Finite Group Databasebeta

← #470669

Subgroups — #470669

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14713C211yesmaximal, characteristic
2#14688C211yesmaximal, characteristic, complemented
3#11324C211yesmaximal, characteristic, complemented
4#11267C211yesmaximal, characteristic, complemented
5#11189C212yesmaximal, complemented
6#11189C212yesmaximal, complemented
7#11123C212yesmaximal, complemented
8#11123C212yesmaximal, complemented
9#11008C211yesmaximal, characteristic
10#11008C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#107C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#390—21no—
19C2×#390—21no—
20C2×#390—21no—
21C2×#390—21no—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#384C2211yescharacteristic
25C2×#383—21no—
26C2×C43C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—21no—
42#1434—21no—
43C23×D4—21no—
44C23×D4—21no—
45C23×D4C2312yes—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#127—24no—
49C2×#127—22no—
50C2×#127—22no—
51C2×#127—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—42no—
71C2×#107—42no—
72C2×#107—42no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#102—42no—
85#390—42no—
86#390—42no—
87#390—42no—
88#390—42no—
89#390—44no—
90#390—44no—
91#390—44no—
92#390—44no—
93#384—42no—
94#383—44no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C43—22no—
108C43—22no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—22no—
114#336—22no—
115#336—42no—
116C25—22no—
117C22×D4—24no—
118C22×D4—24no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4#4114yes—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8#4114yes—
171#127—48no—
172#127—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42D814yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239#107—48no—
240#102—48no—
241#102—48no—
242#102—48no—
243#102—48no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256C24—24no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24C22×D411yescharacteristic
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—48no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—84no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4—84no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
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—42no—
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—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4#12312yes—
382C22×C4#12312yes—
383C22×C4C2×D812yes—
384C22×C4C2×D812yes—
385C22×C4C2×D812yes—
386C22×C4#13512yes—
387C22×C4AGL(1,ℤ/8)12yes—
388C22×C4AGL(1,ℤ/8)12yes—
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4#13712yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)12yes—
397C22×C4H(5,𝔽2)12yes—
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C4H(5,𝔽2)11yescharacteristic
400C22×C42−1+411yescharacteristic
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
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—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C42—24no—
434C42—24no—
435C42—28no—
436C42—44no—
437C42—44no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42—42no—
444C42—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23C2×#12311yescharacteristic
481C23#52511yescharacteristic
482C23#53211yescharacteristic
483C23#53511yescharacteristic
484C23#53511yescharacteristic
485C23#53511yescharacteristic
486C23#53711yescharacteristic
487C23C22×D811yescharacteristic
488C23C2×#13511yescharacteristic
489C23C2×AGL(1,ℤ/8)11yescharacteristic
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23C2×AGL(1,ℤ/8)11yescharacteristic
492C23C2×#13711yescharacteristic
493C23#57511yescharacteristic
494C23#57511yescharacteristic
495D4—44no—
496D4—44no—
497D4—44no—
498D4—44no—
499D4—816no—
500D4—816no—
501D4—82no—
502D4—82no—
503D4—82no—
504D4—82no—
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—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4#48014yes—
581C2×C4#48214yes—
582C2×C4#53812yes—
583C2×C4#53812yes—
584C2×C4#53912yes—
585C2×C4#54011yescharacteristic
586C2×C4#54011yescharacteristic
587C2×C4#54211yescharacteristic
588C2×C4#54212yes—
589C2×C4#54211yescharacteristic
590C2×C4#55112yes—
591C2×C4#55112yes—
592C2×C4#55112yes—
593C2×C4#55312yes—
594C2×C4#55312yes—
595C2×C4#55512yes—
596C2×C4#56012yes—
597C2×C4#56012yes—
598C8—44no—
599C8—82no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—162no—
608C22—162no—
609C22—162no—
610C22—162no—
611C22—162no—
612C22—162no—
613C22—162no—
614C22—162no—
615C22#248211yescharacteristic
616C22#250011yescharacteristic
617C22#260311yescharacteristic
618C22#262411yescharacteristic
619C22#263611yescharacteristic
620C22#266211yescharacteristic
621C22#281811yescharacteristic
622C22C2×#48011yescharacteristic
623C22C2×#48211yescharacteristic
624C22#309411yescharacteristic
625C22#309411yescharacteristic
626C22#310011yescharacteristic
627C22#310511yescharacteristic
628C22#311011yescharacteristic
629C22#311411yescharacteristic
630C22#316911yescharacteristic
631C22#317111yescharacteristic
632C22#318011yescharacteristic
633C22#318211yescharacteristic
634C22#320511yescharacteristic
635C22#320511yescharacteristic
636C22#320511yescharacteristic
637C22#321011yescharacteristic
638C22#321111yescharacteristic
639C22#321211yescharacteristic
640C22#321511yescharacteristic
641C22#321611yescharacteristic
642C22#321711yescharacteristic
643C22#321711yescharacteristic
644C22#321711yescharacteristic
645C22#321911yescharacteristic
646C22#324511yescharacteristic
647C22#324711yescharacteristic
648C22#325111yescharacteristic
649C22#325411yescharacteristic
650C4—24no—
651C4—24no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—162no—
657C4—161no—
658C4—161no—
659C4—161no—
660C4—161no—
661C4#273712yes—
662C4#274912yes—
663C4#275612yes—
664C4#275712yes—
665C2—164no—
666C2#1669311yescharacteristic
667C2#2758211yescharacteristic
668C2#2758811yescharacteristic
669C2#2789111yescharacteristic
670C2#2789711yescharacteristic
671C2#2949311yescharacteristic
672C2#2949811yescharacteristic
673C2#2992011yescharacteristic
674C2#2992511yescharacteristic
675C2#3012511yescharacteristic
676C2#3012811yescharacteristic
677C2#3033611yescharacteristic
678C2#3034311yescharacteristic
679C2#3279611yescharacteristic
680C2#3279911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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. #14688C2semidirect
3. #11324C2semidirect
4. #11267C2semidirect
5. #11189C2semidirect
6. #11189C2semidirect
7. #11123C2semidirect
8. #11123C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
#147132
#146882
#113242
#112672
#111892
#111232
#110082
#107952