Finite Group Databasebeta

← #415846

Subgroups — #415846

Order 512.

2006 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#461C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3C2×#2538C211yesmaximal, characteristic
4C2×#1984C211yesmaximal, characteristic, complemented
5C2×#1956C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic
7#11067C214yesmaximal, complemented
8#11023C214yesmaximal
9#10849C211yesmaximal, characteristic, complemented
10C23×SD16C2212yes—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#461—24no—
16C2×#461—24no—
17C2×#461—24no—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2214yes—
21C22×#102C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2568—24no—
24#2568—24no—
25#2538—24no—
26#2538—24no—
27C2×#400—22no—
28C2×#396—22no—
29C2×#391—21no—
30C2×#391C2214yes—
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382—21no—
35C2×#380—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1984—24no—
38#1956—24no—
39C4×#102—24no—
40C2×#340C2212yes—
41C2×#340C2214yes—
42C2×#336C2214yes—
43C23×Q8D411yescharacteristic
44C23×Q8C2312yes—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C22×SD16—24no—
48C22×SD16—24no—
49C22×SD16—28no—
50C22×SD16—24no—
51C22×SD16—44no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D414yes—
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8—44no—
62C2×C22⋊Q8—44no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4D414yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C42—21no—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#461—232no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8D414yes—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—28no—
99C2×#103—28no—
100C2×#103—44no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107#400—216no—
108#396—216no—
109#391—28no—
110#391—28no—
111#391—28no—
112#390—24no—
113#390—24no—
114#389—24no—
115#389—24no—
116#384—22no—
117#384D414yes—
118#382—28no—
119#380—28no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4D414yes—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—24no—
125C2×#95—24no—
126#340—24no—
127#340—44no—
128#336—28no—
129C25—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—24no—
142C22×Q8—24no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8SD1614yescomplemented
146C22×Q8C2×D414yes—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4C2×D412yes—
156C22×D4#4114yes—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C2×SD16—216no—
174C2×SD16—416no—
175C2×SD16—416no—
176C2×SD16—48no—
177C2×SD16—48no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—24no—
184C22×C8—22no—
185C22×C8—24no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C2×D414yes—
189C22×C8#4114yes—
190C4⋊Q8—28no—
191C4⋊Q8—28no—
192C4⋊Q8—44no—
193C4⋊Q8—44no—
194C22⋊Q8—432no—
195C4×Q8—44no—
196C4×Q8—44no—
197C4×Q8—44no—
198C4×Q8—44no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—44no—
202C4×D4—44no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—84no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42#4112yes—
262C4⋊C8—28no—
263#103—216no—
264#103—416no—
265#102—216no—
266C4×C8—44no—
267#95—28no—
268#95—216no—
269#95—216no—
270#95—24no—
271C24—22no—
272C24—24no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—42no—
281C24—42no—
282C24C22×D411yescharacteristic
283C2×Q8—28no—
284C2×Q8—28no—
285C2×Q8—28no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—28no—
289C2×Q8—28no—
290C2×Q8—42no—
291C2×Q8—44no—
292C2×Q8—42no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—42no—
299C2×Q8—44no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4#12414yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4#12012yes—
373C22×C4#12012yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4C2×SD1614yes—
377C22×C4C2×SD1612yes—
378C22×C4#13512yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4#13712yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C42−1+412yes—
391SD16—432no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—28no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—44no—
403C2×C8—42no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—28no—
409C4⋊C4—416no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—84no—
422C22⋊C4—416no—
423C22⋊C4—48no—
424C22⋊C4—88no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—48no—
430C42#12014yes—
431C42#12114yes—
432C42#12114yes—
433C42#12714yes—
434C23—24no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—81no—
452C23—81no—
453C23—82no—
454C23—81no—
455C23—81no—
456C23—82no—
457C23C2×#12011yescharacteristic
458C23C2×#12011yescharacteristic
459C23C2×#12111yescharacteristic
460C23#54812yes—
461C23#55012yes—
462C23C22×SD1611yescharacteristic
463C23C2×#13511yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×#13711yescharacteristic
466C23#57712yes—
467C23#57812yes—
468Q8—416no—
469Q8—48no—
470Q8—44no—
471Q8—44no—
472Q8—82no—
473Q8—82no—
474Q8—84no—
475D4—24no—
476D4—24no—
477D4—42no—
478D4—44no—
479D4—42no—
480D4—44no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—84no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4—82no—
514C2×C4—84no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—816no—
522C2×C4#39214yes—
523C2×C4#39214yes—
524C2×C4#44814yes—
525C2×C4#44914yes—
526C2×C4#45014yes—
527C2×C4#45214yes—
528C2×C4#46514yes—
529C2×C4#46814yes—
530C2×C4C2×#12011yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×#12111yescharacteristic
533C2×C4#52512yes—
534C2×C4C2×#12411yescharacteristic
535C2×C4#52912yes—
536C2×C4C2×#12711yescharacteristic
537C2×C4#53211yescharacteristic
538C2×C4#53511yescharacteristic
539C2×C4#53611yescharacteristic
540C2×C4#53712yes—
541C2×C4#54212yes—
542C2×C4#54612yes—
543C2×C4#54612yes—
544C2×C4#54612yes—
545C2×C4#54712yes—
546C2×C4#54812yes—
547C2×C4#54812yes—
548C8—44no—
549C8—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22C2×#39211yescharacteristic
562C22#240712yes—
563C22#242612yes—
564C22#243111yescharacteristic
565C22#243611yescharacteristic
566C22C2×#45011yescharacteristic
567C22C2×#44811yescharacteristic
568C22C2×#44911yescharacteristic
569C22C2×#45211yescharacteristic
570C22#300911yescharacteristic
571C22#301111yescharacteristic
572C22#301312yes—
573C22C2×#46511yescharacteristic
574C22C2×#46811yescharacteristic
575C22#306312yes—
576C22#335912yes—
577C22#336012yes—
578C22#336712yes—
579C22#337212yes—
580C22#339212yes—
581C22#339312yes—
582C22#339712yes—
583C22#339812yes—
584C4—24no—
585C4—44no—
586C4—84no—
587C4—82no—
588C4—82no—
589C4—84no—
590C4—84no—
591C4#200018yes—
592C4#242712yes—
593C4#243211yescharacteristic
594C4#243511yescharacteristic
595C4#254012yes—
596C4#262011yescharacteristic
597C4#285011yescharacteristic
598C2—84no—
599C2#1521511yescharacteristic
600C2C2×#200011yescharacteristic
601C2#2339411yescharacteristic
602C2#2418512yes—
603C2#2418712yes—
604C2#2529312yes—
605C2#2530112yes—
606C2#2562711yescharacteristic
607C2#2563211yescharacteristic
608C2#2592911yescharacteristic
609C2#2593411yescharacteristic
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×#2568C2semidirect
4. C2×#1984C2semidirect
5. C2×#1956C2semidirect
7. #11067C2semidirect
9. #10849C2semidirect
145. C22×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C22×#4612
C2×#25682
C2×#25382
C2×#19842
C2×#19562
C2×C4×#1022
#110672
#110232
#108492