Finite Group Databasebeta

← #6395285

Subgroups — #6395285

Order 512.

1990 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3254C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#34155C211yesmaximal, characteristic
4#23142C214yesmaximal, complemented
5#23101C212yesmaximal, complemented
6#23069C212yesmaximal, complemented
7#23035C214yesmaximal, complemented
8#22997C212yesmaximal, complemented
9#22965C212yesmaximal, complemented
10#22152C212yesmaximal, complemented
11#22110C212yesmaximal, complemented
12#21859C211yesmaximal, characteristic
13#21613C212yesmaximal, complemented
14C4×#461C212yesmaximal, complemented
15#21389C211yesmaximal, characteristic, complemented
16#21171C211yesmaximal, characteristic, complemented
17#20550C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20#3254C4116yescomplemented
21C2×#477C2212yes—
22C2×#474C2212yes—
23C2×#468C2214yes—
24C2×#460C2212yes—
25C2×#461C2212yes—
26C2×C4×SD16C2214yes—
27C2×#431C2211yescharacteristic
28C2×#428C2212yes—
29C2×#423C2211yescharacteristic
30#2372—22no—
31#2369C2211yescharacteristic
32#2369C2211yescharacteristic
33#2366C2211yescharacteristic
34#2366C2211yescharacteristic
35#2363—21no—
36#2363—21no—
37#2363C2212yes—
38#2363C2212yes—
39#2360—22no—
40#2360C2214yes—
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#2297—22no—
50#2293—21no—
51#2293C2212yes—
52C2×#384C2211yescharacteristic
53#2274—21no—
54C42×D4—21no—
55C42×D4C2212yes—
56C4×#117C2211yescharacteristic
57C4×#117C2211yescharacteristic
58#1990C2212yes—
59#1988C2211yescharacteristic
60#1988C2211yescharacteristic
61#1981C2211yescharacteristic
62#1981C2211yescharacteristic
63#1973C2214yes—
64#1957C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1947C2212yes—
69#1942C2214yes—
70#1935C2214yes—
71#1934C2214yes—
72#1927C2214yes—
73#1925C2214yes—
74#1921C2214yes—
75#1918C2214yes—
76#1880C2214yes—
77#1877C2214yes—
78#1874C2214yes—
79#1873C2212yes—
80#1873C2212yes—
81#1864C2214yes—
82#1863C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1776C2214yes—
86#1770C2212yes—
87C8×C4⋊C4C2212yes—
88#1767C2212yes—
89C4×C4⋊C8C2212yes—
90#1762C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C4×#103C2214yes—
94#1753C2214yes—
95#1736C2212yes—
96C22×#41—21no—
97C22×SD16C2314yes—
98C22×M16C2312yes—
99#533—24no—
100#533C2×C418yescomplemented
101#532—24no—
102#532C2×C418yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125—21no—
108C2×#124—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2314yes—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117—22no—
125C2×#117D411yescharacteristic
126C2×#117D411yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C42—22no—
133C22×C42C2311yescharacteristic
134#477C2×C4116yescomplemented
135#474C2×C4116yescomplemented
136#468C2×C4132yescomplemented
137#461C2×C4116yescomplemented
138#460C2×C4116yescomplemented
139C4×SD16—48no—
140#431—24no—
141#428—44no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145#423C2×C418yescomplemented
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2312yes—
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2314yes—
162#391—22no—
163#391—22no—
164#391—22no—
165#391C2314yes—
166#390—21no—
167#390—21no—
168#390—22no—
169#390—22no—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389—24no—
179#389C2314yes—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—24no—
187#387—24no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—24no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—24no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228#376—21no—
229#376D412yes—
230C43—21no—
231C43C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2314yes—
240#336C2314yes—
241#336C2314yes—
242C2×#41—28no—
243C2×#41—82no—
244C2×#41—82no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×SD16—44no—
260C2×SD16—44no—
261C2×SD16C22×C4116yes—
262C2×M16—42no—
263C2×M16—42no—
264C2×M16C22×C418yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8C22×C414yes—
272C4⋊Q8C22×C414yes—
273C4⋊Q8C22×C414yes—
274#127—22no—
275#127C22×C414yes—
276#125—28no—
277#124—28no—
278C22⋊Q8—28no—
279C22⋊Q8—24no—
280C22⋊Q8C22×C418yes—
281#121—28no—
282#121—24no—
283#121C22×C418yes—
284C4×Q8—28no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—84no—
289C4×Q8C22×C418yes—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—28no—
293C4×D4—24no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4C22×C418yes—
299#117—28no—
300#117—24no—
301#117—22no—
302#117—22no—
303#117—42no—
304#117—41no—
305#117—41no—
306#117—84no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
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—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—84no—
362C2×C42C22×C414yes—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4114yes—
370C2×C42C2412yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—44no—
375#107C22×C4116yes—
376#106—28no—
377C4⋊C8—22no—
378C4⋊C8—22no—
379C4⋊C8C22×C418yes—
380C4⋊C8C22×C418yes—
381#103—24no—
382#103—24no—
383#103—42no—
384#103—42no—
385#103C22×C4116yes—
386#102—44no—
387#102—44no—
388#102C22×C4116yes—
389C22⋊C8—24no—
390#97—44no—
391C4×C8—24no—
392C4×C8—44no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—22no—
410C24C22×D411yescharacteristic
411#41—816no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—81no—
422C2×Q8—81no—
423C2×Q8C23×C414yes—
424C2×D4—28no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—81no—
435C2×D4—81no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4C23×C414yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4C4×D414yes—
468C22×C4C4×D414yes—
469C22×C4C23×C412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4112yes—
480C22×C4C2511yescharacteristic
481SD16—416no—
482M16—48no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8C23×C418yes—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4#13714yes—
521C4⋊C4C23×C414yes—
522C4⋊C4C23×C414yes—
523C4⋊C4H(5,𝔽2)12yes—
524C4⋊C42−1+412yes—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—28no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—82no—
554C42—82no—
555C42—84no—
556C42—84no—
557C42C4×D414yes—
558C42C4×D414yes—
559C42C23×C412yes—
560C42C23×C412yes—
561C42C2×#4112yes—
562C42C2×#4112yes—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—24no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23C2×C4×D412yes—
577C23C23×D411yescharacteristic
578Q8—24no—
579Q8—24no—
580Q8—48no—
581Q8—42no—
582Q8—42no—
583Q8—84no—
584D4—24no—
585D4—24no—
586D4—48no—
587D4—42no—
588D4—42no—
589D4—84no—
590D4—88no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—416no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—88no—
640C2×C4—88no—
641C2×C4—164no—
642C2×C4C2×C4×D412yes—
643C2×C4C2×C4×D412yes—
644C2×C4C2×C4×D412yes—
645C2×C4C2×C4×D412yes—
646C2×C4C2×C4×D412yes—
647C2×C4C2×#13714yes—
648C2×C4C2×#13712yes—
649C2×C4#57512yes—
650C2×C4#57612yes—
651C2×C4#57712yes—
652C2×C4C24×C411yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4#58511yescharacteristic
657C2×C4#58511yescharacteristic
658C8—28no—
659C8—42no—
660C8—42no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22C22×C4×D411yescharacteristic
676C22#348811yescharacteristic
677C22#348911yescharacteristic
678C22C22×#13711yescharacteristic
679C22C2×#57511yescharacteristic
680C22C2×#57611yescharacteristic
681C22C2×#57711yescharacteristic
682C4—24no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—81no—
695C4—84no—
696C4—81no—
697C4C4×#13714yes—
698C4#294614yes—
699C4C4×H(5,𝔽2)11yescharacteristic
700C4C4×2−1+411yescharacteristic
701C4#351111yescharacteristic
702C4#351211yescharacteristic
703C2—42no—
704C2—84no—
705C2#3426711yescharacteristic
706C2C2×C4×#13711yescharacteristic
707C2C2×#294611yescharacteristic
708C2#6198011yescharacteristic
709C2#6198811yescharacteristic
710C2#6214411yescharacteristic
711C2#6214911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
2. #34214C2semidirect
4. #23142C2semidirect
5. #23101C2semidirect
6. #23069C2semidirect
7. #23035C2semidirect
8. #22997C2semidirect
9. #22965C2semidirect
10. #22152C2semidirect
11. #22110C2semidirect
13. #21613C2semidirect
14. C4×#461C2semidirect
15. #21389C2semidirect
16. #21171C2semidirect
17. #20550C2semidirect
20. #3254C4semidirect
100. #533C2×C4semidirect
134. #477C2×C4semidirect
135. #474C2×C4semidirect
136. #468C2×C4semidirect
137. #461C2×C4semidirect
138. #460C2×C4semidirect
145. #423C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32542
#342142
#341552
#231422
#231012
#230692
#230352
#229972
#229652
#221522
#221102
#218592
#216132
C4×#4612
#213892
#211712
#205502