Finite Group Databasebeta

← #6627140

Subgroups — #6627140

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36573C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic, complemented
3#33297C211yesmaximal, characteristic, complemented
4#33263C211yesmaximal, characteristic, complemented
5#33258C211yesmaximal, characteristic, complemented
6#29207C211yesmaximal, characteristic, complemented
7#29192C211yesmaximal, characteristic
8#29190C211yesmaximal, characteristic
9#29179C211yesmaximal, characteristic, complemented
10#29178C211yesmaximal, characteristic, complemented
11#28960C214yesmaximal, complemented
12#28957C214yesmaximal, complemented
13#28508C211yesmaximal, characteristic
14#25481C214yesmaximal, complemented
15#22979C214yesmaximal, complemented
16#21646C211yesmaximal, characteristic
17#21646C211yesmaximal, characteristic
18C4×#486C211yesmaximal, characteristic
19C4×#446C211yesmaximal, characteristic, complemented
20C2×#486C2211yescharacteristic
21C2×#474C2214yes—
22C2×#446C2212yes—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2582—21no—
30#2580—21no—
31#2580—21no—
32#2580C2212yes—
33#2567C2212yes—
34#2552—21no—
35#2550—21no—
36#2550—21no—
37#2507C2212yes—
38#2507C2212yes—
39#2454—21no—
40#2454—21no—
41#2454—21no—
42#2454—21no—
43#2454C2212yes—
44#2454C2212yes—
45#2451—21no—
46#2451—21no—
47#2369C2212yes—
48#2355—21no—
49Q8×C4⋊C4—21no—
50#2349—21no—
51#2349—21no—
52#2349C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2093C2214yes—
61#2087C2214yes—
62#2086C2214yes—
63#2067C2214yes—
64#2065C2214yes—
65#2059C2214yes—
66#1993C2214yes—
67#1992C2214yes—
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1957C2211yescharacteristic
72#1957C2211yescharacteristic
73#1957C2211yescharacteristic
74#1949C2214yes—
75#1944C2214yes—
76#1943C2214yes—
77#1942C2214yes—
78#1942C2214yes—
79#1936C2214yes—
80#1936C2214yes—
81#1934C2214yes—
82#1921C2214yes—
83#1918C2214yes—
84#1864C2214yes—
85#1862C2214yes—
86#1845C2211yescharacteristic
87#1845C2211yescharacteristic
88#1845C2211yescharacteristic
89#1844C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1771C2212yes—
96#1771C2212yes—
97#1770C2212yes—
98C4×C4⋊C8C2211yescharacteristic
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1762C2214yes—
102#1762C2214yes—
103C4×#102C2214yes—
104C4×#97C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#127C2311yescharacteristic
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—21no—
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#486Q818yescomplemented
133#474—216no—
134#446—28no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#97C2312yes—
147C2×#97C2312yes—
148C2×#97C2312yes—
149C2×#97C2312yes—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2311yescharacteristic
156#399—22no—
157#399—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#396—22no—
164#396—22no—
165#395—22no—
166#395—22no—
167#393—22no—
168#391—22no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—22no—
173#391—22no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390C2312yes—
181#390C2312yes—
182#390C2312yes—
183#389—22no—
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—22no—
200#389—22no—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#387—22no—
213#387—22no—
214#385—22no—
215#385—22no—
216#385—22no—
217#385—22no—
218#385—22no—
219#385—22no—
220#384—22no—
221#384—21no—
222#384—21no—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2312yes—
230#382—21no—
231#382—21no—
232#382—21no—
233#382—21no—
234#382C2312yes—
235#380—22no—
236#380—22no—
237#380—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2311yescharacteristic
254C2×#95—22no—
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269C22×Q8—21no—
270C22×Q8—22no—
271C22×Q8C2×D411yescharacteristic
272C22×D4—21no—
273C22×D4—21no—
274C22×D4C2412yes—
275C23×C4—22no—
276C23×C4—22no—
277C23×C4—22no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—28no—
283C4⋊Q8—22no—
284C4⋊Q8—82no—
285C4⋊Q8—82no—
286C4⋊Q8C2×Q814yes—
287#127—22no—
288#127C2×Q814yes—
289#125—44no—
290#125—82no—
291#125—82no—
292#125—82no—
293#125—82no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—42no—
297C4×Q8—44no—
298C4×Q8—82no—
299C4×Q8—82no—
300C4×D4—28no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—82no—
306C4×D4—82no—
307C4×D4—82no—
308C4×D4—82no—
309C2×C4⋊C4—21no—
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—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
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⋊C4C2×D411yescharacteristic
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⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
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×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#107—28no—
387#107—44no—
388#106—44no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391C4⋊C8#4118yes—
392#102—28no—
393#102—28no—
394#102—44no—
395#102C2×Q8116yes—
396#97—24no—
397#97—24no—
398#97—42no—
399#97—42no—
400C4×C8—24no—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8—42no—
405C4×C8C2×Q814yes—
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#95—22no—
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428C24—22no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—41no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—82no—
437C2×Q8—82no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—24no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4C22×Q814yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
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×C4C22×D411yescharacteristic
478C22×C4C2511yescharacteristic
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×C8C2×#4114yes—
490C2×C8C2×#4114yes—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—44no—
509C4⋊C4—44no—
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—42no—
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⋊C4—82no—
536C4⋊C4C22×Q814yes—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
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—
565C42#13714yes—
566C42C22×Q812yes—
567C42C2×#4114yes—
568C422−1+412yes—
569C23—24no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23C23×D411yescharacteristic
577Q8—44no—
578Q8—48no—
579Q8—164no—
580D4—28no—
581D4—48no—
582D4—48no—
583D4—44no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
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—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—164no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4#53212yes—
641C2×C4#53212yes—
642C2×C4D4×Q812yes—
643C2×C4D4×Q812yes—
644C2×C4C2×#13712yes—
645C2×C4C2×#13712yes—
646C2×C4C2×#13712yes—
647C2×C4#57512yes—
648C2×C4#57612yes—
649C2×C4#57612yes—
650C2×C4C23×Q811yescharacteristic
651C2×C4C22×#4112yes—
652C2×C4C2×2−1+411yescharacteristic
653C2×C4#58512yes—
654C8—24no—
655C8—24no—
656C8—24no—
657C8—42no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C2×#53211yescharacteristic
666C22C2×D4×Q811yescharacteristic
667C22#350011yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57511yescharacteristic
670C22C2×#57611yescharacteristic
671C22C2×#57611yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4#338514yes—
692C4#338914yes—
693C4Q8×#4112yes—
694C4#351512yes—
695C2—84no—
696C2#3656211yescharacteristic
697C2#6070111yescharacteristic
698C2#6071411yescharacteristic
699C2C2×#338511yescharacteristic
700C2C2×#338911yescharacteristic
701C2#6253711yescharacteristic
702C2#6254611yescharacteristic
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. #33297C2semidirect
4. #33263C2semidirect
5. #33258C2semidirect
6. #29207C2semidirect
9. #29179C2semidirect
10. #29178C2semidirect
11. #28960C2semidirect
12. #28957C2semidirect
14. #25481C2semidirect
15. #22979C2semidirect
19. C4×#446C2semidirect
132. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#365732
#365632
#332972
#332632
#332582
#292072
#291922
#291902
#291792
#291782
#289602
#289572
#285082
#254812
#229792
#216462
C4×#4862
C4×#4462