Finite Group Databasebeta

← #847417

Subgroups — #847417

Order 512.

4746 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic
2C2×#2734C212yesmaximal, complemented
3C2×#2434C211yesmaximal, characteristic
4#16603C214yesmaximal, complemented
5#16093C212yesmaximal, complemented
6#15762C212yesmaximal, complemented
7#15318C214yesmaximal, complemented
8#14874C212yesmaximal
9#14728C212yesmaximal
10#14720C212yesmaximal
11#14695C212yesmaximal
12#14450C212yesmaximal, complemented
13#14043C212yesmaximal
14#14039C212yesmaximal
15#13922C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C24×Q8C2211yescharacteristic
18C22×C4⋊Q8—28no—
19C22×C4⋊Q8—216no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2212yes—
24C22×C4⋊Q8C2212yes—
25C22×C4⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2212yes—
29C23×C42C2211yescharacteristic
30#2734—216no—
31#2434—28no—
32C2×#398C2214yes—
33C2×#395C2214yes—
34C2×#395C2212yes—
35C2×#395C2212yes—
36C2×#395C2214yes—
37C2×#395C2214yes—
38C2×#391C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2212yes—
42C2×#389C2214yes—
43C2×#387C2214yes—
44C2×#387C2214yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#385C2212yes—
49C2×#385C2212yes—
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53#1446C2214yes—
54#1446C2214yes—
55#1446C2212yes—
56#1446C2212yes—
57#1444C2214yes—
58#1444C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8—28no—
80C23×Q8—28no—
81C23×Q8—24no—
82C23×Q8—42no—
83C23×Q8—42no—
84C23×Q8—42no—
85C23×Q8—42no—
86C23×Q8—44no—
87C23×Q8—44no—
88C23×Q8D418yes—
89C23×Q8C2312yes—
90C23×Q8C2311yescharacteristic
91C23×Q8C2311yescharacteristic
92C23×Q8C2311yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×Q8C2312yes—
95C23×Q8C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2312yes—
98C24×C4C2312yes—
99C24×C4C2311yescharacteristic
100C24×C4C2311yescharacteristic
101C2×C4⋊Q8—2128no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—28no—
104C2×C4⋊Q8—28no—
105C2×C4⋊Q8—28no—
106C2×C4⋊Q8—28no—
107C2×C4⋊Q8—28no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8—28no—
111C2×C4⋊Q8—216no—
112C2×C4⋊Q8—216no—
113C2×C4⋊Q8—28no—
114C2×C4⋊Q8—44no—
115C2×C4⋊Q8D4116yes—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—42no—
118C2×C22⋊Q8—42no—
119C22×C4⋊C4—28no—
120C22×C4⋊C4—28no—
121C22×C4⋊C4—216no—
122C22×C4⋊C4—216no—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C42—24no—
142C22×C42—28no—
143C22×C42C2312yes—
144C22×C42C2311yescharacteristic
145C22×C42C2312yes—
146C22×C42C2311yescharacteristic
147C22×C42C2312yes—
148#398—216no—
149#395—28no—
150#395—28no—
151#395—216no—
152#395—216no—
153#395—48no—
154#391—24no—
155#391—24no—
156#391—28no—
157#391—44no—
158#389—216no—
159#387—216no—
160#387—216no—
161#385—28no—
162#385—28no—
163#385—44no—
164#385—44no—
165#385Q8116yescomplemented
166#380—42no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4—42no—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C25C2411yescharacteristic
194C22×Q8—232no—
195C22×Q8—232no—
196C22×Q8—28no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—28no—
208C22×Q8—416no—
209C22×Q8—41no—
210C22×Q8—41no—
211C22×Q8—42no—
212C22×Q8—41no—
213C22×Q8—41no—
214C22×Q8—41no—
215C22×Q8—41no—
216C22×Q8—41no—
217C22×Q8—41no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8—41no—
226C22×Q8—41no—
227C22×Q8—42no—
228C22×Q8—416no—
229C22×Q8—432no—
230C22×Q8—432no—
231C22×Q8—432no—
232C22×Q8—432no—
233C22×Q8—416no—
234C22×Q8—416no—
235C22×Q8C2×D414yes—
236C22×Q8C2×D414yes—
237C22×Q8C2×D414yes—
238C22×Q8C2×D418yes—
239C22×Q8C2×D414yes—
240C22×Q8C2×D418yes—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—28no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—28no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—28no—
255C23×C4—24no—
256C23×C4—24no—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4—42no—
260C23×C4—42no—
261C23×C4—44no—
262C23×C4C2×D414yes—
263C23×C4C2×D414yes—
264C23×C4C2×D414yes—
265C23×C4C2×Q818yes—
266C23×C4C2412yes—
267C23×C4C2414yes—
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C23×C4C2414yes—
272C23×C4C2412yes—
273C23×C4C2414yes—
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C23×C4C2414yes—
279C23×C4C2411yescharacteristic
280C4⋊Q8—264no—
281C4⋊Q8—264no—
282C4⋊Q8—264no—
283C4⋊Q8—264no—
284C4⋊Q8—416no—
285C4⋊Q8—416no—
286C22⋊Q8—416no—
287C2×C4⋊C4—264no—
288C2×C4⋊C4—264no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—41no—
314C2×C4⋊C4—41no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2×D4116yes—
325C2×C4⋊C4C2×D4116yes—
326C2×C4⋊C4C2×Q8116yescomplemented
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4C2×Q818yes—
338C2×C22⋊C4C2×Q818yes—
339C2×C42—232no—
340C2×C42—24no—
341C2×C42—28no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42—42no—
351C2×C42C2×D418yes—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—28no—
367#95—28no—
368#95—42no—
369#95—42no—
370#95—44no—
371#95—44no—
372#95—44no—
373#95—44no—
374#95—44no—
375#95C2×Q8116yescomplemented
376C24—22no—
377C24—22no—
378C24—24no—
379C24C22×D412yes—
380C24C22×Q814yes—
381C24H(5,𝔽2)12yes—
382C24H(5,𝔽2)12yes—
383C242−1+414yes—
384C24C2511yescharacteristic
385C2×Q8—216no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—216no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×Q8—416no—
407C2×Q8—416no—
408C2×Q8—4128no—
409C2×Q8—4128no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—48no—
416C2×Q8—48no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—48no—
421C2×Q8—84no—
422C2×Q8—84no—
423C2×Q8#120116yes—
424C22×C4—216no—
425C22×C4—232no—
426C22×C4—216no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—28no—
430C22×C4—216no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—232no—
450C22×C4—216no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—28no—
482C22×C4—28no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—416no—
507C22×C4—416no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D418yes—
513C22×C4C22×D412yes—
514C22×C4C22×D414yes—
515C22×C4C22×D414yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D414yes—
519C22×C4C22×D412yes—
520C22×C4C22×D418yes—
521C22×C4C22×Q814yes—
522C22×C4C22×Q818yes—
523C22×C4C22×Q814yes—
524C22×C4C22×Q818yes—
525C22×C4H(5,𝔽2)14yes—
526C22×C42−1+418yes—
527C4⋊C4—232no—
528C4⋊C4—232no—
529C4⋊C4—232no—
530C4⋊C4—232no—
531C4⋊C4—232no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—84no—
539C4⋊C4—88no—
540C4⋊C4—88no—
541C22⋊C4—216no—
542C22⋊C4—216no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C42—216no—
546C42—216no—
547C42—44no—
548C42—44no—
549C42—84no—
550C23—22no—
551C23—22no—
552C23—28no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—28no—
557C23—28no—
558C23—24no—
559C23—24no—
560C23—28no—
561C23#53414yes—
562C23#54418yes—
563C23D4×Q818yes—
564C23Q8214yes—
565C23#56314yes—
566C23C23×D411yescharacteristic
567C23C23×D412yes—
568C23C23×Q812yes—
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×H(5,𝔽2)12yes—
572C23C2×2−1+412yes—
573C23C2×2−1+414yes—
574Q8—416no—
575Q8—416no—
576Q8—416no—
577Q8—416no—
578Q8—432no—
579Q8—432no—
580Q8—88no—
581C2×C4—264no—
582C2×C4—232no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—416no—
622C2×C4—416no—
623C2×C4—432no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4C2×#12014yes—
660C2×C4C2×#12014yes—
661C2×C4C2×#12014yes—
662C2×C4#53414yes—
663C2×C4D4×Q8116yes—
664C22—28no—
665C22—216no—
666C22—24no—
667C22—24no—
668C22—28no—
669C22—28no—
670C22—28no—
671C22#273418yescomplemented
672C22C22×#12011yescharacteristic
673C22C2×#53411yescharacteristic
674C22C2×#53411yescharacteristic
675C22C2×#54412yes—
676C22C2×D4×Q814yes—
677C22C2×D4×Q812yes—
678C22C2×Q8211yescharacteristic
679C22#348412yes—
680C22#348412yes—
681C22#348614yes—
682C22#348612yes—
683C22#350314yes—
684C22C2×#56311yescharacteristic
685C22#354412yes—
686C22#354412yes—
687C22#355214yes—
688C4—24no—
689C4—24no—
690C4—24no—
691C4—24no—
692C4—216no—
693C4—48no—
694C4—42no—
695C4—42no—
696C4—48no—
697C4—44no—
698C4—44no—
699C4—82no—
700C4—82no—
701C4—84no—
702C2—28no—
703C2#3498011yescharacteristic
704C2Q8×#12012yes—
705C2#3725811yescharacteristic
706C2#3768614yes—
707C2#3959312yes—
708C2#4632512yes—
709C2C2×#273411yescharacteristic
710C2#5915112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2734C2semidirect
4. #16603C2semidirect
5. #16093C2semidirect
6. #15762C2semidirect
7. #15318C2semidirect
12. #14450C2semidirect
165. #385Q8semidirect
326. C2×C4⋊C4C2×Q8semidirect
375. #95C2×Q8semidirect
671. C22#2734semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#27342
C2×#24342
#166032
#160932
#157622
#153182
#148742
#147282
#147202
#146952
#144502
#140432
#140392
#139222