Finite Group Databasebeta

← #412363

Subgroups — #412363

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic
2C2×#2036C212yesmaximal
3C2×#2035C212yesmaximal
4C2×#1960C211yesmaximal, characteristic, complemented
5C2×#1956C211yesmaximal, characteristic, complemented
6C2×#1847C211yesmaximal, characteristic, complemented
7C2×#1750C211yesmaximal, characteristic
8#14340C211yesmaximal, characteristic
9#11124C212yesmaximal, complemented
10#11122C212yesmaximal, complemented
11#10842C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2212yes—
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2621—24no—
26#2621—24no—
27C2×#398—22no—
28C2×#397—22no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35#2036—28no—
36#2035—28no—
37#1960—24no—
38#1956—24no—
39#1847—24no—
40#1750—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45#1439—22no—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8D414yes—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—42no—
65C2×C22⋊Q8—42no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—42no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—42no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—42no—
104C2×#103—42no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8D418yes—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97D414yes—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114#398—216no—
115#397—216no—
116#391—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#384—22no—
121#384—22no—
122#382—22no—
123#382—22no—
124#382—22no—
125#382D414yes—
126#380—24no—
127#380—24no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—24no—
134#340—24no—
135#340—24no—
136#336—24no—
137C25C2×D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8C2×D412yes—
157C22×Q8C2×D412yes—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D414yes—
176C23×C4C2×D412yes—
177C23×C4C2411yescharacteristic
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C2×D414yes—
189C22×C8C2×D414yes—
190C22×C8C2×D414yes—
191C22×C8#4114yes—
192C4⋊Q8—28no—
193C4⋊Q8—28no—
194C4⋊Q8—44no—
195C4⋊Q8—44no—
196C22⋊Q8—48no—
197C22⋊Q8—48no—
198C22⋊Q8—48no—
199C22⋊Q8—48no—
200#117—44no—
201#117—44no—
202#117—44no—
203#117—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—84no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—84no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261#107—48no—
262#106—48no—
263#103—48no—
264#103—48no—
265#103—48no—
266#103—48no—
267C22⋊C8—216no—
268#97—28no—
269C4×C8—44no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—44no—
274#95—44no—
275#95—84no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—24no—
283C24—24no—
284C24#12712yes—
285C24C22×D411yescharacteristic
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4#12014yes—
380C22×C4#12114yes—
381C22×C4#12314yes—
382C22×C4#12712yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D412yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)12yes—
392C22×C42−1+412yes—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8#12118yes—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—48no—
414C4⋊C4—416no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—88no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—88no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42#12114yes—
433C42#12114yes—
434C42#12314yes—
435C42#12414yes—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—44no—
442C23—41no—
443C23—41no—
444C23—44no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—44no—
453C23C2×#12011yescharacteristic
454C23C2×#12111yescharacteristic
455C23C2×#12311yescharacteristic
456C23C2×#12711yescharacteristic
457C23#53211yescharacteristic
458C23#53511yescharacteristic
459C23#53611yescharacteristic
460C23#57714yes—
461C23#57814yes—
462Q8—44no—
463Q8—44no—
464Q8—44no—
465Q8—44no—
466Q8—82no—
467Q8—82no—
468Q8—82no—
469Q8—82no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—82no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4#39418yes—
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×#12112yes—
525C2×C4C2×#12111yescharacteristic
526C2×C4C2×#12311yescharacteristic
527C2×C4C2×#12411yescharacteristic
528C2×C4#53212yes—
529C2×C4#53712yes—
530C2×C4#54212yes—
531C2×C4D4212yes—
532C2×C4D4212yes—
533C2×C4#54712yes—
534C2×C4#54712yes—
535C2×C4#54712yes—
536C2×C4#54812yes—
537C2×C4#55312yes—
538C2×C4#55512yes—
539C8—28no—
540C8—44no—
541C22—44no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—44no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22C2×#39411yescharacteristic
551C22#243211yescharacteristic
552C22#243311yescharacteristic
553C22#250011yescharacteristic
554C22#250111yescharacteristic
555C22#262011yescharacteristic
556C22#285611yescharacteristic
557C22#301012yes—
558C22#301112yes—
559C22#306314yes—
560C22#309714yes—
561C22#315412yes—
562C22#315612yes—
563C22#317814yes—
564C22#322112yes—
565C22#322312yes—
566C22#324214yes—
567C4—28no—
568C4—42no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—162no—
575C4—162no—
576C4#241612yes—
577C4#242212yes—
578C4#249112yes—
579C4#253712yes—
580C2—82no—
581C2#1513011yescharacteristic
582C2#2348412yes—
583C2#2568612yes—
584C2#2573311yescharacteristic
585C2#2573511yescharacteristic
586C2#2790212yes—
587C2#2792212yes—
588C2#2985612yes—
589C2#3318812yes—
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
4. C2×#1960C2semidirect
5. C2×#1956C2semidirect
6. C2×#1847C2semidirect
9. #11124C2semidirect
10. #11122C2semidirect
11. #10842C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
C2×#20362
C2×#20352
C2×#19602
C2×#19562
C2×#18472
C2×#17502
#143402
#111242
#111222
#108422