Finite Group Databasebeta

← #460464

Subgroups — #460464

Order 512.

1584 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
1C2×#2857C211yesmaximal, characteristic
2C2×#2084C212yesmaximal, complemented
3#14686C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11323C212yesmaximal, complemented
6#11207C212yesmaximal, complemented
7#11095C214yesmaximal
8#10712C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8—21no—
11C23×C42C2211yescharacteristic
12C22×#106C2214yes—
13C22×C22⋊C8C2212yes—
14#2857—28no—
15C2×#391—21no—
16C2×#389—21no—
17C2×#389C2212yes—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#380—21no—
22C2×#380—21no—
23C2×#380—21no—
24C2×#380C2212yes—
25C2×#380C2212yes—
26#2084—28no—
27C2×#340C2214yes—
28C2×#340C2214yes—
29C2×#340C2214yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C4⋊#95—21no—
33C4⋊#95—21no—
34C4⋊#95C2212yes—
35C23×Q8—21no—
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4C2311yescharacteristic
39C23×C8C2314yes—
40C2×C4⋊Q8—24no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—42no—
43C2×C22⋊Q8—28no—
44C2×C22⋊Q8—44no—
45C2×C22⋊Q8—44no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42—24no—
61C22×C42—24no—
62C22×C42—42no—
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#106—28no—
68C2×#106—44no—
69C2×#106—44no—
70C2×C22⋊C8—42no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8Q818yescomplemented
73#391—28no—
74#389—28no—
75#389—24no—
76#389D418yes—
77#384—24no—
78#384—24no—
79#384—42no—
80#380—28no—
81#380—24no—
82#380—24no—
83#380—44no—
84#380—44no—
85#380—44no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91#340—28no—
92#340—44no—
93#340—44no—
94#336—24no—
95#336—24no—
96C25C2×D411yescharacteristic
97C22×Q8—24no—
98C22×Q8—42no—
99C22×Q8—42no—
100C22×Q8—42no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8—42no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—41no—
124C23×C4—41no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4SD1614yes—
128C23×C4SD1614yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×Q814yes—
135C23×C4C2411yescharacteristic
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8C2×Q818yes—
143C4⋊Q8—416no—
144C22⋊Q8—432no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D414yes—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—416no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42C2×D414yes—
204C2×C42#4114yes—
205#106—416no—
206C22⋊C8—48no—
207#95—28no—
208#95—28no—
209#95—44no—
210#95—44no—
211#95—44no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24C4⋊Q812yes—
225C24C2×SD1612yes—
226C24C2×SD1612yes—
227C24C22×D411yescharacteristic
228C2×Q8—48no—
229C2×Q8—48no—
230C2×Q8—48no—
231C2×Q8—84no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×Q8—84no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
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#12012yes—
310C22×C4#12012yes—
311C22×C4#12312yes—
312C22×C4#12312yes—
313C22×C4C4⋊Q812yes—
314C22×C4C2×SD1612yes—
315C22×C4C2×SD1612yes—
316C22×C4C2×SD1612yes—
317C22×C4C2×SD1612yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4AGL(1,ℤ/8)12yes—
320C22×C4#13712yes—
321C22×C4#13712yes—
322C22×C4C22×D412yes—
323C22×C4C22×Q812yes—
324C22×C4C2×#4112yes—
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C4H(5,𝔽2)12yes—
327C22×C4H(5,𝔽2)12yes—
328C22×C42−1+411yescharacteristic
329C22×C42−1+412yes—
330C2×C8—28no—
331C2×C8—28no—
332C2×C8—28no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—28no—
340C4⋊C4—48no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—416no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C22⋊C4—416no—
355C22⋊C4—88no—
356C42—28no—
357C42—28no—
358C42—44no—
359C42—44no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23#49814yes—
395C23C2×#12011yescharacteristic
396C23C2×#12311yescharacteristic
397C23C2×C4⋊Q811yescharacteristic
398C23#53411yescharacteristic
399C23#53511yescharacteristic
400C23#53611yescharacteristic
401C23#53711yescharacteristic
402C23C22×SD1611yescharacteristic
403C23C22×SD1611yescharacteristic
404C23C2×AGL(1,ℤ/8)11yescharacteristic
405C23C2×AGL(1,ℤ/8)11yescharacteristic
406C23C2×#13711yescharacteristic
407C23C2×#13711yescharacteristic
408C23#57511yescharacteristic
409C23#57511yescharacteristic
410Q8—816no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—416no—
439C2×C4—416no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
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#44814yes—
475C2×C4#45014yes—
476C2×C4#48114yes—
477C2×C4#48314yes—
478C2×C4#53512yes—
479C2×C4#53712yes—
480C2×C4#54112yes—
481C2×C4#54212yes—
482C2×C4#54312yes—
483C2×C4#54412yes—
484C2×C4#54614yes—
485C2×C4D4×Q814yes—
486C2×C4#55414yes—
487C2×C4#56214yes—
488C8—48no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22#240511yescharacteristic
505C22#243411yescharacteristic
506C22#250211yescharacteristic
507C22#261511yescharacteristic
508C22#262511yescharacteristic
509C22#265911yescharacteristic
510C22#281611yescharacteristic
511C22C2×#45011yescharacteristic
512C22C2×#44811yescharacteristic
513C22#300611yescharacteristic
514C22#300711yescharacteristic
515C22C2×#48311yescharacteristic
516C22C2×#48111yescharacteristic
517C22#309411yescharacteristic
518C22#309511yescharacteristic
519C22C2×#49811yescharacteristic
520C22#316211yescharacteristic
521C22#316511yescharacteristic
522C22#316511yescharacteristic
523C22#318711yescharacteristic
524C22#318711yescharacteristic
525C22#319211yescharacteristic
526C22#319511yescharacteristic
527C22#320611yescharacteristic
528C22#320811yescharacteristic
529C22#321411yescharacteristic
530C22#321611yescharacteristic
531C22#322711yescharacteristic
532C22#322911yescharacteristic
533C22#323511yescharacteristic
534C22#323711yescharacteristic
535C22#324611yescharacteristic
536C22#324611yescharacteristic
537C22#325411yescharacteristic
538C22#325411yescharacteristic
539C4—24no—
540C4—24no—
541C4—24no—
542C4—24no—
543C4—42no—
544C4—84no—
545C4—84no—
546C4—162no—
547C4—162no—
548C4#269214yes—
549C4#270414yes—
550C2—44no—
551C2#1636511yescharacteristic
552C2#2400311yescharacteristic
553C2#2400511yescharacteristic
554C2#2581111yescharacteristic
555C2#2581811yescharacteristic
556C2#2794811yescharacteristic
557C2#2795511yescharacteristic
558C2#2966111yescharacteristic
559C2#2966311yescharacteristic
560C2#2993711yescharacteristic
561C2#2993811yescharacteristic
562C2#3029311yescharacteristic
563C2#3029611yescharacteristic
564C2#3277111yescharacteristic
565C2#3277311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
4. #11384C2semidirect
5. #11323C2semidirect
6. #11207C2semidirect
72. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20842
#146862
#113842
#113232
#112072
#110952
#107122