×C4C2×H(5,𝔽2)12yes—
684C2×C4C2×2−1+411yescharacteristic
685C8—24no—
686C8—42no—
687C8—42no—
688C8—42no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22C2×#53511yescharacteristic
697C22C2×#54811yescharacteristic
698C22#349811yescharacteristic
699C22C22×#13711yescharacteristic
700C22C2×#57511yescharacteristic
701C22C2×#57711yescharacteristic
702C22C2×#57811yescharacteristic
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—44no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#336914yes—
723C4#337714yes—
724C4#351912yes—
725C4#354812yes—
726C2—162no—
727C2#3916911yescharacteristic
728C2#6080411yescharacteristic
729C2#6082411yescharacteristic
730C2C2×#336911yescharacteristic
731C2C2×#337711yescharacteristic
732C2#6242411yescharacteristic
733C2#6242711yescharacteristic
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
2. #36563C2semidirect
4. #30904C2semidirect
5. #30901C2semidirect
6. #30220C2semidirect
8. #29191C2semidirect
10. #29141C2semidirect
13. #28440C2semidirect
14. #24741C2semidirect
15. #24207C2semidirect
16. #22270C2semidirect
18. #21691C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#365632
#332302
#309042
#309012
#302202
#302192
#291912
#291902
#291412
#284442
#284402
#247412
#242072
#222702
#222682
#216912
#213402