Finite Group Databasebeta

← #644026

Subgroups — #644026

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17081C212yesmaximal, complemented
2#17076C212yesmaximal, complemented
3#16906C214yesmaximal, complemented
4#15235C212yesmaximal, complemented
5#15231C212yesmaximal, complemented
6#15221C212yesmaximal, complemented
7#15146C212yesmaximal, complemented
8#14767C214yesmaximal
9#14686C211yesmaximal, characteristic
10#14347C212yesmaximal
11#14275C212yesmaximal
12#14130C212yesmaximal
13#13881C212yesmaximal
14C4×#380C211yesmaximal, characteristic
15C2×C4⋊#95C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#400C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#385C2214yes—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2212yes—
26C2×#382C2212yes—
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C22×#95C2212yes—
36C22×#95C2212yes—
37#1445C2214yes—
38#1444C2214yes—
39#1444C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69#1434C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C22×C4⋊C4—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C42—42no—
106C22×C42—42no—
107C22×C42Q818yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#400—48no—
115#389—44no—
116#389—44no—
117#385—48no—
118#384—42no—
119#384—42no—
120#382—44no—
121#382—44no—
122#380—28no—
123#380—24no—
124#380—24no—
125#380—28no—
126#380—44no—
127C4×C4⋊C4—42no—
128C4×C4⋊C4—42no—
129C4×C22⋊C4—28no—
130C2×#95—28no—
131C2×#95—44no—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25C2411yescharacteristic
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4—41no—
180C23×C4—41no—
181C23×C4—41no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4C2×Q814yes—
191C23×C4C2×Q814yes—
192C23×C4C2×Q814yes—
193C23×C4#4114yes—
194C23×C4#4114yes—
195C23×C4#4114yes—
196C23×C4#4114yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—416no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—416no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—41no—
285C2×C42C2×Q814yes—
286C2×C42C2×Q814yes—
287C2×C42C2×Q814yes—
288C2×C42C2×Q814yes—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24C22×Q812yes—
327C24C2×#4112yes—
328C24C2×#4112yes—
329C24C2×#4112yes—
330C24C2×#4112yes—
331C24H(5,𝔽2)12yes—
332C242−1+412yes—
333C24C2511yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
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—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4#125116yes—
446C22×C4#125116yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q812yes—
452C22×C4C22×Q812yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—88no—
482C4⋊C4—88no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C42—28no—
487C42—28no—
488C42—44no—
489C42—44no—
490C42—48no—
491C42—48no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
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—
520C23C2×#12514yes—
521C23C2×#12514yes—
522C23#54414yes—
523C23#55514yes—
524C23#55614yes—
525C23#55614yes—
526C23#55914yes—
527C23C23×Q811yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4111yescharacteristic
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23C2×2−1+411yescharacteristic
539C23C2×2−1+411yescharacteristic
540C23#58511yescharacteristic
541C23#58511yescharacteristic
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—232no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
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—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—816no—
633C2×C4—816no—
634C2×C4C2×#12514yes—
635C2×C4C2×#12514yes—
636C2×C4C2×#12514yes—
637C2×C4C2×#12514yes—
638C2×C4#53814yes—
639C2×C4#53814yes—
640C2×C4#53914yes—
641C2×C4#53914yes—
642C2×C4#54314yes—
643C2×C4#54414yes—
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—
659C22#257118yes—
660C22C22×#12511yescharacteristic
661C22C22×#12511yescharacteristic
662C22C2×#53811yescharacteristic
663C22C2×#53811yescharacteristic
664C22C2×#53911yescharacteristic
665C22C2×#53911yescharacteristic
666C22C2×#54311yescharacteristic
667C22C2×#54411yescharacteristic
668C22C2×#54411yescharacteristic
669C22C2×#54411yescharacteristic
670C22#345811yescharacteristic
671C22C2×#55511yescharacteristic
672C22C2×#55611yescharacteristic
673C22C2×#55611yescharacteristic
674C22C2×#55911yescharacteristic
675C22#350511yescharacteristic
676C22#350511yescharacteristic
677C22#350511yescharacteristic
678C22#350511yescharacteristic
679C22#350611yescharacteristic
680C22#350611yescharacteristic
681C22#350711yescharacteristic
682C22#350711yescharacteristic
683C22#350811yescharacteristic
684C22#350911yescharacteristic
685C22#351111yescharacteristic
686C22#351111yescharacteristic
687C22#353611yescharacteristic
688C22#353611yescharacteristic
689C22#353711yescharacteristic
690C22#353811yescharacteristic
691C22#355811yescharacteristic
692C22#355911yescharacteristic
693C22#356111yescharacteristic
694C22#356311yescharacteristic
695C4—24no—
696C4—24no—
697C4—24no—
698C4—24no—
699C4—42no—
700C4—82no—
701C4—84no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—84no—
706C4—84no—
707C4—84no—
708C4#264618yes—
709C2—28no—
710C2C2×#264611yescharacteristic
711C2C2×#257111yescharacteristic
712C2#3502011yescharacteristic
713C2#3518111yescharacteristic
714C2#3528611yescharacteristic
715C2#3747711yescharacteristic
716C2#3756111yescharacteristic
717C2#3950311yescharacteristic
718C2#3950411yescharacteristic
719C2#3951111yescharacteristic
720C2#3951211yescharacteristic
721C2#3975911yescharacteristic
722C2#3976011yescharacteristic
723C2#3976711yescharacteristic
724C2#3976811yescharacteristic
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. #17081C2semidirect
2. #17076C2semidirect
3. #16906C2semidirect
4. #15235C2semidirect
5. #15231C2semidirect
6. #15221C2semidirect
7. #15146C2semidirect

Maximal subgroups

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

subgroupindex
#170812
#170762
#169062
#152352
#152312
#152212
#151462
#147672
#146862
#143472
#142752
#141302
#138812
C4×#3802
C2×C4⋊#952