Finite Group Databasebeta

← #517180

Subgroups — #517180

Order 512.

3122 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2C2×#2329C212yesmaximal
3#14702C212yesmaximal
4#14697C212yesmaximal
5#14353C211yesmaximal, characteristic
6#14275C212yesmaximal
7#14237C214yesmaximal, complemented
8#14236C214yesmaximal, complemented
9#14214C214yesmaximal, complemented
10#14211C214yesmaximal, complemented
11#13778C212yesmaximal
12C4×#380C212yesmaximal
13C2×C4×#95C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C23×C42C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2659C4116yescomplemented
19C2×#398C2214yes—
20C2×#395C2214yes—
21#2329—216no—
22C2×#391C2214yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#389C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#387C2214yes—
30C2×#385C2214yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C4×#95C2214yes—
67C4×#95C2214yes—
68C4×#95C2214yes—
69C23×Q8C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8C2×C4116yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C42—24no—
97C22×C42—24no—
98C22×C42—42no—
99C22×C42—42no—
100C22×C42—42no—
101C22×C42—42no—
102C22×C42—42no—
103C22×C42—42no—
104C22×C42Q818yescomplemented
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2314yes—
112C22×C42C2312yes—
113#398C2×C4132yescomplemented
114#395C2×C4132yescomplemented
115#391—216no—
116#391—44no—
117#391—44no—
118#389—216no—
119#389—44no—
120#389—44no—
121#387—216no—
122#387—216no—
123#385—216no—
124#380—28no—
125#380—28no—
126#380—28no—
127#380C2×C4116yes—
128#380C2×C4116yes—
129C4×C4⋊C4—44no—
130C4×C4⋊C4—44no—
131C4×C22⋊C4—216no—
132C2×#95—24no—
133C2×#95C2×C418yes—
134C2×#95D418yes—
135C2×#95Q818yescomplemented
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C25C2411yescharacteristic
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8C22×C418yes—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4—41no—
180C23×C4—41no—
181C23×C4—41no—
182C23×C4—41no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4—41no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4C22×C414yes—
200C23×C4C22×C414yes—
201C23×C4C22×C414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×Q814yes—
206C23×C4C2×Q814yes—
207C23×C4C2×Q814yes—
208C23×C4C2×Q814yes—
209C23×C4C2×Q814yes—
210C23×C4C2×Q814yes—
211C23×C4#4114yes—
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C22⋊Q8—264no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4C22×C4116yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C418yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C418yes—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4C22×C4116yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—416no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—416no—
295C2×C42—416no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C2×Q818yes—
301C2×C42C2×Q818yes—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95C22×C4116yes—
319#95C22×C4116yes—
320#95C22×C4116yes—
321#95C2×D4116yes—
322#95C2×Q8116yes—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24C23×C412yes—
332C24C22×D412yes—
333C24C22×Q812yes—
334C24C22×Q812yes—
335C24C2×#4112yes—
336C24C2×#4112yes—
337C24C2×#4112yes—
338C24C2511yescharacteristic
339C2×Q8—216no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8—84no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4C4×D4116yes—
454C22×C4C4×Q8116yes—
455C22×C4C4×Q8116yes—
456C22×C4C4⋊Q8116yes—
457C22×C4C23×C418yes—
458C22×C4C23×C418yes—
459C22×C4C23×C414yes—
460C22×C4C23×C414yes—
461C22×C4C23×C414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×Q814yes—
466C22×C4C22×Q814yes—
467C22×C4C22×Q814yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C22×Q812yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+412yes—
479C4⋊C4—216no—
480C4⋊C4—232no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C22⋊C4—232no—
494C22⋊C4—416no—
495C42—216no—
496C42—48no—
497C42—48no—
498C42—48no—
499C42—48no—
500C42—48no—
501C42—88no—
502C42—88no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23C2×C4×D414yes—
532C23C2×C4×Q814yes—
533C23C2×C4×Q814yes—
534C23C2×C4⋊Q814yes—
535C23#53214yes—
536C23#53314yes—
537C23#53314yes—
538C23C24×C411yescharacteristic
539C23C23×D411yescharacteristic
540C23C23×Q811yescharacteristic
541C23C23×Q811yescharacteristic
542C23C22×#4111yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×2−1+411yescharacteristic
549C23#58511yescharacteristic
550C23#58511yescharacteristic
551C23#58511yescharacteristic
552C23#58511yescharacteristic
553Q8—816no—
554C2×C4—232no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
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—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—416no—
607C2×C4—416no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4C2×C4×D418yes—
640C2×C4C2×C4×Q818yes—
641C2×C4C2×C4×Q818yes—
642C2×C4C2×C4⋊Q818yes—
643C2×C4#54314yes—
644C2×C4#54414yes—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22C4×C4⋊Q818yes—
660C22C22×C4×D411yescharacteristic
661C22C22×C4×Q811yescharacteristic
662C22C22×C4×Q811yescharacteristic
663C22C4×H(5,𝔽2)11yescharacteristic
664C22C4×H(5,𝔽2)11yescharacteristic
665C22C4×H(5,𝔽2)11yescharacteristic
666C22C4×2−1+411yescharacteristic
667C22C22×C4⋊Q811yescharacteristic
668C22C2×#53211yescharacteristic
669C22C2×#53311yescharacteristic
670C22C2×#53311yescharacteristic
671C22C2×#54311yescharacteristic
672C22C2×#54411yescharacteristic
673C22#345811yescharacteristic
674C22#345811yescharacteristic
675C22#345911yescharacteristic
676C22#345911yescharacteristic
677C22#346011yescharacteristic
678C22#346111yescharacteristic
679C22#349011yescharacteristic
680C22#349011yescharacteristic
681C22#349111yescharacteristic
682C22#349111yescharacteristic
683C22#349211yescharacteristic
684C22#349211yescharacteristic
685C22#349311yescharacteristic
686C22#349311yescharacteristic
687C22#349611yescharacteristic
688C22#349611yescharacteristic
689C22#349611yescharacteristic
690C22#349711yescharacteristic
691C22#350111yescharacteristic
692C22#350111yescharacteristic
693C22#350111yescharacteristic
694C22#350211yescharacteristic
695C4—28no—
696C4—28no—
697C4—28no—
698C4—28no—
699C4—24no—
700C4—24no—
701C4—44no—
702C4—44no—
703C4—48no—
704C4—48no—
705C4—84no—
706C4—84no—
707C2—28no—
708C2C2×C4×C4⋊Q811yescharacteristic
709C2C4×#54311yescharacteristic
710C2C4×#54411yescharacteristic
711C2#3427011yescharacteristic
712C2#3427111yescharacteristic
713C2#3427311yescharacteristic
714C2#3427411yescharacteristic
715C2#3500211yescharacteristic
716C2#3500311yescharacteristic
717C2#3500411yescharacteristic
718C2#3501211yescharacteristic
719C2#3501311yescharacteristic
720C2#3505211yescharacteristic
721C2#3512011yescharacteristic
722C2#3516711yescharacteristic
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
7. #14237C2semidirect
8. #14236C2semidirect
9. #14214C2semidirect
10. #14211C2semidirect
18. #2659C4semidirect
104. C22×C42Q8semidirect
113. #398C2×C4semidirect
114. #395C2×C4semidirect
135. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
C2×#23292
#147022
#146972
#143532
#142752
#142372
#142362
#142142
#142112
#137782
C4×#3802
C2×C4×#952