Finite Group Databasebeta

← #449007

Subgroups — #449007

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#2071C211yesmaximal, characteristic
4#13910C211yesmaximal, characteristic, complemented
5#11346C211yesmaximal, characteristic, complemented
6#11207C212yesmaximal, complemented
7#11066C212yesmaximal
8#11066C212yesmaximal
9#10931C211yesmaximal, characteristic
10#10885C212yesmaximal, complemented
11#10884C211yesmaximal, characteristic, complemented
12C22×#126—21no—
13C22×#102C2212yes—
14C22×#102C2212yes—
15#2512—24no—
16#2512—24no—
17C2×#397—21no—
18C2×#396—21no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2211yescharacteristic
25C2×#388—21no—
26C2×#384—21no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2100—24no—
33#2071—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1444—21no—
46C4⋊#95—21no—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#126—28no—
51C2×#126—44no—
52C2×#126—44no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—42no—
83C2×#102—42no—
84#397—28no—
85#396—28no—
86#390—22no—
87#390—22no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390D414yes—
92#389—22no—
93#389—24no—
94#389—24no—
95#389—42no—
96#389—42no—
97#389D414yes—
98#388—28no—
99#384—22no—
100#384—22no—
101#384—22no—
102#384—22no—
103#384—44no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#336—22no—
123#336—22no—
124#336—42no—
125#336—42no—
126C25—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C2×D412yes—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
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—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161#126—432no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
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—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
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—24no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4SD1614yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4#4114yes—
209C2×C4⋊C4#4114yes—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245#102—48no—
246#102—48no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—44no—
253#95—44no—
254C24—22no—
255C24—41no—
256C24—41no—
257C24—42no—
258C24—42no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—42no—
265C24C22×D411yescharacteristic
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
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#12012yes—
339C22×C4#12012yes—
340C22×C4C2×SD1612yes—
341C22×C4C2×SD1612yes—
342C22×C4C2×SD1612yes—
343C22×C4AGL(1,ℤ/8)12yes—
344C22×C4AGL(1,ℤ/8)12yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4112yes—
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+412yes—
352C22×C42−1+412yes—
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—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—48no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—88no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—88no—
391C4⋊C4#12618yescomplemented
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C42—24no—
399C42—24no—
400C42—48no—
401C42—48no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—42no—
407C42—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—82no—
416C23—81no—
417C23—82no—
418C23—81no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—81no—
424C23—82no—
425C23—81no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23C2×#12011yescharacteristic
432C23#52511yescharacteristic
433C23#53611yescharacteristic
434C23#54612yes—
435C23#54812yes—
436C23C22×SD1611yescharacteristic
437C23C2×AGL(1,ℤ/8)11yescharacteristic
438C23C2×AGL(1,ℤ/8)11yescharacteristic
439C23#57511yescharacteristic
440C23#57612yes—
441C23#57812yes—
442D4—44no—
443D4—44no—
444D4—82no—
445D4—82no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—28no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#45014yes—
509C2×C4C2×#12612yes—
510C2×C4#53211yescharacteristic
511C2×C4#53211yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#53611yescharacteristic
514C2×C4#53912yes—
515C2×C4#54112yes—
516C2×C4#54212yes—
517C2×C4#54712yes—
518C2×C4#54712yes—
519C2×C4#54712yes—
520C2×C4#55012yes—
521C2×C4#55212yes—
522C2×C4#55312yes—
523C2×C4#55612yes—
524C2×C4#56312yes—
525C8—82no—
526C8—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—162no—
535C22—162no—
536C22—162no—
537C22—162no—
538C22#240611yescharacteristic
539C22#243111yescharacteristic
540C22#243211yescharacteristic
541C22#248112yes—
542C22#271612yes—
543C22C2×#45011yescharacteristic
544C22#300611yescharacteristic
545C22#300811yescharacteristic
546C22#301111yescharacteristic
547C22#310311yescharacteristic
548C22#310911yescharacteristic
549C22#312412yes—
550C22#322711yescharacteristic
551C22#323411yescharacteristic
552C22#324312yes—
553C22#329812yes—
554C22#330912yes—
555C22#331812yes—
556C22#333012yes—
557C22#336212yes—
558C22#336612yes—
559C22#336712yes—
560C22#337712yes—
561C4—24no—
562C4—42no—
563C4—48no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—82no—
568C4—82no—
569C4—81no—
570C4—82no—
571C4—162no—
572C4#249612yes—
573C4#251212yes—
574C4#271812yes—
575C4#272112yes—
576C2—162no—
577C2#1592811yescharacteristic
578C2#2406111yescharacteristic
579C2#2407311yescharacteristic
580C2#2557111yescharacteristic
581C2#2558111yescharacteristic
582C2#2564911yescharacteristic
583C2#2566111yescharacteristic
584C2#2755412yes—
585C2#2757112yes—
586C2#3223012yes—
587C2#3225712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
4. #13910C2semidirect
5. #11346C2semidirect
6. #11207C2semidirect
10. #10885C2semidirect
11. #10884C2semidirect
391. C4⋊C4#126semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#21002
C2×#20712
#139102
#113462
#112072
#110662
#109312
#108852
#108842