Finite Group Databasebeta

← #6395242

Subgroups — #6395242

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3249C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#34145C211yesmaximal, characteristic
4#23133C214yesmaximal
5#23102C212yesmaximal, complemented
6#23069C212yesmaximal, complemented
7#23028C214yesmaximal
8#23004C212yesmaximal, complemented
9#22971C212yesmaximal, complemented
10#22154C212yesmaximal, complemented
11#22119C212yesmaximal, complemented
12#21860C211yesmaximal, characteristic
13#21616C212yesmaximal, complemented
14#21506C212yesmaximal, complemented
15#21385C211yesmaximal, characteristic
16C2×#1948C211yesmaximal, characteristic
17#20547C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20#3249C4116yescomplemented
21C2×#477C2212yes—
22C2×#477C2212yes—
23C2×#467C2214yes—
24C2×#462C2212yes—
25C2×#462C2212yes—
26C2×#441C2214yes—
27C2×#431C2211yescharacteristic
28C22×#106C2212yes—
29C22×C4⋊C8C2211yescharacteristic
30#2370—21no—
31#2370—21no—
32#2369C2211yescharacteristic
33#2369C2211yescharacteristic
34#2368—22no—
35#2365C2211yescharacteristic
36#2365C2211yescharacteristic
37#2360—22no—
38#2360C2212yes—
39#2360C2214yes—
40#2360C2212yes—
41#2355—21no—
42#2355C2212yes—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4C2212yes—
45#2330—22no—
46#2317—21no—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2212yes—
49#2296—21no—
50#2295—22no—
51#2295C2212yes—
52C2×#384C2211yescharacteristic
53#2274—21no—
54C42×Q8C2212yes—
55C4×#117—21no—
56C4×#117C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#1990C2212yes—
59#1987C2211yescharacteristic
60#1987C2211yescharacteristic
61#1981C2211yescharacteristic
62#1981C2211yescharacteristic
63#1971C2214yes—
64#1956C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1948C2212yes—
69#1943C2214yes—
70#1935C2214yes—
71#1935C2214yes—
72#1927C2214yes—
73#1927C2214yes—
74#1918C2214yes—
75#1918C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1872C2212yes—
79#1872C2214yes—
80#1872C2212yes—
81#1865C2214yes—
82#1865C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1776C2214yes—
86#1770C2212yes—
87#1770C2212yes—
88#1767C2212yes—
89#1767C2212yes—
90#1763C2214yes—
91#1763C2214yes—
92C4×#103C2212yes—
93C4×#103C2212yes—
94#1752C2214yes—
95C2×#340C2212yes—
96C22×Q16C2314yes—
97C23×C8C2312yes—
98#533—24no—
99#533—24no—
100#533C2×C418yes—
101#533C2×C418yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#125—21no—
107C2×#125—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×#117—21no—
121C2×#117—21no—
122C2×#117D411yescharacteristic
123C2×#117D411yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131#477C2×C4116yescomplemented
132#477C2×C4116yescomplemented
133#467C2×C4132yes—
134#462C2×C4116yescomplemented
135#462C2×C4116yescomplemented
136#441—48no—
137#431—24no—
138C2×#107C2314yes—
139C2×#106—44no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2×C418yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×C22⋊C8C2312yes—
156C2×#97C2314yes—
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2312yes—
164#391C2312yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—24no—
173#389—24no—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#389C2312yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—24no—
188#385—22no—
189#385—22no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—24no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376—22no—
228#376D412yes—
229C2×#95—21no—
230C43—22no—
231C43C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2314yes—
240#336C2314yes—
241#336C2314yes—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8C2412yes—
245C22×Q8C2412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×Q16—44no—
253C2×Q16—44no—
254C2×Q16C22×C4116yes—
255C22×C8—42no—
256C22×C8—42no—
257C22×C8C22×C418yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8C22×C414yes—
266C4⋊Q8C22×C414yes—
267C4⋊Q8C22×C414yes—
268C4⋊Q8C22×C414yes—
269#125—28no—
270#125—28no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8C22×C418yes—
276C22⋊Q8C22×C418yes—
277C4×Q8—28no—
278C4×Q8—28no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—42no—
284C4×Q8—42no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—84no—
288C4×Q8C22×C418yes—
289C4×Q8C22×C418yes—
290#117—28no—
291#117—28no—
292#117—24no—
293#117—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—41no—
320C2×C4⋊C4—41no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C22×C414yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42#4114yes—
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107C22×C4116yes—
366#106—28no—
367#106—44no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8C22×C418yes—
371C4⋊C8C22×C418yes—
372#103—24no—
373#103—24no—
374#103—42no—
375#103—42no—
376#103—44no—
377#103—44no—
378#103C22×C4116yes—
379#103C22×C4116yes—
380C22⋊C8—24no—
381#97—44no—
382#97—44no—
383C4×C8—24no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24C22×D411yescharacteristic
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—81no—
420C2×Q8—81no—
421C2×Q8C23×C414yes—
422C2×Q8C23×C414yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4C4×D414yes—
448C22×C4C4×D414yes—
449C22×C4#13714yes—
450C22×C4#13714yes—
451C22×C4C23×C412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2511yescharacteristic
463Q16—416no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—48no—
473C2×C8C23×C418yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—24no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4C23×C414yes—
512C4⋊C4C23×C414yes—
513C4⋊C4H(5,𝔽2)12yes—
514C4⋊C42−1+412yes—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C42—28no—
520C42—28no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—42no—
528C42—42no—
529C42—44no—
530C42—44no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42—82no—
536C42—82no—
537C42—84no—
538C42—84no—
539C42C4×D414yes—
540C42C4×D414yes—
541C42C23×C412yes—
542C42C23×C412yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23C2×C4×D412yes—
550C23C2×#13712yes—
551C23C2×#13712yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—24no—
555Q8—24no—
556Q8—24no—
557Q8—48no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—84no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—164no—
616C2×C4#441116yescomplemented
617C2×C4C2×C4×D412yes—
618C2×C4C2×C4×D412yes—
619C2×C4C2×C4×D412yes—
620C2×C4C2×C4×D412yes—
621C2×C4C2×C4×D412yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13712yes—
625C2×C4C2×#13712yes—
626C2×C4C24×C411yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C2×C4#58511yescharacteristic
631C2×C4#58511yescharacteristic
632C8—28no—
633C8—42no—
634C8—42no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22C2×#44114yes—
642C22C22×C4×D411yescharacteristic
643C22#348811yescharacteristic
644C22#348911yescharacteristic
645C22C22×#13711yescharacteristic
646C22C22×#13711yescharacteristic
647C22#358511yescharacteristic
648C22#358611yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—44no—
662C4—81no—
663C4—84no—
664C4—81no—
665C4C2×#44114yes—
666C4C2×#44114yes—
667C4C4×H(5,𝔽2)11yescharacteristic
668C4C4×2−1+411yescharacteristic
669C4#351111yescharacteristic
670C4#351211yescharacteristic
671C2—24no—
672C2#3426711yescharacteristic
673C2C22×#44111yescharacteristic
674C2#6017011yescharacteristic
675C2#6196011yescharacteristic
676C2#6196511yescharacteristic
677C2#6210811yescharacteristic
678C2#6211311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×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
5. #23102C2semidirect
6. #23069C2semidirect
8. #23004C2semidirect
9. #22971C2semidirect
10. #22154C2semidirect
11. #22119C2semidirect
13. #21616C2semidirect
14. #21506C2semidirect
20. #3249C4semidirect
131. #477C2×C4semidirect
132. #477C2×C4semidirect
134. #462C2×C4semidirect
135. #462C2×C4semidirect
616. C2×C4#441semidirect

Maximal subgroups

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

subgroupindex
C2×#32492
#342142
#341452
#231332
#231022
#230692
#230282
#230042
#229712
#221542
#221192
#218602
#216162
#215062
#213852
C2×#19482
#205472