Finite Group Databasebeta

← #6490703

Subgroups — #6490703

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#33155C214yesmaximal
4#30100C214yesmaximal, complemented
5#29281C214yesmaximal, complemented
6#29219C211yesmaximal, characteristic, complemented
7#29218C211yesmaximal, characteristic, complemented
8#28688C211yesmaximal, characteristic, complemented
9#28688C211yesmaximal, characteristic, complemented
10#27962C211yesmaximal, characteristic
11#27962C211yesmaximal, characteristic
12#27623C214yesmaximal
13#25813C211yesmaximal, characteristic
14#25811C211yesmaximal, characteristic
15#22248C211yesmaximal, characteristic, complemented
16#22247C211yesmaximal, characteristic, complemented
17#22246C211yesmaximal, characteristic, complemented
18#22246C211yesmaximal, characteristic, complemented
19#20782C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#498C2211yescharacteristic
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2647C2212yes—
30#2628C2212yes—
31#2511—21no—
32#2511—21no—
33#2492—21no—
34#2492—21no—
35#2451—21no—
36#2451—21no—
37#2423—21no—
38#2423—21no—
39#2369—21no—
40#2369—21no—
41#2365—21no—
42#2365—21no—
43#2345—21no—
44#2345—21no—
45Q8×C22⋊C4—21no—
46Q8×C22⋊C4—21no—
47#2331C2212yes—
48#2331C2212yes—
49#2331C2212yes—
50#2316C2212yes—
51#2314C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58#2285C2211yescharacteristic
59#2274C2212yes—
60C4×#117C2211yescharacteristic
61#2101C2214yes—
62#2101C2214yes—
63#2100C2214yes—
64#2099C2214yes—
65#2085C2214yes—
66#2085C2214yes—
67#2058C2214yes—
68#2057C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1988C2214yes—
78#1987C2214yes—
79#1949C2214yes—
80#1949C2214yes—
81#1921C2214yes—
82#1921C2214yes—
83#1920C2214yes—
84#1920C2214yes—
85#1919C2214yes—
86#1919C2214yes—
87#1918C2214yes—
88#1918C2214yes—
89#1853C2214yes—
90#1852C2214yes—
91#1840C2212yes—
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1832C2212yes—
97#1825C2214yes—
98#1825C2214yes—
99#1818C2212yes—
100#1798C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1744C2211yescharacteristic
106#1744C2211yescharacteristic
107C23×Q8—21no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134#500—24no—
135#500—24no—
136#498—24no—
137#498—24no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150#400—22no—
151#400—22no—
152#400—22no—
153#399—22no—
154#398—22no—
155#398—22no—
156#395—22no—
157#393—22no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—22no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#388—22no—
195#388—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2312yes—
211#382C2312yes—
212#382C2312yes—
213#382C2312yes—
214#380—21no—
215#380—21no—
216#380—21no—
217#380—21no—
218#380—22no—
219#380—22no—
220#380C2312yes—
221#380C2312yes—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4D411yescharacteristic
233C4×C4⋊C4D411yescharacteristic
234C4×C4⋊C4D411yescharacteristic
235C4×C4⋊C4D411yescharacteristic
236C4×C4⋊C4C2312yes—
237C4×C4⋊C4C2312yes—
238C4×C4⋊C4C2312yes—
239C4×C4⋊C4C2312yes—
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2312yes—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—22no—
244C4×C22⋊C4D411yescharacteristic
245C4×C22⋊C4D411yescharacteristic
246C4×C22⋊C4D411yescharacteristic
247C4×C22⋊C4D411yescharacteristic
248#376C2311yescharacteristic
249#376C2311yescharacteristic
250C43C2311yescharacteristic
251C43C2311yescharacteristic
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—28no—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—22no—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—22no—
279C22×Q8—82no—
280C22×Q8—82no—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—28no—
290C4⋊Q8—28no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—28no—
300C4⋊Q8—82no—
301C4⋊Q8—82no—
302C4⋊Q8#4114yes—
303C4⋊Q8#4114yes—
304C22⋊Q8—82no—
305C22⋊Q8—82no—
306C4×Q8—82no—
307C4×Q8—82no—
308C4×Q8—82no—
309C4×Q8—82no—
310C4×Q8—82no—
311C4×Q8—82no—
312#117—24no—
313#117—24no—
314#117—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C22⋊C4C2×D411yescharacteristic
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#107—28no—
389#106—28no—
390#106—28no—
391C4⋊C8—24no—
392C4⋊C8—24no—
393C22⋊C8—24no—
394C22⋊C8—24no—
395C4×C8—22no—
396C4×C8—22no—
397C4×C8—22no—
398C4×C8—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429C24C22×D411yescharacteristic
430C2×Q8—28no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—82no—
445C2×Q8—82no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—816no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—28no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C42−1+414yes—
483C22×C4C2511yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4C2×#4114yes—
536C4⋊C4C2×#4114yes—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—44no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—21no—
555C42—21no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—44no—
560C42—82no—
561C42—82no—
562C42—82no—
563C42—82no—
564C42—82no—
565C42—82no—
566C42C2×#4112yes—
567C42C2×#4112yes—
568C422−1+414yes—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#53612yes—
576C23C23×D411yescharacteristic
577Q8—48no—
578Q8—48no—
579Q8—88no—
580Q8—84no—
581Q8—84no—
582Q8—164no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—88no—
642C2×C4#52512yes—
643C2×C4#52512yes—
644C2×C4#53612yes—
645C2×C4#53612yes—
646C2×C4#53612yes—
647C2×C4#57612yes—
648C2×C4#57712yes—
649C2×C4#57712yes—
650C2×C4#57812yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×2−1+412yes—
654C2×C4C2×2−1+412yes—
655C8—42no—
656C8—42no—
657C8—42no—
658C8—42no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22C2×#52511yescharacteristic
667C22C2×#53611yescharacteristic
668C22C2×#53611yescharacteristic
669C22C2×#57611yescharacteristic
670C22C2×#57711yescharacteristic
671C22C2×#57711yescharacteristic
672C22C2×#57811yescharacteristic
673C4—22no—
674C4—22no—
675C4—22no—
676C4—41no—
677C4—44no—
678C4—44no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4#311814yes—
693C4#312114yes—
694C4#352312yes—
695C4#352312yes—
696C2—42no—
697C2#3524811yescharacteristic
698C2C2×#311811yescharacteristic
699C2C2×#312111yescharacteristic
700C2#6087711yescharacteristic
701C2#6087811yescharacteristic
702C2#6088011yescharacteristic
703C2#6088111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #30100C2semidirect
5. #29281C2semidirect
6. #29219C2semidirect
7. #29218C2semidirect
8. #28688C2semidirect
9. #28688C2semidirect
15. #22248C2semidirect
16. #22247C2semidirect
17. #22246C2semidirect
18. #22246C2semidirect

Maximal subgroups

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

subgroupindex
#352832
#342162
#331552
#301002
#292812
#292192
#292182
#286882
#279622
#276232
#258132
#258112
#222482
#222472
#222462
#207822