Finite Group Databasebeta

← #4462543

Subgroups — #4462543

Order 512.

3332 conjugacy classes of subgroups.

The subgroup lattice has 782 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic, complemented
2C22×#399C211yesmaximal, characteristic, complemented
3#17176C214yesmaximal, complemented
4#17136C214yesmaximal, complemented
5#17126C214yesmaximal, complemented
6#17101C214yesmaximal, complemented
7#14912C212yesmaximal, complemented
8#14911C212yesmaximal, complemented
9#14822C211yesmaximal, characteristic, complemented
10#14734C211yesmaximal, characteristic, complemented
11#14729C212yesmaximal, complemented
12#14725C212yesmaximal, complemented
13#14719C211yesmaximal, characteristic, complemented
14#14716C211yesmaximal, characteristic
15#14339C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×#120C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25#2618—28no—
26C2×#400C2212yescomplemented
27C2×#399—28no—
28C2×#399C2214yes—
29C2×#399C2214yes—
30C2×#399C2212yes—
31C2×#398C2212yescomplemented
32C2×#396C2212yescomplemented
33C2×#396C2212yescomplemented
34C2×#396C2214yescomplemented
35C2×#394C2214yes—
36C2×#394C2212yescomplemented
37C2×#394C2212yescomplemented
38C2×#392C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#385C2212yes—
44C2×#381C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×#379C2212yes—
48C22×#95C2211yescharacteristic
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2212yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2212yescomplemented
58#1440C2214yes—
59#1440C2214yescomplemented
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yescomplemented
67#1439C2214yescomplemented
68#1439C2214yescomplemented
69#1438C2214yes—
70#1438C2214yes—
71#1438C2214yes—
72#1438C2214yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C23×D4C2314yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#123—28no—
88C2×#123—44no—
89C2×#123—44no—
90C2×#121—24no—
91C2×#121—44no—
92C2×#120—44no—
93C2×#120—44no—
94C2×#120—44no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4D418yescomplemented
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—44no—
117C22×C22⋊C4—44no—
118C22×C22⋊C4—44no—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137#400—44no—
138#399—216no—
139#399—216no—
140#399—44no—
141#398—28no—
142#396—28no—
143#396—28no—
144#396—48no—
145#394—28no—
146#394—48no—
147#394—44no—
148#392—44no—
149#386—28no—
150#386—44no—
151#386—44no—
152#386—44no—
153#385—44no—
154#381—44no—
155#380—24no—
156#380—44no—
157#379—44no—
158C2×#95—24no—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C25—22no—
179C25—24no—
180C25—24no—
181C25—22no—
182C25—24no—
183C25—22no—
184C25—42no—
185C25—42no—
186C25—42no—
187C25C2412yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—22no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—42no—
207C22×D4—44no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2412yes—
246#123—432no—
247#121—416no—
248#120—432no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—416no—
275C2×C4⋊C4—416no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D418yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—416no—
324C2×C22⋊C4—416no—
325C2×C22⋊C4—416no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
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⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—42no—
354#95—42no—
355#95—44no—
356#95—42no—
357#95—42no—
358#95—44no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—28no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—24no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—24no—
385C24—416no—
386C24—416no—
387C24—48no—
388C24—48no—
389C24—44no—
390C24—44no—
391C24—44no—
392C24—44no—
393C24—44no—
394C24—44no—
395C24—48no—
396C24—48no—
397C24—88no—
398C24—84no—
399C24—84no—
400C24—88no—
401C24C22×D412yes—
402C24H(5,𝔽2)14yes—
403C24H(5,𝔽2)14yes—
404C24H(5,𝔽2)12yes—
405C242−1+414yes—
406C242−1+412yes—
407C24C2511yescharacteristic
408C2×D4—44no—
409C2×D4—416no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—88no—
423C2×D4—88no—
424C2×D4—88no—
425C2×D4—88no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—48no—
500C22×C4—416no—
501C22×C4—48no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—88no—
512C22×C4—88no—
513C22×C4—88no—
514C22×C4C22×D414yes—
515C22×C4C22×D414yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4H(5,𝔽2)14yes—
519C22×C4H(5,𝔽2)14yes—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—416no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—88no—
531C4⋊C4—88no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—416no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—416no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—88no—
546C22⋊C4—88no—
547C22⋊C4—88no—
548C22⋊C4—88no—
549C22⋊C4—88no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—28no—
555C23—24no—
556C23—24no—
557C23—28no—
558C23—28no—
559C23—24no—
560C23—24no—
561C23—28no—
562C23—24no—
563C23—28no—
564C23—28no—
565C23—24no—
566C23—24no—
567C23—416no—
568C23—416no—
569C23—416no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—42no—
585C23—42no—
586C23—44no—
587C23—44no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—832no—
612C23—832no—
613C23—88no—
614C23—88no—
615C23—88no—
616C23—88no—
617C23#53414yes—
618C23#53514yes—
619C23#53714yes—
620C23#56018yes—
621C23#56218yes—
622C23C23×D411yescharacteristic
623C23C2×H(5,𝔽2)12yes—
624C23C2×H(5,𝔽2)12yes—
625C23C2×H(5,𝔽2)11yescharacteristic
626C23C2×H(5,𝔽2)12yes—
627C23C2×H(5,𝔽2)12yes—
628C23C2×2−1+411yescharacteristic
629C23C2×2−1+412yes—
630C23#58512yes—
631D4—88no—
632D4—832no—
633D4—816no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—28no—
650C2×C4—24no—
651C2×C4—416no—
652C2×C4—416no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—416no—
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—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—88no—
706C2×C4—88no—
707C2×C4—88no—
708C22—24no—
709C22—216no—
710C22—216no—
711C22—24no—
712C22—24no—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—44no—
717C22—48no—
718C22—48no—
719C22—44no—
720C22—44no—
721C22—832no—
722C22—82no—
723C22—84no—
724C22—84no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—84no—
731C22—84no—
732C22—84no—
733C22—84no—
734C22—82no—
735C22—82no—
736C22—82no—
737C22—82no—
738C22—82no—
739C22#281318yes—
740C22C2×#53411yescharacteristic
741C22C2×#53511yescharacteristic
742C22C2×#53711yescharacteristic
743C22#348411yescharacteristic
744C22#348411yescharacteristic
745C22#348611yescharacteristic
746C22#348711yescharacteristic
747C22C2×#56012yes—
748C22C2×#56212yes—
749C22#353414yes—
750C22#353412yes—
751C22#353514yes—
752C22#353512yes—
753C22#353614yes—
754C22#353612yes—
755C22#353812yes—
756C22#354314yes—
757C4—44no—
758C4—44no—
759C4—42no—
760C4—44no—
761C4—42no—
762C4—44no—
763C4—82no—
764C4—84no—
765C4—84no—
766C4—82no—
767C4—82no—
768C4—82no—
769C2—28no—
770C2—84no—
771C2—162no—
772C2C2×#281311yescharacteristic
773C2#3670711yescharacteristic
774C2#3671011yescharacteristic
775C2#3671911yescharacteristic
776C2#3672211yescharacteristic
777C2#3718611yescharacteristic
778C2#3723211yescharacteristic
779C2#5254714yes—
780C2#5254814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14716C2
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×#2618C2semidirect
2. C22×#399C2semidirect
3. #17176C2semidirect
4. #17136C2semidirect
5. #17126C2semidirect
6. #17101C2semidirect
7. #14912C2semidirect
8. #14911C2semidirect
9. #14822C2semidirect
10. #14734C2semidirect
11. #14729C2semidirect
12. #14725C2semidirect
13. #14719C2semidirect
15. #14339C2semidirect
26. C2×#400C22semidirect
31. C2×#398C22semidirect
32. C2×#396C22semidirect
33. C2×#396C22semidirect
34. C2×#396C22semidirect
36. C2×#394C22semidirect
37. C2×#394C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
59. #1440C22semidirect
66. #1439C22semidirect
67. #1439C22semidirect
68. #1439C22semidirect
100. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C22×#3992
#171762
#171362
#171262
#171012
#149122
#149112
#148222
#147342
#147292
#147252
#147192
#147162
#143392