Finite Group Databasebeta

← #409833

Subgroups — #409833

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14686C211yesmaximal, characteristic
2C4×#380C211yesmaximal, characteristic
3#11373C211yesmaximal, characteristic, complemented
4#11373C211yesmaximal, characteristic, complemented
5#11324C211yesmaximal, characteristic, complemented
6#11267C211yesmaximal, characteristic, complemented
7#11199C212yesmaximal
8#11096C212yesmaximal
9#10907C212yesmaximal, complemented
10#10806C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic
12C23×C42C2211yescharacteristic
13C22×#107C2212yes—
14C22×C22⋊C8C2212yes—
15C2×#384C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#380—21no—
18C2×#380—21no—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#380C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4—21no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C4⋊#95—21no—
36C4⋊#95—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41C22⋊#95—21no—
42C4×#95—21no—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C42—24no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42—42no—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#107—24no—
72C2×#107—42no—
73C2×#107—42no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—42no—
77#384—22no—
78#384—22no—
79#384—42no—
80#380—22no—
81#380—22no—
82#380—22no—
83#380—42no—
84#380—42no—
85#380—44no—
86#380—44no—
87#380Q814yes—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—42no—
90C4×C22⋊C4—44no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—24no—
110#336—22no—
111#336—22no—
112C25C2×D411yescharacteristic
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4D814yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×Q812yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
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×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8#4114yes—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
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—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
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—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C2×Q812yes—
200C2×C4⋊C4C2×Q812yes—
201C2×C4⋊C4#4112yes—
202C2×C4⋊C4#4112yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—48no—
222C2×C42—48no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42#4114yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234#107—48no—
235C22⋊C8—48no—
236#95—24no—
237#95—24no—
238#95—42no—
239#95—44no—
240#95—44no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—44no—
245#95—42no—
246#95—42no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24C22⋊Q812yes—
260C24C2×D812yes—
261C24#13712yes—
262C24C22×D411yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—28no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
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—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4C22⋊Q812yes—
360C22×C4#12312yes—
361C22×C4#12312yes—
362C22×C4#12312yes—
363C22×C4#12312yes—
364C22×C4C2×D812yes—
365C22×C4C2×D812yes—
366C22×C4#13512yes—
367C22×C4AGL(1,ℤ/8)12yes—
368C22×C4#13712yes—
369C22×C4#13712yes—
370C22×C4C22×Q811yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4112yes—
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)12yes—
379C22×C4H(5,𝔽2)12yes—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
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—42no—
409C4⋊C4—42no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—88no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—44no—
422C42—44no—
423C42—84no—
424C42—84no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23#47414yes—
460C23C2×C22⋊Q811yescharacteristic
461C23C2×#12311yescharacteristic
462C23C2×#12311yescharacteristic
463C23#52511yescharacteristic
464C23#53211yescharacteristic
465C23#53411yescharacteristic
466C23#53511yescharacteristic
467C23C22×D811yescharacteristic
468C23C2×#13511yescharacteristic
469C23C2×AGL(1,ℤ/8)11yescharacteristic
470C23C2×#13711yescharacteristic
471C23C2×#13711yescharacteristic
472C23C2×#13711yescharacteristic
473C23#57511yescharacteristic
474C23#57511yescharacteristic
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
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—416no—
500C2×C4—416no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
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—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4#48014yes—
556C2×C4#48014yes—
557C2×C4#48214yes—
558C2×C4#52912yes—
559C2×C4#52911yescharacteristic
560C2×C4#52912yes—
561C2×C4#52911yescharacteristic
562C2×C4#53311yescharacteristic
563C2×C4#53311yescharacteristic
564C2×C4#53812yes—
565C2×C4#54012yes—
566C2×C4#55112yes—
567C2×C4#55112yes—
568C2×C4#55112yes—
569C2×C4#55312yes—
570C2×C4#55312yes—
571C2×C4#55412yes—
572C2×C4#55412yes—
573C2×C4#55512yes—
574C8—44no—
575C8—82no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22#239511yescharacteristic
592C22#248311yescharacteristic
593C22#248711yescharacteristic
594C22#250011yescharacteristic
595C22#256411yescharacteristic
596C22#263511yescharacteristic
597C22#285511yescharacteristic
598C22C2×#47411yescharacteristic
599C22#307611yescharacteristic
600C22#307811yescharacteristic
601C22#307911yescharacteristic
602C22C2×#48011yescharacteristic
603C22C2×#48011yescharacteristic
604C22C2×#48211yescharacteristic
605C22#309111yescharacteristic
606C22#309211yescharacteristic
607C22#309511yescharacteristic
608C22#309511yescharacteristic
609C22#309511yescharacteristic
610C22#310011yescharacteristic
611C22#310711yescharacteristic
612C22#311311yescharacteristic
613C22#311511yescharacteristic
614C22#316911yescharacteristic
615C22#317211yescharacteristic
616C22#318111yescharacteristic
617C22#318211yescharacteristic
618C22#318611yescharacteristic
619C22#319011yescharacteristic
620C22#319411yescharacteristic
621C22#319511yescharacteristic
622C22#320511yescharacteristic
623C22#321011yescharacteristic
624C22#321811yescharacteristic
625C22#321911yescharacteristic
626C4—24no—
627C4—24no—
628C4—24no—
629C4—42no—
630C4—42no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—162no—
636C4—162no—
637C4#252212yes—
638C4#253112yes—
639C4#257412yes—
640C4#257812yes—
641C2—44no—
642C2#1506311yescharacteristic
643C2#2375411yescharacteristic
644C2#2376311yescharacteristic
645C2#2760011yescharacteristic
646C2#2760211yescharacteristic
647C2#2767711yescharacteristic
648C2#2768511yescharacteristic
649C2#2789111yescharacteristic
650C2#2789611yescharacteristic
651C2#2915711yescharacteristic
652C2#2916511yescharacteristic
653C2#3010511yescharacteristic
654C2#3011611yescharacteristic
655C2#3314611yescharacteristic
656C2#3315411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11373C2semidirect
4. #11373C2semidirect
5. #11324C2semidirect
6. #11267C2semidirect
9. #10907C2semidirect
10. #10806C2semidirect

Maximal subgroups

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

subgroupindex
#146862
C4×#3802
#113732
#113242
#112672
#111992
#110962
#109072
#108062
#107122