Finite Group Databasebeta

← #622147

Subgroups — #622147

Order 512.

3212 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic, complemented
2C2×#2854C211yesmaximal, characteristic
3C2×#2625C211yesmaximal, characteristic, complemented
4C2×#2620C211yesmaximal, characteristic
5C2×#2624C211yesmaximal, characteristic
6C2×#2296C211yesmaximal, characteristic, complemented
7#15808C214yesmaximal, complemented
8#15765C214yesmaximal, complemented
9#15549C214yesmaximal, complemented
10#15384C214yesmaximal, complemented
11#15193C212yesmaximal, complemented
12#15148C212yesmaximal, complemented
13#14153C212yesmaximal, complemented
14#13910C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2855—28no—
24#2854—28no—
25#2625—28no—
26#2624—28no—
27#2620—28no—
28C2×#400C2212yes—
29C2×#399C2212yes—
30C2×#399C2214yes—
31C2×#398C2212yes—
32C2×#397C2214yes—
33C2×#397C2212yes—
34C2×#396C2214yes—
35C2×#394C2214yes—
36C2×#394C2214yes—
37#2296—28no—
38C2×#391C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2212yes—
41C2×#390C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2212yes—
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#389C2212yes—
48C2×#389C2212yes—
49C2×#389C2212yes—
50C2×#388C2214yes—
51C2×#387C2214yes—
52C2×#386C2212yes—
53C2×#385C2212yes—
54C2×#385C2212yes—
55C2×#385C2212yes—
56C2×#384C2211yescharacteristic
57C2×#384C2214yes—
58C2×#381C2212yes—
59C2×#380C2211yescharacteristic
60C2×#380C2211yescharacteristic
61C2×C4×C4⋊C4C2212yes—
62C2×C4×C4⋊C4C2212yes—
63C2×C4×C4⋊C4C2212yes—
64#1445C2214yes—
65#1445C2214yes—
66#1445C2214yes—
67#1444C2214yes—
68#1444C2214yes—
69#1444C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#124—24no—
88C2×#124—28no—
89C2×#123—28no—
90C2×#123—48no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—28no—
101C2×#117D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C42—24no—
132C22×C42—24no—
133C22×C42—24no—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2314yes—
139C22×C42C2311yescharacteristic
140#400—28no—
141#399—28no—
142#399—48no—
143#398—28no—
144#397—216no—
145#397—28no—
146#396—216no—
147#394—216no—
148#394—216no—
149#391—28no—
150#391—28no—
151#391—24no—
152#390—28no—
153#390—28no—
154#390—24no—
155#389—28no—
156#389—28no—
157#389—28no—
158#389—28no—
159#389—28no—
160#389—28no—
161#388—216no—
162#387—216no—
163#386—28no—
164#385—28no—
165#385—28no—
166#385—28no—
167#384—24no—
168#384—48no—
169#381—28no—
170#380D418yes—
171#380D418yes—
172C4×C4⋊C4—28no—
173C4×C4⋊C4—28no—
174C4×C4⋊C4—28no—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C25C2412yes—
194C25C2411yescharacteristic
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—44no—
209C22×D4—44no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238#124—416no—
239#123—432no—
240C22⋊Q8—416no—
241#121—416no—
242#117—232no—
243#117—416no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C2×D418yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D418yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D418yes—
327C2×C22⋊C4C2×D418yes—
328C2×C22⋊C4C2×D418yes—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—28no—
340C2×C42—44no—
341C2×C42—44no—
342C2×C42—416no—
343C2×C42C2×D414yes—
344C2×C42C2×D414yes—
345C2×C42#4118yes—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363#95C2×D418yes—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C2511yescharacteristic
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—88no—
397C2×D4—88no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4#127116yes—
478C22×C4C22×D418yes—
479C22×C4C22×D418yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C42−1+414yes—
493C22×C42−1+414yes—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C4⋊C4—88no—
511C22⋊C4—216no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C42—28no—
524C42—416no—
525C42—48no—
526C42—48no—
527C42—48no—
528C42#123116yescomplemented
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23C2×#12714yes—
571C23C23×D411yescharacteristic
572C23C23×D411yescharacteristic
573C23C23×D411yescharacteristic
574C23C22×#4112yes—
575C23C2×H(5,𝔽2)12yes—
576C23C2×2−1+412yes—
577C23C2×2−1+412yes—
578C23#58512yes—
579C23#58512yes—
580Q8—88no—
581D4—816no—
582D4—88no—
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—232no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4C2×#12318yes—
661C2×C4C2×#12314yes—
662C2×C4C2×#12714yes—
663C2×C4C2×#12714yes—
664C2×C4#53614yes—
665C2×C4#54718yes—
666C2×C4#54718yes—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—82no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—82no—
684C22—82no—
685C22C22×#12311yescharacteristic
686C22C22×#12711yescharacteristic
687C22C2×#53611yescharacteristic
688C22#346012yes—
689C22#346112yes—
690C22C2×#54712yes—
691C22C2×#54712yes—
692C22#348411yescharacteristic
693C22#348511yescharacteristic
694C22#348611yescharacteristic
695C22#348711yescharacteristic
696C22#348812yes—
697C22#348812yes—
698C22#348912yes—
699C22#348912yes—
700C22#351112yes—
701C22#351212yes—
702C22#355012yes—
703C22#355412yes—
704C22#355512yes—
705C22#355712yes—
706C4—28no—
707C4—24no—
708C4—44no—
709C4—48no—
710C4—48no—
711C4—44no—
712C4—44no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4#250118yes—
719C2—44no—
720C2—162no—
721C2C2×#250111yescharacteristic
722C2#3525512yes—
723C2#3556511yescharacteristic
724C2#3563411yescharacteristic
725C2#3727512yes—
726C2#3779412yes—
727C2#3779512yes—
728C2#3779812yes—
729C2#3779912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2855C2semidirect
3. C2×#2625C2semidirect
6. C2×#2296C2semidirect
7. #15808C2semidirect
8. #15765C2semidirect
9. #15549C2semidirect
10. #15384C2semidirect
11. #15193C2semidirect
12. #15148C2semidirect
13. #14153C2semidirect
14. #13910C2semidirect
15. #13777C2semidirect
528. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#28542
C2×#26252
C2×#26202
C2×#26242
C2×#22962
#158082
#157652
#155492
#153842
#151932
#151482
#141532
#139102
#137772