Finite Group Databasebeta

← #803745

Subgroups — #803745

Order 512.

2928 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2446C212yesmaximal, complemented
2C2×#2429C211yesmaximal, characteristic
3C2×#2423C212yesmaximal
4C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#17064C212yesmaximal
6#15804C214yesmaximal, complemented
7#15765C214yesmaximal
8#15199C211yesmaximal, characteristic, complemented
9#15193C211yesmaximal, characteristic, complemented
10#15145C212yesmaximal, complemented
11#15135C211yesmaximal, characteristic
12#15082C212yesmaximal, complemented
13#14231C212yesmaximal
14#13997C212yesmaximal
15#13910C212yesmaximal, complemented
16C4×#399C211yesmaximal, characteristic
17#13817C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2211yescharacteristic
23#2446—216no—
24#2429—28no—
25#2423—216no—
26C2×#400C2214yes—
27C2×#399C2214yes—
28C2×#399C2212yes—
29C2×#399C2211yescharacteristic
30C2×#398C2214yes—
31C2×#398C2214yes—
32C2×#398C2212yes—
33C2×#396C2212yes—
34Q8×C4⋊C4—28no—
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#389C2211yescharacteristic
44C2×#389C2212yes—
45C2×#388C2214yes—
46C2×#388C2212yes—
47C2×#387C2214yes—
48C2×#387C2214yes—
49C2×#387C2214yes—
50C2×#384C2211yescharacteristic
51C2×#384C2212yes—
52C2×#384C2212yes—
53C2×#384C2214yes—
54C2×#380C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C4⋊C4C2212yes—
59C2×C4×C22⋊C4C2211yescharacteristic
60C2×C4×C22⋊C4C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2211yescharacteristic
64#1445C2214yes—
65#1445C2212yes—
66#1445C2214yes—
67#1444C2212yes—
68#1444C2211yescharacteristic
69#1444C2212yes—
70#1444C2212yes—
71#1444C2214yes—
72#1444C2214yes—
73#1444C2214yes—
74#1444C2212yes—
75#1439C2212yes—
76#1439C2212yes—
77#1439C2211yescharacteristic
78#1439C2214yes—
79#1437C2214yes—
80#1436C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2214yes—
83C4⋊#95C2214yes—
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8—28no—
100C2×C4×Q8D418yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#400—216no—
138#399—216no—
139#399—44no—
140#399Q818yescomplemented
141#398—28no—
142#398—216no—
143#398—216no—
144#396—28no—
145#391—28no—
146#391—24no—
147#391—28no—
148#391D4116yes—
149#389—216no—
150#389—28no—
151#389—28no—
152#389—24no—
153#389—28no—
154#389—28no—
155#388—28no—
156#388—216no—
157#387—216no—
158#387—216no—
159#387—216no—
160#384—24no—
161#384—28no—
162#384—28no—
163#384—48no—
164#380—28no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—28no—
168C4×C4⋊C4—28no—
169C4×C22⋊C4—28no—
170C4×C22⋊C4—24no—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2311yescharacteristic
197C25C2411yescharacteristic
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8C2×D414yes—
207C22×Q8C2×D418yes—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C4⋊Q8—416no—
230C22⋊Q8—416no—
231C22⋊Q8—416no—
232C4×Q8—232no—
233C4×Q8—432no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×Q818yes—
289C2×C4⋊C4C2×Q814yes—
290C2×C4⋊C4#4118yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C2×Q818yes—
309C2×C22⋊C4C2×Q814yes—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—28no—
323C2×C42—28no—
324C2×C42—44no—
325C2×C42C2×D414yes—
326C2×C42C2×D418yes—
327C2×C42C2×D418yes—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—42no—
351#95C2×D4116yes—
352#95C2×D4116yes—
353#95C2×Q814yes—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—21no—
364C24C22×Q812yes—
365C24C2511yescharacteristic
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—216no—
369C2×Q8—216no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8#127116yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4C22×D414yes—
453C22×C4C22×D418yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D418yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D414yes—
462C22×C4C22×Q812yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q812yes—
465C22×C4C22×Q814yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C42−1+412yes—
470C22×C42−1+414yes—
471C22×C42−1+412yes—
472C22×C42−1+412yes—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—28no—
476C4⋊C4—216no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—416no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—88no—
491C4⋊C4—88no—
492C4⋊C4—84no—
493C4⋊C4—88no—
494C4⋊C4#123116yes—
495C22⋊C4—28no—
496C22⋊C4—216no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—88no—
502C42—216no—
503C42—28no—
504C42—216no—
505C42—416no—
506C42—416no—
507C42—48no—
508C42—44no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23C23×D411yescharacteristic
533C23C23×D412yes—
534C23C23×Q811yescharacteristic
535C23C22×#4112yes—
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23C2×2−1+411yescharacteristic
539C23C2×2−1+412yes—
540C23C2×2−1+411yescharacteristic
541C23#58512yes—
542C23#58511yescharacteristic
543Q8—232no—
544Q8—88no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—88no—
629C2×C4C2×#12318yes—
630C2×C4C2×#12314yes—
631C2×C4C2×#12718yes—
632C2×C4C2×#12714yes—
633C2×C4#53614yes—
634C2×C4#54718yes—
635C2×C4#54718yes—
636C2×C4D4×Q814yes—
637C2×C4D4×Q818yes—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—82no—
644C22—81no—
645C22—81no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22C22×#12311yescharacteristic
650C22C22×#12711yescharacteristic
651C22C2×#53611yescharacteristic
652C22C2×#54712yes—
653C22C2×#54712yes—
654C22C2×D4×Q811yescharacteristic
655C22C2×D4×Q812yes—
656C22Q8×#4112yes—
657C22#348711yescharacteristic
658C22#348712yes—
659C22#348812yes—
660C22#348811yescharacteristic
661C22#349912yes—
662C22#349911yescharacteristic
663C22#350211yescharacteristic
664C22#350311yescharacteristic
665C22#351512yes—
666C22#351612yes—
667C22#352012yes—
668C22#355011yescharacteristic
669C22#355512yes—
670C22#355511yescharacteristic
671C22#356312yes—
672C4—28no—
673C4—24no—
674C4—28no—
675C4—28no—
676C4—48no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—42no—
682C4—81no—
683C4—82no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4#250118yes—
688C2—82no—
689C2C2×#250111yescharacteristic
690C2#3599811yescharacteristic
691C2#3606611yescharacteristic
692C2Q8×#12311yescharacteristic
693C2Q8×#12711yescharacteristic
694C2#3959711yescharacteristic
695C2#4287911yescharacteristic
696C2#4507712yes—
697C2#4523812yes—
698C2#5906712yes—
699C2#5911312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2446C2semidirect
4. C2×Q8×C4⋊C4C2semidirect
6. #15804C2semidirect
8. #15199C2semidirect
9. #15193C2semidirect
10. #15145C2semidirect
12. #15082C2semidirect
15. #13910C2semidirect
17. #13817C2semidirect
140. #399Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24462
C2×#24292
C2×#24232
C2×Q8×C4⋊C42
#170642
#158042
#157652
#151992
#151932
#151452
#151352
#150822
#142312
#139972
#139102
C4×#3992
#138172