Finite Group Databasebeta

← #396499

Subgroups — #396499

Order 512.

2032 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2310C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#2095C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal
5C2×#1787C212yesmaximal
6#14340C211yesmaximal, characteristic
7#11124C212yesmaximal, complemented
8#11122C212yesmaximal, complemented
9#11052C211yesmaximal, characteristic, complemented
10#11052C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2212yes—
22#2310—24no—
23#2310—24no—
24C2×#391—22no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#381—22no—
32C2×#380C2212yes—
33#2096—24no—
34#2095—24no—
35#1854—28no—
36#1787—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1441—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C23×Q8—22no—
46C23×Q8—22no—
47C23×Q8—22no—
48C23×Q8—22no—
49C23×Q8—21no—
50C23×Q8—22no—
51C23×Q8—24no—
52C23×Q8—24no—
53C23×Q8—42no—
54C23×Q8—42no—
55C23×Q8Q814yes—
56C23×Q8C2311yescharacteristic
57C23×Q8C2311yescharacteristic
58C24×C4—21no—
59C24×C4C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2312yes—
62C2×#117—24no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—24no—
68C2×#117—42no—
69C2×#117—42no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D412yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—42no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8D418yes—
103#391—216no—
104#391—22no—
105#391—22no—
106#391—22no—
107#391—22no—
108#391—22no—
109#391—22no—
110#391D414yes—
111#391D414yes—
112#384—22no—
113#384—22no—
114#382—42no—
115#381—216no—
116#380—24no—
117#380—24no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—24no—
125#340—24no—
126#340—24no—
127#336—22no—
128#336—22no—
129#336—42no—
130C25C2×D411yescharacteristic
131C22×Q8—24no—
132C22×Q8—28no—
133C22×Q8—28no—
134C22×Q8—24no—
135C22×Q8—28no—
136C22×Q8—28no—
137C22×Q8—24no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—216no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—48no—
154C22×Q8—48no—
155C22×Q8—48no—
156C22×Q8—48no—
157C22×Q8—48no—
158C22×Q8—48no—
159C22×Q8—48no—
160C22×Q8—48no—
161C22×Q8—48no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—48no—
165C22×Q8—42no—
166C22×Q8—48no—
167C22×Q8—42no—
168C22×Q8C2×D412yes—
169C22×Q8C2×D412yes—
170C22×Q8C2×Q812yes—
171C22×Q8C2×Q812yes—
172C22×Q8#4112yes—
173C22×Q8#4112yes—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D414yes—
197C23×C4C2×Q812yes—
198C23×C4#4114yes—
199C23×C4#4112yes—
200C23×C4C2411yescharacteristic
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8C2×D414yes—
214C22×C8C2×D414yes—
215#117—48no—
216#117—48no—
217#117—48no—
218#117—48no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—84no—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—84no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263#107—48no—
264#106—48no—
265#103—48no—
266#103—48no—
267#103—48no—
268#103—48no—
269C22⋊C8—216no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—44no—
275#95—44no—
276#95—84no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—24no—
282C24—24no—
283C24C22⋊Q812yes—
284C24#12312yes—
285C24#12712yes—
286C24C22×D411yescharacteristic
287C2×Q8—216no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—432no—
307C2×Q8—432no—
308C2×Q8—48no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—48no—
312C2×Q8—48no—
313C2×Q8—432no—
314C2×Q8—432no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—48no—
328C2×Q8C22⋊Q814yes—
329C2×Q8C22⋊Q814yes—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—216no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—48no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4#12014yes—
397C22×C4C22⋊Q818yes—
398C22×C4C22⋊Q812yes—
399C22×C4#12318yes—
400C22×C4#12312yes—
401C22×C4#12712yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C2×#4112yes—
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)12yes—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—48no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—88no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—88no—
443C42—24no—
444C42—24no—
445C42—48no—
446C42—42no—
447C42—42no—
448C23—28no—
449C23—28no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23#39914yes—
465C23C2×#12011yescharacteristic
466C23C2×C22⋊Q811yescharacteristic
467C23C2×#12311yescharacteristic
468C23#52511yescharacteristic
469C23C2×#12711yescharacteristic
470C23#53411yescharacteristic
471C23#53511yescharacteristic
472C23#57714yes—
473C23#57814yes—
474Q8—28no—
475Q8—28no—
476Q8—432no—
477Q8—44no—
478Q8—44no—
479Q8—44no—
480Q8—44no—
481Q8—44no—
482Q8—44no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—432no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—82no—
536C2×C4#39914yes—
537C2×C4C2×C22⋊Q811yescharacteristic
538C2×C4C2×C22⋊Q811yescharacteristic
539C2×C4C2×C22⋊Q812yes—
540C2×C4C2×#12312yes—
541C2×C4#52511yescharacteristic
542C2×C4#52511yescharacteristic
543C2×C4#53812yes—
544C2×C4#54312yes—
545C2×C4D4212yes—
546C2×C4D4212yes—
547C2×C4#54612yes—
548C2×C4#54612yes—
549C2×C4#54612yes—
550C2×C4#54612yes—
551C2×C4#55112yes—
552C2×C4#55112yes—
553C8—44no—
554C8—44no—
555C22—24no—
556C22—28no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22C2×#39911yescharacteristic
565C22#240511yescharacteristic
566C22#243311yescharacteristic
567C22#248211yescharacteristic
568C22#253911yescharacteristic
569C22#262611yescharacteristic
570C22#284811yescharacteristic
571C22#301012yes—
572C22#301112yes—
573C22#308214yes—
574C22#309714yes—
575C22#312114yes—
576C22#315412yes—
577C22#315612yes—
578C22#320114yes—
579C22#322112yes—
580C22#322312yes—
581C4—24no—
582C4—44no—
583C4—44no—
584C4—44no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—162no—
589C4—162no—
590C4#241812yes—
591C4#241812yes—
592C4#243712yes—
593C4#243712yes—
594C2—44no—
595C2#1473411yescharacteristic
596C2#2373612yes—
597C2#2401412yes—
598C2#2573311yescharacteristic
599C2#2573511yescharacteristic
600C2#2759612yes—
601C2#2832412yes—
602C2#2996312yes—
603C2#3296412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2096C2semidirect
3. C2×#2095C2semidirect
7. #11124C2semidirect
8. #11122C2semidirect
9. #11052C2semidirect
10. #11052C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23102
C2×#20962
C2×#20952
C2×#18542
C2×#17872
#143402
#111242
#111222
#110522
#107132