Finite Group Databasebeta

← #627985

Subgroups — #627985

Order 512.

3690 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic, complemented
2C2×#2621C211yesmaximal, characteristic, complemented
3C2×#2629C211yesmaximal, characteristic, complemented
4C2×#2293C211yesmaximal, characteristic
5#15913C214yesmaximal
6#15912C214yesmaximal, complemented
7#15880C214yesmaximal, complemented
8#15848C214yesmaximal, complemented
9#15179C212yesmaximal, complemented
10#15177C212yesmaximal
11#14714C211yesmaximal, characteristic, complemented
12#14709C211yesmaximal, characteristic, complemented
13#14340C211yesmaximal, characteristic, complemented
14#14017C212yesmaximal, complemented
15#13969C212yesmaximal, complemented
16C2×#541—28no—
17C2×#541—28no—
18C2×#541—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#126C2214yes—
21C22×#125C2214yes—
22C22×#125C2211yescharacteristic
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2212yes—
25C22×C4×Q8C2212yes—
26C22×C4×D4C2211yescharacteristic
27C22×#117C2211yescharacteristic
28C22×#117C2211yescharacteristic
29#2629—28no—
30#2621—28no—
31C2×#398C2214yes—
32C2×#398C2214yes—
33C2×#397C2214yes—
34C2×#396C2214yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38#2293—28no—
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C43C2211yescharacteristic
52#1444C2214yes—
53#1444C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81#541—2128no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8D418yescomplemented
85C2×#126—216no—
86C2×#126—216no—
87C2×#126—216no—
88C2×#125—216no—
89C2×#125—216no—
90C2×#125—216no—
91C2×#125—24no—
92C2×#125—24no—
93C2×#125D418yescomplemented
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—44no—
100C2×C4×Q8—28no—
101C2×C4×Q8—28no—
102C2×C4×Q8—28no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×#117—24no—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117D418yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2314yes—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#398—216no—
145#398—216no—
146#397—216no—
147#396—216no—
148#391—28no—
149#391—28no—
150#391—44no—
151#390—28no—
152#389—28no—
153#389—28no—
154#388—28no—
155#388—28no—
156#384—24no—
157#382—24no—
158#380—28no—
159#380—44no—
160C4×C4⋊C4—24no—
161C4×C22⋊C4—24no—
162C4×C22⋊C4—24no—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C43—24no—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8C2×D418yes—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—42no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4C2×D414yes—
210C23×C4#4114yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C4⋊Q8—232no—
225#126—2128no—
226#125—2128no—
227#125—232no—
228C22⋊Q8—264no—
229C22⋊Q8—432no—
230C4×Q8—264no—
231C4×D4—232no—
232#117—232no—
233#117—416no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
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—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D4116yes—
275C2×C4⋊C4C2×D4116yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D418yes—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42#4114yes—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24C22×D412yes—
370C24C2×#4112yes—
371C24C2511yescharacteristic
372C2×Q8—216no—
373C2×Q8—216no—
374C2×Q8—216no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×D4—28no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—44no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4C22×D414yes—
467C22×C4C22×D418yes—
468C22×C4C22×D418yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+414yes—
483C22×C42−1+414yes—
484C22×C42−1+412yes—
485C4⋊C4—28no—
486C4⋊C4—232no—
487C4⋊C4—232no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—232no—
491C4⋊C4—232no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—232no—
502C22⋊C4—216no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C42—232no—
510C42—28no—
511C42—28no—
512C42—48no—
513C42—48no—
514C42—44no—
515C42—44no—
516C42#120116yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C23×D411yescharacteristic
564C23C23×D411yescharacteristic
565C23C23×D411yescharacteristic
566C23C22×#4111yescharacteristic
567C23C22×#4111yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×2−1+411yescharacteristic
570C23C2×2−1+412yes—
571C23C2×2−1+412yes—
572C23#58512yes—
573C23#58512yes—
574Q8—432no—
575D4—416no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
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—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4C2×#12018yes—
657C2×C4C2×#12014yes—
658C2×C4#52514yes—
659C2×C4#53714yes—
660C2×C4#54114yescomplemented
661C2×C4#54614yes—
662C2×C4#54714yes—
663C2×C4#54814yes—
664C2×C4#55014yes—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—44no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—82no—
687C22C22×#12011yescharacteristic
688C22C2×#52511yescharacteristic
689C22C2×#53711yescharacteristic
690C22C2×#54111yescharacteristic
691C22#345511yescharacteristic
692C22#346112yes—
693C22C2×#54611yescharacteristic
694C22C2×#54711yescharacteristic
695C22C2×#54811yescharacteristic
696C22C2×#55011yescharacteristic
697C22#348912yes—
698C22#348912yes—
699C22#349512yes—
700C22#349712yes—
701C22#349712yes—
702C22#349912yes—
703C22#350012yes—
704C22#350911yescharacteristic
705C22#351211yescharacteristic
706C22#355212yes—
707C22#355512yes—
708C22#356212yes—
709C22#356312yes—
710C4—28no—
711C4—24no—
712C4—44no—
713C4—44no—
714C4—48no—
715C4—42no—
716C4—42no—
717C4—48no—
718C4—42no—
719C4—82no—
720C4—84no—
721C4—84no—
722C4#240718yes—
723C2—82no—
724C2—82no—
725C2C2×#240711yescharacteristic
726C2#3530911yescharacteristic
727C2#3531011yescharacteristic
728C2#3566012yes—
729C2#3737812yes—
730C2#3937311yescharacteristic
731C2#3937911yescharacteristic
732C2#3938211yescharacteristic
733C2#3938511yescharacteristic
734C2#3941712yes—
735C2#3942712yes—
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. C22×#541C2semidirect
2. C2×#2621C2semidirect
3. C2×#2629C2semidirect
6. #15912C2semidirect
7. #15880C2semidirect
8. #15848C2semidirect
9. #15179C2semidirect
11. #14714C2semidirect
12. #14709C2semidirect
13. #14340C2semidirect
14. #14017C2semidirect
15. #13969C2semidirect
84. C2×C4⋊Q8D4semidirect
93. C2×#125D4semidirect
660. C2×C4#541semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#26212
C2×#26292
C2×#22932
#159132
#159122
#158802
#158482
#151792
#151772
#147142
#147092
#143402
#140172
#139692