Finite Group Databasebeta

← #674409

Subgroups — #674409

Order 512.

4414 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2417C214yesmaximal, complemented
4C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
5C2×C4×#127C211yesmaximal, characteristic
6#15505C218yesmaximal, complemented
7#15177C214yesmaximal, complemented
8#15145C212yesmaximal, complemented
9#15134C212yesmaximal, complemented
10#14714C211yesmaximal, characteristic, complemented
11#14025C212yesmaximal
12#13906C211yesmaximal, characteristic, complemented
13#13906C212yesmaximal, complemented
14C4×#389C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2214yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C22×C4×D4C2214yes—
23C22×C4×D4C2211yescharacteristic
24C23×C4⋊C4C2212yes—
25#2542—28no—
26#2542—28no—
27#2417—232no—
28C2×#399C2214yes—
29C2×#399C2214yes—
30C2×#398C2218yes—
31C2×#396C2218yes—
32D4×C4⋊C4—28no—
33C4×#127—28no—
34C2×#391C2214yes—
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2212yes—
39C2×#390C2214yes—
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#386C2218yes—
44C2×#385C2214yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2212yes—
48C2×#384C2211yescharacteristic
49C2×#380C2212yes—
50C2×#380C2214yes—
51C2×#380C2218yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C43C2211yescharacteristic
56#1444C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1439C2214yes—
60#1438C2218yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C4×#95C2212yes—
68C23×Q8C2312yes—
69C23×D4C2311yescharacteristic
70C23×D4C2314yes—
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2314yes—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×#127—24no—
79C2×#127—24no—
80C2×#127Q818yescomplemented
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—48no—
87C2×C22⋊Q8—48no—
88C2×C4×D4—28no—
89C2×C4×D4—216no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—28no—
93C2×C4×D4—216no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—44no—
98C2×C4×D4—48no—
99C2×C4×D4D418yes—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—216no—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2318yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#399—216no—
135#399—216no—
136#398—232no—
137#396—232no—
138#391—216no—
139#390—24no—
140#390—28no—
141#390—24no—
142#390—216no—
143#390—28no—
144#390—44no—
145#389—28no—
146#389D4116yescomplemented
147#386—232no—
148#385—216no—
149#384—24no—
150#384—28no—
151#384—24no—
152#384—28no—
153#380—28no—
154#380—216no—
155#380—232no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—44no—
158C4×C4⋊C4D418yescomplemented
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C43—24no—
169C25C2412yes—
170C25C2412yes—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—28no—
174C22×Q8—24no—
175C22×Q8—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—28no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—28no—
189C22×D4—22no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4C2×D414yes—
193C22×D4C2×Q814yes—
194C22×D4C2×Q8116yes—
195C22×D4C2×Q814yes—
196C22×D4#4114yes—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—216no—
209C23×C4—44no—
210C23×C4—48no—
211C23×C4C2×D418yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C4⋊Q8—416no—
225#127—232no—
226C22⋊Q8—464no—
227C22⋊Q8—432no—
228C22⋊Q8—432no—
229C4×D4—232no—
230C4×D4—464no—
231C4×D4—432no—
232C4×D4—416no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—432no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D4116yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D418yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—216no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—216no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4C2×D418yes—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—42no—
315C2×C42C2×D418yes—
316C2×C42C2×D418yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×Q814yes—
320C2×C42#4114yes—
321#95—24no—
322#95—216no—
323#95—28no—
324#95—216no—
325#95—216no—
326#95—28no—
327#95—44no—
328#95C2×D418yes—
329#95C2×D4116yes—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24C22×D414yes—
349C24C22×Q814yes—
350C24C22×Q814yes—
351C24C2×#4114yes—
352C24C2511yescharacteristic
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—416no—
357C2×Q8—416no—
358C2×D4—232no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—232no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4C22⋊Q8116yescomplemented
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—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—216no—
409C22×C4—232no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—416no—
426C22×C4—432no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—416no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—816no—
437C22×C4—88no—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D418yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D418yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+414yes—
461C22×C42−1+412yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—216no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—432no—
468C4⋊C4—416no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—416no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—84no—
477C4⋊C4#127116yescomplemented
478C22⋊C4—216no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—432no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C22⋊C4—816no—
487C42—28no—
488C42—216no—
489C42—216no—
490C42—48no—
491C42—44no—
492C42—48no—
493C42#121116yes—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—24no—
510C23—28no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23C2×C22⋊Q818yes—
540C23C23×D411yescharacteristic
541C23C23×D411yescharacteristic
542C23C23×D412yes—
543C23C23×Q811yescharacteristic
544C23C22×#4111yescharacteristic
545C23C22×#4111yescharacteristic
546C23C2×H(5,𝔽2)12yes—
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+412yes—
550C23#58512yes—
551Q8—816no—
552D4—232no—
553D4—464no—
554D4—416no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—432no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—832no—
610C2×C4—816no—
611C2×C4C2×#12114yes—
612C2×C4C2×#12114yes—
613C2×C4C2×#12114yes—
614C2×C4C2×C22⋊Q814yes—
615C2×C4C2×#12718yes—
616C2×C4C2×#12714yes—
617C2×C4#53514yes—
618C2×C4#53714yes—
619C2×C4D4218yes—
620C2×C4#54614yes—
621C2×C4D4×Q818yes—
622C2×C4D4×Q814yes—
623C2×C4#55018yes—
624C2×C4#55214yes—
625C2×C4#55414yes—
626C22—28no—
627C22—216no—
628C22—44no—
629C22—44no—
630C22—48no—
631C22—44no—
632C22—44no—
633C22—48no—
634C22—48no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—48no—
639C22—44no—
640C22—44no—
641C22—48no—
642C22—82no—
643C22—82no—
644C22—84no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22C22×#12111yescharacteristic
649C22C22×C22⋊Q811yescharacteristic
650C22C22×#12711yescharacteristic
651C22C2×#53511yescharacteristic
652C22C2×#53711yescharacteristic
653C22C2×D4212yes—
654C22C2×#54611yescharacteristic
655C22C2×D4×Q811yescharacteristic
656C22C2×D4×Q812yes—
657C22C2×#55012yes—
658C22D4×#4112yes—
659C22C2×#55211yescharacteristic
660C22C2×#55411yescharacteristic
661C22#348912yes—
662C22#349612yes—
663C22#349911yescharacteristic
664C22#349912yes—
665C22#350011yescharacteristic
666C22#350112yes—
667C22#350712yes—
668C22#352012yes—
669C22#354712yes—
670C22#354812yes—
671C4—24no—
672C4—28no—
673C4—28no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—48no—
678C4—82no—
679C4—84no—
680C4—84no—
681C4#241718yescomplemented
682C4#254118yes—
683C2—48no—
684C2—84no—
685C2C2×#241711yescharacteristic
686C2C2×#254111yescharacteristic
687C2#3562111yescharacteristic
688C2D4×C22⋊Q812yes—
689C2#3626011yescharacteristic
690C2Q8×#12711yescharacteristic
691C2#3895111yescharacteristic
692C2#3934012yes—
693C2#3961811yescharacteristic
694C2#5944412yes—
695C2#5982312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2542C2semidirect
2. C2×#2542C2semidirect
3. C2×#2417C2semidirect
4. C2×D4×C4⋊C4C2semidirect
6. #15505C2semidirect
7. #15177C2semidirect
8. #15145C2semidirect
9. #15134C2semidirect
10. #14714C2semidirect
12. #13906C2semidirect
13. #13906C2semidirect
14. C4×#389C2semidirect
80. C2×#127Q8semidirect
146. #389D4semidirect
158. C4×C4⋊C4D4semidirect
378. C2×D4C22⋊Q8semidirect
477. C4⋊C4#127semidirect
681. C4#2417semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24172
C2×D4×C4⋊C42
C2×C4×#1272
#155052
#151772
#151452
#151342
#147142
#140252
#139062
C4×#3892