Finite Group Databasebeta

← #6854405

Subgroups — #6854405

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36573C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic, complemented
3#33316C211yesmaximal, characteristic, complemented
4#30971C214yesmaximal, complemented
5#30963C214yesmaximal, complemented
6#30307C211yesmaximal, characteristic, complemented
7#30307C211yesmaximal, characteristic, complemented
8#29207C211yesmaximal, characteristic, complemented
9#29178C211yesmaximal, characteristic, complemented
10#29177C211yesmaximal, characteristic, complemented
11#29139C211yesmaximal, characteristic
12#28514C211yesmaximal, characteristic
13#28514C211yesmaximal, characteristic
14#26059C214yesmaximal, complemented
15#25069C214yesmaximal, complemented
16#22220C211yesmaximal, characteristic
17#22219C211yesmaximal, characteristic
18#21646C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×#501C2214yes—
21C2×#486C2211yescharacteristic
22C2×#478C2214yes—
23C2×#478C2214yes—
24#2868C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2582—21no—
31#2580—21no—
32#2580—21no—
33#2580C2212yes—
34#2567—21no—
35#2552C2212yes—
36#2550C2212yes—
37#2550C2212yes—
38#2507—21no—
39#2507—21no—
40#2454—21no—
41#2454—21no—
42#2454—21no—
43#2454—21no—
44#2454C2212yes—
45#2454C2212yes—
46#2451—21no—
47#2451—21no—
48#2369—21no—
49#2355C2212yes—
50Q8×C4⋊C4—21no—
51#2349—21no—
52#2349—21no—
53#2349C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C42×Q8C2211yescharacteristic
61#2092C2214yes—
62#2092C2214yes—
63#2088C2214yes—
64#2086C2214yes—
65#2085C2214yes—
66#2085C2214yes—
67#2059C2214yes—
68#2059C2214yes—
69#2059C2214yes—
70#2023C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1985C2212yes—
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1949C2214yes—
79#1940C2212yes—
80#1936C2214yes—
81#1926C2214yes—
82#1925C2214yes—
83#1919C2214yes—
84#1918C2214yes—
85#1917C2212yes—
86#1917C2212yes—
87#1864C2214yes—
88#1862C2214yes—
89#1847C2214yes—
90#1845C2211yescharacteristic
91#1845C2211yescharacteristic
92#1845C2211yescharacteristic
93#1844C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1802C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×#127C2311yescharacteristic
112C2×#125—21no—
113C2×#125—21no—
114C2×#125C2312yes—
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132#501—48no—
133#486Q818yescomplemented
134#478—48no—
135#478—48no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#97C2312yes—
150C2×#97C2312yes—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155#399—22no—
156#399—22no—
157#399—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#396—22no—
163#396—22no—
164#395—22no—
165#395—22no—
166#393—22no—
167#391—22no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—22no—
172#391—22no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
182#389—22no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
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#389—21no—
202#389—22no—
203#389—22no—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#387—22no—
212#387—22no—
213#385—22no—
214#385—22no—
215#385—22no—
216#385—22no—
217#385—22no—
218#385—22no—
219#384—22no—
220#384—21no—
221#384—21no—
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2312yes—
229#382—21no—
230#382—21no—
231#382C2312yes—
232#382C2312yes—
233#382C2312yes—
234#380—22no—
235#380—22no—
236#380—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2311yescharacteristic
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C2×#95—22no—
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8—22no—
272C22×Q8C2×D411yescharacteristic
273C22×D4—21no—
274C22×D4—21no—
275C22×D4C2412yes—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4—22no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C4⋊Q8—22no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C4⋊Q8—82no—
287C4⋊Q8C2×Q814yes—
288#127—22no—
289#127C2×Q814yes—
290#125—44no—
291#125—44no—
292#125—44no—
293#125—82no—
294#125—82no—
295C4×Q8—24no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—44no—
299C4×Q8—82no—
300C4×Q8—82no—
301C4×D4—28no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—82no—
307C4×D4—82no—
308C4×D4—82no—
309C4×D4—82no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
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⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
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⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—28no—
387#107—44no—
388#107—44no—
389#106—28no—
390#106—44no—
391#106—44no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395#102—28no—
396#102—44no—
397#102—44no—
398#102C2×Q8116yes—
399#97—24no—
400#97—24no—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8C2×Q814yes—
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
427C24—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—82no—
436C2×Q8—82no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4C22×Q814yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8C22×Q814yes—
489C2×C8H(5,𝔽2)14yes—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
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⋊C4C22×Q814yes—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—44no—
563C42—82no—
564C42—82no—
565C42C22×Q812yes—
566C422−1+414yes—
567C422−1+412yes—
568C23—24no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23C23×D411yescharacteristic
576Q8—44no—
577Q8—48no—
578Q8—164no—
579D4—28no—
580D4—48no—
581D4—48no—
582D4—44no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
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—21no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
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—164no—
639C2×C4—164no—
640C2×C4—164no—
641C2×C4#53612yes—
642C2×C4#53612yes—
643C2×C4D4×Q812yes—
644C2×C4D4×Q812yes—
645C2×C4#57512yes—
646C2×C4#57512yes—
647C2×C4#57612yes—
648C2×C4#57712yes—
649C2×C4C23×Q811yescharacteristic
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4C2×2−1+412yes—
652C2×C4C2×2−1+411yescharacteristic
653C8—24no—
654C8—24no—
655C8—42no—
656C8—42no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22C2×#53611yescharacteristic
665C22C2×D4×Q811yescharacteristic
666C22#349811yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57511yescharacteristic
669C22C2×#57611yescharacteristic
670C22C2×#57711yescharacteristic
671C4—22no—
672C4—22no—
673C4—44no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4#338814yes—
691C4#338914yes—
692C4#350312yes—
693C4#354412yes—
694C2—84no—
695C2#3961711yescharacteristic
696C2#6086211yescharacteristic
697C2#6087711yescharacteristic
698C2C2×#338811yescharacteristic
699C2C2×#338911yescharacteristic
700C2#6242611yescharacteristic
701C2#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
3. #33316C2semidirect
4. #30971C2semidirect
5. #30963C2semidirect
6. #30307C2semidirect
7. #30307C2semidirect
8. #29207C2semidirect
9. #29178C2semidirect
10. #29177C2semidirect
14. #26059C2semidirect
15. #25069C2semidirect
19. #21340C2semidirect
133. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#365732
#365632
#333162
#309712
#309632
#303072
#292072
#291782
#291772
#291392
#285142
#260592
#250692
#222202
#222192
#216462
#213402