Finite Group Databasebeta

← #405415

Subgroups — #405415

Order 512.

1956 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#1872C212yesmaximal
3C2×#1872C212yesmaximal
4C2×C4⋊#95C211yesmaximal, characteristic
5#11267C212yesmaximal, complemented
6#11190C214yesmaximal, complemented
7#11010C212yesmaximal, complemented
8#10712C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#103C2214yes—
15C22×#103C2214yes—
16C22×C22⋊C8C2212yes—
17#2857—28no—
18C2×#391C2212yes—
19C2×#389—21no—
20C2×#389—21no—
21C2×#384C2211yescharacteristic
22C2×#380—21no—
23C2×#380—21no—
24C22×#95—21no—
25C22×#95—21no—
26#1872—28no—
27#1872—28no—
28C2×#340C2214yes—
29C2×#340C2214yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C4⋊#95—21no—
33C4⋊#95—21no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C23×Q8C2312yes—
37C24×C4—21no—
38C24×C4—21no—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2314yes—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8D418yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—42no—
63C22×C4⋊C4—42no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4D414yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42—42no—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#103—28no—
78C2×#103—28no—
79C2×#103—28no—
80C2×#103—44no—
81C2×#103—44no—
82C2×#103—44no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86#391—24no—
87#391—24no—
88#389—28no—
89#389—28no—
90#384—24no—
91#380—28no—
92#380—28no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—44no—
100C2×#95—44no—
101#340—28no—
102#340—28no—
103#336—24no—
104#336—42no—
105C25C2×D411yescharacteristic
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8C2×D414yes—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4SD1614yes—
151C23×C4Q1614yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D414yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4114yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C22×C8—24no—
163C22×C8—24no—
164C22×C8—24no—
165C22×C8—24no—
166C22×C8—24no—
167C22×C8—24no—
168C22×C8—24no—
169C4⋊Q8—416no—
170C22⋊Q8—216no—
171C22⋊Q8—216no—
172C22⋊Q8—48no—
173C22⋊Q8—48no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—416no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4#4114yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—416no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42#4114yes—
228#103—416no—
229#103—416no—
230C22⋊C8—48no—
231#95—28no—
232#95—28no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24#12012yes—
246C24#12112yes—
247C24#12412yes—
248C24C2×SD1612yes—
249C24C2×Q1612yes—
250C24#13712yes—
251C24#13712yes—
252C24C22×D411yescharacteristic
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—28no—
256C2×Q8—28no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—44no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—48no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4#12012yes—
358C22×C4#12118yes—
359C22×C4#12112yes—
360C22×C4#12318yes—
361C22×C4#12312yes—
362C22×C4#12312yes—
363C22×C4#12412yes—
364C22×C4C2×SD1612yes—
365C22×C4C2×SD1612yes—
366C22×C4C2×Q1612yes—
367C22×C4C2×Q1612yes—
368C22×C4AGL(1,ℤ/8)12yes—
369C22×C4#13712yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C2×#4112yes—
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4112yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C4H(5,𝔽2)12yes—
378C2×C8—44no—
379C2×C8—44no—
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—44no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—416no—
400C22⋊C4—416no—
401C42—28no—
402C42—44no—
403C42—44no—
404C42—44no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23#39414yes—
434C23#44814yes—
435C23#45114yes—
436C23#46114yes—
437C23#46214yes—
438C23#48714yes—
439C23#49114yes—
440C23C2×#12011yescharacteristic
441C23C2×#12111yescharacteristic
442C23C2×#12311yescharacteristic
443C23#52511yescharacteristic
444C23C2×#12411yescharacteristic
445C23#53411yescharacteristic
446C23#53511yescharacteristic
447C23C22×SD1611yescharacteristic
448C23C22×Q1611yescharacteristic
449C23C2×AGL(1,ℤ/8)11yescharacteristic
450C23C2×#13711yescharacteristic
451C23C2×#13711yescharacteristic
452C23C2×#13711yescharacteristic
453C23#57511yescharacteristic
454C23#57511yescharacteristic
455Q8—48no—
456Q8—48no—
457Q8—84no—
458Q8—84no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—216no—
478C2×C4—216no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—416no—
486C2×C4—416no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—816no—
508C2×C4—816no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#39414yes—
526C2×C4#48314yes—
527C2×C4#48414yes—
528C2×C4C2×#12112yes—
529C2×C4C2×#12312yes—
530C2×C4#52912yes—
531C2×C4#53212yes—
532C2×C4#53412yes—
533C2×C4#53812yes—
534C2×C4#54614yes—
535C2×C4#54614yes—
536C2×C4#55314yes—
537C2×C4#55314yes—
538C8—84no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22#204318yes—
554C22C2×#39411yescharacteristic
555C22#240411yescharacteristic
556C22#240511yescharacteristic
557C22#250411yescharacteristic
558C22#259111yescharacteristic
559C22#261711yescharacteristic
560C22#285411yescharacteristic
561C22C2×#44811yescharacteristic
562C22C2×#45111yescharacteristic
563C22#300611yescharacteristic
564C22#300711yescharacteristic
565C22C2×#46111yescharacteristic
566C22C2×#46211yescharacteristic
567C22#303611yescharacteristic
568C22#303711yescharacteristic
569C22C2×#48311yescharacteristic
570C22C2×#48411yescharacteristic
571C22#309511yescharacteristic
572C22#309511yescharacteristic
573C22#310111yescharacteristic
574C22#310411yescharacteristic
575C22#311011yescharacteristic
576C22#311311yescharacteristic
577C22C2×#48711yescharacteristic
578C22C2×#49111yescharacteristic
579C22#314111yescharacteristic
580C22#314211yescharacteristic
581C22#318711yescharacteristic
582C22#318911yescharacteristic
583C22#319311yescharacteristic
584C22#319511yescharacteristic
585C22#320811yescharacteristic
586C22#320911yescharacteristic
587C22#321511yescharacteristic
588C22#321511yescharacteristic
589C4—24no—
590C4—24no—
591C4—24no—
592C4—42no—
593C4—42no—
594C4—84no—
595C4—84no—
596C4—84no—
597C4—84no—
598C4#244414yes—
599C4#254314yes—
600C2—28no—
601C2#1491311yescharacteristic
602C2C2×#204311yescharacteristic
603C2#2350811yescharacteristic
604C2#2393311yescharacteristic
605C2#2393411yescharacteristic
606C2#2400311yescharacteristic
607C2#2400611yescharacteristic
608C2#2797211yescharacteristic
609C2#2798311yescharacteristic
610C2#2932911yescharacteristic
611C2#2933511yescharacteristic
612C2#2971411yescharacteristic
613C2#2971511yescharacteristic
614C2#3310811yescharacteristic
615C2#3310911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #11267C2semidirect
6. #11190C2semidirect
7. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#18722
C2×C4⋊#952
#112672
#111902
#110102
#107122