Finite Group Databasebeta

← #626841

Subgroups — #626841

Order 512.

3834 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
1C22×#544C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3C2×#2659C211yesmaximal, characteristic
4C2×#2627C211yesmaximal, characteristic
5#15850C212yesmaximal, complemented
6#15849C212yesmaximal, complemented
7#15848C214yesmaximal, complemented
8#15783C212yesmaximal, complemented
9#15762C212yesmaximal, complemented
10#15318C214yesmaximal, complemented
11#14758C214yesmaximal
12#14702C211yesmaximal, characteristic
13#14281C211yesmaximal, characteristic
14#13878C214yesmaximal
15C2×#1436C211yesmaximal, characteristic
16C2×#544—28no—
17C2×#544—28no—
18C2×#544—28no—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125C2212yes—
23C22×#125C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×#117C2212yes—
29C23×C4⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic
31#2857—28no—
32#2659—28no—
33#2627—28no—
34C2×#400C2212yes—
35C2×#398C2212yes—
36C2×#398C2212yes—
37C2×#398C2214yes—
38C2×#395C2212yes—
39C2×#395C2212yes—
40C2×#395C2212yes—
41C2×#395C2214yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2212yes—
45C2×#389C2214yes—
46C2×#385C2212yes—
47C2×#384C2211yescharacteristic
48C2×#381C2212yes—
49C2×#380C2214yes—
50C2×#380C2214yes—
51C2×#380C2211yescharacteristic
52C22×#95C2211yescharacteristic
53C22×#95C2212yes—
54C22×#95C2212yes—
55#1446C2212yes—
56#1446C2214yes—
57#1445C2212yes—
58#1444C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2212yes—
74#1436C2212yes—
75#1436C2212yes—
76#1436C2212yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90#544—2128no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—44no—
99C2×#125—28no—
100C2×#125—28no—
101C2×#125—28no—
102C2×#125—28no—
103C2×#125—28no—
104C2×#125—28no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—28no—
115C2×C22⋊Q8D418yes—
116C2×C22⋊Q8D418yes—
117C2×#117—28no—
118C2×#117—28no—
119C2×#117—28no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4D418yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2314yes—
139C22×C4⋊C4C2314yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2314yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2312yes—
146C22×C42—24no—
147C22×C42—44no—
148C22×C42C2311yescharacteristic
149C22×C42C2312yes—
150C22×C42C2312yes—
151C22×C42C2312yes—
152C22×C42C2311yescharacteristic
153#400—28no—
154#398—216no—
155#398—28no—
156#398—28no—
157#395—28no—
158#395—28no—
159#395—28no—
160#395—48no—
161#391—24no—
162#391—24no—
163#391—44no—
164#389—216no—
165#385—28no—
166#384—24no—
167#381—28no—
168#380—24no—
169#380—216no—
170#380—216no—
171C2×#95—24no—
172C2×#95—44no—
173C2×#95Q8116yescomplemented
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C25C2411yescharacteristic
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8—44no—
214C22×Q8—44no—
215C22×Q8C2×D414yes—
216C22×Q8C2×D414yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×Q818yes—
244C23×C4C2×Q818yes—
245C23×C4C2×Q818yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2414yes—
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2412yes—
259C23×C4C2412yes—
260C4⋊Q8—264no—
261C4⋊Q8—264no—
262C4⋊Q8—416no—
263#125—264no—
264#125—264no—
265C22⋊Q8—264no—
266C22⋊Q8—232no—
267C22⋊Q8—232no—
268C22⋊Q8—416no—
269#117—264no—
270C2×C4⋊C4—232no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D418yes—
323C2×C4⋊C4C2×D418yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2×D418yes—
327C2×C4⋊C4C2×D418yes—
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—28no—
337C2×C22⋊C4C2×D418yes—
338C2×C22⋊C4C2×D418yes—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—24no—
350C2×C42—416no—
351C2×C42—42no—
352C2×C42—42no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—24no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95—44no—
373#95C2×Q8116yes—
374#95C2×Q8116yes—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C22×Q814yes—
383C242−1+414yes—
384C24C2511yescharacteristic
385C2×Q8—216no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—48no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—82no—
395C2×Q8—82no—
396C2×Q8—84no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×Q8—88no—
404C2×Q8—88no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—216no—
421C22×C4—22no—
422C22×C4—24no—
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—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—216no—
455C22×C4—216no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4#120116yes—
493C22×C4C4⋊Q8116yescomplemented
494C22×C4C22×D414yes—
495C22×C4C22×D414yes—
496C22×C4C22×D414yes—
497C22×C4C22×D414yes—
498C22×C4C22×D414yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C22×Q814yes—
507C22×C4C22×Q818yes—
508C22×C4C22×Q818yes—
509C22×C4C22×Q814yes—
510C22×C4H(5,𝔽2)14yes—
511C22×C42−1+414yes—
512C4⋊C4—216no—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—216no—
518C4⋊C4—216no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—416no—
524C4⋊C4—416no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—84no—
528C4⋊C4—88no—
529C4⋊C4—88no—
530C4⋊C4—88no—
531C22⋊C4—216no—
532C22⋊C4—216no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C42—28no—
536C42—216no—
537C42—216no—
538C42—48no—
539C42—44no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23C2×#12014yes—
557C23C2×C4⋊Q814yes—
558C23#53614yes—
559C23D4×Q818yes—
560C23D4×Q818yes—
561C23C23×D411yescharacteristic
562C23C23×D411yescharacteristic
563C23C23×D411yescharacteristic
564C23C23×Q812yes—
565C23C2×H(5,𝔽2)12yes—
566C23C2×2−1+412yes—
567C23C2×2−1+412yes—
568C23#58512yes—
569C23#58512yes—
570Q8—88no—
571Q8—88no—
572Q8—88no—
573Q8—816no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—232no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—416no—
612C2×C4—432no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—88no—
652C2×C4—816no—
653C2×C4C2×#12014yes—
654C2×C4C2×#12014yes—
655C2×C4C2×C4⋊Q818yes—
656C2×C4#54418yescomplemented
657C22—24no—
658C22—28no—
659C22—28no—
660C22—24no—
661C22—28no—
662C22—28no—
663C22—28no—
664C22—28no—
665C22#243418yes—
666C22C22×#12011yescharacteristic
667C22C22×C4⋊Q811yescharacteristic
668C22C2×#53611yescharacteristic
669C22C2×#54412yes—
670C22#345812yes—
671C22C2×D4×Q812yes—
672C22C2×D4×Q812yes—
673C22#348411yescharacteristic
674C22#348511yescharacteristic
675C22#348912yes—
676C22#349612yes—
677C22#349612yes—
678C22#349712yes—
679C22#349911yescharacteristic
680C22#350011yescharacteristic
681C22#350112yes—
682C22#350212yes—
683C22#353512yes—
684C22#353812yes—
685C22#356312yes—
686C22#356412yes—
687C4—24no—
688C4—24no—
689C4—28no—
690C4—28no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—48no—
698C4—82no—
699C4—82no—
700C4—84no—
701C2—28no—
702C2C2×#243411yescharacteristic
703C2#3516712yes—
704C2#3534711yescharacteristic
705C2#3566011yescharacteristic
706C2#3674212yes—
707C2#3768512yes—
708C2#3768912yes—
709C2#3936412yes—
710C2#3936812yes—
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
5. #15850C2semidirect
6. #15849C2semidirect
7. #15848C2semidirect
8. #15783C2semidirect
9. #15762C2semidirect
10. #15318C2semidirect
173. C2×#95Q8semidirect
493. C22×C4C4⋊Q8semidirect
656. C2×C4#544semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#28572
C2×#26592
C2×#26272
#158502
#158492
#158482
#157832
#157622
#153182
#147582
#147022
#142812
#138782
C2×#14362