Finite Group Databasebeta

← #6445750

Subgroups — #6445750

Order 512.

3296 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#3166C211yesmaximal, characteristic, complemented
3C2×#3077C211yesmaximal, characteristic
4#34953C211yesmaximal, characteristic, complemented
5#25851C214yesmaximal, complemented
6#25850C214yesmaximal, complemented
7#25843C218yesmaximal, complemented
8#23553C212yesmaximal, complemented
9#23552C212yesmaximal, complemented
10#21314C211yesmaximal, characteristic, complemented
11#21303C211yesmaximal, characteristic, complemented
12#20873C212yesmaximal
13#20873C212yesmaximal
14C4×#418C211yesmaximal, characteristic
15C2×#554—28no—
16C2×#554—22no—
17C2×#554—22no—
18C2×#554—28no—
19C2×#554C2214yes—
20C2×#554C2214yes—
21C2×#533C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8—21no—
27C22×C22⋊Q8—21no—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×C4×D4C2211yescharacteristic
32C23×C4⋊C4—21no—
33#3166—28no—
34C2×#501C2212yescomplemented
35C2×#501C2212yescomplemented
36C2×#499C2214yescomplemented
37#3077—28no—
38C2×#478C2214yes—
39C2×#476C2212yes—
40C2×#474C2212yes—
41C2×#428C2212yes—
42C2×#428C2212yes—
43C2×#428C2212yes—
44C2×#423C2211yescharacteristic
45C2×#418C2212yes—
46C2×#418C2212yes—
47C2×C4×M16C2211yescharacteristic
48#2584C2214yescomplemented
49#2583—21no—
50#2583—21no—
51#2567C2212yescomplemented
52#2546C2212yescomplemented
53#2502—21no—
54#2434—21no—
55#2399—22no—
56C4×C4⋊Q8—22no—
57#2296C2212yes—
58#2296C2212yes—
59C4×#117C2211yescharacteristic
60#2059C2218yescomplemented
61#2059C2218yescomplemented
62#2058C2218yescomplemented
63#2057C2218yescomplemented
64#1989C2214yescomplemented
65#1989C2214yescomplemented
66#1985C2214yescomplemented
67#1984C2214yescomplemented
68#1864C2218yes—
69#1864C2218yes—
70#1862C2218yes—
71#1862C2218yes—
72#1762C2214yes—
73C4×#102C2214yes—
74#1736C2212yes—
75#1736C2212yes—
76#1736C2212yes—
77#1728C2212yes—
78#1728C2212yes—
79C23×Q8—21no—
80C23×Q8D411yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C24×C4—21no—
85C22×M16C2312yes—
86C22×M16C2312yes—
87#554—232no—
88#554—232no—
89#554—216no—
90#554—216no—
91#554—216no—
92#554—216no—
93#533—24no—
94#533—24no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2314yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—24no—
103C2×#125—24no—
104C2×#125—22no—
105C2×#125—21no—
106C2×#125—21no—
107C2×#125—24no—
108C2×#125D418yes—
109C2×#125C2312yes—
110C2×#125C2314yes—
111C2×#125C2312yes—
112C2×#125C2312yes—
113C2×C22⋊Q8—216no—
114C2×C22⋊Q8—216no—
115C2×C22⋊Q8—216no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8D412yes—
129C2×C4×Q8D412yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—24no—
136C2×C4×D4—22no—
137C2×C4×D4—22no—
138C2×C4×D4—21no—
139C2×C4×D4—21no—
140C2×C4×D4—24no—
141C2×C4×D4D418yes—
142C2×C4×D4C2314yes—
143C2×C4×D4C2314yes—
144C2×C4×D4C2312yes—
145C2×C4×D4C2312yes—
146C2×#117C2312yes—
147C2×#117C2312yes—
148C22×C4⋊C4—216no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—24no—
155C22×C4⋊C4—24no—
156C22×C4⋊C4D411yescharacteristic
157C22×C4⋊C4D411yescharacteristic
158C22×C4⋊C4D411yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2311yescharacteristic
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C42C2311yescharacteristic
166C22×C42C2311yescharacteristic
167C22×C42C2311yescharacteristic
168#501—28no—
169#501—28no—
170#499—216no—
171#478—216no—
172#476—28no—
173#474—28no—
174#428—28no—
175#428—28no—
176#428—28no—
177C2×#107C2314yes—
178C2×#107C2314yes—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×#106C2314yes—
182C2×#106C2314yes—
183#423—24no—
184C2×C4⋊C8C2312yes—
185C2×C4⋊C8C2312yes—
186#418—28no—
187#418Q8116yescomplemented
188C2×#102C2314yes—
189C2×#102C2314yes—
190C2×#102C2318yes—
191C4×M16—24no—
192C2×#97C2312yes—
193C2×C4×C8C2312yes—
194#398—22no—
195#398—22no—
196#398—24no—
197#395—24no—
198#395—22no—
199#395—22no—
200#391—22no—
201#391—22no—
202#389—24no—
203#389—24no—
204#389C2314yes—
205#389C2314yes—
206#389C2314yes—
207#389C2314yes—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#384—22no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—24no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219#376—21no—
220C43—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225#336C2318yes—
226C25—21no—
227C22×Q8—28no—
228C22×Q8—24no—
229C22×Q8—24no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8—44no—
233C22×Q8—84no—
234C22×Q8C2×D412yes—
235C22×Q8C2×D412yes—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—24no—
239C22×D4C2×D414yes—
240C22×D4C2×Q818yes—
241C22×D4C2412yes—
242C23×C4—28no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—28no—
247C23×C4—24no—
248C23×C4—28no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—48no—
253C23×C4—84no—
254C23×C4—84no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—28no—
266C2×M16C2×Q818yes—
267C22×C8C2414yes—
268C22×C8C2414yes—
269C4⋊Q8—216no—
270C4⋊Q8—28no—
271C4⋊Q8—28no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—48no—
277#125—216no—
278#125—28no—
279#125—28no—
280#125—28no—
281#125—28no—
282#125—24no—
283#125—24no—
284#125—24no—
285#125—24no—
286#125C2×D418yes—
287#125C2×D418yes—
288C22⋊Q8—232no—
289C22⋊Q8—232no—
290C22⋊Q8—232no—
291C22⋊Q8—232no—
292C22⋊Q8—232no—
293C22⋊Q8—232no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296C4×Q8—28no—
297C4×Q8—28no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×D4—216no—
305C4×D4—216no—
306C4×D4—28no—
307C4×D4—28no—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—28no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4C2×D418yes—
315C4×D4C2×D418yes—
316#117—24no—
317#117—24no—
318#117—24no—
319#117C2×Q818yes—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—232no—
323C2×C4⋊C4—232no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2414yes—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D414yes—
388C2×C42C2×D414yes—
389C2×C42C2412yes—
390C2×C42C2412yes—
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#107—28no—
394#107—28no—
395#107—28no—
396#106—28no—
397#106—28no—
398#106—28no—
399C4⋊C8—24no—
400C4⋊C8—24no—
401#102—28no—
402#102—28no—
403#102C2×Q8132yescomplemented
404#97—24no—
405C4×C8—24no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412C24—24no—
413C24—28no—
414C24—22no—
415C24—44no—
416C24—44no—
417C24C22×D411yescharacteristic
418C2×Q8—216no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—416no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—82no—
430C2×Q8—88no—
431C2×Q8—816no—
432C2×D4—24no—
433C2×D4—28no—
434C2×D4—28no—
435C2×D4—216no—
436C2×D4—22no—
437C2×D4—24no—
438C2×D4C4⋊Q8116yescomplemented
439C2×D4C22×D414yes—
440C2×D4C22×Q818yes—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—216no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—216no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—432no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—88no—
473C22×C4—816no—
474C22×C4—816no—
475C22×C4#12014yes—
476C22×C4#12014yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×Q814yes—
489C22×C42−1+414yes—
490C22×C4C2511yescharacteristic
491M16—216no—
492M16—48no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—28no—
497C2×C8C22×Q818yes—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—48no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C22×D414yes—
531C4⋊C4C22×D414yes—
532C4⋊C4C22×D414yes—
533C4⋊C4C22×Q818yes—
534C22⋊C4—216no—
535C22⋊C4—216no—
536C22⋊C4—28no—
537C22⋊C4—28no—
538C22⋊C4—28no—
539C22⋊C4—44no—
540C42—24no—
541C42—24no—
542C42—28no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—44no—
548C42—44no—
549C42—82no—
550C42#12014yes—
551C42#12014yes—
552C42C22×D412yes—
553C42C22×D412yes—
554C42C22×D412yes—
555C42C22×D412yes—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—216no—
560C23—28no—
561C23—48no—
562C23—48no—
563C23—432no—
564C23—44no—
565C23—44no—
566C23C2×#12012yes—
567C23D4×Q814yes—
568C23C23×D411yescharacteristic
569Q8—48no—
570Q8—48no—
571Q8—84no—
572Q8—84no—
573Q8—816no—
574D4—24no—
575D4—216no—
576D4—28no—
577D4C2×C4⋊Q818yes—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—432no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—88no—
615C2×C4—88no—
616C2×C4—816no—
617C2×C4C2×#12012yes—
618C2×C4C2×#12012yes—
619C2×C4C2×#12012yes—
620C2×C4C2×#12012yes—
621C2×C4C2×#12012yes—
622C2×C4C2×C4⋊Q814yes—
623C2×C4#53614yes—
624C2×C4D4×Q814yes—
625C2×C4D4×Q818yes—
626C2×C4#57514yes—
627C2×C4C23×D411yescharacteristic
628C2×C4C23×D411yescharacteristic
629C2×C4C23×Q812yes—
630C2×C4C2×2−1+412yes—
631C8—28no—
632C8—44no—
633C22—28no—
634C22—22no—
635C22—22no—
636C22—28no—
637C22—432no—
638C22—416no—
639C22—44no—
640C22—44no—
641C22#243414yes—
642C22C22×#12011yescharacteristic
643C22C2×D4×Q812yes—
644C22C2×#57512yes—
645C22#358512yes—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—48no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—44no—
658C4—82no—
659C4—82no—
660C4#243414yes—
661C4C22×C4⋊Q811yescharacteristic
662C4C2×#53611yescharacteristic
663C4C2×D4×Q812yes—
664C2—24no—
665C2—48no—
666C2C2×#243411yescharacteristic
667C2#6035911yescharacteristic
668C2#6036711yescharacteristic
669C2#6136714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3166C2semidirect
4. #34953C2semidirect
5. #25851C2semidirect
6. #25850C2semidirect
7. #25843C2semidirect
8. #23553C2semidirect
9. #23552C2semidirect
10. #21314C2semidirect
11. #21303C2semidirect
34. C2×#501C22semidirect
35. C2×#501C22semidirect
36. C2×#499C22semidirect
48. #2584C22semidirect
51. #2567C22semidirect
52. #2546C22semidirect
60. #2059C22semidirect
61. #2059C22semidirect
62. #2058C22semidirect
63. #2057C22semidirect
64. #1989C22semidirect
65. #1989C22semidirect
66. #1985C22semidirect
67. #1984C22semidirect
187. #418Q8semidirect
403. #102C2×Q8semidirect
438. C2×D4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#31662
C2×#30772
#349532
#258512
#258502
#258432
#235532
#235522
#213142
#213032
#208732
C4×#4182