Finite Group Databasebeta

← #432740

Subgroups — #432740

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#2071C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic, complemented
5#11380C211yesmaximal, characteristic, complemented
6#11292C212yesmaximal, complemented
7#11267C212yesmaximal, complemented
8#11189C212yesmaximal
9#10931C211yesmaximal, characteristic
10#10884C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C22⋊Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17#2538—24no—
18#2538—24no—
19C2×#396—21no—
20C2×#396—21no—
21C2×#391—21no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2099—24no—
31#2071—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1441—21no—
43#1441—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C22⋊Q8—28no—
53C2×C22⋊Q8—44no—
54C2×C22⋊Q8—44no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—42no—
92#396—28no—
93#396—28no—
94#391—44no—
95#390—22no—
96#390—22no—
97#390—22no—
98#390D414yes—
99#389—22no—
100#389—22no—
101#389—22no—
102#389D414yes—
103#384—22no—
104#384—22no—
105#384—22no—
106#384—22no—
107#380—28no—
108C4×C4⋊C4—42no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#336—22no—
123#336—22no—
124#336—24no—
125#336—42no—
126C25—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8D814yes—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C2×D412yes—
141C22×D4#4114yes—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
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—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22⋊Q8—432no—
176C4×Q8—48no—
177C4×Q8—48no—
178C4×D4—48no—
179C4×D4—48no—
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—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4#4114yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42#4114yes—
257#102—48no—
258#102—48no—
259#95—24no—
260#95—24no—
261#95—42no—
262#95—44no—
263#95—44no—
264#95—82no—
265#95—82no—
266C24—22no—
267C24—24no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24C22×D411yescharacteristic
278C2×Q8—28no—
279C2×Q8—28no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×D4—24no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4#12414yes—
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—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
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×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4#12012yes—
367C22×C4#12012yes—
368C22×C4#12312yes—
369C22×C4#12312yes—
370C22×C4C2×D812yes—
371C22×C4C2×D812yes—
372C22×C4C2×D812yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4#13712yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382C22×C42−1+412yes—
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—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—42no—
432C42—42no—
433C42—84no—
434C23—24no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—44no—
441C23—42no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—81no—
448C23—81no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—81no—
455C23—81no—
456C23—82no—
457C23C2×#12011yescharacteristic
458C23C2×#12311yescharacteristic
459C23#53411yescharacteristic
460C23#54812yes—
461C23#54812yes—
462C23C22×D811yescharacteristic
463C23C2×AGL(1,ℤ/8)11yescharacteristic
464C23C2×#13711yescharacteristic
465C23#57511yescharacteristic
466C23#57712yes—
467C23#57712yes—
468Q8—416no—
469D4—28no—
470D4—44no—
471D4—44no—
472D4—44no—
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—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—816no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4#44714yes—
540C2×C4#48014yes—
541C2×C4C2×#12411yescharacteristic
542C2×C4#52912yes—
543C2×C4#53211yescharacteristic
544C2×C4#53611yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53912yes—
547C2×C4#53912yes—
548C2×C4#54112yes—
549C2×C4#54612yes—
550C2×C4#54612yes—
551C2×C4#54712yes—
552C2×C4#54812yes—
553C2×C4#55112yes—
554C2×C4#55212yes—
555C2×C4#55612yes—
556C2×C4#56312yes—
557C8—82no—
558C8—82no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—84no—
563C22—82no—
564C22—84no—
565C22—82no—
566C22—84no—
567C22—84no—
568C22—82no—
569C22—82no—
570C22#240511yescharacteristic
571C22#242112yes—
572C22#243211yescharacteristic
573C22#243611yescharacteristic
574C22#272812yes—
575C22C2×#44711yescharacteristic
576C22#300711yescharacteristic
577C22#300911yescharacteristic
578C22#301011yescharacteristic
579C22C2×#48011yescharacteristic
580C22#309411yescharacteristic
581C22#309912yes—
582C22#318611yescharacteristic
583C22#319311yescharacteristic
584C22#320412yes—
585C22#335812yes—
586C22#335812yes—
587C22#336812yes—
588C22#336912yes—
589C22#337212yes—
590C22#337312yes—
591C22#337612yes—
592C22#337612yes—
593C4—24no—
594C4—24no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—84no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—81no—
603C4—162no—
604C4#248812yes—
605C4#256012yes—
606C4#268912yes—
607C4#269012yes—
608C2—84no—
609C2#1546511yescharacteristic
610C2#2400211yescharacteristic
611C2#2401211yescharacteristic
612C2#2499012yes—
613C2#2500612yes—
614C2#2564511yescharacteristic
615C2#2566211yescharacteristic
616C2#2593311yescharacteristic
617C2#2594511yescharacteristic
618C2#3254412yes—
619C2#3256112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2099C2semidirect
4. #13926C2semidirect
5. #11380C2semidirect
6. #11292C2semidirect
7. #11267C2semidirect
10. #10884C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#20992
C2×#20712
#139262
#113802
#112922
#112672
#111892
#109312
#108842
#107882