Finite Group Databasebeta

← #803738

Subgroups — #803738

Order 512.

3344 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2447C212yesmaximal, complemented
2C2×#2427C211yesmaximal, characteristic
3C2×#2420C212yesmaximal
4C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
5#17059C212yesmaximal, complemented
6#15549C214yesmaximal, complemented
7#15539C214yesmaximal, complemented
8#15240C211yesmaximal, characteristic, complemented
9#15193C211yesmaximal, characteristic, complemented
10#15135C211yesmaximal, characteristic, complemented
11#14839C212yesmaximal, complemented
12#14777C212yesmaximal, complemented
13#13997C212yesmaximal, complemented
14#13954C212yesmaximal, complemented
15#13904C212yesmaximal, complemented
16C4×#399C211yesmaximal, characteristic, complemented
17#13820C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×D4C2212yes—
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24#2447—216no—
25#2427—28no—
26#2420—216no—
27C2×#399C2211yescharacteristic
28C2×#399C2212yes—
29C2×#397C2214yes—
30C2×#397C2212yes—
31C2×#397C2214yes—
32C2×#397C2214yes—
33C2×#394C2214yes—
34C2×#394C2212yes—
35D4×C4⋊C4—28no—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2211yescharacteristic
44C2×#386C2214yes—
45C2×#386C2212yes—
46C2×#385C2214yes—
47C2×#385C2214yes—
48C2×#384C2211yescharacteristic
49C2×#382C2212yes—
50C2×#381C2214yes—
51C2×#381C2214yes—
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×#380C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C22⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2212yes—
60C2×C4×C22⋊C4C2212yes—
61#1445C2212yes—
62#1445C2211yescharacteristic
63#1445C2212yes—
64#1445C2212yes—
65#1445C2212yes—
66#1445C2214yes—
67#1445C2212yes—
68#1445C2214yes—
69#1445C2214yes—
70#1444C2212yes—
71#1444C2211yescharacteristic
72#1444C2212yes—
73#1444C2214yes—
74#1444C2214yes—
75#1439C2212yes—
76#1439C2212yes—
77#1439C2211yescharacteristic
78#1439C2214yes—
79#1437C2214yes—
80#1437C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2214yes—
83C4⋊#95C2214yes—
84C4⋊#95C2214yes—
85C22⋊#95C2212yes—
86C4×#95C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×D4C2312yes—
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2314yes—
92C24×C4C2312yes—
93C2×#124—24no—
94C2×#124—28no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—28no—
97C2×#121—28no—
98C2×#121—24no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C2×C4×D4D418yes—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2314yes—
140#399—24no—
141#399—44no—
142#397—216no—
143#397—28no—
144#397—216no—
145#397—216no—
146#394—216no—
147#394—28no—
148#391—28no—
149#391—24no—
150#391—28no—
151#390D4116yes—
152#389—28no—
153#389—28no—
154#389—24no—
155#389—28no—
156#386—28no—
157#386—216no—
158#385—216no—
159#385—216no—
160#384—24no—
161#382—28no—
162#381—216no—
163#381—216no—
164#380—28no—
165#380—28no—
166#380—28no—
167C4×C4⋊C4—28no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—28no—
170C4×C22⋊C4—28no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—44no—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2311yescharacteristic
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C25C2411yescharacteristic
200C25C2412yes—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×D4—24no—
207C22×D4—28no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—42no—
213C22×D4—44no—
214C22×D4C2×D414yes—
215C22×D4C2×D418yes—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—28no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—28no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D418yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248#124—416no—
249C22⋊Q8—416no—
250#121—416no—
251C4×D4—232no—
252C4×D4—432no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—432no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4#4114yes—
292C2×C4⋊C4#4118yes—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—41no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4C2×D4116yes—
326C2×C22⋊C4C2×D4116yes—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—28no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42C2×D414yes—
339C2×C42C2×D418yes—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
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—24no—
358#95—24no—
359#95—28no—
360#95—28no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—42no—
365#95—42no—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24—21no—
376C24—22no—
377C24—21no—
378C24—22no—
379C24—21no—
380C24—21no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24—21no—
385C24C22×D418yes—
386C24C22×D414yes—
387C242−1+412yes—
388C24C2511yescharacteristic
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—216no—
397C2×D4—216no—
398C2×D4—416no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—88no—
402C2×D4—84no—
403C2×D4#127116yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—416no—
484C22×C4—416no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—88no—
491C22×C4—88no—
492C22×C4—88no—
493C22×C4C22×D414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D418yes—
497C22×C4C22×D414yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D414yes—
501C22×C4C2×#4114yes—
502C22×C4C2×#4118yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4114yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C42−1+412yes—
507C22×C42−1+412yes—
508C4⋊C4—28no—
509C4⋊C4—216no—
510C4⋊C4—48no—
511C4⋊C4—416no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—84no—
519C4⋊C4—88no—
520C4⋊C4—88no—
521C4⋊C4—84no—
522C4⋊C4#123116yes—
523C22⋊C4—216no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—416no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—88no—
536C42—28no—
537C42—216no—
538C42—48no—
539C42—44no—
540C42—88no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—24no—
549C23—28no—
550C23—24no—
551C23—28no—
552C23—24no—
553C23—24no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—44no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—41no—
578C23—42no—
579C23—41no—
580C23—42no—
581C23—41no—
582C23—42no—
583C23—41no—
584C23—42no—
585C23—42no—
586C23C2×#12718yes—
587C23C23×D411yescharacteristic
588C23C23×D412yes—
589C23C22×#4112yes—
590C23C22×#4111yescharacteristic
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×2−1+411yescharacteristic
593C23C2×2−1+411yescharacteristic
594C23C2×2−1+411yescharacteristic
595C23#58512yes—
596C23#58512yes—
597C23#58511yescharacteristic
598Q8—88no—
599D4—232no—
600D4—816no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—22no—
624C2×C4—432no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—41no—
682C2×C4—832no—
683C2×C4—88no—
684C2×C4C2×#12318yes—
685C2×C4C2×#12314yes—
686C2×C4C2×#12714yes—
687C2×C4#53614yes—
688C2×C4#54614yes—
689C2×C4#54718yes—
690C2×C4#54718yes—
691C22—28no—
692C22—28no—
693C22—28no—
694C22—48no—
695C22—48no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—48no—
700C22—48no—
701C22—44no—
702C22—44no—
703C22—42no—
704C22—44no—
705C22—42no—
706C22—42no—
707C22—82no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22—82no—
713C22C22×#12311yescharacteristic
714C22C22×#12711yescharacteristic
715C22C2×#53611yescharacteristic
716C22C2×#54611yescharacteristic
717C22C2×#54712yes—
718C22C2×#54712yes—
719C22D4×#4112yes—
720C22#347512yes—
721C22#348512yes—
722C22#348511yescharacteristic
723C22#348912yes—
724C22#348911yescharacteristic
725C22#349612yes—
726C22#350011yescharacteristic
727C22#351312yes—
728C22#351612yes—
729C22#351811yescharacteristic
730C22#352311yescharacteristic
731C22#352412yes—
732C22#355211yescharacteristic
733C22#355411yescharacteristic
734C22#355712yes—
735C22#356512yes—
736C4—24no—
737C4—28no—
738C4—48no—
739C4—44no—
740C4—42no—
741C4—44no—
742C4—44no—
743C4—81no—
744C4—84no—
745C4—81no—
746C4—82no—
747C4—82no—
748C4—81no—
749C4—82no—
750C4#250118yes—
751C2—48no—
752C2—82no—
753C2C2×#250111yescharacteristic
754C2#3598311yescharacteristic
755C2#3606011yescharacteristic
756C2#3615511yescharacteristic
757C2#3626011yescharacteristic
758C2#3891111yescharacteristic
759C2#4264711yescharacteristic
760C2#4510012yes—
761C2#4510712yes—
762C2#5907312yes—
763C2#5911912yes—
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×#2447C2semidirect
4. C2×D4×C4⋊C4C2semidirect
5. #17059C2semidirect
6. #15549C2semidirect
7. #15539C2semidirect
8. #15240C2semidirect
9. #15193C2semidirect
10. #15135C2semidirect
11. #14839C2semidirect
12. #14777C2semidirect
13. #13997C2semidirect
14. #13954C2semidirect
15. #13904C2semidirect
16. C4×#399C2semidirect
17. #13820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24472
C2×#24272
C2×#24202
C2×D4×C4⋊C42
#170592
#155492
#155392
#152402
#151932
#151352
#148392
#147772
#139972
#139542
#139042
C4×#3992
#138202