Finite Group Databasebeta

← #471562

Subgroups — #471562

Order 512.

1392 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×#2068C212yesmaximal, complemented
2C2×#2036C212yesmaximal
3C2×#1956C211yesmaximal, characteristic, complemented
4C2×#1923C211yesmaximal, characteristic
5#15178C211yesmaximal, characteristic
6#15135C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11067C212yesmaximal, complemented
9#11010C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10862C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#399—21no—
24C2×#391C2212yes—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#2068—28no—
30#2036—28no—
31#1956—24no—
32#1923—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1445—21no—
39#1444—21no—
40#1444—21no—
41#1444—21no—
42#1439—21no—
43#1439—21no—
44#1439—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C23×Q8C2312yes—
48C24×C4C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8—42no—
61C2×C22⋊Q8—42no—
62C2×C22⋊Q8—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—42no—
79C2×#107—42no—
80C2×#107D418yes—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—42no—
99C2×C4×C8—42no—
100C2×C4×C8D414yes—
101#399—44no—
102#391—24no—
103#391—42no—
104#389—24no—
105#389—42no—
106#384—22no—
107#384—22no—
108#380—24no—
109#380—42no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—42no—
125#336—22no—
126#336—22no—
127C25C2×D411yescharacteristic
128C22×Q8—22no—
129C22×Q8—22no—
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#4114yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D414yes—
182C4⋊Q8—28no—
183C4⋊Q8—48no—
184C4⋊Q8—44no—
185C22⋊Q8—48no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
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—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4Q1614yes—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42C2×D412yes—
241C2×C42#4112yes—
242#107—48no—
243C4⋊C8—44no—
244#103—48no—
245#103—48no—
246#103—48no—
247#103—48no—
248C22⋊C8—44no—
249C22⋊C8—44no—
250C4×C8—44no—
251#95—22no—
252#95—22no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—42no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24C2×Q1612yes—
272C24C22×D411yescharacteristic
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
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—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
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—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4C2×Q1612yes—
368C22×C4C2×Q1612yes—
369C22×C4#13512yes—
370C22×C4AGL(1,ℤ/8)12yes—
371C22×C4#13712yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C2×#4112yes—
375C22×C4C2×#4111yescharacteristic
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)12yes—
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—41no—
397C2×C8—42no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8#12714yes—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—416no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—84no—
418C4⋊C4—88no—
419C4⋊C4—84no—
420C22⋊C4—28no—
421C22⋊C4—416no—
422C22⋊C4—44no—
423C22⋊C4—88no—
424C42—22no—
425C42—22no—
426C42—24no—
427C42—41no—
428C42—41no—
429C42—42no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—42no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23C2×#12111yescharacteristic
454C23#53511yescharacteristic
455C23#53611yescharacteristic
456C23D4212yes—
457C23#54612yes—
458C23C22×Q1611yescharacteristic
459C23C2×#13511yescharacteristic
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23C2×#13711yescharacteristic
462C23#57712yes—
463C23#57812yes—
464Q8—44no—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—82no—
469Q8—82no—
470Q8—82no—
471Q8—82no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
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#46714yes—
530C2×C4#46814yes—
531C2×C4#52511yescharacteristic
532C2×C4C2×#12711yescharacteristic
533C2×C4#53412yes—
534C2×C4#53411yescharacteristic
535C2×C4#53511yescharacteristic
536C2×C4#53711yescharacteristic
537C2×C4#53712yes—
538C2×C4#53711yescharacteristic
539C2×C4#53812yes—
540C2×C4#53912yes—
541C2×C4#54211yescharacteristic
542C2×C4#54211yescharacteristic
543C2×C4#54612yes—
544C2×C4#54812yes—
545C2×C4#55012yes—
546C2×C4#55112yes—
547C2×C4#56012yes—
548C2×C4#56212yes—
549C8—24no—
550C8—42no—
551C8—81no—
552C8—81no—
553C22—44no—
554C22—42no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—82no—
559C22—81no—
560C22—82no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22#253411yescharacteristic
565C22#254111yescharacteristic
566C22#262011yescharacteristic
567C22#267812yes—
568C22#268112yes—
569C22C2×#46711yescharacteristic
570C22C2×#46811yescharacteristic
571C22#305711yescharacteristic
572C22#305911yescharacteristic
573C22#320711yescharacteristic
574C22#320911yescharacteristic
575C22#322512yes—
576C22#323011yescharacteristic
577C22#323111yescharacteristic
578C22#324212yes—
579C22#327212yes—
580C22#327812yes—
581C22#328312yes—
582C22D4×Q1612yes—
583C22#329912yes—
584C22#330112yes—
585C22#330812yes—
586C22#331212yes—
587C4—24no—
588C4—24no—
589C4—42no—
590C4—84no—
591C4—84no—
592C4—81no—
593C4—81no—
594C4—84no—
595C4—162no—
596C4#254011yescharacteristic
597C4#263611yescharacteristic
598C4#265611yescharacteristic
599C4#270512yes—
600C4#273012yes—
601C4#281311yescharacteristic
602C2—82no—
603C2#1680111yescharacteristic
604C2#2809511yescharacteristic
605C2#2810511yescharacteristic
606C2#2848611yescharacteristic
607C2#2849011yescharacteristic
608C2#2984411yescharacteristic
609C2#2984811yescharacteristic
610C2#3098712yes—
611C2#3099012yes—
612C2#3111912yes—
613C2#3112212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2068C2semidirect
3. C2×#1956C2semidirect
7. #11357C2semidirect
8. #11067C2semidirect
9. #11010C2semidirect
11. #10862C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20682
C2×#20362
C2×#19562
C2×#19232
#151782
#151352
#113572
#110672
#110102
#109702
#108622