Finite Group Databasebeta

← #398599

Subgroups — #398599

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×#2025C211yesmaximal, characteristic, complemented
3C2×#1934C212yesmaximal, complemented
4C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
5#14698C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic, complemented
7#11268C211yesmaximal, characteristic, complemented
8#11200C212yesmaximal, complemented
9#11051C211yesmaximal, characteristic, complemented
10#10969C212yesmaximal
11#10880C212yesmaximal, complemented
12C22×#125C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2212yes—
19#2851—28no—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C2212yes—
23C2×#387—22no—
24C2×#386—22no—
25C2×#385—22no—
26C2×#382C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29#2025—24no—
30#1934—28no—
31C4×C22⋊C8—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40#1437—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4D412yes—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#125—24no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—42no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—24no—
56C2×#121—42no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—24no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—42no—
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—42no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C4×C8—24no—
86C2×C4×C8—24no—
87C2×C4×C8—42no—
88#390—24no—
89#390—22no—
90#390—22no—
91#389—24no—
92#389D418yescomplemented
93#387—216no—
94#386—48no—
95#385—216no—
96#382—24no—
97#380—24no—
98#380—24no—
99#380—24no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—42no—
112#336—22no—
113#336D414yescomplemented
114C25—21no—
115C25C2×D411yescharacteristic
116C22×D4—24no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—42no—
127C22×D4—44no—
128C22×D4—42no—
129C22×D4—44no—
130C22×D4#4112yes—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4#4114yes—
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×C8C2×D414yes—
170C22×C8#4114yes—
171#125—48no—
172#125—48no—
173#121—48no—
174#121—48no—
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—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4#4112yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—48no—
227C2×C42—48no—
228C2×C42—42no—
229C2×C42C2×D414yes—
230C2×C42C2×D412yes—
231C2×C42#4112yes—
232#102—216no—
233#102—48no—
234C22⋊C8—216no—
235C4×C8—48no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—44no—
240#95—42no—
241#95—44no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24#12412yes—
263C24#13514yes—
264C24C22×D411yescharacteristic
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—48no—
270C2×D4—416no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—84no—
284C2×D4—88no—
285C2×D4—84no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—216no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—48no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4#12014yes—
355C22×C4#12114yes—
356C22×C4#12412yes—
357C22×C4#13514yes—
358C22×C4AGL(1,ℤ/8)14yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C2×#4112yes—
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4H(5,𝔽2)12yes—
367C22×C42−1+411yescharacteristic
368C22×C42−1+411yescharacteristic
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
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—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—416no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—416no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C22⋊C4—48no—
400C22⋊C4—416no—
401C22⋊C4—88no—
402C22⋊C4—84no—
403C22⋊C4—88no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—48no—
409C23—22no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—42no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—82no—
442C23—82no—
443C23—81no—
444C23—81no—
445C23—82no—
446C23—82no—
447C23—81no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23#48814yes—
455C23C2×#12011yescharacteristic
456C23C2×#12111yescharacteristic
457C23#52511yescharacteristic
458C23C2×#12411yescharacteristic
459C23#53211yescharacteristic
460C23#53411yescharacteristic
461C23#53611yescharacteristic
462C23C2×#13512yes—
463C23C2×#13512yes—
464C23C2×AGL(1,ℤ/8)12yes—
465C23#57512yes—
466D4—44no—
467D4—44no—
468D4—816no—
469D4—82no—
470D4—82no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—432no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#44918yes—
526C2×C4#52512yes—
527C2×C4#52512yes—
528C2×C4#52912yes—
529C2×C4#53212yes—
530C2×C4#53912yes—
531C2×C4#54212yes—
532C2×C4D4212yes—
533C2×C4#54612yes—
534C2×C4#54711yescharacteristic
535C2×C4#54812yes—
536C2×C4#54812yes—
537C2×C4#54811yescharacteristic
538C2×C4#55112yes—
539C2×C4#55112yes—
540C2×C4#55211yescharacteristic
541C2×C4#55611yescharacteristic
542C8—44no—
543C8—44no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—84no—
556C22—84no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—161no—
561C22—161no—
562C22—161no—
563C22—161no—
564C22—162no—
565C22—162no—
566C22#238811yescharacteristic
567C22#241011yescharacteristic
568C22#241511yescharacteristic
569C22#243611yescharacteristic
570C22#256111yescharacteristic
571C22#266311yescharacteristic
572C22#285011yescharacteristic
573C22C2×#44912yes—
574C22#300212yes—
575C22#305012yes—
576C22#305512yes—
577C22#310212yes—
578C22#310512yes—
579C22C2×#48811yescharacteristic
580C22#313111yescharacteristic
581C22#313811yescharacteristic
582C22#314211yescharacteristic
583C22#316812yes—
584C22#317112yes—
585C22#318812yes—
586C22#319112yes—
587C22#322812yes—
588C22#323212yes—
589C4—28no—
590C4—44no—
591C4—42no—
592C4—84no—
593C4—84no—
594C4—82no—
595C4—161no—
596C4—161no—
597C4#246112yes—
598C4#246312yes—
599C4#247712yes—
600C4#256012yes—
601C2—44no—
602C2—162no—
603C2#1476511yescharacteristic
604C2#2356912yes—
605C2#2438212yes—
606C2#2470412yes—
607C2#2590511yescharacteristic
608C2#2590711yescharacteristic
609C2#2910411yescharacteristic
610C2#2910811yescharacteristic
611C2#3036012yes—
612C2#3304411yescharacteristic
613C2#3305811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
1. C2×#2851C2semidirect
2. C2×#2025C2semidirect
3. C2×#1934C2semidirect
4. C2×C4×C22⋊C8C2semidirect
6. #11385C2semidirect
7. #11268C2semidirect
8. #11200C2semidirect
9. #11051C2semidirect
11. #10880C2semidirect
92. #389D4semidirect
113. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#20252
C2×#19342
C2×C4×C22⋊C82
#146982
#113852
#112682
#112002
#110512
#109692
#108802