Finite Group Databasebeta

← #1395717

Subgroups — #1395717

Order 512.

2622 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2491C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#16676C212yesmaximal, complemented
4#16510C212yesmaximal, complemented
5#16375C214yesmaximal, complemented
6#16294C214yesmaximal, complemented
7#15148C211yesmaximal, characteristic, complemented
8#15144C211yesmaximal, characteristic, complemented
9#15128C212yesmaximal, complemented
10#15126C212yesmaximal
11#15125C211yesmaximal, characteristic, complemented
12#15078C212yesmaximal, complemented
13#14288C212yesmaximal, complemented
14#14022C212yesmaximal, complemented
15#13973C212yesmaximal, complemented
16#13921C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2212yes—
21C22×#121C2211yescharacteristic
22#2491—28no—
23C2×#399C2211yescharacteristic
24C2×#399C2212yes—
25C2×#395C2214yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#393C2212yes—
29C2×#393C2212yes—
30C2×#393C2214yes—
31#2369—28no—
32C2×#391C2214yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#388C2212yes—
41C2×#388C2214yes—
42C2×#383C2212yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52#1446C2212yes—
53#1446C2214yes—
54#1445C2211yescharacteristic
55#1445C2212yes—
56#1444C2214yes—
57#1444C2211yescharacteristic
58#1443C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2212yes—
68#1439C2211yescharacteristic
69#1439C2212yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1438C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2214yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2312yes—
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—44no—
94C2×#124—28no—
95C2×#124—28no—
96C2×#124—44no—
97C2×#121—24no—
98C2×#121—44no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2314yes—
133#399—24no—
134#399—44no—
135#395—48no—
136#394—28no—
137#394—48no—
138#393—28no—
139#393—44no—
140#393—48no—
141#391—216no—
142#390—24no—
143#390—28no—
144#390—24no—
145#390—44no—
146#389—28no—
147#389—28no—
148#389—44no—
149#388—28no—
150#388—48no—
151#383—44no—
152#380—24no—
153#380—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—44no—
158C4×C4⋊C4D418yes—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—44no—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—28no—
192C22×Q8—24no—
193C22×Q8—42no—
194C22×Q8—44no—
195C22×Q8—42no—
196C22×Q8—44no—
197C22×Q8—42no—
198C22×Q8—44no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—44no—
207C22×D4—42no—
208C22×D4—44no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C4⋊Q8—232no—
230C4⋊Q8—416no—
231#124—432no—
232#121—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4#4118yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
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#4114yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C2×D418yes—
325C2×C42C2×D414yes—
326#95—22no—
327#95—22no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—42no—
349#95—44no—
350#95—44no—
351#95C2×D418yes—
352C24—21no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24C2×#4112yes—
374C24C2511yescharacteristic
375C2×Q8—216no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—84no—
379C2×Q8—84no—
380C2×Q8—84no—
381C2×Q8—88no—
382C2×Q8—88no—
383C2×Q8—88no—
384C2×Q8—88no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—84no—
393C2×D4—88no—
394C2×D4—88no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
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—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4118yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C42−1+414yes—
487C22×C42−1+412yes—
488C22×C42−1+412yes—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—88no—
501C4⋊C4—88no—
502C22⋊C4—28no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—84no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C42—28no—
518C42—216no—
519C42—48no—
520C42—44no—
521C42—48no—
522C42—88no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23C23×D411yescharacteristic
570C23C22×#4112yes—
571C23C22×#4111yescharacteristic
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)11yescharacteristic
575C23C2×H(5,𝔽2)12yes—
576C23C2×2−1+411yescharacteristic
577C23C2×2−1+411yescharacteristic
578C23C2×2−1+412yes—
579C23#58511yescharacteristic
580Q8—816no—
581Q8—816no—
582D4—816no—
583D4—88no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—44no—
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—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—88no—
668C2×C4#52514yes—
669C2×C4#53514yes—
670C2×C4#53614yes—
671C2×C4#54218yes—
672C2×C4#54218yes—
673C2×C4#54614yes—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—44no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—82no—
683C22—81no—
684C22—82no—
685C22—84no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—84no—
693C22—82no—
694C22—84no—
695C22—84no—
696C22C2×#52511yescharacteristic
697C22C2×#53511yescharacteristic
698C22C2×#53611yescharacteristic
699C22C2×#54212yes—
700C22C2×#54212yes—
701C22C2×#54611yescharacteristic
702C22#348212yes—
703C22#348611yescharacteristic
704C22#348811yescharacteristic
705C22#349811yescharacteristic
706C22#350711yescharacteristic
707C22#351111yescharacteristic
708C22#351412yes—
709C22#351912yes—
710C22#351912yes—
711C22#352212yes—
712C22#353612yes—
713C22#354112yes—
714C22#354412yes—
715C22#354711yescharacteristic
716C22#354811yescharacteristic
717C22#354912yes—
718C22#355012yes—
719C4—24no—
720C4—44no—
721C4—42no—
722C4—44no—
723C4—44no—
724C4—42no—
725C4—48no—
726C4—42no—
727C4—44no—
728C4—81no—
729C4—82no—
730C4—82no—
731C4—84no—
732C4—82no—
733C4#263218yes—
734C2—82no—
735C2—162no—
736C2C2×#263211yescharacteristic
737C2#3600411yescharacteristic
738C2#3642211yescharacteristic
739C2#3895311yescharacteristic
740C2#3947011yescharacteristic
741C2#4010212yes—
742C2#4022212yes—
743C2#4258512yes—
744C2#4273112yes—
745C2#4278911yescharacteristic
746C2#4281011yescharacteristic
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
2. C2×#2369C2semidirect
3. #16676C2semidirect
4. #16510C2semidirect
5. #16375C2semidirect
6. #16294C2semidirect
7. #15148C2semidirect
8. #15144C2semidirect
9. #15128C2semidirect
11. #15125C2semidirect
12. #15078C2semidirect
13. #14288C2semidirect
14. #14022C2semidirect
15. #13973C2semidirect
16. #13921C2semidirect
17. C4×#399C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24912
C2×#23692
#166762
#165102
#163752
#162942
#151482
#151442
#151282
#151262
#151252
#150782
#142882
#140222
#139732
#139212
C4×#3992