Finite Group Databasebeta

← #674623

Subgroups — #674623

Order 512.

3458 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C212yesmaximal
2C2×#2547C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic, complemented
4C2×D4×C4⋊C4C211yesmaximal, characteristic
5C2×C4×#124C211yesmaximal, characteristic
6#15472C218yesmaximal, complemented
7#15237C212yesmaximal, complemented
8#15188C212yesmaximal, complemented
9#15132C214yesmaximal, complemented
10#14714C211yesmaximal, characteristic, complemented
11#14027C212yesmaximal
12#13926C211yesmaximal, characteristic, complemented
13#13907C214yesmaximal, complemented
14C4×#389C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4⋊Q8C2212yes—
17C22×#124C2212yes—
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23#2547—216no—
24#2547—28no—
25#2546—28no—
26C2×#398C2214yes—
27C2×#397C2218yes—
28C2×#397C2214yes—
29D4×C4⋊C4—28no—
30C4×#124—28no—
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#390C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#385C2214yes—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2214yes—
48C2×#380C2218yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2214yes—
53C2×C43C2211yescharacteristic
54#1443C2218yes—
55#1442C2218yes—
56#1441C2218yes—
57#1441C2218yes—
58#1441C2214yes—
59#1440C2218yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2214yes—
66C22⋊#95C2218yes—
67C4×#95C2212yes—
68C23×Q8C2312yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2314yes—
73C2×C4⋊Q8—28no—
74C2×C4⋊Q8—28no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—44no—
79C2×#124—24no—
80C2×#124—24no—
81C2×#124—28no—
82C2×#124—28no—
83C2×#124—28no—
84C2×#124D418yes—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×D4—28no—
89C2×C4×D4—24no—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×C4×D4—44no—
93C2×C4×D4D418yes—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—216no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#398—216no—
125#397—232no—
126#397—216no—
127#391—28no—
128#391—28no—
129#391—24no—
130#391—28no—
131#391—28no—
132#391—44no—
133#390—24no—
134#389—28no—
135#389—28no—
136#389—28no—
137#389—28no—
138#388—216no—
139#385—216no—
140#384—24no—
141#382—24no—
142#380—216no—
143#380—28no—
144#380—232no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4D418yes—
148C4×C22⋊C4—216no—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2318yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2312yes—
164C43—24no—
165C25C2412yes—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8C2×D414yes—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4C2×D414yes—
185C22×D4C2×D414yes—
186C22×D4#4114yes—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—216no—
194C23×C4—44no—
195C23×C4—48no—
196C23×C4C2×D418yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C4⋊Q8—432no—
211C4⋊Q8—416no—
212#124—232no—
213#124—432no—
214C4×Q8—416no—
215C4×D4—232no—
216C4×D4—432no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—432no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4#4118yes—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D418yes—
272C2×C22⋊C4C2×D4116yes—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—42no—
292C2×C42C2×D418yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42#4114yes—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—28no—
302#95—216no—
303#95—216no—
304#95—216no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—48no—
310#95—48no—
311#95C2×D418yes—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24C22×D414yes—
321C24C22×D414yes—
322C24C2×#4114yes—
323C24C2511yescharacteristic
324C2×Q8—28no—
325C2×Q8—28no—
326C2×Q8—28no—
327C2×Q8—44no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—44no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—44no—
335C2×Q8—48no—
336C2×Q8—44no—
337C2×Q8—48no—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4#121116yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—216no—
377C22×C4—232no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—416no—
403C22×C4—432no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—88no—
409C22×C4—816no—
410C22×C4C22×D414yes—
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4118yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4114yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+414yes—
430C22×C42−1+414yes—
431C22×C42−1+412yes—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—416no—
435C4⋊C4—416no—
436C4⋊C4—416no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—416no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—84no—
446C4⋊C4#124116yescomplemented
447C22⋊C4—232no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—432no—
452C42—28no—
453C42—216no—
454C42—48no—
455C42—48no—
456C42—48no—
457C42—44no—
458C42—48no—
459C42#121116yes—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—24no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23C2×#12118yes—
483C23C23×D411yescharacteristic
484C23C23×D411yescharacteristic
485C23C23×D411yescharacteristic
486C23C22×#4111yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4112yes—
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×2−1+412yes—
491C23C2×2−1+411yescharacteristic
492C23C2×2−1+412yes—
493C23#58512yes—
494Q8—416no—
495Q8—816no—
496D4—232no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—432no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
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—832no—
562C2×C4C2×#12114yes—
563C2×C4C2×#12114yes—
564C2×C4C2×#12114yes—
565C2×C4C2×#12114yes—
566C2×C4C2×#12418yes—
567C2×C4C2×#12414yes—
568C2×C4#53614yes—
569C2×C4#53714yes—
570C2×C4D4214yes—
571C2×C4#54714yes—
572C2×C4#54718yes—
573C2×C4#54814yes—
574C2×C4#55314yes—
575C2×C4#55618yes—
576C22—28no—
577C22—216no—
578C22—44no—
579C22—44no—
580C22—48no—
581C22—48no—
582C22—44no—
583C22—48no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22C22×#12111yescharacteristic
588C22C22×#12111yescharacteristic
589C22C22×#12411yescharacteristic
590C22C2×#53611yescharacteristic
591C22C2×#53711yescharacteristic
592C22C2×D4211yescharacteristic
593C22C2×#54712yes—
594C22C2×#54711yescharacteristic
595C22C2×#54811yescharacteristic
596C22D4×#4112yes—
597C22C2×#55311yescharacteristic
598C22C2×#55612yes—
599C22#348212yes—
600C22#348912yes—
601C22#349912yes—
602C22#349911yescharacteristic
603C22#350011yescharacteristic
604C22#350712yes—
605C22#351412yes—
606C22#351512yes—
607C22#352312yes—
608C22#355212yes—
609C22#355512yes—
610C4—28no—
611C4—24no—
612C4—44no—
613C4—44no—
614C4—44no—
615C4—44no—
616C4—44no—
617C4—84no—
618C4—82no—
619C4—84no—
620C4—84no—
621C4#241618yes—
622C4#254718yescomplemented
623C2—48no—
624C2C2×#241611yescharacteristic
625C2C2×#254711yescharacteristic
626C2#3562111yescharacteristic
627C2D4×#12411yescharacteristic
628C2#3625912yes—
629C2#3644611yescharacteristic
630C2#3892312yes—
631C2#3917311yescharacteristic
632C2#3936011yescharacteristic
633C2#5942812yes—
634C2#5943712yes—
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
3. C2×#2546C2semidirect
6. #15472C2semidirect
7. #15237C2semidirect
8. #15188C2semidirect
9. #15132C2semidirect
10. #14714C2semidirect
12. #13926C2semidirect
13. #13907C2semidirect
14. C4×#389C2semidirect
446. C4⋊C4#124semidirect
622. C4#2547semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#25462
C2×D4×C4⋊C42
C2×C4×#1242
#154722
#152372
#151882
#151322
#147142
#140272
#139262
#139072
C4×#3892