Finite Group Databasebeta

← #400856

Subgroups — #400856

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2071C211yesmaximal, characteristic
2C2×#1936C212yesmaximal
3C2×#1767C211yesmaximal, characteristic, complemented
4#14712C211yesmaximal, characteristic, complemented
5#14700C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic, complemented
7#11267C211yesmaximal, characteristic, complemented
8#11228C212yesmaximal, complemented
9#11076C212yesmaximal
10#11049C211yesmaximal, characteristic
11#10881C212yesmaximal, complemented
12C22×C4⋊C8C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15C22×#97C2212yes—
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#389—21no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#388—21no—
24C2×#388—21no—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C43C2211yescharacteristic
28#2071—24no—
29#1936—28no—
30#1767—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1437—21no—
40#1437—21no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C4⋊#95C2211yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—42no—
82C2×#97—24no—
83C2×#97—24no—
84C2×#97—42no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390—24no—
89#390D414yes—
90#389—24no—
91#389—24no—
92#389—24no—
93#389—44no—
94#389—44no—
95#388—44no—
96#388—44no—
97#384—24no—
98#384—22no—
99#384—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C43—22no—
109C43—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—42no—
116#336—22no—
117#336D414yes—
118C25—21no—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4C2×D412yes—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C2×D414yes—
154C22×C8#4114yes—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4#4114yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4114yes—
241C4⋊C8—216no—
242#102—216no—
243#102—48no—
244#97—48no—
245#95—42no—
246#95—42no—
247#95—44no—
248#95—44no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—44no—
253C24—22no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24C22×D411yescharacteristic
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
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—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#12412yes—
355C22×C4#12412yes—
356C22×C4AGL(1,ℤ/8)12yes—
357C22×C4AGL(1,ℤ/8)12yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4AGL(1,ℤ/8)12yes—
360C22×C4#13712yes—
361C22×C4#13712yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+412yes—
371C22×C42−1+411yescharacteristic
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4—88no—
405C4⋊C4—84no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C42—24no—
413C42—24no—
414C42—22no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—24no—
419C42—22no—
420C42—44no—
421C42—44no—
422C42—44no—
423C42—44no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23C2×#12111yescharacteristic
460C23C2×#12111yescharacteristic
461C23#52511yescharacteristic
462C23C2×#12411yescharacteristic
463C23#53211yescharacteristic
464C23#53611yescharacteristic
465C23#53711yescharacteristic
466C23C2×AGL(1,ℤ/8)11yescharacteristic
467C23C2×AGL(1,ℤ/8)11yescharacteristic
468C23C2×AGL(1,ℤ/8)11yescharacteristic
469C23C2×AGL(1,ℤ/8)11yescharacteristic
470C23C2×#13711yescharacteristic
471C23C2×#13711yescharacteristic
472C23#57511yescharacteristic
473C23#57511yescharacteristic
474D4—44no—
475D4—44no—
476D4—82no—
477D4—82no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4#46814yes—
553C2×C4#46914yes—
554C2×C4#48914yes—
555C2×C4#52512yes—
556C2×C4#52512yes—
557C2×C4#52912yes—
558C2×C4#52912yes—
559C2×C4#53912yes—
560C2×C4#54112yes—
561C2×C4D4211yescharacteristic
562C2×C4#54712yes—
563C2×C4#54712yes—
564C2×C4#54711yescharacteristic
565C2×C4#54811yescharacteristic
566C2×C4#55212yes—
567C2×C4#55311yescharacteristic
568C2×C4#55612yes—
569C2×C4#55612yes—
570C2×C4#55912yes—
571C8—44no—
572C8—44no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—161no—
581C22—161no—
582C22—161no—
583C22—161no—
584C22—161no—
585C22—161no—
586C22—161no—
587C22—161no—
588C22#239211yescharacteristic
589C22#241311yescharacteristic
590C22#241611yescharacteristic
591C22#254711yescharacteristic
592C22#259211yescharacteristic
593C22#266411yescharacteristic
594C22#285011yescharacteristic
595C22#303611yescharacteristic
596C22#303611yescharacteristic
597C22#303811yescharacteristic
598C22#303911yescharacteristic
599C22C2×#46811yescharacteristic
600C22C2×#46911yescharacteristic
601C22#305511yescharacteristic
602C22#305611yescharacteristic
603C22#310911yescharacteristic
604C22#311211yescharacteristic
605C22#311411yescharacteristic
606C22#311711yescharacteristic
607C22C2×#48911yescharacteristic
608C22#313511yescharacteristic
609C22#313811yescharacteristic
610C22#314111yescharacteristic
611C22#318011yescharacteristic
612C22#318111yescharacteristic
613C22#318211yescharacteristic
614C22#318411yescharacteristic
615C22#323411yescharacteristic
616C22#323411yescharacteristic
617C22#323811yescharacteristic
618C22#323911yescharacteristic
619C22#325111yescharacteristic
620C22#325211yescharacteristic
621C22#325311yescharacteristic
622C22#325411yescharacteristic
623C4—24no—
624C4—24no—
625C4—24no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—82no—
630C4—84no—
631C4—161no—
632C4—161no—
633C4—162no—
634C4#245512yes—
635C4#246912yes—
636C4#247812yes—
637C4#259612yes—
638C2—162no—
639C2#1480811yescharacteristic
640C2#2368211yescharacteristic
641C2#2368311yescharacteristic
642C2#2460711yescharacteristic
643C2#2460811yescharacteristic
644C2#2475811yescharacteristic
645C2#2475911yescharacteristic
646C2#2872211yescharacteristic
647C2#2872611yescharacteristic
648C2#2935211yescharacteristic
649C2#2935411yescharacteristic
650C2#3038311yescharacteristic
651C2#3038411yescharacteristic
652C2#3304811yescharacteristic
653C2#3305111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1767C2semidirect
4. #14712C2semidirect
6. #11357C2semidirect
7. #11267C2semidirect
8. #11228C2semidirect
11. #10881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20712
C2×#19362
C2×#17672
#147122
#147002
#113572
#112672
#112282
#110762
#110492
#108812