Finite Group Databasebeta

← #411898

Subgroups — #411898

Order 512.

2342 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3C2×#2538C211yesmaximal, characteristic
4C2×#2001C214yesmaximal, complemented
5C2×#1956C211yesmaximal, characteristic, complemented
6C2×#1954C211yesmaximal, characteristic
7C2×C4×#103C211yesmaximal, characteristic, complemented
8#10970C214yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10C23×Q16C2212yes—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×C22⋊Q8C2214yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#462—24no—
17C2×#462—24no—
18C2×#462—24no—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2214yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2214yes—
24C22×C4×C8C2211yescharacteristic
25#2542—24no—
26#2542—24no—
27#2538—24no—
28#2538—24no—
29C2×#399—22no—
30C2×#396—22no—
31C2×#390—21no—
32C2×#390—21no—
33C2×#389—21no—
34C2×#384—21no—
35C2×#384C2211yescharacteristic
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38#2001—216no—
39#1956—24no—
40#1954—24no—
41C4×#103—24no—
42C2×#340C2214yes—
43C2×#340C2212yes—
44C23×Q8C2312yes—
45C23×Q8C2311yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4C2312yes—
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—28no—
51C22×Q16—24no—
52C22×Q16—44no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D414yes—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—28no—
66C2×C22⋊Q8—28no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—44no—
69C2×C22⋊Q8—44no—
70C2×C22⋊Q8—44no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8D414yes—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4D412yes—
98C22×C42—21no—
99C22×C42—21no—
100C22×C42C2311yescharacteristic
101#462—232no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8D414yes—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—28no—
111C2×#103—28no—
112C2×#103—44no—
113C2×#103D418yescomplemented
114C2×C22⋊C8—28no—
115C2×C22⋊C8—28no—
116C2×C22⋊C8—44no—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120#399—216no—
121#396—216no—
122#390—28no—
123#390—28no—
124#389—28no—
125#384—28no—
126#384—22no—
127#384D414yes—
128#380—24no—
129#380—24no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4D414yes—
132C2×#95—22no—
133C2×#95—21no—
134#340—28no—
135#340—24no—
136C25C2×D412yes—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—24no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8C2×D414yes—
155C22×Q8C2×D412yes—
156C22×Q8C2×D414yes—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4Q1614yescomplemented
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C2×Q16—216no—
184C2×Q16—416no—
185C2×Q16—416no—
186C2×Q16—48no—
187C2×Q16—48no—
188C22×C8—24no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—24no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8C2×D414yes—
199C22×C8C2×D414yes—
200C4⋊Q8—28no—
201C4⋊Q8—28no—
202C4⋊Q8—44no—
203C4⋊Q8—44no—
204C22⋊Q8—416no—
205C22⋊Q8—416no—
206C22⋊Q8—416no—
207C22⋊Q8—416no—
208C22⋊Q8—48no—
209C22⋊Q8—48no—
210C22⋊Q8—416no—
211C4×Q8—28no—
212C4×Q8—28no—
213C4×Q8—44no—
214C4×Q8—44no—
215C4×D4—44no—
216C4×D4—44no—
217C4×D4—44no—
218C4×D4—44no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—216no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4#4114yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—84no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42#4112yes—
279C4⋊C8—28no—
280#103—216no—
281#103—216no—
282#103—416no—
283C22⋊C8—416no—
284C4×C8—44no—
285#95—28no—
286#95—24no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24C2×Q1614yes—
298C24C22×D411yescharacteristic
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—28no—
312C2×Q8—28no—
313C2×Q8—41no—
314C2×Q8—44no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—44no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—44no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8#12714yes—
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—28no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4#12012yes—
395C22×C4#12012yes—
396C22×C4#12012yes—
397C22×C4#12012yes—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4C2×Q1612yes—
401C22×C4#13512yes—
402C22×C4#13712yes—
403C22×C4#13712yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C4H(5,𝔽2)12yes—
414Q16—432no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—28no—
419C2×C8—24no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—44no—
425C2×C8—42no—
426C2×C8—42no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—28no—
433C4⋊C4—432no—
434C4⋊C4—416no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—84no—
445C4⋊C4#12118yes—
446C22⋊C4—432no—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—88no—
450C42—48no—
451C42—48no—
452C42#12014yes—
453C42#12114yes—
454C42#12114yes—
455C42#12714yes—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—24no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—44no—
479C23C2×#12011yescharacteristic
480C23C2×#12011yescharacteristic
481C23C2×#12111yescharacteristic
482C23D4212yes—
483C23#54612yes—
484C23C22×Q1611yescharacteristic
485C23C2×#13511yescharacteristic
486C23C2×#13711yescharacteristic
487C23C2×#13711yescharacteristic
488C23#57712yes—
489C23#57812yes—
490Q8—24no—
491Q8—24no—
492Q8—42no—
493Q8—44no—
494Q8—42no—
495Q8—44no—
496Q8—48no—
497Q8—44no—
498Q8—44no—
499Q8—82no—
500Q8—82no—
501Q8—84no—
502D4—416no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—84no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4#39214yes—
546C2×C4#39214yes—
547C2×C4#45114yes—
548C2×C4#45114yes—
549C2×C4#45214yes—
550C2×C4#45214yes—
551C2×C4#46714yes—
552C2×C4#47014yes—
553C2×C4C2×#12011yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12112yes—
556C2×C4C2×#12111yescharacteristic
557C2×C4C2×#12711yescharacteristic
558C2×C4C2×#12711yescharacteristic
559C2×C4#53212yes—
560C2×C4#53211yescharacteristic
561C2×C4#53511yescharacteristic
562C2×C4#53511yescharacteristic
563C2×C4#53512yes—
564C2×C4#54012yes—
565C2×C4D4212yes—
566C2×C4D4212yes—
567C2×C4D4212yes—
568C2×C4#54612yes—
569C2×C4#55012yes—
570C2×C4#55012yes—
571C8—44no—
572C8—44no—
573C22—48no—
574C22—44no—
575C22—44no—
576C22—48no—
577C22—44no—
578C22—82no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22—82no—
584C22C2×#39211yescharacteristic
585C22#240812yes—
586C22#242812yes—
587C22#243111yescharacteristic
588C22#243311yescharacteristic
589C22C2×#45111yescharacteristic
590C22C2×#45111yescharacteristic
591C22C2×#45211yescharacteristic
592C22C2×#45211yescharacteristic
593C22#300911yescharacteristic
594C22#301111yescharacteristic
595C22#301312yes—
596C22C2×#46711yescharacteristic
597C22C2×#47011yescharacteristic
598C22#306312yes—
599C22#327612yes—
600C22#327812yes—
601C22#328412yes—
602C22D4×Q1612yes—
603C22#329912yes—
604C22#330212yes—
605C22#332812yes—
606C22#333112yes—
607C4—24no—
608C4—44no—
609C4—48no—
610C4—82no—
611C4—84no—
612C4—82no—
613C4—84no—
614C4#200118yescomplemented
615C4#242812yes—
616C4#243111yescharacteristic
617C4#243311yescharacteristic
618C4#254112yes—
619C4#261911yescharacteristic
620C4#285611yescharacteristic
621C2—84no—
622C2#1512411yescharacteristic
623C2C2×#200111yescharacteristic
624C2#2339411yescharacteristic
625C2#2424112yes—
626C2#2424312yes—
627C2#2538612yes—
628C2#2539412yes—
629C2#2557011yescharacteristic
630C2#2557311yescharacteristic
631C2#2575511yescharacteristic
632C2#2576011yescharacteristic
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
1. C22×#462C2semidirect
4. C2×#2001C2semidirect
5. C2×#1956C2semidirect
7. C2×C4×#103C2semidirect
8. #10970C2semidirect
9. #10849C2semidirect
113. C2×#103D4semidirect
163. C22×D4Q16semidirect
614. C4#2001semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C2×#25422
C2×#25382
C2×#20012
C2×#19562
C2×#19542
C2×C4×#1032
#109702
#108492