Finite Group Databasebeta

← #400364

Subgroups — #400364

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1918C211yesmaximal, characteristic, complemented
4C2×#1852C212yesmaximal, complemented
5#15129C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic, complemented
7#11199C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic
10#10806C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C22×C4×D4—21no—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C22×#107C2212yes—
16C22×#107C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C2×#394—21no—
20D4×C4⋊C4—24no—
21D4×C4⋊C4—24no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#385—22no—
25C2×#384C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2083—24no—
31#1918—24no—
32#1852—28no—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1441—21no—
44#1440—22no—
45C4⋊#95C2211yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4—22no—
48C24×C4C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4—28no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—44no—
58C2×C4×D4—44no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—42no—
87C2×#107D414yescomplemented
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—42no—
94#394—44no—
95#389—28no—
96#389—22no—
97#389—22no—
98#385—216no—
99#384—22no—
100#384D414yes—
101#380—24no—
102#380—24no—
103#380—24no—
104#380—24no—
105#380—24no—
106#380—42no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D414yes—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—42no—
124#336—22no—
125#336—22no—
126C25C2×D412yes—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4D814yes—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—28no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—82no—
156C23×C4—82no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4#4114yes—
161C23×C4#4114yes—
162C23×C4C2411yescharacteristic
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C2×D412yes—
179C22×C8#4114yes—
180C4×D4—28no—
181C4×D4—28no—
182C4×D4—216no—
183C4×D4—432no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4#4112yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—82no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253#107—28no—
254#107—48no—
255C22⋊C8—216no—
256C22⋊C8—48no—
257#95—28no—
258#95—24no—
259#95—44no—
260#95—42no—
261#95—44no—
262#95—42no—
263#95—81no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24#12314yes—
277C24C2×D814yes—
278C24#13714yes—
279C24C22×D411yescharacteristic
280C2×D4—28no—
281C2×D4—216no—
282C2×D4—28no—
283C2×D4—216no—
284C2×D4—28no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
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—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—416no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—416no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4#12314yes—
373C22×C4C2×D812yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C42−1+412yes—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—42no—
402C4⋊C4—28no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—416no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—48no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4#12114yes—
429C22⋊C4—216no—
430C22⋊C4—416no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C42—22no—
434C42—22no—
435C42—24no—
436C42—48no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23#48018yes—
472C23C2×#12011yescharacteristic
473C23C2×#12312yes—
474C23D4211yescharacteristic
475C23#54611yescharacteristic
476C23#54712yes—
477C23C22×D811yescharacteristic
478C23C2×AGL(1,ℤ/8)11yescharacteristic
479C23C2×#13712yes—
480C23#57611yescharacteristic
481C23#57711yescharacteristic
482C23#57812yes—
483D4—232no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—832no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#44714yes—
548C2×C4C2×#12111yescharacteristic
549C2×C4#52511yescharacteristic
550C2×C4#52511yescharacteristic
551C2×C4#52512yes—
552C2×C4#52512yes—
553C2×C4#52912yes—
554C2×C4#53211yescharacteristic
555C2×C4#53511yescharacteristic
556C2×C4#53611yescharacteristic
557C2×C4#53912yes—
558C2×C4#53912yes—
559C2×C4D4211yescharacteristic
560C2×C4#54611yescharacteristic
561C2×C4#54812yes—
562C2×C4#54812yes—
563C2×C4#55112yes—
564C2×C4#55212yes—
565C2×C4#55312yes—
566C8—42no—
567C8—44no—
568C8—81no—
569C22—28no—
570C22—28no—
571C22—28no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22#240811yescharacteristic
585C22#241912yes—
586C22#243211yescharacteristic
587C22#244312yes—
588C22#284911yescharacteristic
589C22C2×#44711yescharacteristic
590C22#300411yescharacteristic
591C22#301111yescharacteristic
592C22#301211yescharacteristic
593C22C2×#48012yes—
594C22#309212yes—
595C22#309712yes—
596C22#309812yes—
597C22#327012yes—
598C22#327211yescharacteristic
599C22D4×D811yescharacteristic
600C22#329411yescharacteristic
601C22#331212yes—
602C22#332511yescharacteristic
603C22#333412yes—
604C22#334412yes—
605C22#334612yes—
606C22#335312yes—
607C4—24no—
608C4—44no—
609C4—44no—
610C4—42no—
611C4—82no—
612C4—82no—
613C4—84no—
614C4—162no—
615C4—161no—
616C4#239211yescharacteristic
617C4#246012yes—
618C4#246312yes—
619C4#253611yescharacteristic
620C4#260311yescharacteristic
621C4#263311yescharacteristic
622C2—48no—
623C2#1480111yescharacteristic
624C2#2423611yescharacteristic
625C2#2425511yescharacteristic
626C2#2493312yes—
627C2#2495312yes—
628C2#2570311yescharacteristic
629C2#2571211yescharacteristic
630C2#2612612yes—
631C2#2614812yes—
632C2#3297811yescharacteristic
633C2#3298911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
3. C2×#1918C2semidirect
4. C2×#1852C2semidirect
6. #11373C2semidirect
7. #11199C2semidirect
8. #11096C2semidirect
10. #10806C2semidirect
11. #10730C2semidirect
87. C2×#107D4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#20832
C2×#19182
C2×#18522
#151292
#113732
#111992
#110962
#109642
#108062
#107302