Finite Group Databasebeta

← #676570

Subgroups — #676570

Order 512.

2154 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16134C214yesmaximal, complemented
2#15490C214yesmaximal
3#15220C212yesmaximal
4#15195C212yesmaximal
5#15146C212yesmaximal, complemented
6#15139C212yesmaximal, complemented
7#14686C211yesmaximal, characteristic
8#14151C211yesmaximal, characteristic, complemented
9#14051C212yesmaximal, complemented
10#13973C212yesmaximal
11#13973C212yesmaximal
12#13936C212yesmaximal, complemented
13#13911C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15C4⋊C42C211yesmaximal, characteristic, complemented
16C4×#380C211yesmaximal, characteristic
17C4×#380C211yesmaximal, characteristic
18C23×C42C2211yescharacteristic
19C2×#400C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#384C2212yes—
27C2×#382C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43#1444C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2211yescharacteristic
74#1434C2212yes—
75#1434C2212yes—
76#1434C2214yes—
77#1434C2214yes—
78#1434C2214yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C42—24no—
109C22×C42—24no—
110C22×C42—24no—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120#400—48no—
121#389—44no—
122#389—44no—
123#389—44no—
124#389—44no—
125#384—24no—
126#384—24no—
127#384—44no—
128#382—28no—
129#380—24no—
130#380—28no—
131#380—24no—
132#380—24no—
133#380—44no—
134#380—44no—
135#380Q818yes—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—44no—
142C4×C22⋊C4—44no—
143C4×C22⋊C4—44no—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C25C2411yescharacteristic
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—41no—
194C23×C4—41no—
195C23×C4C2×Q814yes—
196C23×C4#4114yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
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—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×Q814yes—
261C2×C4⋊C4C2×Q814yes—
262C2×C4⋊C4#4114yes—
263C2×C4⋊C4#4114yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×Q818yes—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—416no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42#4114yes—
305C2×C42#4114yes—
306C2×C42#4114yes—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C2×Q818yes—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24C22×Q812yes—
351C24C2×#4112yes—
352C242−1+412yes—
353C24C2511yescharacteristic
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—28no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
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—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4C22×Q812yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q814yes—
466C22×C4C22×Q812yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+412yes—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—88no—
509C42—44no—
510C42—44no—
511C42—44no—
512C42—48no—
513C42—48no—
514C42—48no—
515C42—48no—
516C42—88no—
517C42#126116yes—
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—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23#55714yes—
553C23C23×Q811yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4111yescharacteristic
556C23C22×#4111yescharacteristic
557C23C22×#4111yescharacteristic
558C23C22×#4111yescharacteristic
559C23C22×#4111yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×2−1+411yescharacteristic
565C23C2×2−1+411yescharacteristic
566C23#58511yescharacteristic
567C23#58511yescharacteristic
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—416no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—416no—
625C2×C4—416no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4C2×#12614yes—
669C2×C4C2×#12614yes—
670C2×C4C2×#12614yes—
671C2×C4#52914yes—
672C2×C4#53314yes—
673C2×C4#54014yes—
674C2×C4#54214yes—
675C2×C4#55214yes—
676C2×C4#55214yes—
677C2×C4#55314yes—
678C2×C4#55314yes—
679C2×C4#55414yes—
680C2×C4#55414yes—
681C2×C4#55514yes—
682C2×C4#55514yes—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22C22×#12611yescharacteristic
699C22C2×#52911yescharacteristic
700C22C2×#53311yescharacteristic
701C22C2×#54011yescharacteristic
702C22C2×#54211yescharacteristic
703C22C2×#55211yescharacteristic
704C22C2×#55211yescharacteristic
705C22C2×#55311yescharacteristic
706C22C2×#55311yescharacteristic
707C22C2×#55411yescharacteristic
708C22C2×#55411yescharacteristic
709C22C2×#55511yescharacteristic
710C22C2×#55511yescharacteristic
711C22C2×#55711yescharacteristic
712C22Q8×#4111yescharacteristic
713C22#348011yescharacteristic
714C22#348311yescharacteristic
715C22#349211yescharacteristic
716C22#349311yescharacteristic
717C22#349411yescharacteristic
718C22#349411yescharacteristic
719C22#349511yescharacteristic
720C22#349511yescharacteristic
721C22#350111yescharacteristic
722C22#350311yescharacteristic
723C22#350611yescharacteristic
724C22#350811yescharacteristic
725C22#351411yescharacteristic
726C22#351511yescharacteristic
727C22#352011yescharacteristic
728C22#352011yescharacteristic
729C22#354611yescharacteristic
730C22#354711yescharacteristic
731C22#354911yescharacteristic
732C22#355211yescharacteristic
733C4—24no—
734C4—44no—
735C4—44no—
736C4—44no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—44no—
741C4—82no—
742C4—84no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—84no—
747C4#249518yes—
748C4#249718yes—
749C2—44no—
750C2C2×#249711yescharacteristic
751C2C2×#249511yescharacteristic
752C2#3554411yescharacteristic
753C2#3628711yescharacteristic
754C2#3646011yescharacteristic
755C2#3655211yescharacteristic
756C2#3659911yescharacteristic
757C2#3893011yescharacteristic
758C2#3914611yescharacteristic
759C2#3960811yescharacteristic
760C2#3973511yescharacteristic
761C2#5922711yescharacteristic
762C2#5923011yescharacteristic
763C2#5923311yescharacteristic
764C2#5932111yescharacteristic
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. #16134C2semidirect
5. #15146C2semidirect
6. #15139C2semidirect
8. #14151C2semidirect
9. #14051C2semidirect
12. #13936C2semidirect
13. #13911C2semidirect
14. #13910C2semidirect
15. C4⋊C42C2semidirect

Maximal subgroups

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

subgroupindex
#161342
#154902
#152202
#151952
#151462
#151392
#146862
#141512
#140512
#139732
#139362
#139112
#139102
C4⋊C422
C4×#3802