Finite Group Databasebeta

← #6481372

Subgroups — #6481372

Order 512.

3110 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#3236C211yesmaximal, characteristic, complemented
3C2×#3113C211yesmaximal, characteristic, complemented
4#35172C211yesmaximal, characteristic
5#32984C212yesmaximal, complemented
6#32982C212yesmaximal, complemented
7#29599C212yesmaximal, complemented
8#29597C212yesmaximal, complemented
9#25715C214yesmaximal, complemented
10#25584C214yesmaximal, complemented
11#24139C218yesmaximal, complemented
12#21838C211yesmaximal, characteristic, complemented
13#21790C211yesmaximal, characteristic
14#20740C211yesmaximal, 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#3113—28no—
37C2×#484C2212yes—
38C2×#483C2212yes—
39C2×#483C2212yes—
40C2×#479C2214yescomplemented
41C2×#475C2212yescomplemented
42C2×#462C2212yescomplemented
43C2×#451C2212yes—
44C2×#441C2214yescomplemented
45C2×#432C2211yescharacteristic
46C2×#432C2211yescharacteristic
47#2848C2212yes—
48#2848C2212yes—
49#2626—21no—
50#2626—21no—
51#2580—22no—
52#2566—22no—
53#2563C2212yes—
54#2542C2212yes—
55#2538C2214yes—
56#2317—21no—
57#2309—21no—
58#2285C2211yescharacteristic
59#2036C2218yescomplemented
60#2035C2218yescomplemented
61#2001C2218yescomplemented
62#2001C2218yescomplemented
63#1992C2214yescomplemented
64#1989C2214yescomplemented
65#1955C2214yes—
66#1955C2214yes—
67#1873C2218yescomplemented
68#1872C2218yescomplemented
69#1854C2218yescomplemented
70#1853C2218yescomplemented
71#1824C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1803C2214yescomplemented
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×#123—216no—
109C2×#123—24no—
110C2×#123—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2314yes—
120C2×C22⋊Q8C2314yes—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2314yes—
123C2×C4×Q8—24no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8D418yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4D412yes—
134C2×C4×D4D412yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×#117—216no—
138C2×#117—22no—
139C2×#117—21no—
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#483—28no—
172#483—28no—
173#479—216no—
174#475—28no—
175#462—28no—
176#451—28no—
177#441—216no—
178#432—24no—
179#432—24no—
180C2×#107C2314yes—
181C2×#106C2314yes—
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×#97C2312yes—
196#399—22no—
197#399—22no—
198#396—24no—
199#396—24no—
200#393—22no—
201#393—22no—
202#390—22no—
203#390—22no—
204#390—24no—
205#389—24no—
206#385—24no—
207#385—24no—
208#385—22no—
209#385—22no—
210#384—22no—
211#384C2312yes—
212#382—22no—
213#382—21no—
214#380—21no—
215#380C2314yes—
216#380C2314yes—
217#380C2312yes—
218#379—22no—
219#379—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2312yes—
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—24no—
236C22×D4—42no—
237C22×D4C2×D412yes—
238C22×D4C2×D412yes—
239C23×C4—28no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—44no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C2×Q16—28no—
263C2×Q16—48no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270#126—232no—
271#126—232no—
272#125—232no—
273#125—232no—
274#125—28no—
275#125—28no—
276#125—24no—
277#125—24no—
278#125—44no—
279#125C2×D418yes—
280#125C2×D418yes—
281#123—232no—
282#123—232no—
283C22⋊Q8—216no—
284C22⋊Q8—216no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C22⋊Q8—28no—
288C22⋊Q8—28no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—44no—
292C22⋊Q8—48no—
293C22⋊Q8—48no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—44no—
301C4×Q8C2×D418yes—
302C4×Q8C2×D418yes—
303C4×D4—28no—
304C4×D4—28no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—24no—
309#117—216no—
310#117—232no—
311#117—232no—
312#117—28no—
313#117—28no—
314#117—44no—
315#117—44no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—216no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—28no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C42—28no—
379C2×C42—28no—
380C2×C42—24no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D414yes—
388C2×C42C2×D414yes—
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391#107—28no—
392#106—28no—
393#106—28no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396#103—28no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—48no—
401C22⋊C8—24no—
402C22⋊C8—24no—
403C22⋊C8—48no—
404C22⋊C8—48no—
405#97—24no—
406#97—24no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24#13714yes—
421C24C22×D412yes—
422C24C22×D412yes—
423C24C22×D411yescharacteristic
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—28no—
431C2×Q8—48no—
432C2×Q8—44no—
433C2×Q8—48no—
434C2×Q8—42no—
435C2×Q8—82no—
436C2×Q8C22×D414yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—416no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—216no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—44no—
479C22×C4—432no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4#12014yes—
488C22×C4#12014yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C2511yescharacteristic
499Q16—416no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—28no—
503C2×C8—28no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—216no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—28no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—28no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—82no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4C22×D414yes—
534C4⋊C4C22×D414yes—
535C4⋊C4C22×D414yes—
536C22⋊C4—216no—
537C22⋊C4—216no—
538C22⋊C4—216no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C42—216no—
550C42—28no—
551C42—28no—
552C42—44no—
553C42—44no—
554C42#12014yes—
555C42#12014yes—
556C42C22×D412yes—
557C42C22×D412yes—
558C42C22×D412yes—
559C42C22×D412yes—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—24no—
565C23—28no—
566C23—22no—
567C23—24no—
568C23—22no—
569C23—24no—
570C23—22no—
571C23—22no—
572C23—48no—
573C23—44no—
574C23—48no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—84no—
580C23—84no—
581C23C2×#12012yes—
582C23C2×#13714yes—
583C23C2×#13712yes—
584C23#57512yes—
585C23C23×D411yescharacteristic
586Q8—24no—
587Q8—24no—
588Q8—48no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—84no—
593D4—48no—
594D4—48no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—432no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4C2×#12012yes—
641C2×C4C2×#12012yes—
642C2×C4C2×#12012yes—
643C2×C4C2×#12012yes—
644C2×C4C2×#12012yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C23×D411yescharacteristic
647C2×C4#58514yes—
648C8—44no—
649C8—44no—
650C22—24no—
651C22—24no—
652C22—22no—
653C22—22no—
654C22—28no—
655C22—24no—
656C22—416no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—88no—
666C22#300714yes—
667C22C22×#12011yescharacteristic
668C22#348512yes—
669C22C22×#13711yescharacteristic
670C22C2×#57511yescharacteristic
671C22#358612yes—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—84no—
684C4—82no—
685C4—82no—
686C4#344912yes—
687C4#348712yes—
688C2—24no—
689C2—44no—
690C2—82no—
691C2#3514911yescharacteristic
692C2C2×#300711yescharacteristic
693C2#6036811yescharacteristic
694C2#6170512yes—
695C2#6171112yes—
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×#3113C2semidirect
5. #32984C2semidirect
6. #32982C2semidirect
7. #29599C2semidirect
8. #29597C2semidirect
9. #25715C2semidirect
10. #25584C2semidirect
11. #24139C2semidirect
12. #21838C2semidirect
14. #20740C2semidirect
34. C2×#501C22semidirect
35. C2×#491C22semidirect
40. C2×#479C22semidirect
41. C2×#475C22semidirect
42. C2×#462C22semidirect
44. C2×#441C22semidirect
59. #2036C22semidirect
60. #2035C22semidirect
61. #2001C22semidirect
62. #2001C22semidirect
63. #1992C22semidirect
64. #1989C22semidirect
67. #1873C22semidirect
68. #1872C22semidirect
69. #1854C22semidirect
70. #1853C22semidirect
74. #1803C22semidirect
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×#31132
#351722
#329842
#329822
#295992
#295972
#257152
#255842
#241392
#218382
#217902
#207402