Finite Group Databasebeta

← #732677

Subgroups — #732677

Order 512.

2728 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2352C211yesmaximal, characteristic
3#16980C212yesmaximal, complemented
4#16602C212yesmaximal, complemented
5#15861C214yesmaximal, complemented
6#15476C214yesmaximal
7#15125C212yesmaximal, complemented
8#15125C212yesmaximal, complemented
9#15125C212yesmaximal, complemented
10#15045C212yesmaximal, complemented
11#14348C211yesmaximal, characteristic, complemented
12#13955C211yesmaximal, characteristic, complemented
13#13951C212yesmaximal, complemented
14#13907C212yesmaximal, complemented
15#13906C211yesmaximal, characteristic
16#13847C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2551—28no—
24C2×#395C2214yes—
25C2×#393C2214yes—
26C2×#393C2214yes—
27#2352—28no—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2212yes—
33C2×#387C2212yes—
34C2×#386C2212yes—
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2211yescharacteristic
49#1446C2214yes—
50#1446C2214yes—
51#1445C2212yes—
52#1442C2212yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1439C2211yescharacteristic
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2211yescharacteristic
66#1439C2211yescharacteristic
67#1439C2211yescharacteristic
68#1439C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2212yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C23×Q8C2314yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—48no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4Q818yescomplemented
98C2×#117—24no—
99C2×#117—28no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#395—216no—
135#393—48no—
136#393—48no—
137#390—44no—
138#389—28no—
139#389—28no—
140#389—28no—
141#388—28no—
142#387—28no—
143#386—28no—
144#384—24no—
145#382—24no—
146#380—24no—
147#380—24no—
148#380—48no—
149#380—44no—
150#380—44no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4D418yes—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—44no—
156C4×C22⋊C4—44no—
157#376—24no—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C22×Q8—28no—
189C22×Q8—28no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—41no—
200C22×D4—41no—
201C22×D4C2×Q814yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4C2×Q814yes—
216C23×C4C2×Q814yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C4⋊Q8—432no—
233C4⋊Q8—432no—
234C4×D4—232no—
235#117—416no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4#4118yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4C2×Q814yes—
304C2×C22⋊C4C2×Q814yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—44no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42C2×D418yes—
318C2×C42C2×D414yes—
319C2×C42C2×Q814yes—
320C2×C42#4114yes—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—42no—
337#95—42no—
338#95—42no—
339#95—42no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95C2×D418yes—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24C22×Q812yes—
368C24C22×Q812yes—
369C24H(5,𝔽2)12yes—
370C242−1+412yes—
371C24C2511yescharacteristic
372C2×Q8—416no—
373C2×Q8—416no—
374C2×Q8—88no—
375C2×Q8—88no—
376C2×Q8—88no—
377C2×Q8—88no—
378C2×Q8—88no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×Q812yes—
478C22×C4C22×Q812yes—
479C22×C4C22×Q812yes—
480C22×C4C22×Q812yes—
481C22×C4C22×Q812yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4118yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4114yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C42−1+414yes—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—416no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—84no—
503C4⋊C4—88no—
504C4⋊C4#124116yes—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—84no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C42—48no—
517C42—48no—
518C42—48no—
519C42—48no—
520C42—84no—
521C42C22⋊Q8116yes—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23#54414yes—
568C23C23×D411yescharacteristic
569C23C23×Q811yescharacteristic
570C23C22×#4111yescharacteristic
571C23C22×#4112yes—
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)12yes—
575C23C2×2−1+411yescharacteristic
576C23C2×2−1+412yes—
577C23#58511yescharacteristic
578C23#58511yescharacteristic
579Q8—832no—
580D4—416no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—416no—
625C2×C4—416no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—816no—
665C2×C4C2×C22⋊Q814yes—
666C2×C4C2×C22⋊Q814yes—
667C2×C4C2×C22⋊Q814yes—
668C2×C4C2×#12418yes—
669C2×C4C2×#12414yes—
670C2×C4#53514yes—
671C2×C4#55418yes—
672C2×C4#55518yes—
673C22—24no—
674C22—28no—
675C22—44no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—82no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—82no—
694C22—81no—
695C22C22×C22⋊Q811yescharacteristic
696C22C22×#12411yescharacteristic
697C22C2×#53511yescharacteristic
698C22C2×#54411yescharacteristic
699C22#345811yescharacteristic
700C22C2×#55412yes—
701C22C2×#55512yes—
702C22#348411yescharacteristic
703C22#348611yescharacteristic
704C22#348811yescharacteristic
705C22#348911yescharacteristic
706C22#349411yescharacteristic
707C22#349511yescharacteristic
708C22#350312yes—
709C22#351112yes—
710C22#351412yes—
711C22#351812yes—
712C22#352012yes—
713C22#352212yes—
714C22#353712yes—
715C22#353812yes—
716C22#354112yes—
717C22#354212yes—
718C4—28no—
719C4—24no—
720C4—24no—
721C4—44no—
722C4—42no—
723C4—42no—
724C4—44no—
725C4—44no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—84no—
731C4—84no—
732C4#256318yes—
733C2—44no—
734C2—82no—
735C2C2×#256311yescharacteristic
736C2#3525311yescharacteristic
737C2#3557411yescharacteristic
738C2#3593911yescharacteristic
739C2#3594811yescharacteristic
740C2#3809812yes—
741C2#3810512yes—
742C2#4248011yescharacteristic
743C2#4280811yescharacteristic
744C2#4507812yes—
745C2#4520812yes—
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
1. C2×#2551C2semidirect
3. #16980C2semidirect
4. #16602C2semidirect
5. #15861C2semidirect
7. #15125C2semidirect
8. #15125C2semidirect
9. #15125C2semidirect
10. #15045C2semidirect
11. #14348C2semidirect
12. #13955C2semidirect
13. #13951C2semidirect
14. #13907C2semidirect
17. #13783C2semidirect
97. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#23522
#169802
#166022
#158612
#154762
#151252
#150452
#143482
#139552
#139512
#139072
#139062
#138472
#137832