Finite Group Databasebeta

← #732676

Subgroups — #732676

Order 512.

2790 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#17079C212yesmaximal, complemented
4#17008C212yesmaximal, complemented
5#15475C214yesmaximal, complemented
6#15431C214yesmaximal, complemented
7#15216C212yesmaximal, complemented
8#15181C212yesmaximal
9#15125C212yesmaximal, complemented
10#14865C212yesmaximal
11#14348C211yesmaximal, characteristic, complemented
12#13911C211yesmaximal, characteristic, complemented
13#13907C212yesmaximal, complemented
14#13907C212yesmaximal, complemented
15#13905C211yesmaximal, characteristic
16#13847C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2212yes—
23#2550—28no—
24C2×#398C2212yes—
25C2×#395C2212yes—
26C2×#393C2214yes—
27D4×C4⋊C4—28no—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#385C2214yes—
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2214yes—
41C2×#380C2214yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2211yescharacteristic
49#1446C2214yes—
50#1445C2211yescharacteristic
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2214yes—
70#1438C2211yescharacteristic
71#1438C2212yes—
72#1438C2211yescharacteristic
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—44no—
91C2×#125—24no—
92C2×#125—44no—
93C2×C4×D4—24no—
94C2×C4×D4—28no—
95C2×C4×D4—24no—
96C2×C4×D4—48no—
97C2×C4×D4Q818yescomplemented
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#398—28no—
134#395—28no—
135#393—48no—
136#391—28no—
137#389—24no—
138#389—24no—
139#389—28no—
140#389—28no—
141#389—28no—
142#385—216no—
143#384—24no—
144#382—24no—
145#382—44no—
146#380—28no—
147#380—48no—
148#380—48no—
149#380—44no—
150#380—44no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4D418yes—
154C4×C22⋊C4—44no—
155C4×C22⋊C4—44no—
156#376—24no—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25C2412yes—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—44no—
190C22×Q8—42no—
191C22×Q8—44no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4C2×Q814yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×Q818yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C4⋊Q8—416no—
228#125—416no—
229C4×D4—232no—
230C4×D4—432no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—432no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×Q814yes—
276C2×C4⋊C4#4118yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4C2×Q818yes—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—44no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42C2×Q814yes—
315C2×C42#4114yes—
316#95—22no—
317#95—22no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95C2×D418yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24C22×Q814yes—
358C24H(5,𝔽2)14yes—
359C242−1+414yes—
360C24C2511yescharacteristic
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—84no—
364C2×Q8—88no—
365C2×Q8—88no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—28no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
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—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
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—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×Q814yes—
466C22×C4C22×Q812yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4118yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C42−1+412yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—88no—
494C4⋊C4#124116yes—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C42—48no—
504C42—48no—
505C42—44no—
506C42—44no—
507C42—48no—
508C42C22⋊Q8116yes—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23#54418yes—
544C23C23×D411yescharacteristic
545C23C23×Q811yescharacteristic
546C23C22×#4111yescharacteristic
547C23C22×#4112yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)12yes—
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×2−1+412yes—
552C23C2×2−1+411yescharacteristic
553C23#58511yescharacteristic
554C23#58511yescharacteristic
555Q8—816no—
556D4—232no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—432no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—832no—
643C2×C4C2×C22⋊Q814yes—
644C2×C4C2×C22⋊Q814yes—
645C2×C4C2×C22⋊Q814yes—
646C2×C4C2×#12418yes—
647C2×C4C2×#12414yes—
648C2×C4#53514yes—
649C2×C4#55418yes—
650C2×C4#55518yes—
651C22—28no—
652C22—28no—
653C22—28no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22C22×C22⋊Q811yescharacteristic
667C22C22×#12411yescharacteristic
668C22C2×#53511yescharacteristic
669C22C2×#54412yes—
670C22C2×#55412yes—
671C22C2×#55512yes—
672C22#348411yescharacteristic
673C22#348611yescharacteristic
674C22#348811yescharacteristic
675C22#348911yescharacteristic
676C22#349212yes—
677C22#350311yescharacteristic
678C22#350411yescharacteristic
679C22#350611yescharacteristic
680C22#350711yescharacteristic
681C22#351312yes—
682C22#351612yes—
683C22#351912yes—
684C22#352312yes—
685C22#353612yes—
686C22#353612yes—
687C22#354112yes—
688C22#354212yes—
689C4—28no—
690C4—24no—
691C4—24no—
692C4—42no—
693C4—44no—
694C4—44no—
695C4—42no—
696C4—81no—
697C4—82no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4—84no—
702C4—81no—
703C4—82no—
704C4#256318yes—
705C2—48no—
706C2C2×#256311yescharacteristic
707C2#3522411yescharacteristic
708C2#3551911yescharacteristic
709C2#3590311yescharacteristic
710C2#3591011yescharacteristic
711C2#3809912yes—
712C2#3810112yes—
713C2#4247811yescharacteristic
714C2#4281011yescharacteristic
715C2#4492512yes—
716C2#4512212yes—
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×#2550C2semidirect
3. #17079C2semidirect
4. #17008C2semidirect
5. #15475C2semidirect
6. #15431C2semidirect
7. #15216C2semidirect
9. #15125C2semidirect
11. #14348C2semidirect
12. #13911C2semidirect
13. #13907C2semidirect
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×#25502
C2×D4×C4⋊C42
#170792
#170082
#154752
#154312
#152162
#151812
#151252
#148652
#143482
#139112
#139072
#139052
#138472
#137832