Finite Group Databasebeta

← #412355

Subgroups — #412355

Order 512.

1894 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2619C211yesmaximal, characteristic
2C2×#2034C212yesmaximal, complemented
3C2×#2033C212yesmaximal, complemented
4C2×#1961C211yesmaximal, characteristic, complemented
5C2×#1957C211yesmaximal, characteristic, complemented
6C2×#1846C211yesmaximal, characteristic, complemented
7C2×#1750C211yesmaximal, characteristic, complemented
8#14340C211yesmaximal, characteristic, complemented
9#11123C212yesmaximal, complemented
10#11121C212yesmaximal, complemented
11#10843C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#121C2212yes—
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2619—24no—
26#2619—24no—
27C2×#394—22no—
28C2×#392—22no—
29C2×#390C2212yes—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34#2034—28no—
35#2033—28no—
36#1961—24no—
37#1957—24no—
38#1846—24no—
39#1750—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44#1441—22no—
45#1439—22no—
46C4⋊#95C2212yes—
47C23×D4—22no—
48C23×D4—22no—
49C23×D4C2312yes—
50C24×C4C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—22no—
59C2×#127—22no—
60C2×#127D414yes—
61C2×#121—24no—
62C2×#121—24no—
63C2×#121—24no—
64C2×#121—24no—
65C2×#121—42no—
66C2×#121—42no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—42no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—42no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—42no—
101C2×#102—42no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8D414yescomplemented
107C2×C22⋊C8D414yescomplemented
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8D414yescomplemented
114#394—216no—
115#392—216no—
116#390—24no—
117#390—24no—
118#384—22no—
119#384—22no—
120#382—22no—
121#382—22no—
122#382—22no—
123#382D414yes—
124#380—24no—
125#380—24no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—24no—
130#340—24no—
131#340—24no—
132#336—22no—
133#336—22no—
134C25—22no—
135C25C2×D411yescharacteristic
136C22×D4—28no—
137C22×D4—28no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—84no—
156C22×D4—84no—
157C22×D4—84no—
158C22×D4—84no—
159C22×D4C2×D414yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2411yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8#4112yes—
208C22×C8#4112yes—
209#127—28no—
210#127—28no—
211#127—44no—
212#127—44no—
213#121—48no—
214#121—48no—
215#121—48no—
216#121—48no—
217#117—44no—
218#117—44no—
219#117—44no—
220#117—44no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
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—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—84no—
243C2×C4⋊C4C2×D414yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—84no—
253C2×C22⋊C4—84no—
254C2×C22⋊C4—84no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C2×C42#4112yes—
267#107—48no—
268#106—48no—
269#102—48no—
270#102—48no—
271#102—48no—
272#102—48no—
273C22⋊C8—28no—
274C22⋊C8—28no—
275#97—44no—
276C4×C8—28no—
277#95—28no—
278#95—44no—
279#95—44no—
280#95—84no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24—44no—
300C24#12712yes—
301C24C22×D411yescharacteristic
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—416no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—416no—
332C2×D4—816no—
333C2×D4—816no—
334C2×D4—88no—
335C2×D4—88no—
336C2×D4—816no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—816no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
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—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
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—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
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×C4#12114yes—
398C22×C4#12314yes—
399C22×C4#12712yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413C22×C42−1+411yescharacteristic
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8#12114yes—
441C2×C8#12114yes—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—48no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—88no—
456C22⋊C4—416no—
457C22⋊C4—48no—
458C22⋊C4—816no—
459C22⋊C4—88no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42#12114yes—
465C42#12114yes—
466C42#12314yes—
467C42#12414yes—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—44no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—44no—
480C23—44no—
481C23—41no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—48no—
489C23—48no—
490C23—84no—
491C23—84no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23—82no—
505C23—82no—
506C23—84no—
507C23—84no—
508C23C2×#12011yescharacteristic
509C23C2×#12111yescharacteristic
510C23C2×#12311yescharacteristic
511C23C2×#12711yescharacteristic
512C23#53211yescharacteristic
513C23#53511yescharacteristic
514C23#53611yescharacteristic
515C23#57614yes—
516C23#57714yes—
517D4—44no—
518D4—44no—
519D4—44no—
520D4—44no—
521D4—832no—
522D4—832no—
523D4—82no—
524D4—82no—
525D4—82no—
526D4—82no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—84no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—84no—
562C2×C4—82no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4#39418yes—
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12111yescharacteristic
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×#12311yescharacteristic
572C2×C4C2×#12411yescharacteristic
573C2×C4#53211yescharacteristic
574C2×C4#53211yescharacteristic
575C2×C4#53711yescharacteristic
576C2×C4#53711yescharacteristic
577C2×C4#54211yescharacteristic
578C2×C4#54211yescharacteristic
579C2×C4D4211yescharacteristic
580C2×C4D4211yescharacteristic
581C2×C4D4211yescharacteristic
582C2×C4D4211yescharacteristic
583C2×C4#54711yescharacteristic
584C2×C4#54711yescharacteristic
585C2×C4#54711yescharacteristic
586C2×C4#54711yescharacteristic
587C2×C4#54711yescharacteristic
588C2×C4#54711yescharacteristic
589C2×C4#54811yescharacteristic
590C2×C4#54811yescharacteristic
591C2×C4#55311yescharacteristic
592C2×C4#55311yescharacteristic
593C2×C4#55511yescharacteristic
594C2×C4#55511yescharacteristic
595C8—24no—
596C8—24no—
597C8—42no—
598C8—42no—
599C22—44no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—84no—
609C22—88no—
610C22—84no—
611C22—84no—
612C22—88no—
613C22—162no—
614C22—162no—
615C22—164no—
616C22—162no—
617C22—162no—
618C22—164no—
619C22C2×#39411yescharacteristic
620C22#243211yescharacteristic
621C22#243311yescharacteristic
622C22#250011yescharacteristic
623C22#250111yescharacteristic
624C22#262011yescharacteristic
625C22#285611yescharacteristic
626C22#301214yes—
627C22#305712yes—
628C22#305812yes—
629C22#309614yes—
630C22#315412yes—
631C22#315512yes—
632C22#317412yes—
633C22#317612yes—
634C22#322414yes—
635C22#324114yes—
636C4—28no—
637C4—42no—
638C4—84no—
639C4—82no—
640C4—162no—
641C4—162no—
642C4#241611yescharacteristic
643C4#241611yescharacteristic
644C4#242211yescharacteristic
645C4#242211yescharacteristic
646C4#249111yescharacteristic
647C4#249111yescharacteristic
648C4#253711yescharacteristic
649C4#253711yescharacteristic
650C2—82no—
651C2—164no—
652C2#1513011yescharacteristic
653C2#2348612yes—
654C2#2568212yes—
655C2#2573612yes—
656C2#2790112yes—
657C2#2792312yes—
658C2#2984712yes—
659C2#3317811yescharacteristic
660C2#3317911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2619C2
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×#2034C2semidirect
3. C2×#2033C2semidirect
4. C2×#1961C2semidirect
5. C2×#1957C2semidirect
6. C2×#1846C2semidirect
7. C2×#1750C2semidirect
8. #14340C2semidirect
9. #11123C2semidirect
10. #11121C2semidirect
11. #10843C2semidirect
106. C2×C22⋊C8D4semidirect
107. C2×C22⋊C8D4semidirect
113. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26192
C2×#20342
C2×#20332
C2×#19612
C2×#19572
C2×#18462
C2×#17502
#143402
#111232
#111212
#108432