Finite Group Databasebeta

← #674686

Subgroups — #674686

Order 512.

3446 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2C2×#2504C212yesmaximal, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
4#15554C218yesmaximal, complemented
5#15233C212yesmaximal, complemented
6#15219C212yesmaximal, complemented
7#15132C214yesmaximal, complemented
8#14687C211yesmaximal, characteristic, complemented
9#13970C214yesmaximal, complemented
10#13911C211yesmaximal, characteristic, complemented
11#13906C211yesmaximal, characteristic, complemented
12#13881C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C22×#121C2212yes—
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2212yes—
21#2537—28no—
22#2504—216no—
23C2×#397C2218yes—
24C2×#394C2214yes—
25D4×C4⋊C4—28no—
26C2×#390C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#386C2218yes—
32C2×#385C2214yes—
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#382C2212yes—
36C2×#382C2212yes—
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2214yes—
47C2×C43C2211yescharacteristic
48#1443C2218yes—
49#1442C2218yes—
50#1442C2214yes—
51#1440C2218yes—
52#1440C2218yes—
53#1439C2214yes—
54#1438C2218yes—
55#1437C2214yes—
56#1437C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2214yes—
59C4⋊#95C2214yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2214yes—
64C4⋊#95C2211yescharacteristic
65C22⋊#95C2218yes—
66C23×D4C2311yescharacteristic
67C23×D4C2314yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2312yes—
71C24×C4C2314yes—
72C2×#121—28no—
73C2×#121—28no—
74C2×#121—216no—
75C2×#121—28no—
76C2×#121—44no—
77C2×C4×D4—28no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—48no—
84C2×C4×D4D418yes—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#397—232no—
118#394—216no—
119#390—28no—
120#390—28no—
121#390—24no—
122#390D418yescomplemented
123#389—28no—
124#386—232no—
125#385—216no—
126#384—24no—
127#384—24no—
128#382—28no—
129#382—28no—
130#382—44no—
131#380—28no—
132#380—28no—
133#380—28no—
134#380—48no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—24no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—48no—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C2×#95C2318yes—
152C2×#95C2314yes—
153C43—24no—
154C25C2412yes—
155C25C2412yes—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—216no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—24no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—44no—
169C22×D4—48no—
170C22×D4C2×D414yes—
171C22×D4C2×D414yes—
172C22×D4#4118yes—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—28no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—44no—
187C23×C4—48no—
188C23×C4—48no—
189C23×C4C2×D418yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202#121—432no—
203#121—432no—
204C4×D4—232no—
205C4×D4—432no—
206C4×D4—416no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—432no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D414yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4C2×D418yes—
267C2×C22⋊C4C2×D4116yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—28no—
280C2×C42—24no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C2×D414yes—
284C2×C42C2×D414yes—
285C2×C42#4114yes—
286C2×C42#4114yes—
287C2×C42#4114yes—
288#95—28no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—216no—
293#95—216no—
294#95—28no—
295#95—24no—
296#95—48no—
297#95—44no—
298#95—44no—
299#95—48no—
300#95—48no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24C22×D414yes—
321C24C22×D414yes—
322C24C2×#4118yes—
323C24C2511yescharacteristic
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—216no—
330C2×D4—216no—
331C2×D4—416no—
332C2×D4—432no—
333C2×D4—416no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—88no—
339C2×D4—816no—
340C2×D4#124116yes—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—416no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—416no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—816no—
407C22×C4—88no—
408C22×C4—816no—
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4C22×D414yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)14yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+412yes—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—416no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C22⋊C4—216no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—416no—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—816no—
450C42—28no—
451C42—28no—
452C42—48no—
453C42—48no—
454C42—48no—
455C42—48no—
456C42#123116yes—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23C2×#12418yes—
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C22×#4111yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4112yes—
508C23C22×#4111yescharacteristic
509C23C2×H(5,𝔽2)12yes—
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×2−1+411yescharacteristic
512C23#58512yes—
513C23#58512yes—
514D4—232no—
515D4—416no—
516D4—832no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—432no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—832no—
576C2×C4—816no—
577C2×C4C2×#12314yes—
578C2×C4C2×#12314yes—
579C2×C4C2×#12314yes—
580C2×C4C2×#12414yes—
581C2×C4#53214yes—
582C2×C4#53514yes—
583C2×C4#54214yes—
584C2×C4D4214yes—
585C2×C4#54714yes—
586C2×C4#54714yes—
587C2×C4#55314yes—
588C2×C4#55318yes—
589C2×C4#55318yes—
590C22—28no—
591C22—28no—
592C22—28no—
593C22—48no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—48no—
598C22—48no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—48no—
603C22—48no—
604C22—44no—
605C22—44no—
606C22—84no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—84no—
611C22—82no—
612C22C22×#12311yescharacteristic
613C22C22×#12411yescharacteristic
614C22C2×#53211yescharacteristic
615C22C2×#53511yescharacteristic
616C22C2×#54211yescharacteristic
617C22C2×D4211yescharacteristic
618C22C2×#54711yescharacteristic
619C22C2×#54711yescharacteristic
620C22D4×#4112yes—
621C22C2×#55312yes—
622C22C2×#55311yescharacteristic
623C22C2×#55312yes—
624C22#347512yes—
625C22#349312yes—
626C22#349612yes—
627C22#350011yescharacteristic
628C22#350011yescharacteristic
629C22#350612yes—
630C22#351212yes—
631C22#351412yes—
632C22#351512yes—
633C22#354612yes—
634C22#354712yes—
635C4—24no—
636C4—44no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—84no—
641C4—82no—
642C4—84no—
643C4—84no—
644C4—82no—
645C4#242218yes—
646C4#250418yescomplemented
647C2—48no—
648C2—84no—
649C2C2×#242211yescharacteristic
650C2C2×#250411yescharacteristic
651C2#3561811yescharacteristic
652C2D4×#12411yescharacteristic
653C2#3616411yescharacteristic
654C2#3646812yes—
655C2#3919312yes—
656C2#3936211yescharacteristic
657C2#3949111yescharacteristic
658C2#5921212yes—
659C2#5921312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
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×#2504C2semidirect
3. C2×D4×C4⋊C4C2semidirect
4. #15554C2semidirect
5. #15233C2semidirect
6. #15219C2semidirect
7. #15132C2semidirect
8. #14687C2semidirect
9. #13970C2semidirect
10. #13911C2semidirect
11. #13906C2semidirect
12. #13881C2semidirect
13. C4×#390C2semidirect
14. C4×#390C2semidirect
122. #390D4semidirect
646. C4#2504semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#25042
C2×D4×C4⋊C42
#155542
#152332
#152192
#151322
#146872
#139702
#139112
#139062
#138812
C4×#3902