Finite Group Databasebeta

← #766198

Subgroups — #766198

Order 512.

3662 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2396C211yesmaximal, characteristic, complemented
2C2×#2309C211yesmaximal, characteristic
3#15820C212yesmaximal, complemented
4#15775C214yesmaximal, complemented
5#15662C212yesmaximal, complemented
6#15467C212yesmaximal, complemented
7#15297C214yesmaximal, complemented
8#15284C212yesmaximal, complemented
9#15157C212yesmaximal, complemented
10#15147C212yesmaximal, complemented
11#14022C212yesmaximal, complemented
12#14022C212yesmaximal, complemented
13#13939C212yesmaximal, complemented
14#13902C212yesmaximal
15#13777C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic, complemented
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#2396—28no—
21C2×#400C2211yescharacteristic, complemented
22C2×#399C2211yescharacteristic
23C2×#398C2212yescomplemented
24C2×#397C2214yescomplemented
25C2×#396C2212yes—
26C2×#395C2211yescharacteristic, complemented
27C2×#393C2211yescharacteristic
28C2×#393C2214yescomplemented
29#2309—28no—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#388C2214yescomplemented
39C2×#388C2214yescomplemented
40C2×#380C2212yes—
41C2×#379C2212yes—
42C2×#379C2212yes—
43C2×#379C2214yes—
44C2×#379C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic, complemented
46C2×C4×C4⋊C4C2212yescomplemented
47C2×C4×C4⋊C4C2211yescharacteristic, complemented
48C2×C4×C4⋊C4C2211yescharacteristic, complemented
49C2×C4×C22⋊C4C2212yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1438C2214yes—
63#1438C2214yes—
64#1437C2212yes—
65#1437C2212yes—
66#1437C2214yescomplemented
67#1437C2214yescomplemented
68#1436C2212yescomplemented
69#1436C2212yescomplemented
70C4⋊#95C2212yescomplemented
71C4⋊#95C2212yescomplemented
72C4⋊#95C2212yescomplemented
73C4⋊#95C2212yescomplemented
74C4⋊#95C2212yescomplemented
75#1434C2214yescomplemented
76#1434C2214yescomplemented
77C26C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4—28no—
80C23×D4—24no—
81C23×D4—28no—
82C23×D4—42no—
83C23×D4—42no—
84C23×D4C2312yes—
85C23×D4C2312yes—
86C24×C4C2311yescharacteristic
87C2×C4×Q8—24no—
88C2×C4×Q8—42no—
89C2×C4×Q8—42no—
90C2×#117—24no—
91C2×#117—28no—
92C2×#117—42no—
93C2×#117—42no—
94C2×#117D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130#400—24no—
131#399—24no—
132#398—28no—
133#397—48no—
134#396—28no—
135#395—24no—
136#393—24no—
137#393—48no—
138#390—28no—
139#390—28no—
140#390—28no—
141#390—28no—
142#390—28no—
143#390—28no—
144#390—44no—
145#390—44no—
146#388—48no—
147#388—48no—
148#380—28no—
149#379—28no—
150#379—28no—
151#379—48no—
152#379—48no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—42no—
156C4×C4⋊C4—42no—
157C4×C22⋊C4—28no—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C25—24no—
178C25—28no—
179C25—28no—
180C25—42no—
181C25—42no—
182C25—44no—
183C25—44no—
184C25—44no—
185C25—44no—
186C25C2414yes—
187C25C2411yescharacteristic
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8—42no—
193C22×D4—232no—
194C22×D4—28no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—28no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—432no—
204C22×D4—432no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—416no—
208C22×D4—416no—
209C22×D4#4118yes—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4C2×D414yes—
214C23×C4#4118yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C4×Q8—416no—
229#117—416no—
230#117—416no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4C2×D414yes—
313C2×C22⋊C4C2×D414yes—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—42no—
344#95—42no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—216no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—416no—
374C24—416no—
375C24—48no—
376C24—416no—
377C24—416no—
378C24—48no—
379C24—88no—
380C24—88no—
381C24—88no—
382C24—88no—
383C24—88no—
384C24—88no—
385C24—88no—
386C24—88no—
387C24—84no—
388C24—84no—
389C24—88no—
390C24—88no—
391C24—84no—
392C24—84no—
393C24C22×D412yes—
394C24C2×#4118yes—
395C24C2×#4114yes—
396C24C2511yescharacteristic
397C2×Q8—48no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—216no—
405C2×D4—216no—
406C2×D4—416no—
407C2×D4—416no—
408C2×D4—48no—
409C2×D4—4128no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—216no—
421C22×C4—216no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
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—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4#124116yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C42—28no—
528C42—28no—
529C42—48no—
530C42—48no—
531C42—48no—
532C42—84no—
533C42—84no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—432no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—832no—
591C23—832no—
592C23—832no—
593C23—832no—
594C23—832no—
595C23—832no—
596C23—816no—
597C23—816no—
598C23C2×#12414yes—
599C23C23×D411yescharacteristic
600C23C22×#4112yes—
601C23C22×#4112yes—
602C23C2×H(5,𝔽2)11yescharacteristic
603C23C2×H(5,𝔽2)11yescharacteristic
604C23C2×H(5,𝔽2)12yes—
605C23C2×H(5,𝔽2)12yes—
606C23#58512yes—
607C23#58512yes—
608Q8—416no—
609D4—432no—
610D4—432no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—232no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4#52514yes—
686C2×C4C2×#12414yes—
687C2×C4C2×#12414yes—
688C2×C4#53414yes—
689C2×C4#55118yes—
690C2×C4#55118yes—
691C22—24no—
692C22—28no—
693C22—44no—
694C22—44no—
695C22—42no—
696C22—42no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—48no—
701C22—48no—
702C22—48no—
703C22—48no—
704C22—48no—
705C22—48no—
706C22—48no—
707C22—864no—
708C22—84no—
709C22—84no—
710C22—84no—
711C22—84no—
712C22—84no—
713C22—84no—
714C22—84no—
715C22—84no—
716C22C2×#52511yescharacteristic
717C22C22×#12411yescharacteristic
718C22C2×#53411yescharacteristic
719C22C2×#55112yes—
720C22C2×#55112yes—
721C22#348812yes—
722C22#348812yes—
723C22#350612yes—
724C22#350612yes—
725C22#350812yes—
726C22#351012yes—
727C22#351312yes—
728C22#351412yes—
729C22#351912yes—
730C22#352112yes—
731C22#353711yescharacteristic
732C22#354011yescharacteristic
733C22#354212yes—
734C22#354612yes—
735C22#354911yescharacteristic
736C22#355111yescharacteristic
737C4—24no—
738C4—42no—
739C4—42no—
740C4—44no—
741C4—42no—
742C4—42no—
743C4—44no—
744C4—84no—
745C4—82no—
746C4—84no—
747C4—82no—
748C4—84no—
749C4#256118yes—
750C2—44no—
751C2—88no—
752C2C2×#256111yescharacteristic
753C2#3552212yes—
754C2#3555612yes—
755C2#3726911yescharacteristic
756C2#3727311yescharacteristic
757C2#4495912yes—
758C2#4496112yes—
759C2#4536512yes—
760C2#4536712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2396C2semidirect
3. #15820C2semidirect
4. #15775C2semidirect
5. #15662C2semidirect
6. #15467C2semidirect
7. #15297C2semidirect
8. #15284C2semidirect
9. #15157C2semidirect
10. #15147C2semidirect
11. #14022C2semidirect
12. #14022C2semidirect
13. #13939C2semidirect
15. #13777C2semidirect
17. C22×C4×Q8C22semidirect
21. C2×#400C22semidirect
23. C2×#398C22semidirect
24. C2×#397C22semidirect
26. C2×#395C22semidirect
28. C2×#393C22semidirect
38. C2×#388C22semidirect
39. C2×#388C22semidirect
45. C2×C4×C4⋊C4C22semidirect
46. C2×C4×C4⋊C4C22semidirect
47. C2×C4×C4⋊C4C22semidirect
48. C2×C4×C4⋊C4C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
66. #1437C22semidirect
67. #1437C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. C4⋊#95C22semidirect
71. C4⋊#95C22semidirect
72. C4⋊#95C22semidirect
73. C4⋊#95C22semidirect
74. C4⋊#95C22semidirect
75. #1434C22semidirect
76. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
C2×#23092
#158202
#157752
#156622
#154672
#152972
#152842
#151572
#151472
#140222
#139392
#139022
#137772