Finite Group Databasebeta

← #461398

Subgroups — #461398

Order 512.

1678 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
1C2×#1787C212yesmaximal
2#15195C211yesmaximal, characteristic
3#13922C211yesmaximal, characteristic
4#11200C212yesmaximal
5#11190C212yesmaximal, complemented
6#11190C212yesmaximal, complemented
7#11077C212yesmaximal, complemented
8#11050C212yesmaximal, complemented
9#11042C211yesmaximal, characteristic
10C24×Q8C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2212yes—
17C2×#391C2212yes—
18C2×#391C2212yes—
19C2×#384C2211yescharacteristic
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#1787—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30#1446—21no—
31#1446—21no—
32#1442—21no—
33#1442—21no—
34#1439—21no—
35#1439—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C22⋊#95—21no—
39C22⋊#95—21no—
40C23×Q8—21no—
41C23×Q8—22no—
42C23×Q8—24no—
43C23×Q8—24no—
44C23×Q8—41no—
45C23×Q8—41no—
46C23×Q8—41no—
47C23×Q8—41no—
48C23×Q8—42no—
49C23×Q8—42no—
50C23×Q8D414yes—
51C23×Q8C2312yes—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—41no—
68C2×C4⋊C8—42no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84#391—24no—
85#391—24no—
86#391—42no—
87#391—42no—
88#384—41no—
89#384—41no—
90#380—24no—
91#380—24no—
92#380—41no—
93#380—41no—
94C4×C22⋊C4—41no—
95C4×C22⋊C4—41no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—41no—
113#336—42no—
114C25C2×D411yescharacteristic
115C22×Q8—24no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—216no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8—48no—
131C22×Q8—44no—
132C22×Q8—44no—
133C22×Q8—48no—
134C22×Q8—48no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—44no—
138C22×Q8—44no—
139C22×Q8—44no—
140C22×Q8—48no—
141C22×Q8—44no—
142C22×Q8—48no—
143C22×Q8—48no—
144C22×Q8—48no—
145C22×Q8—48no—
146C22×Q8—48no—
147C22×Q8—44no—
148C22×Q8—44no—
149C22×Q8C2×D414yes—
150C22×Q8#4114yes—
151C22×Q8#4114yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—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—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4Q1614yes—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C4⋊C8—44no—
234#103—48no—
235#103—48no—
236#103—48no—
237#103—48no—
238C22⋊C8—48no—
239#95—41no—
240#95—41no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—42no—
246#95—42no—
247#95—44no—
248#95—42no—
249#95—82no—
250#95—82no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24#12112yes—
260C24C2×Q1612yes—
261C24#13712yes—
262C24C22×D411yescharacteristic
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—416no—
279C2×Q8—48no—
280C2×Q8—416no—
281C2×Q8—432no—
282C2×Q8—432no—
283C2×Q8—416no—
284C2×Q8—416no—
285C2×Q8—416no—
286C2×Q8—48no—
287C2×Q8—416no—
288C2×Q8—48no—
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—
301C2×Q8#12318yes—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—416no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—416no—
338C22×C4—48no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
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—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4#12112yes—
375C22×C4C2×Q1612yes—
376C22×C4C2×Q1612yes—
377C22×C4#13712yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4111yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C4H(5,𝔽2)12yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
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—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—84no—
419C22⋊C4—88no—
420C42—42no—
421C42—42no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C23—22no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23#46214yes—
449C23C2×#12111yescharacteristic
450C23#53411yescharacteristic
451C23#53511yescharacteristic
452C23#54612yes—
453C23#54612yes—
454C23C22×Q1611yescharacteristic
455C23C2×#13711yescharacteristic
456C23C2×#13711yescharacteristic
457C23#57511yescharacteristic
458C23#57712yes—
459C23#57712yes—
460Q8—28no—
461Q8—28no—
462Q8—432no—
463Q8—44no—
464Q8—44no—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—44no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—432no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
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—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4C2×#12312yes—
530C2×C4#53411yescharacteristic
531C2×C4#53412yes—
532C2×C4#53411yescharacteristic
533C2×C4#53812yes—
534C2×C4#53812yes—
535C2×C4#53812yes—
536C2×C4#53812yes—
537C2×C4#53811yescharacteristic
538C2×C4#53811yescharacteristic
539C2×C4#55112yes—
540C2×C4#55112yes—
541C2×C4#55111yescharacteristic
542C2×C4#55112yes—
543C2×C4#55111yescharacteristic
544C2×C4#56012yes—
545C2×C4#56011yescharacteristic
546C2×C4H(3,𝔽4)11yescharacteristic
547C8—82no—
548C8—82no—
549C22—24no—
550C22—28no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22#241212yes—
561C22#261711yescharacteristic
562C22#268411yescharacteristic
563C22#268512yes—
564C22#269711yescharacteristic
565C22C2×#46211yescharacteristic
566C22#303711yescharacteristic
567C22#304312yes—
568C22#318911yescharacteristic
569C22#319511yescharacteristic
570C22#320011yescharacteristic
571C22#320011yescharacteristic
572C22#320911yescharacteristic
573C22#321511yescharacteristic
574C22#322112yes—
575C22#329912yes—
576C22#329912yes—
577C22#331612yes—
578C22#331612yes—
579C22#332012yes—
580C22#332112yes—
581C22#332612yes—
582C22#332912yes—
583C4—42no—
584C4—48no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—81no—
589C4—162no—
590C4—162no—
591C4—162no—
592C4#248311yescharacteristic
593C4#262212yes—
594C4#273512yes—
595C4#273511yescharacteristic
596C4#273611yescharacteristic
597C4#281211yescharacteristic
598C2—44no—
599C2#1639111yescharacteristic
600C2#2453212yes—
601C2#2454112yes—
602C2#2971511yescharacteristic
603C2#2972211yescharacteristic
604C2#3126311yescharacteristic
605C2#3127211yescharacteristic
606C2#3129212yes—
607C2#3130412yes—
608C2#3166911yescharacteristic
609C2#3168211yescharacteristic
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
5. #11190C2semidirect
6. #11190C2semidirect
7. #11077C2semidirect
8. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#17872
#151952
#139222
#112002
#111902
#110772
#110502
#110422