Finite Group Databasebeta

← #1395639

Subgroups — #1395639

Order 512.

3912 conjugacy classes of subgroups.

The subgroup lattice has 781 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#2420C212yesmaximal
3C2×#2422C211yesmaximal, characteristic
4#16364C214yesmaximal, complemented
5#16263C214yesmaximal, complemented
6#16262C212yesmaximal, complemented
7#16251C212yesmaximal, complemented
8#15230C211yesmaximal, characteristic, complemented
9#15228C212yesmaximal, complemented
10#15216C212yesmaximal, complemented
11#15158C211yesmaximal, characteristic, complemented
12#15125C211yesmaximal, characteristic, complemented
13#15073C212yesmaximal, complemented
14#14256C212yesmaximal, complemented
15#13916C211yesmaximal, characteristic, complemented
16#13916C212yesmaximal, complemented
17C4×#393C211yesmaximal, characteristic, complemented
18C2×#554—216no—
19C2×#554—28no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2212yes—
22C22×#123C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2214yes—
25C22×#121C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C22×C4×D4C2212yes—
29C23×C4⋊C4C2212yes—
30#2422—28no—
31#2420—216no—
32C2×#397C2212yes—
33C2×#397C2214yes—
34C2×#395C2212yes—
35C2×#394C2212yes—
36C2×#394C2214yes—
37C2×#393C2211yescharacteristic
38C2×#393C2212yes—
39C2×#390C2211yescharacteristic
40C2×#390C2212yes—
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#386C2214yes—
44C2×#385C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382C2212yes—
47C2×#381C2214yes—
48C2×#381C2212yes—
49C2×#381C2214yes—
50C2×#380C2212yes—
51C2×#380C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54#1446C2212yes—
55#1446C2211yescharacteristic
56#1446C2214yes—
57#1444C2214yes—
58#1444C2214yes—
59#1442C2212yes—
60#1442C2211yescharacteristic
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2211yescharacteristic
76#1439C2214yes—
77#1439C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2214yes—
85C4×#95C2212yes—
86C4×#95C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C23×D4C2312yes—
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C24×C4C2314yes—
94#554—2128no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×#125—28no—
98C2×#125—28no—
99C2×#125—28no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—44no—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—216no—
106C2×C22⋊Q8—216no—
107C2×C22⋊Q8—216no—
108C2×#121—28no—
109C2×#121—24no—
110C2×C4×Q8—24no—
111C2×C4×Q8—28no—
112C2×C4×D4—28no—
113C2×C4×D4—28no—
114C2×C4×D4—28no—
115C2×C4×D4D4116yes—
116C2×C4×D4D418yes—
117C22×C4⋊C4—216no—
118C22×C4⋊C4—28no—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2311yescharacteristic
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2314yes—
145C22×C22⋊C4C2312yes—
146C22×C42C2312yes—
147C22×C42C2311yescharacteristic
148C22×C42C2312yes—
149C22×C42C2311yescharacteristic
150#397—216no—
151#397—28no—
152#395—44no—
153#394—216no—
154#394—28no—
155#393—24no—
156#393—44no—
157#390—28no—
158#390—24no—
159#389—28no—
160#389—24no—
161#386—216no—
162#385—28no—
163#384—24no—
164#382—28no—
165#381—216no—
166#381—48no—
167#381—44no—
168#380—28no—
169#380—24no—
170C4×C22⋊C4—28no—
171C4×C22⋊C4—24no—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C25C2411yescharacteristic
200C25C2412yes—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—28no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—44no—
210C22×Q8—42no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—28no—
214C22×D4—22no—
215C22×D4—24no—
216C22×D4—22no—
217C22×D4—42no—
218C22×D4—44no—
219C22×D4C2×D418yes—
220C22×D4C2×D414yes—
221C23×C4—28no—
222C23×C4—28no—
223C23×C4—28no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4C2×D418yes—
237C23×C4C2×D4116yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C4⋊Q8—232no—
254#125—264no—
255#123—432no—
256C22⋊Q8—2128no—
257C22⋊Q8—264no—
258#121—416no—
259C4×Q8—232no—
260C4×D4—264no—
261C4×D4—232no—
262C2×C4⋊C4—264no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D418yes—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—41no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4C2×D418yes—
335C2×C22⋊C4C2×D4116yes—
336C2×C22⋊C4#4114yes—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42C2×D418yes—
346C2×C42C2×D414yes—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—28no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—24no—
360#95—22no—
361#95—28no—
362#95—24no—
363#95—42no—
364#95—44no—
365#95—42no—
366#95—44no—
367#95—44no—
368#95—42no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95—42no—
373#95—44no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—21no—
384C24—22no—
385C24—21no—
386C24—21no—
387C24—21no—
388C24—21no—
389C24—22no—
390C24—21no—
391C24—22no—
392C24—22no—
393C24C22×D414yes—
394C24C22×D418yes—
395C24C2×#4112yes—
396C24C2511yescharacteristic
397C2×Q8—28no—
398C2×Q8—216no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—88no—
404C2×Q8—84no—
405C2×Q8—88no—
406C2×Q8—84no—
407C2×D4—216no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—216no—
411C2×D4—48no—
412C2×D4—416no—
413C2×D4—48no—
414C2×D4—84no—
415C2×D4—88no—
416C2×D4#120116yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—216no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—232no—
455C22×C4—216no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
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—24no—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—48no—
495C22×C4—416no—
496C22×C4—416no—
497C22×C4—416no—
498C22×C4—416no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—48no—
504C22×C4—88no—
505C22×C4—88no—
506C22×C4—88no—
507C22×C4—88no—
508C22×C4C22×D412yes—
509C22×C4C22×D414yes—
510C22×C4C22×D418yes—
511C22×C4C22×D414yes—
512C22×C4C22×D414yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D414yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C42−1+412yes—
521C22×C42−1+414yes—
522C22×C42−1+412yes—
523C4⋊C4—28no—
524C4⋊C4—232no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—232no—
528C4⋊C4—28no—
529C4⋊C4—216no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—84no—
536C4⋊C4—88no—
537C22⋊C4—232no—
538C22⋊C4—216no—
539C22⋊C4—28no—
540C22⋊C4—44no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—88no—
548C22⋊C4—88no—
549C22⋊C4—88no—
550C42—216no—
551C42—28no—
552C42—44no—
553C42—48no—
554C23—28no—
555C23—24no—
556C23—24no—
557C23—28no—
558C23—24no—
559C23—28no—
560C23—28no—
561C23—24no—
562C23—24no—
563C23—28no—
564C23—28no—
565C23—24no—
566C23—22no—
567C23—24no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—41no—
582C23—42no—
583C23—41no—
584C23—41no—
585C23—42no—
586C23—42no—
587C23—41no—
588C23—41no—
589C23—42no—
590C23—42no—
591C23—41no—
592C23—42no—
593C23—41no—
594C23—41no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23C2×#12018yes—
600C23C23×D411yescharacteristic
601C23C23×D412yes—
602C23C22×#4111yescharacteristic
603C23C2×H(5,𝔽2)11yescharacteristic
604C23C2×H(5,𝔽2)11yescharacteristic
605C23C2×2−1+411yescharacteristic
606C23C2×2−1+411yescharacteristic
607C23C2×2−1+412yes—
608C23#58512yes—
609C23#58511yescharacteristic
610C23#58512yes—
611Q8—416no—
612Q8—816no—
613D4—232no—
614D4—816no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—432no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—42no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—832no—
700C2×C4—88no—
701C2×C4C2×#12014yes—
702C2×C4#53618yes—
703C2×C4#53614yes—
704C2×C4#53614yes—
705C2×C4#53618yes—
706C2×C4#54614yes—
707C22—28no—
708C22—28no—
709C22—28no—
710C22—48no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—48no—
715C22—44no—
716C22—48no—
717C22—48no—
718C22—44no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—44no—
723C22—81no—
724C22—81no—
725C22—81no—
726C22—82no—
727C22—81no—
728C22—82no—
729C22C22×#12011yescharacteristic
730C22C2×#53612yes—
731C22C2×#53611yescharacteristic
732C22C2×#53611yescharacteristic
733C22C2×#53612yes—
734C22C2×#54611yescharacteristic
735C22D4×#4112yes—
736C22#348511yescharacteristic
737C22#348512yes—
738C22#348811yescharacteristic
739C22#348812yes—
740C22#349612yes—
741C22#350011yescharacteristic
742C22#350711yescharacteristic
743C22#351111yescharacteristic
744C22#351512yes—
745C22#354712yes—
746C22#354711yescharacteristic
747C22#354811yescharacteristic
748C22#355712yes—
749C22#355912yes—
750C22#356512yes—
751C22#356512yes—
752C4—24no—
753C4—42no—
754C4—44no—
755C4—44no—
756C4—44no—
757C4—48no—
758C4—42no—
759C4—42no—
760C4—82no—
761C4—81no—
762C4—82no—
763C4—82no—
764C4—82no—
765C4—84no—
766C4#261418yes—
767C2—48no—
768C2—82no—
769C2C2×#261411yescharacteristic
770C2#3575011yescharacteristic
771C2#3621211yescharacteristic
772C2#3895511yescharacteristic
773C2#3947011yescharacteristic
774C2#3988012yes—
775C2#3995212yes—
776C2#4243912yes—
777C2#4264711yescharacteristic
778C2#4268112yes—
779C2#4268211yescharacteristic
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. C22×#554C2semidirect
4. #16364C2semidirect
5. #16263C2semidirect
6. #16262C2semidirect
7. #16251C2semidirect
8. #15230C2semidirect
9. #15228C2semidirect
10. #15216C2semidirect
11. #15158C2semidirect
12. #15125C2semidirect
13. #15073C2semidirect
14. #14256C2semidirect
15. #13916C2semidirect
16. #13916C2semidirect
17. C4×#393C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#24202
C2×#24222
#163642
#162632
#162622
#162512
#152302
#152282
#152162
#151582
#151252
#150732
#142562
#139162
C4×#3932