Finite Group Databasebeta

← #616709

Subgroups — #616709

Order 512.

2732 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2C2×C42×Q8C211yesmaximal, characteristic, complemented
3#15202C211yesmaximal, characteristic, complemented
4#15200C211yesmaximal, characteristic, complemented
5#15157C211yesmaximal, characteristic
6#15148C211yesmaximal, characteristic
7#15078C212yesmaximal, complemented
8#15072C214yesmaximal, complemented
9#14875C212yesmaximal, complemented
10#14700C212yesmaximal
11#14699C212yesmaximal, complemented
12#14231C212yesmaximal, complemented
13#14188C212yesmaximal, complemented
14#14079C214yesmaximal, complemented
15#14045C212yesmaximal, complemented
16#14022C212yesmaximal
17#13755C211yesmaximal, characteristic, complemented
18C22×#126C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#2860—28no—
23C2×#399C2212yes—
24C2×#393C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2212yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#387C2214yes—
42C42×Q8—28no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×#376C2212yes—
51C2×C43C2212yes—
52C2×C43C2211yescharacteristic
53#1446C2212yes—
54#1445C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81#1434C2212yes—
82#1434C2214yes—
83#1434C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C2×#126—28no—
88C2×#126—48no—
89C2×C4×Q8—24no—
90C2×C4×Q8—28no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—28no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#399—28no—
137#393—28no—
138#390—28no—
139#390—24no—
140#390—24no—
141#390—24no—
142#390—28no—
143#389—28no—
144#389—24no—
145#389—24no—
146#389—24no—
147#389—28no—
148#388—216no—
149#388—28no—
150#388—28no—
151#388—28no—
152#388—48no—
153#388—48no—
154#387—48no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162#376—44no—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C43—28no—
187C43D418yes—
188C25C2412yes—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8#4114yes—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—22no—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215#126—432no—
216C4×Q8—232no—
217C4×Q8—232no—
218C4×Q8—232no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
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—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4#4118yes—
258C2×C4⋊C4#4114yes—
259C2×C4⋊C4#4118yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42—28no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42#4118yes—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—42no—
341#95—42no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24C2511yescharacteristic
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—216no—
362C2×Q8—28no—
363C2×Q8—216no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—48no—
368C2×D4—48no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4118yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C22×C42−1+412yes—
462C22×C42−1+412yes—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—416no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—88no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—88no—
489C42—216no—
490C42—216no—
491C42—28no—
492C42—28no—
493C42—28no—
494C42—216no—
495C42—28no—
496C42—28no—
497C42—28no—
498C42—28no—
499C42—416no—
500C42—48no—
501C42—48no—
502C42—48no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C23×D411yescharacteristic
527C23C22×#4111yescharacteristic
528C23C22×#4112yes—
529C23C22×#4112yes—
530C23C22×#4111yescharacteristic
531C23C22×#4112yes—
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×2−1+411yescharacteristic
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+411yescharacteristic
537Q8—232no—
538D4—88no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4#52514yes—
624C2×C4#52514yes—
625C2×C4#52514yes—
626C2×C4#52918yes—
627C2×C4#53214yes—
628C2×C4#53514yes—
629C2×C4#53614yes—
630C2×C4#53714yes—
631C2×C4#55218yes—
632C2×C4#55618yes—
633C22—82no—
634C22—84no—
635C22—84no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—84no—
642C22—84no—
643C22—82no—
644C22C2×#52511yescharacteristic
645C22C2×#52511yescharacteristic
646C22C2×#52511yescharacteristic
647C22C2×#52912yes—
648C22C2×#53211yescharacteristic
649C22#344412yes—
650C22C2×#53511yescharacteristic
651C22C2×#53611yescharacteristic
652C22C2×#53711yescharacteristic
653C22C2×#55212yes—
654C22C2×#55612yes—
655C22#348112yes—
656C22#348112yes—
657C22#348212yes—
658C22#348312yes—
659C22#350812yes—
660C22#351112yes—
661C22#352211yescharacteristic
662C22#352212yes—
663C22#352212yes—
664C22#352211yescharacteristic
665C22#352311yescharacteristic
666C22#352311yescharacteristic
667C4—24no—
668C4—28no—
669C4—24no—
670C4—24no—
671C4—24no—
672C4—48no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—44no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4#260418yes—
683C2—162no—
684C2C2×#260411yescharacteristic
685C2#3496411yescharacteristic
686C2#3497711yescharacteristic
687C2#3524711yescharacteristic
688C2#3524911yescharacteristic
689C2#3529211yescharacteristic
690C2#3630012yes—
691C2#3631212yes—
692C2#3637012yes—
693C2#3637112yes—
694C2#3662111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C42×Q8C2semidirect
3. #15202C2semidirect
4. #15200C2semidirect
7. #15078C2semidirect
8. #15072C2semidirect
9. #14875C2semidirect
11. #14699C2semidirect
12. #14231C2semidirect
13. #14188C2semidirect
14. #14079C2semidirect
15. #14045C2semidirect
17. #13755C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28602
C2×C42×Q82
#152022
#152002
#151572
#151482
#150782
#150722
#148752
#147002
#146992
#142312
#141882
#140792
#140452
#140222
#137552