Finite Group Databasebeta

← #477167

Subgroups — #477167

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2035C212yesmaximal
2C2×#2024C212yesmaximal, complemented
3#15178C211yesmaximal, characteristic
4#15125C211yesmaximal, characteristic
5#11257C212yesmaximal, complemented
6#11147C211yesmaximal, characteristic
7#11122C212yesmaximal, complemented
8#11010C212yesmaximal, complemented
9#10970C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×#106C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C2×#393—21no—
21C2×#391C2212yes—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#2035—28no—
25#2024—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1446—21no—
32#1444—21no—
33#1444—21no—
34#1442—21no—
35#1439—21no—
36#1439—21no—
37#1439—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×Q8—21no—
42C23×Q8C2312yes—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—42no—
51C2×C4⋊Q8—42no—
52C2×C4⋊Q8—41no—
53C2×C4⋊Q8—41no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#106—24no—
74C2×#106—42no—
75C2×#106—42no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—41no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—41no—
92#393—44no—
93#391—24no—
94#391—42no—
95#380—24no—
96#380—42no—
97C4×C22⋊C4—41no—
98C4×C22⋊C4—41no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110#340—24no—
111#340—42no—
112#340D418yes—
113#336—24no—
114#336—42no—
115C25C2×D411yescharacteristic
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—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—44no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—82no—
134C22×Q8#4114yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D414yes—
167C4⋊Q8—44no—
168C4⋊Q8—48no—
169C4⋊Q8—44no—
170C4⋊Q8—48no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173C22⋊Q8—48no—
174C22⋊Q8—48no—
175C22⋊Q8—48no—
176C22⋊Q8—48no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D414yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4SD1614yes—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224#106—48no—
225#103—48no—
226#103—48no—
227#103—48no—
228#103—48no—
229C22⋊C8—44no—
230C22⋊C8—44no—
231#95—41no—
232#95—41no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—42no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—82no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24C2×SD1612yes—
253C24C22×D411yescharacteristic
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—48no—
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—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—88no—
284C2×Q8—84no—
285C2×Q8—88no—
286C2×Q8—84no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
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—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4C2×SD1612yes—
353C22×C4C2×SD1612yes—
354C22×C4#13712yes—
355C22×C4#13712yes—
356C22×C4C22×D412yes—
357C22×C4C2×#4112yes—
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+412yes—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—416no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—416no—
397C22⋊C4—88no—
398C42—42no—
399C42—42no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23#53411yescharacteristic
428C23#53511yescharacteristic
429C23#53611yescharacteristic
430C23D4212yes—
431C23#54612yes—
432C23C22×SD1611yescharacteristic
433C23C2×#13711yescharacteristic
434C23C2×#13711yescharacteristic
435C23#57511yescharacteristic
436C23#57712yes—
437C23#57812yes—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—44no—
442Q8—816no—
443Q8—82no—
444Q8—82no—
445Q8—82no—
446Q8—82no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
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—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4#53512yes—
505C2×C4#53512yes—
506C2×C4#53712yes—
507C2×C4#53812yes—
508C2×C4#53812yes—
509C2×C4#53912yes—
510C2×C4D4211yescharacteristic
511C2×C4#54612yes—
512C2×C4#54812yes—
513C2×C4#55111yescharacteristic
514C2×C4#56011yescharacteristic
515C2×C4H(3,𝔽4)11yescharacteristic
516C2×C4H(3,𝔽4)12yes—
517C2×C4#56212yes—
518C2×C4#56212yes—
519C2×C4#56212yes—
520C8—44no—
521C8—82no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—44no—
526C22—44no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—82no—
531C22—82no—
532C22—81no—
533C22#266911yescharacteristic
534C22#267012yes—
535C22#268112yes—
536C22#268411yescharacteristic
537C22#281611yescharacteristic
538C22#318711yescharacteristic
539C22#319511yescharacteristic
540C22#320011yescharacteristic
541C22#320211yescharacteristic
542C22#320611yescharacteristic
543C22#321611yescharacteristic
544C22#322512yes—
545C22#322911yescharacteristic
546C22#323711yescharacteristic
547C22#324212yes—
548C22#326712yes—
549C22#327012yes—
550C22#327612yes—
551C22D4×SD1612yes—
552C22#329812yes—
553C22#331312yes—
554C22#331612yes—
555C22#332912yes—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—84no—
561C4—84no—
562C4—81no—
563C4—84no—
564C4—162no—
565C4#267612yes—
566C4#269612yes—
567C4#281912yes—
568C4#282412yes—
569C2—82no—
570C2#1710511yescharacteristic
571C2#3050011yescharacteristic
572C2#3051711yescharacteristic
573C2#3052212yes—
574C2#3053712yes—
575C2#3112012yes—
576C2#3114312yes—
577C2#3126211yescharacteristic
578C2#3127211yescharacteristic
579C2#3277111yescharacteristic
580C2#3278011yescharacteristic
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
2. C2×#2024C2semidirect
5. #11257C2semidirect
7. #11122C2semidirect
8. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20352
C2×#20242
#151782
#151252
#112572
#111472
#111222
#110102
#109702