Finite Group Databasebeta

← #705310

Subgroups — #705310

Order 512.

2662 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2495C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3#16389C214yesmaximal, complemented
4#16063C214yesmaximal, complemented
5#15992C214yesmaximal
6#15241C211yesmaximal, characteristic, complemented
7#15237C211yesmaximal, characteristic, complemented
8#15233C211yesmaximal, characteristic
9#15195C211yesmaximal, characteristic
10#14829C214yesmaximal
11#14261C211yesmaximal, characteristic, complemented
12#14000C211yesmaximal, characteristic, complemented
13#13978C211yesmaximal, characteristic, complemented
14#13976C211yesmaximal, characteristic, complemented
15#13942C211yesmaximal, characteristic, complemented
16#13922C211yesmaximal, characteristic
17#13922C211yesmaximal, characteristic
18#13842C211yesmaximal, characteristic
19#13786C211yesmaximal, characteristic, complemented
20C24×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#2495—28no—
24C2×#400C2212yes—
25C2×#397C2212yes—
26Q8×C22⋊C4—28no—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#388C2212yes—
42C2×#384C2211yescharacteristic
43C2×#383C2212yes—
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×#376C2211yescharacteristic
57#1446C2214yes—
58#1446C2214yes—
59#1444C2214yes—
60#1443C2214yes—
61#1443C2214yes—
62#1443C2212yes—
63#1443C2214yes—
64#1443C2214yes—
65#1443C2214yes—
66#1442C2212yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1440C2212yes—
74#1440C2214yes—
75#1439C2212yes—
76#1439C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87#1434C2214yes—
88#1434C2214yes—
89C22⋊#95C2214yes—
90C22⋊#95C2214yes—
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8—24no—
94C23×Q8—24no—
95C23×Q8—24no—
96C23×Q8—24no—
97C23×Q8—24no—
98C23×Q8—42no—
99C23×Q8—42no—
100C23×Q8C2311yescharacteristic
101C23×Q8C2312yes—
102C23×Q8C2311yescharacteristic
103C24×C4C2311yescharacteristic
104C24×C4C2312yes—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—42no—
110C2×C4×Q8—42no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#400—28no—
140#397—28no—
141#391—24no—
142#391—24no—
143#391—24no—
144#391—24no—
145#391—24no—
146#391—24no—
147#391—24no—
148#391—24no—
149#391—28no—
150#391—28no—
151#391—42no—
152#391—42no—
153#389—24no—
154#389—24no—
155#388—28no—
156#384—42no—
157#383—28no—
158#382—24no—
159#382—42no—
160#380—24no—
161#380—24no—
162#380—24no—
163#380—28no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—42no—
166C4×C22⋊C4—28no—
167C4×C22⋊C4—42no—
168C4×C22⋊C4—42no—
169C4×C22⋊C4D418yes—
170#376—42no—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2312yes—
204C2×#95C2314yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C2×#95C2311yescharacteristic
208C2×#95C2311yescharacteristic
209C25C2411yescharacteristic
210C22×Q8—232no—
211C22×Q8—22no—
212C22×Q8—24no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—416no—
229C22×Q8—416no—
230C22×Q8—416no—
231C22×Q8—416no—
232C22×Q8—416no—
233C22×Q8—416no—
234C22×Q8#4114yes—
235C22×Q8#4114yes—
236C22×Q8#4114yes—
237C22×Q8#4114yes—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4C2×D414yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C23×C4C2412yes—
267C23×C4C2412yes—
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270C4×Q8—416no—
271C4×Q8—416no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4C2×D414yes—
321C2×C22⋊C4C2×D414yes—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42—44no—
346C2×C42C2×D414yes—
347C2×C42C2×D414yes—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—24no—
367#95—24no—
368#95—28no—
369#95—42no—
370#95—42no—
371#95—44no—
372#95—44no—
373#95—44no—
374#95—44no—
375#95—44no—
376#95—42no—
377#95—42no—
378#95—42no—
379#95—42no—
380#95—42no—
381#95—44no—
382#95—44no—
383#95—44no—
384#95C2×D414yes—
385#95C2×D414yes—
386C24—21no—
387C24—21no—
388C24—21no—
389C24—21no—
390C24—21no—
391C24—21no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24C22×D412yes—
396C242−1+414yes—
397C24C2511yescharacteristic
398C2×Q8—216no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—48no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—4128no—
414C2×Q8—48no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8#126116yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—216no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—24no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—24no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—22no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4—48no—
518C22×C4—48no—
519C22×C4—44no—
520C22×C4—44no—
521C22×C4—44no—
522C22×C4—44no—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C2×#4114yes—
530C22×C4C2×#4114yes—
531C22×C4C2×#4114yes—
532C22×C4C2×#4114yes—
533C22×C4C2×#4112yes—
534C22×C4C2×#4112yes—
535C22×C4C2×#4112yes—
536C22×C4C2×#4112yes—
537C22×C4H(5,𝔽2)12yes—
538C22×C4H(5,𝔽2)12yes—
539C22×C42−1+414yes—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—84no—
550C4⋊C4—84no—
551C4⋊C4—88no—
552C22⋊C4—28no—
553C22⋊C4—44no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—84no—
559C42—28no—
560C42—28no—
561C42—48no—
562C42—48no—
563C42—48no—
564C42—44no—
565C42—44no—
566C42—44no—
567C42—44no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23#53614yes—
591C23C23×D411yescharacteristic
592C23C22×#4111yescharacteristic
593C23C22×#4111yescharacteristic
594C23C22×#4111yescharacteristic
595C23C22×#4111yescharacteristic
596C23C2×H(5,𝔽2)11yescharacteristic
597C23C2×H(5,𝔽2)11yescharacteristic
598C23C2×2−1+412yes—
599C23C2×2−1+412yes—
600C23#58512yes—
601C23#58512yes—
602Q8—232no—
603Q8—432no—
604Q8—416no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—432no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—416no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—44no—
701C2×C4—42no—
702C2×C4—44no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4#52514yes—
706C2×C4C2×#12618yes—
707C2×C4C2×#12614yes—
708C2×C4#53814yes—
709C2×C4#54614yes—
710C2×C4#54614yes—
711C2×C4#55114yes—
712C2×C4#55114yes—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—44no—
717C22—42no—
718C22—44no—
719C22—42no—
720C22—42no—
721C22—44no—
722C22—44no—
723C22—42no—
724C22C2×#52511yescharacteristic
725C22C22×#12611yescharacteristic
726C22C2×#53611yescharacteristic
727C22#344911yescharacteristic
728C22C2×#53811yescharacteristic
729C22C2×#54611yescharacteristic
730C22C2×#54611yescharacteristic
731C22C2×#55111yescharacteristic
732C22C2×#55111yescharacteristic
733C22#349911yescharacteristic
734C22#350011yescharacteristic
735C22#351612yes—
736C22#351612yes—
737C22#351612yes—
738C22#351612yes—
739C22#351612yes—
740C22#351812yes—
741C22#352312yes—
742C22#352412yes—
743C22#353812yes—
744C22#354012yes—
745C22#354112yes—
746C22#354712yes—
747C4—28no—
748C4—42no—
749C4—42no—
750C4—44no—
751C4—44no—
752C4—42no—
753C4—42no—
754C4—42no—
755C4—82no—
756C4—82no—
757C4—82no—
758C4—82no—
759C4—82no—
760C4—82no—
761C4—82no—
762C4—82no—
763C4—82no—
764C4#248018yes—
765C2—44no—
766C2C2×#248011yescharacteristic
767C2#3524611yescharacteristic
768C2#3566911yescharacteristic
769C2#3588312yes—
770C2#3808111yescharacteristic
771C2#3808511yescharacteristic
772C2#3809511yescharacteristic
773C2#3894611yescharacteristic
774C2#4286712yes—
775C2#4532012yes—
776C2#4542012yes—
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
3. #16389C2semidirect
4. #16063C2semidirect
6. #15241C2semidirect
7. #15237C2semidirect
11. #14261C2semidirect
12. #14000C2semidirect
13. #13978C2semidirect
14. #13976C2semidirect
15. #13942C2semidirect
19. #13786C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24952
C2×Q8×C22⋊C42
#163892
#160632
#159922
#152412
#152372
#152332
#151952
#148292
#142612
#140002
#139782
#139762
#139422
#139222
#138422
#137862