Finite Group Databasebeta

← #404767

Subgroups — #404767

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2023C211yesmaximal, characteristic
2C2×#1744C211yesmaximal, characteristic, complemented
3#14714C211yesmaximal, characteristic, complemented
4#14700C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11315C212yesmaximal, complemented
7#11267C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal
9#11049C211yesmaximal, characteristic
10#10925C212yesmaximal
11#10880C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#102C2212yes—
14C22×#102C2212yes—
15C22×C4×C8C2212yes—
16C2×#391—21no—
17C2×#391—21no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388—21no—
23C2×#388—21no—
24C2×#384C2211yescharacteristic
25C2×C43C2211yescharacteristic
26#2023—24no—
27#1744—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1437—21no—
38#1437—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—42no—
80C2×#102—42no—
81C2×C4×C8—24no—
82C2×C4×C8—24no—
83C2×C4×C8—24no—
84#391—44no—
85#391—44no—
86#390—22no—
87#390—22no—
88#390—22no—
89#390—22no—
90#390—24no—
91#389—24no—
92#388—44no—
93#388—44no—
94#384—42no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C43—22no—
105C43—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—22no—
114#336D414yes—
115C25—21no—
116C22×Q8—24no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4#4112yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
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×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C2×D414yes—
158C22×C8#4114yes—
159C4⋊Q8—48no—
160C4⋊Q8—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4#4114yes—
196C2×C4⋊C4#4112yes—
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—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42D814yes—
237C2×C42SD1614yes—
238C2×C42C2×D412yes—
239C2×C42#4112yes—
240C2×C42#4114yes—
241#102—48no—
242#102—48no—
243C4×C8—216no—
244#95—44no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—44no—
249#95—44no—
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×Q8—48no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—84no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8—84no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—41no—
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—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#12412yes—
358C22×C4#12412yes—
359C22×C4C2×D812yes—
360C22×C4C2×D812yes—
361C22×C4C2×D812yes—
362C22×C4C2×SD1612yes—
363C22×C4C2×SD1612yes—
364C22×C4C2×SD1612yes—
365C22×C4#13512yes—
366C22×C4#13512yes—
367C22×C4AGL(1,ℤ/8)12yes—
368C22×C4#13712yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4112yes—
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4H(5,𝔽2)12yes—
377C22×C42−1+411yescharacteristic
378C22×C42−1+411yescharacteristic
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—48no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—84no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—28no—
422C42—28no—
423C42—44no—
424C42—44no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
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—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23C2×#12111yescharacteristic
465C23#52511yescharacteristic
466C23#52511yescharacteristic
467C23C2×#12411yescharacteristic
468C23#53211yescharacteristic
469C23#53511yescharacteristic
470C23#53611yescharacteristic
471C23C22×D811yescharacteristic
472C23C22×SD1611yescharacteristic
473C23C2×#13511yescharacteristic
474C23C2×#13511yescharacteristic
475C23C2×AGL(1,ℤ/8)11yescharacteristic
476C23C2×#13711yescharacteristic
477C23#57511yescharacteristic
478C23#57511yescharacteristic
479Q8—816no—
480D4—44no—
481D4—44no—
482D4—82no—
483D4—82no—
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—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4#46514yes—
558C2×C4#46614yes—
559C2×C4#48614yes—
560C2×C4#48914yes—
561C2×C4#52512yes—
562C2×C4#52912yes—
563C2×C4#52912yes—
564C2×C4#52912yes—
565C2×C4#54212yes—
566C2×C4#54212yes—
567C2×C4#54612yes—
568C2×C4#54711yescharacteristic
569C2×C4#54811yescharacteristic
570C2×C4#55112yes—
571C2×C4#55112yes—
572C2×C4#55212yes—
573C2×C4#55211yescharacteristic
574C2×C4#55512yes—
575C2×C4#55612yes—
576C2×C4#55611yescharacteristic
577C8—44no—
578C8—44no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—161no—
587C22—161no—
588C22—161no—
589C22—161no—
590C22—161no—
591C22—161no—
592C22—161no—
593C22—161no—
594C22#239811yescharacteristic
595C22#241111yescharacteristic
596C22#241511yescharacteristic
597C22#260411yescharacteristic
598C22#261011yescharacteristic
599C22#264011yescharacteristic
600C22#285011yescharacteristic
601C22C2×#46511yescharacteristic
602C22C2×#46611yescharacteristic
603C22#305511yescharacteristic
604C22#305611yescharacteristic
605C22#310011yescharacteristic
606C22#310011yescharacteristic
607C22#310111yescharacteristic
608C22#310311yescharacteristic
609C22#310511yescharacteristic
610C22#310611yescharacteristic
611C22#310711yescharacteristic
612C22#310811yescharacteristic
613C22C2×#48611yescharacteristic
614C22#313111yescharacteristic
615C22C2×#48911yescharacteristic
616C22#313511yescharacteristic
617C22#316711yescharacteristic
618C22#316911yescharacteristic
619C22#318011yescharacteristic
620C22#318111yescharacteristic
621C22#320511yescharacteristic
622C22#320611yescharacteristic
623C22#321111yescharacteristic
624C22#321111yescharacteristic
625C22#322611yescharacteristic
626C22#322711yescharacteristic
627C22#323311yescharacteristic
628C22#323311yescharacteristic
629C4—24no—
630C4—24no—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—84no—
637C4—161no—
638C4—162no—
639C4—161no—
640C4#247412yes—
641C4#247712yes—
642C4#252712yes—
643C4#253212yes—
644C2—162no—
645C2#1490211yescharacteristic
646C2#2381511yescharacteristic
647C2#2381711yescharacteristic
648C2#2450011yescharacteristic
649C2#2450311yescharacteristic
650C2#2470011yescharacteristic
651C2#2470111yescharacteristic
652C2#2951311yescharacteristic
653C2#2951411yescharacteristic
654C2#2955211yescharacteristic
655C2#2955411yescharacteristic
656C2#3020311yescharacteristic
657C2#3020811yescharacteristic
658C2#3304211yescharacteristic
659C2#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
2. C2×#1744C2semidirect
3. #14714C2semidirect
5. #11384C2semidirect
6. #11315C2semidirect
7. #11267C2semidirect
11. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20232
C2×#17442
#147142
#147002
#113842
#113152
#112672
#111892
#110492
#109252
#108802