Finite Group Databasebeta

← #470745

Subgroups — #470745

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14686C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic
3#11378C212yesmaximal, complemented
4#11357C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11323C211yesmaximal, characteristic, complemented
7#11315C212yesmaximal, complemented
8#11267C211yesmaximal, characteristic, complemented
9#11200C212yesmaximal
10#11095C212yesmaximal
11#10712C211yesmaximal, characteristic
12C23×C42C2211yescharacteristic
13C22×#106C2212yes—
14C22×C22⋊C8C2212yes—
15C2×#384C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#381—21no—
18C2×#380—21no—
19C2×#380—21no—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C4⋊#95—21no—
34C4⋊#95—21no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—21no—
42C22⋊#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⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C42—24no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42—42no—
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#106—24no—
70C2×#106—42no—
71C2×#106—42no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—42no—
75#384—22no—
76#384—22no—
77#384—42no—
78#381—44no—
79#380—24no—
80#380—42no—
81#380—42no—
82#380—42no—
83#380—44no—
84#380—44no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—24no—
99#340—42no—
100#340—42no—
101#340—42no—
102#340—42no—
103#340—42no—
104#340—42no—
105#336—22no—
106#336—22no—
107#336—42no—
108C25C2×D411yescharacteristic
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4SD1614yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4#4114yes—
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×C8—22no—
158C22×C8#4114yes—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
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—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
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—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
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×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—48no—
219C2×C42—48no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42#4112yes—
223C2×C42#4112yes—
224#106—48no—
225C22⋊C8—48no—
226#95—42no—
227#95—42no—
228#95—44no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—44no—
233#95—42no—
234#95—42no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—42no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24#12312yes—
252C24C2×SD1612yes—
253C24#13712yes—
254C24C22×D411yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—28no—
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—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
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—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4#12312yes—
354C22×C4#12312yes—
355C22×C4#12312yes—
356C22×C4C2×SD1612yes—
357C22×C4C2×SD1612yes—
358C22×C4#13512yes—
359C22×C4AGL(1,ℤ/8)12yes—
360C22×C4AGL(1,ℤ/8)12yes—
361C22×C4#13712yes—
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)12yes—
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+412yes—
371C22×C42−1+411yescharacteristic
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
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—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
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—84no—
405C4⋊C4—84no—
406C22⋊C4—84no—
407C22⋊C4—84no—
408C22⋊C4—88no—
409C42—24no—
410C42—24no—
411C42—44no—
412C42—44no—
413C42—42no—
414C42—42no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23#48314yes—
450C23C2×#12311yescharacteristic
451C23C2×#12311yescharacteristic
452C23#53211yescharacteristic
453C23#53411yescharacteristic
454C23#53511yescharacteristic
455C23#53611yescharacteristic
456C23#53711yescharacteristic
457C23C22×SD1611yescharacteristic
458C23C2×#13511yescharacteristic
459C23C2×AGL(1,ℤ/8)11yescharacteristic
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23C2×#13711yescharacteristic
462C23C2×#13711yescharacteristic
463C23#57511yescharacteristic
464C23#57511yescharacteristic
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
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—416no—
487C2×C4—416no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4#48114yes—
547C2×C4#48514yes—
548C2×C4#53812yes—
549C2×C4#53912yes—
550C2×C4#53912yes—
551C2×C4#54012yes—
552C2×C4#54011yescharacteristic
553C2×C4#54011yescharacteristic
554C2×C4#54211yescharacteristic
555C2×C4#54211yescharacteristic
556C2×C4#55112yes—
557C2×C4#55112yes—
558C2×C4#55112yes—
559C2×C4#55212yes—
560C2×C4#55212yes—
561C2×C4#55612yes—
562C2×C4#56212yes—
563C2×C4#56312yes—
564C8—44no—
565C8—82no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22#248311yescharacteristic
582C22#249811yescharacteristic
583C22#249911yescharacteristic
584C22#262311yescharacteristic
585C22#262411yescharacteristic
586C22#266111yescharacteristic
587C22#281411yescharacteristic
588C22C2×#48311yescharacteristic
589C22C2×#48511yescharacteristic
590C22C2×#48111yescharacteristic
591C22#309111yescharacteristic
592C22#309211yescharacteristic
593C22#309411yescharacteristic
594C22#309411yescharacteristic
595C22#309511yescharacteristic
596C22#316711yescharacteristic
597C22#317111yescharacteristic
598C22#318111yescharacteristic
599C22#318211yescharacteristic
600C22#318711yescharacteristic
601C22#318811yescharacteristic
602C22#319311yescharacteristic
603C22#319611yescharacteristic
604C22#320811yescharacteristic
605C22#321211yescharacteristic
606C22#321411yescharacteristic
607C22#321811yescharacteristic
608C22#322811yescharacteristic
609C22#322911yescharacteristic
610C22#323911yescharacteristic
611C22#324011yescharacteristic
612C22#324611yescharacteristic
613C22#324811yescharacteristic
614C22#325211yescharacteristic
615C22#325311yescharacteristic
616C4—24no—
617C4—24no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—162no—
622C4—161no—
623C4—161no—
624C4—161no—
625C4—162no—
626C4—161no—
627C4#273812yes—
628C4#274012yes—
629C4#274312yes—
630C4#274512yes—
631C2—44no—
632C2#1669811yescharacteristic
633C2#2760111yescharacteristic
634C2#2760411yescharacteristic
635C2#2785511yescharacteristic
636C2#2786311yescharacteristic
637C2#2787611yescharacteristic
638C2#2788211yescharacteristic
639C2#2990811yescharacteristic
640C2#2991011yescharacteristic
641C2#2992111yescharacteristic
642C2#2992611yescharacteristic
643C2#3031911yescharacteristic
644C2#3032511yescharacteristic
645C2#3273911yescharacteristic
646C2#3274011yescharacteristic
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. #11378C2semidirect
4. #11357C2semidirect
5. #11357C2semidirect
6. #11323C2semidirect
7. #11315C2semidirect
8. #11267C2semidirect

Maximal subgroups

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

subgroupindex
#146862
#146852
#113782
#113572
#113232
#113152
#112672
#112002
#110952
#107122