Finite Group Databasebeta

← #6477077

Subgroups — #6477077

Order 512.

2096 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#33147C212yesmaximal, complemented
4#32954C212yesmaximal, complemented
5#30168C214yesmaximal, complemented
6#29297C212yesmaximal, complemented
7#29105C212yesmaximal, complemented
8#27647C212yesmaximal, complemented
9#27601C212yesmaximal
10#24703C214yesmaximal, complemented
11#24301C212yesmaximal, complemented
12#24003C212yesmaximal
13#22078C214yesmaximal, complemented
14#20714C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#483C2214yes—
21C2×#475C2214yescomplemented
22C2×#431C2212yes—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857—21no—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2857C2212yes—
31C4×C22⋊Q8—21no—
32C4×C22⋊Q8—21no—
33C4×C22⋊Q8—21no—
34C4×C22⋊Q8—21no—
35C4×C22⋊Q8C2212yes—
36C4×C22⋊Q8C2212yes—
37C4×C22⋊Q8C2212yes—
38C4×C22⋊Q8C2212yes—
39C2×#391—21no—
40C2×#391C2211yescharacteristic
41C2×#389—21no—
42C2×#389C2212yes—
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#385—21no—
46C2×#384C2211yescharacteristic
47C2×#380—21no—
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53#2084C2214yescomplemented
54#2077C2218yes—
55#2072C2214yescomplemented
56#2058C2214yescomplemented
57#2035C2218yes—
58#2024C2214yescomplemented
59#1988C2214yescomplemented
60#1948C2214yescomplemented
61#1941C2218yescomplemented
62#1933C2218yescomplemented
63#1919C2218yescomplemented
64#1872C2214yes—
65#1865C2218yescomplemented
66#1853C2214yes—
67#1853C2214yes—
68#1853C2214yes—
69#1824C2214yes—
70#1823C2214yes—
71#1817C2212yes—
72#1803C2214yescomplemented
73#1797C2212yes—
74#1797C2212yes—
75#1787C2214yes—
76#1779C2212yes—
77C4×C22⋊C8C2212yes—
78C4×C22⋊C8C2212yes—
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—44no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—48no—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—44no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—22no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—44no—
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#483—216no—
135#475—48no—
136#431—44no—
137C2×#106C2318yes—
138C2×#106C2318yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2318yes—
141C2×#103C2318yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C4×C8C2314yes—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—21no—
152#391C2312yes—
153#391C2312yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389C2312yes—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#384—21no—
175#384—21no—
176#384C2312yes—
177#384C2312yes—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380—22no—
183#380—22no—
184#380—21no—
185#380—21no—
186#380C2312yes—
187#380C2314yes—
188#380C2312yes—
189#380C2314yes—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—22no—
208C2×#95—22no—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215C25C2×D411yescharacteristic
216C22×Q8—24no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—42no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4SD1618yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C22×C8C2418yes—
260C4⋊Q8—28no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—48no—
270C22⋊Q8—48no—
271C22⋊Q8—48no—
272C22⋊Q8—48no—
273C22⋊Q8#4118yes—
274C4×Q8—28no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—44no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—48no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—44no—
340C2×C42—44no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#106—216no—
358#106—48no—
359C4⋊C8—28no—
360#103—216no—
361#103—48no—
362C22⋊C8—28no—
363C22⋊C8—28no—
364C22⋊C8—44no—
365C4×C8—44no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—41no—
385C24—41no—
386C24C2×SD1614yes—
387C24C22×D411yescharacteristic
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—416no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—816no—
404C2×Q8C2×#4114yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—416no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4C2×SD1614yes—
478C22×C4C2×SD1614yes—
479C22×C4C2×SD1614yes—
480C22×C4C2×SD1614yes—
481C22×C4C2×SD1614yes—
482C22×C4C2×SD1614yes—
483C22×C4#13714yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4114yes—
496C22×C42−1+414yes—
497C22×C4C2511yescharacteristic
498C2×C8—28no—
499C2×C8—28no—
500C2×C8—28no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—22no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4C2×#4114yes—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—44no—
548C42—44no—
549C42—44no—
550C42—44no—
551C42H(5,𝔽2)14yes—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—28no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—28no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23#52512yes—
574C23#53612yes—
575C23C22×SD1612yes—
576C23C22×SD1612yes—
577C23C22×SD1612yes—
578C23C2×#13712yes—
579C23#57512yes—
580C23C23×D411yescharacteristic
581Q8—28no—
582Q8—48no—
583Q8—44no—
584Q8—88no—
585Q8—88no—
586C2×C4—28no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4—88no—
650C2×C4#52512yes—
651C2×C4#53412yes—
652C2×C4#53412yes—
653C2×C4#53612yes—
654C2×C4C22×SD1612yes—
655C2×C4C22×SD1612yes—
656C2×C4C22×SD1612yes—
657C2×C4C22×SD1612yes—
658C2×C4C2×#13712yes—
659C2×C4C2×#13712yes—
660C2×C4#57512yes—
661C2×C4C22×#4112yes—
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4C2×2−1+412yes—
664C8—48no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—416no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22#310314yes—
686C22#322914yes—
687C22C2×#52511yescharacteristic
688C22C2×#53411yescharacteristic
689C22C2×#53611yescharacteristic
690C22C23×SD1611yescharacteristic
691C22C22×#13711yescharacteristic
692C22C2×#57511yescharacteristic
693C22C2×#57511yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—41no—
703C4—44no—
704C4—41no—
705C4—44no—
706C4—84no—
707C4—82no—
708C4—84no—
709C4—82no—
710C4#318714yes—
711C4#352014yes—
712C2—24no—
713C2—24no—
714C2—42no—
715C2#3508411yescharacteristic
716C2C2×#310311yescharacteristic
717C2#6056211yescharacteristic
718C2C2×#318711yescharacteristic
719C2#6075111yescharacteristic
720C2C2×#322911yescharacteristic
721C2#6086611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33147C2semidirect
4. #32954C2semidirect
5. #30168C2semidirect
6. #29297C2semidirect
7. #29105C2semidirect
8. #27647C2semidirect
10. #24703C2semidirect
11. #24301C2semidirect
13. #22078C2semidirect
21. C2×#475C22semidirect
53. #2084C22semidirect
55. #2072C22semidirect
56. #2058C22semidirect
58. #2024C22semidirect
59. #1988C22semidirect
60. #1948C22semidirect
61. #1941C22semidirect
62. #1933C22semidirect
63. #1919C22semidirect
65. #1865C22semidirect
72. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×C4×C22⋊Q82
#331472
#329542
#301682
#292972
#291052
#276472
#276012
#247032
#243012
#240032
#220782
#207142