Finite Group Databasebeta

← #6481379

Subgroups — #6481379

Order 512.

3116 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic, complemented
3C2×#3104C211yesmaximal, characteristic, complemented
4#35240C211yesmaximal, characteristic
5#32984C212yesmaximal, complemented
6#32983C212yesmaximal, complemented
7#29599C212yesmaximal, complemented
8#29597C212yesmaximal, complemented
9#25716C214yesmaximal, complemented
10#25584C214yesmaximal, complemented
11#24127C218yesmaximal, complemented
12#21836C211yesmaximal, characteristic, complemented
13#21788C211yesmaximal, characteristic
14#20741C211yesmaximal, characteristic, complemented
15C2×#552—28no—
16C2×#552—28no—
17C2×#552—22no—
18C2×#552—22no—
19C2×#552C2214yes—
20C2×#552C2214yes—
21C2×#533C2211yescharacteristic
22C22×#126—21no—
23C22×#125—21no—
24C22×#125C2211yescharacteristic
25C22×#123—21no—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117—21no—
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3236—28no—
34C2×#501C2212yescomplemented
35C2×#491C2212yescomplemented
36#3104—28no—
37C2×#484C2212yes—
38C2×#484C2212yes—
39C2×#483C2212yes—
40C2×#479C2214yescomplemented
41C2×#477C2212yescomplemented
42C2×#462C2212yescomplemented
43C2×#451C2212yes—
44C2×C4×Q16C2214yescomplemented
45C2×#432C2211yescharacteristic
46C2×#431C2211yescharacteristic
47#2848C2212yes—
48#2848C2212yes—
49#2633—22no—
50#2547C2212yes—
51#2542C2212yes—
52#2536C2214yes—
53#2502—21no—
54#2482—21no—
55#2482—21no—
56#2435—21no—
57#2326—22no—
58#2283C2211yescharacteristic
59#2036C2218yescomplemented
60#2035C2218yescomplemented
61#2001C2218yescomplemented
62#2001C2218yescomplemented
63#1991C2214yescomplemented
64#1989C2214yescomplemented
65#1955C2214yes—
66#1954C2214yes—
67#1873C2218yescomplemented
68#1872C2218yescomplemented
69#1854C2218yescomplemented
70#1852C2218yescomplemented
71#1824C2212yes—
72#1824C2212yes—
73#1823C2212yes—
74#1805C2214yescomplemented
75#1787C2212yes—
76#1779C2212yes—
77#1779C2212yes—
78#1763C2214yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4D411yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×Q16C2314yes—
85#552—232no—
86#552—232no—
87#552—216no—
88#552—216no—
89#552—216no—
90#552—216no—
91#533—24no—
92#533—24no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#126—216no—
96C2×#126—24no—
97C2×#126—24no—
98C2×#125—216no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125—21no—
102C2×#125—21no—
103C2×#125—24no—
104C2×#125—21no—
105C2×#125D418yes—
106C2×#125C2312yes—
107C2×#125C2312yes—
108C2×#124—22no—
109C2×#123—216no—
110C2×#123—24no—
111C2×#123—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2314yes—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2314yes—
123C2×C22⋊Q8C2314yes—
124C2×C4×Q8—24no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8D418yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4D412yes—
135C2×C4×D4D412yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×#117—216no—
139C2×#117—22no—
140C2×#117—21no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117—24no—
144C2×#117—24no—
145C2×#117C2314yes—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—22no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4—24no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4C2314yes—
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4D411yescharacteristic
164C22×C22⋊C4D411yescharacteristic
165C22×C42—21no—
166C22×C42—21no—
167C22×C42C2311yescharacteristic
168#501—28no—
169#491—28no—
170#484—28no—
171#484—28no—
172#483—28no—
173#479—216no—
174#477—28no—
175#462—28no—
176#451—28no—
177C4×Q16—216no—
178#432—24no—
179#431—24no—
180C2×#107C2314yes—
181C2×#107C2314yes—
182C2×#106C2314yes—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2314yes—
188C2×#103C2314yes—
189C2×#103C2318yes—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2318yes—
193C2×C22⋊C8C2318yes—
194C2×#97C2312yes—
195C2×C4×C8C2312yes—
196#399—22no—
197#399—22no—
198#398—24no—
199#397—22no—
200#397—22no—
201#394—24no—
202#391—22no—
203#390—22no—
204#388—24no—
205#388—24no—
206#385—24no—
207#385—24no—
208#385—22no—
209#385—22no—
210#384—22no—
211#383—21no—
212#382C2312yes—
213#381—21no—
214#380C2312yes—
215#380C2314yes—
216#380C2314yes—
217#379—22no—
218#379—22no—
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221#376—21no—
222#340C2318yes—
223#340C2318yes—
224#340C2318yes—
225C25C2×D411yescharacteristic
226C25C2×D411yescharacteristic
227C22×Q8—22no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—44no—
231C22×Q8C2×D414yes—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C22×D4—24no—
235C22×D4—44no—
236C22×D4C2×D412yes—
237C22×D4C2×D412yes—
238C23×C4—28no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—48no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×Q16—28no—
261C2×Q16—48no—
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268#126—232no—
269#126—232no—
270#125—232no—
271#125—232no—
272#125—28no—
273#125—28no—
274#125—24no—
275#125—24no—
276#125—44no—
277#125C2×D418yes—
278#125C2×D418yes—
279#124—48no—
280#123—232no—
281#123—232no—
282C22⋊Q8—216no—
283C22⋊Q8—216no—
284C22⋊Q8—28no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C22⋊Q8—28no—
288C22⋊Q8—24no—
289C22⋊Q8—24no—
290C22⋊Q8—44no—
291C22⋊Q8—48no—
292C22⋊Q8—48no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—44no—
300C4×Q8C2×D418yes—
301C4×Q8C2×D418yes—
302C4×D4—28no—
303C4×D4—28no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308#117—216no—
309#117—232no—
310#117—232no—
311#117—28no—
312#117—28no—
313#117—44no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—216no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—28no—
357C2×C22⋊C4—28no—
358C2×C22⋊C4—28no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C42—28no—
375C2×C42—28no—
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D414yes—
384C2×C42C2×D414yes—
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#107—28no—
389#106—28no—
390C4⋊C8—24no—
391C4⋊C8—24no—
392#103—28no—
393#103—28no—
394#103—28no—
395#103—28no—
396#103—48no—
397C22⋊C8—24no—
398C22⋊C8—24no—
399C22⋊C8—48no—
400C22⋊C8—48no—
401#97—24no—
402C4×C8—24no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409C24—22no—
410C24—24no—
411C24—22no—
412C24—24no—
413C24—24no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24C22×D412yes—
418C24C22×D412yes—
419C24C22×D411yescharacteristic
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—28no—
427C2×Q8—48no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—44no—
431C2×Q8—42no—
432C2×Q8C22×D414yes—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—416no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—216no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—432no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4#12014yes—
481C22×C4#12014yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C2511yescharacteristic
492Q16—416no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—28no—
496C2×C8—28no—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—28no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4C22×D414yes—
525C4⋊C4C22×D414yes—
526C4⋊C4C22×D414yes—
527C22⋊C4—216no—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—48no—
535C22⋊C4—44no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—216no—
541C42—28no—
542C42—28no—
543C42—44no—
544C42—44no—
545C42#12014yes—
546C42#12014yes—
547C42C22×D412yes—
548C42C22×D412yes—
549C42C22×D412yes—
550C42C22×D412yes—
551C23—22no—
552C23—22no—
553C23—28no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—24no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—416no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—88no—
570C23C2×#12012yes—
571C23#57514yes—
572C23#57814yes—
573C23C23×D411yescharacteristic
574Q8—24no—
575Q8—24no—
576Q8—48no—
577Q8—44no—
578Q8—44no—
579Q8—44no—
580Q8—84no—
581D4—48no—
582D4—48no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—432no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4C2×#12012yes—
625C2×C4C2×#12012yes—
626C2×C4C2×#12012yes—
627C2×C4C2×#12012yes—
628C2×C4C2×#12012yes—
629C2×C4C23×D411yescharacteristic
630C2×C4C23×D411yescharacteristic
631C2×C4#58514yes—
632C8—44no—
633C8—44no—
634C22—28no—
635C22—22no—
636C22—22no—
637C22—28no—
638C22—24no—
639C22—416no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—88no—
649C22#301114yes—
650C22C22×#12011yescharacteristic
651C22#348512yes—
652C22C2×#57512yes—
653C22C2×#57812yes—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—84no—
666C4—82no—
667C4—82no—
668C4#344912yes—
669C4#348712yes—
670C2—24no—
671C2—44no—
672C2—82no—
673C2#3514911yescharacteristic
674C2#6035911yescharacteristic
675C2C2×#301111yescharacteristic
676C2#6173814yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3236C2semidirect
3. C2×#3104C2semidirect
5. #32984C2semidirect
6. #32983C2semidirect
7. #29599C2semidirect
8. #29597C2semidirect
9. #25716C2semidirect
10. #25584C2semidirect
11. #24127C2semidirect
12. #21836C2semidirect
14. #20741C2semidirect
34. C2×#501C22semidirect
35. C2×#491C22semidirect
40. C2×#479C22semidirect
41. C2×#477C22semidirect
42. C2×#462C22semidirect
44. C2×C4×Q16C22semidirect
59. #2036C22semidirect
60. #2035C22semidirect
61. #2001C22semidirect
62. #2001C22semidirect
63. #1991C22semidirect
64. #1989C22semidirect
67. #1873C22semidirect
68. #1872C22semidirect
69. #1854C22semidirect
70. #1852C22semidirect
74. #1805C22semidirect
78. #1763C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#32362
C2×#31042
#352402
#329842
#329832
#295992
#295972
#257162
#255842
#241272
#218362
#217882
#207412