Finite Group Databasebeta

← #478411

Subgroups — #478411

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15178C211yesmaximal, characteristic
2#15150C211yesmaximal, characteristic
3#11315C212yesmaximal, complemented
4#11190C212yesmaximal, complemented
5#11147C211yesmaximal, characteristic
6#11050C212yesmaximal, complemented
7#11010C212yesmaximal, complemented
8#10970C212yesmaximal
9#10970C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#397—21no—
19C2×#397—21no—
20C2×#391C2212yes—
21C2×#384C2211yescharacteristic
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1445—21no—
31#1445—21no—
32#1444—21no—
33#1444—21no—
34#1439—21no—
35#1439—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C23×Q8C2312yes—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—42no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—42no—
50C2×C22⋊Q8—42no—
51C2×C22⋊Q8—42no—
52C2×C22⋊Q8—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#103—24no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—41no—
82C2×C22⋊C8—41no—
83#397—44no—
84#397—44no—
85#391—24no—
86#391—42no—
87#384—41no—
88#384—41no—
89#380—42no—
90#380—42no—
91C4×C22⋊C4—41no—
92C4×C22⋊C4—41no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104#340—24no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336—42no—
109#336—42no—
110C25C2×D411yescharacteristic
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8#4114yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—44no—
158C4⋊Q8—48no—
159C4⋊Q8—44no—
160C22⋊Q8—48no—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163C22⋊Q8—48no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4#4114yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4SD1614yes—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215#103—48no—
216#103—48no—
217#103—48no—
218#103—48no—
219C22⋊C8—44no—
220C22⋊C8—44no—
221#95—41no—
222#95—41no—
223#95—44no—
224#95—44no—
225#95—42no—
226#95—44no—
227#95—42no—
228#95—42no—
229#95—44no—
230#95—42no—
231#95—44no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—22no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—22no—
241C24—22no—
242C24C2×SD1612yes—
243C24C22×D411yescharacteristic
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
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×C4C2×SD1612yes—
338C22×C4C2×SD1612yes—
339C22×C4AGL(1,ℤ/8)12yes—
340C22×C4#13712yes—
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C4H(5,𝔽2)11yescharacteristic
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—416no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—416no—
383C22⋊C4—88no—
384C42—42no—
385C42—42no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—24no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23#53411yescharacteristic
414C23#53511yescharacteristic
415C23#53511yescharacteristic
416C23#54612yes—
417C23#54612yes—
418C23C22×SD1611yescharacteristic
419C23C2×AGL(1,ℤ/8)11yescharacteristic
420C23C2×#13711yescharacteristic
421C23#57511yescharacteristic
422C23#57812yes—
423C23#57812yes—
424Q8—44no—
425Q8—44no—
426Q8—44no—
427Q8—44no—
428Q8—82no—
429Q8—82no—
430Q8—82no—
431Q8—82no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4#53812yes—
490C2×C4#53812yes—
491C2×C4#53812yes—
492C2×C4#53812yes—
493C2×C4#54012yes—
494C2×C4#54012yes—
495C2×C4#55111yescharacteristic
496C2×C4#55112yes—
497C2×C4#55312yes—
498C2×C4#55911yescharacteristic
499C2×C4#56011yescharacteristic
500C2×C4#56012yes—
501C2×C4#56012yes—
502C2×C4#56012yes—
503C2×C4#56012yes—
504C2×C4H(3,𝔽4)11yescharacteristic
505C8—82no—
506C8—82no—
507C22—42no—
508C22—42no—
509C22—44no—
510C22—44no—
511C22—42no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22—81no—
518C22#268411yescharacteristic
519C22#268512yes—
520C22#270312yes—
521C22#270911yescharacteristic
522C22#281511yescharacteristic
523C22#318711yescharacteristic
524C22#319311yescharacteristic
525C22#320211yescharacteristic
526C22#320211yescharacteristic
527C22#320611yescharacteristic
528C22#320811yescharacteristic
529C22#321411yescharacteristic
530C22#321611yescharacteristic
531C22#322312yes—
532C22#322312yes—
533C22#329812yes—
534C22#329812yes—
535C22#330812yes—
536C22#331212yes—
537C22#331312yes—
538C22#331512yes—
539C22#332812yes—
540C22#333012yes—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—84no—
546C4—81no—
547C4—84no—
548C4—162no—
549C4—162no—
550C4#274912yes—
551C4#279712yes—
552C4#282512yes—
553C4#282812yes—
554C2—82no—
555C2#1714711yescharacteristic
556C2#3126211yescharacteristic
557C2#3127311yescharacteristic
558C2#3128812yes—
559C2#3130912yes—
560C2#3183612yes—
561C2#3185612yes—
562C2#3202711yescharacteristic
563C2#3203911yescharacteristic
564C2#3276011yescharacteristic
565C2#3276811yescharacteristic
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
3. #11315C2semidirect
4. #11190C2semidirect
6. #11050C2semidirect
7. #11010C2semidirect

Maximal subgroups

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

subgroupindex
#151782
#151502
#113152
#111902
#111472
#110502
#110102
#109702