Finite Group Databasebeta

← #552761

Subgroups — #552761

Order 512.

3214 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊C4C212yesmaximal
2C2×#2310C211yesmaximal, characteristic
3#14674C212yesmaximal
4#14501C212yesmaximal, complemented
5#14467C212yesmaximal, complemented
6#14466C214yesmaximal, complemented
7#14340C211yesmaximal, characteristic
8#14295C214yesmaximal
9#14165C212yesmaximal
10#14017C212yesmaximal, complemented
11#14017C212yesmaximal, complemented
12#13978C212yesmaximal, complemented
13#13976C212yesmaximal, complemented
14#13951C212yesmaximal
15#13852C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4×Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21Q8×C22⋊C4—216no—
22#2310C4116yescomplemented
23C2×#391C2214yes—
24C2×#391C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#380C2212yes—
38C2×#380C2214yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2212yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2211yescharacteristic
55#1441C2211yescharacteristic
56#1439C2212yes—
57#1439C2212yes—
58#1439C2211yescharacteristic
59#1439C2211yescharacteristic
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C4×#95C2212yes—
77C23×Q8—42no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8C2×C418yes—
82C23×Q8D418yes—
83C23×Q8Q818yes—
84C23×Q8C2312yes—
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4×Q8—28no—
89C2×C4×Q8—28no—
90C2×C4×Q8—44no—
91C2×C4×Q8—44no—
92C2×C4×Q8—44no—
93C2×C4×Q8—44no—
94C2×#117—42no—
95C2×#117—42no—
96C2×#117—42no—
97C2×#117—42no—
98C2×#117C2×C418yes—
99C2×#117C2×C418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2314yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127#391—28no—
128#391—28no—
129#391—216no—
130#391—216no—
131#391C2×C4116yes—
132#391C2×C4116yes—
133#391C2×C4116yes—
134#391C2×C4116yes—
135#389—44no—
136#389—44no—
137#384—42no—
138#382—42no—
139#381C2×C4116yes—
140#381C2×C4116yes—
141#380—28no—
142#380—216no—
143C4×C4⋊C4—42no—
144C4×C4⋊C4—42no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—216no—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C25C2411yescharacteristic
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—416no—
185C22×Q8—416no—
186C22×Q8—416no—
187C22×Q8—416no—
188C22×Q8—416no—
189C22×Q8—416no—
190C22×Q8C4⋊C4164yescomplemented
191C22×Q8C22×C418yes—
192C22×Q8C22×C418yes—
193C22×Q8C2×D418yes—
194C22×Q8C2×D418yes—
195C22×Q8C2×Q818yes—
196C22×Q8C2×Q818yes—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4C22×C414yes—
206C23×C4C22×C418yes—
207C23×C4C2×D414yes—
208C23×C4C2×D418yes—
209C23×C4C2×Q814yes—
210C23×C4C2×Q818yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C4×Q8—432no—
226C4×Q8—432no—
227#117—416no—
228#117—416no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
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—28no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4C22×C414yes—
283C2×C22⋊C4C22×C414yes—
284C2×C22⋊C4C22×C418yes—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C22×C414yes—
304C2×C42C22×C414yes—
305C2×C42C22×C418yes—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—44no—
323#95—41no—
324#95—42no—
325#95—41no—
326#95—42no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95C22×C4116yes—
331#95C22×C4116yes—
332#95C22×C4116yes—
333#95C22×C4116yes—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24C23×C412yes—
343C24C22×D412yes—
344C24C22×Q812yes—
345C24C2511yescharacteristic
346C2×Q8—216no—
347C2×Q8—216no—
348C2×Q8—216no—
349C2×Q8—216no—
350C2×Q8—216no—
351C2×Q8—216no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—4128no—
359C2×Q8C2×C4⋊C4132yes—
360C2×Q8C2×C4⋊C4132yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4C2×C4⋊C4116yes—
444C22×C4C2×C4⋊C4132yes—
445C22×C4C23×C412yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4C23×C412yes—
449C22×C4C23×C418yes—
450C22×C4C23×C418yes—
451C22×C4C23×C412yes—
452C22×C4C23×C412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D418yes—
455C22×C4C22×D412yes—
456C22×C4C22×Q812yes—
457C22×C4C22×Q818yes—
458C22×C4C22×Q812yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+414yes—
464C22×C42−1+414yes—
465C4⋊C4—216no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—88no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C22⋊C4—216no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C42—28no—
486C42—28no—
487C42—48no—
488C42—416no—
489C42—416no—
490C42—48no—
491C42—48no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23C22×C4⋊C414yes—
515C23C24×C411yescharacteristic
516C23C23×D411yescharacteristic
517C23C23×Q811yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×2−1+412yes—
523C23C2×2−1+412yes—
524C23#58512yes—
525C23#58512yes—
526Q8—264no—
527Q8—432no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—232no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—432no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4C22×C4⋊C414yes—
612C2×C4C22×C4⋊C4116yes—
613C2×C4C22×C4⋊C414yes—
614C2×C4#51814yes—
615C2×C4#51814yes—
616C2×C4#53414yes—
617C2×C4#53414yes—
618C2×C4#54314yes—
619C2×C4#54314yes—
620C22—24no—
621C22—28no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22C23×C4⋊C411yescharacteristic
632C22C2×#51811yescharacteristic
633C22C2×#51811yescharacteristic
634C22C4×2−1+412yes—
635C22C4×2−1+412yes—
636C22C2×#53411yescharacteristic
637C22C2×#53411yescharacteristic
638C22C2×#54311yescharacteristic
639C22C2×#54311yescharacteristic
640C22#348612yes—
641C22#348712yes—
642C22#350212yes—
643C22#350212yes—
644C22#353811yescharacteristic
645C22#354011yescharacteristic
646C22#354112yes—
647C22#354712yes—
648C22#355011yescharacteristic
649C22#355111yescharacteristic
650C22#355212yes—
651C22#355412yes—
652C22#356112yes—
653C22#356212yes—
654C4—24no—
655C4—216no—
656C4—44no—
657C4—44no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—48no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4—84no—
667C4#231518yes—
668C2—44no—
669C2C2×#231511yescharacteristic
670C2#3424912yes—
671C2#3441211yescharacteristic
672C2#3441311yescharacteristic
673C2#3443912yes—
674C2#4280612yes—
675C2#4280711yescharacteristic
676C2#4281612yes—
677C2#4281711yescharacteristic
678C2#4281811yescharacteristic
679C2#4288311yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #14501C2semidirect
5. #14467C2semidirect
6. #14466C2semidirect
10. #14017C2semidirect
11. #14017C2semidirect
12. #13978C2semidirect
13. #13976C2semidirect
22. #2310C4semidirect
190. C22×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C42
C2×#23102
#146742
#145012
#144672
#144662
#143402
#142952
#141652
#140172
#139782
#139762
#139512
#138522