Finite Group Databasebeta

← #6469383

Subgroups — #6469383

Order 512.

2118 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#532C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#29392C214yesmaximal, complemented
4#29273C212yesmaximal, complemented
5#28202C212yesmaximal, complemented
6#27645C212yesmaximal, complemented
7#27611C212yesmaximal, complemented
8#24299C212yesmaximal, complemented
9#24061C212yesmaximal
10#23837C212yesmaximal, complemented
11#23815C212yesmaximal, complemented
12C4×#476C214yesmaximal, complemented
13C4×#481C214yesmaximal, complemented
14#20777C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×C4×#41—21no—
18C22×C4×Q8—21no—
19C2×#481C2214yes—
20C2×#476C2214yes—
21C4×C4⋊Q8—21no—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8C2212yes—
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2212yes—
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2212yes—
31C4×#127C2211yescharacteristic
32C4×#127C2211yescharacteristic
33C4×#124—21no—
34C4×#124—21no—
35C4×#121—21no—
36C4×#121—21no—
37C4×#121C2212yes—
38C4×#121C2212yes—
39#2297—21no—
40C2×#391—21no—
41#2293—21no—
42#2293C2212yes—
43C2×#389—21no—
44C2×#389C2212yes—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C42×D4—21no—
48C42×D4C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C43C2211yescharacteristic
52#2099C2214yes—
53#2084C2214yes—
54#2084C2214yes—
55#2071C2214yes—
56#2058C2214yes—
57#2058C2214yes—
58#2041C2214yes—
59#2023C2214yes—
60#1987C2214yes—
61#1948C2214yes—
62#1926C2218yes—
63#1920C2218yes—
64#1919C2218yes—
65#1874C2214yes—
66#1864C2218yes—
67#1853C2214yes—
68#1839C2212yes—
69#1831C2212yes—
70#1824C2214yes—
71#1817C2212yes—
72#1802C2214yes—
73#1797C2212yes—
74C4×#106C2218yes—
75C4×C4⋊C8C2212yes—
76C4×#102C2218yes—
77C4×C22⋊C8C2212yes—
78#1744C2212yes—
79C22×#41—21no—
80C23×Q8—21no—
81#532—28no—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#124—21no—
90C2×#121—21no—
91C2×#121C2312yes—
92C4×#41—48no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8—48no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117—21no—
111C2×#117—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C42—22no—
119C22×C42—22no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#481—216no—
128#476—216no—
129C2×#106C2318yes—
130C2×#106C2318yes—
131C2×C4⋊C8C2314yes—
132C2×#102C2318yes—
133C2×#102C2318yes—
134C2×C22⋊C8C2314yes—
135C2×C4×C8C2314yes—
136#391—22no—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—21no—
141#391—21no—
142#390—21no—
143#390—21no—
144#390—21no—
145#390—21no—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389C2314yes—
157#389C2314yes—
158#389C2312yes—
159#389C2312yes—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#386—22no—
165#386—22no—
166#384—21no—
167#384—21no—
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#384C2312yes—
171#384C2312yes—
172#382—21no—
173#380—21no—
174#380C2312yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C2×#95—22no—
196C2×#95—22no—
197C43D411yescharacteristic
198C43D411yescharacteristic
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2318yes—
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C2×#41—28no—
214C2×#41—44no—
215C2×#41—44no—
216C22×Q8—28no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—44no—
221C22×Q8—44no—
222C22×D4—22no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C22×C8C2418yes—
239C4⋊Q8—28no—
240C4⋊Q8—28no—
241C4⋊Q8—28no—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8#4118yescomplemented
245#127—24no—
246#124—28no—
247#121—28no—
248#121—24no—
249#121#4118yes—
250C4×Q8—48no—
251C4×Q8—48no—
252C4×Q8—48no—
253C4×Q8—44no—
254C4×Q8—44no—
255C4×Q8—48no—
256C4×D4—28no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—28no—
260C4×D4—28no—
261C4×D4—44no—
262C4×D4—48no—
263C4×D4—48no—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—44no—
267#117—44no—
268#117—48no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—48no—
318C2×C42—48no—
319C2×C42SD1618yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#106—216no—
349#106—216no—
350C4⋊C8—28no—
351#102—216no—
352#102—216no—
353C22⋊C8—28no—
354C4×C8—28no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365C24—22no—
366C24C22×D411yescharacteristic
367#41—432no—
368C2×Q8—24no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—44no—
372C2×Q8—432no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—44no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—28no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4C2×#4114yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4C2×SD1614yes—
429C22×C4C2×SD1614yes—
430C22×C4C2×SD1614yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4114yes—
445C22×C4C2511yescharacteristic
446C2×C8—28no—
447C2×C8—28no—
448C2×C8—28no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4C2×#4118yes—
472C4⋊C4C2×#4114yes—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—48no—
504C42—44no—
505C42—44no—
506C42—44no—
507C42—48no—
508C42C2×SD1614yes—
509C42C2×SD1614yes—
510C42C2×SD1614yes—
511C42C2×SD1614yes—
512C42AGL(1,ℤ/8)14yes—
513C42C2×#4114yes—
514C422−1+414yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23#52512yes—
527C23C22×SD1612yes—
528C23#57512yes—
529C23C23×D411yescharacteristic
530Q8—416no—
531Q8—416no—
532Q8—48no—
533D4—28no—
534D4—48no—
535D4—48no—
536D4—416no—
537D4—44no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—416no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—88no—
587C2×C4—88no—
588C2×C4—88no—
589C2×C4#52512yes—
590C2×C4#52512yes—
591C2×C4#52512yes—
592C2×C4#53612yes—
593C2×C4#53612yes—
594C2×C4C22×SD1612yes—
595C2×C4C22×SD1612yes—
596C2×C4C22×SD1612yes—
597C2×C4C22×SD1612yes—
598C2×C4C22×SD1612yes—
599C2×C4C22×SD1612yes—
600C2×C4C2×AGL(1,ℤ/8)12yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4#57512yes—
604C2×C4C22×#4112yes—
605C2×C4C22×#4112yes—
606C2×C4C2×2−1+412yes—
607C8—48no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22#310114yes—
622C22C2×#52511yescharacteristic
623C22C2×#52511yescharacteristic
624C22C2×#53611yescharacteristic
625C22C23×SD1611yescharacteristic
626C22C22×AGL(1,ℤ/8)11yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57511yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—44no—
639C4—48no—
640C4—41no—
641C4—41no—
642C4—82no—
643C4—84no—
644C4—82no—
645C4#310314yes—
646C4#322714yes—
647C4#348214yes—
648C2—24no—
649C2—84no—
650C2#3497111yescharacteristic
651C2C2×#310111yescharacteristic
652C2C2×#310311yescharacteristic
653C2#6055811yescharacteristic
654C2#6056111yescharacteristic
655C2C2×#322711yescharacteristic
656C2#6085411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. #29392C2semidirect
4. #29273C2semidirect
5. #28202C2semidirect
6. #27645C2semidirect
7. #27611C2semidirect
8. #24299C2semidirect
10. #23837C2semidirect
11. #23815C2semidirect
12. C4×#476C2semidirect
13. C4×#481C2semidirect
14. #20777C2semidirect
244. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
C4×#5322
C2×C4×C4⋊Q82
#293922
#292732
#282022
#276452
#276112
#242992
#240612
#238372
#238152
C4×#4762
C4×#4812
#207772