Finite Group Databasebeta

← #472488

Subgroups — #472488

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1970C211yesmaximal, characteristic
2C2×#1916C211yesmaximal, characteristic, complemented
3#15134C212yesmaximal, complemented
4#11328C211yesmaximal, characteristic, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11076C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#11049C212yesmaximal, complemented
9#11022C211yesmaximal, characteristic
10#11022C211yesmaximal, characteristic
11#10841C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#396—22no—
20C2×#396—22no—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#384C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1970—24no—
28#1916—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1439—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C23×D4C2312yes—
42C22×D8—22no—
43C22×D8—22no—
44C22×D8—42no—
45C22×D8—42no—
46C22×D8Q814yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—22no—
69C2×#107—42no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—42no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—42no—
75C2×#102—42no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×C4×C8—42no—
82C2×C4×C8D414yes—
83#396—48no—
84#396—48no—
85#390—24no—
86#390—24no—
87#390—24no—
88#390—42no—
89#390—42no—
90#390—42no—
91#389—42no—
92#389—42no—
93#384—24no—
94#384—42no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D411yescharacteristic
101C2×#95D411yescharacteristic
102#340—22no—
103#340—22no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111C25—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4C2×Q814yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C2×D8—28no—
137C2×D8—48no—
138C2×D8—44no—
139C2×D8—44no—
140C2×D8—48no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8C2×Q812yes—
160C22×C8#4112yes—
161C2×C4⋊C4—22no—
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—24no—
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—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4D814yescomplemented
192C2×C4⋊C4#4114yes—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
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⋊C4—44no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C2×D412yes—
222C2×C42#4112yes—
223#107—44no—
224C4⋊C8—44no—
225#102—48no—
226#102—48no—
227#102—48no—
228C4×C8—44no—
229#95—22no—
230#95—22no—
231#95—44no—
232#95—44no—
233#95—44no—
234C24—24no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24C22×D411yescharacteristic
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
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—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4C2×D812yes—
325C22×C4C2×D812yes—
326C22×C4C2×D812yes—
327C22×C4#13512yes—
328C22×C4AGL(1,ℤ/8)14yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×Q811yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4112yes—
333C22×C4C2×#4111yescharacteristic
334C22×C4H(5,𝔽2)12yes—
335C22×C4H(5,𝔽2)12yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C42−1+411yescharacteristic
339D8—416no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8C22⋊Q814yes—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—88no—
375C4⋊C4—88no—
376C4⋊C4—88no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C22⋊C4—88no—
382C22⋊C4—88no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—42no—
391C42—42no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23C2×#12111yescharacteristic
428C23#53512yes—
429C23#54612yes—
430C23D4×Q811yescharacteristic
431C23#55011yescharacteristic
432C23C22×D811yescharacteristic
433C23C2×#13511yescharacteristic
434C23C2×AGL(1,ℤ/8)12yes—
435C23#57611yescharacteristic
436C23#57712yes—
437C23#57811yescharacteristic
438D4—44no—
439D4—44no—
440D4—44no—
441D4—44no—
442D4—82no—
443D4—82no—
444D4—82no—
445D4—82no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—44no—
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—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4#46614yes—
496C2×C4#46914yes—
497C2×C4C2×C22⋊Q811yescharacteristic
498C2×C4#53211yescharacteristic
499C2×C4#53411yescharacteristic
500C2×C4#53511yescharacteristic
501C2×C4#53511yescharacteristic
502C2×C4#53711yescharacteristic
503C2×C4#53812yes—
504C2×C4#53812yes—
505C2×C4#54012yes—
506C2×C4#54212yes—
507C2×C4#54312yes—
508C2×C4#55312yes—
509C2×C4#55312yes—
510C2×C4#55412yes—
511C2×C4#55411yescharacteristic
512C2×C4#55611yescharacteristic
513C2×C4#56012yes—
514C2×C4#56212yes—
515C8—24no—
516C8—42no—
517C8—81no—
518C8—81no—
519C22—84no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—162no—
527C22—162no—
528C22—162no—
529C22—162no—
530C22—162no—
531C22—162no—
532C22—162no—
533C22—162no—
534C22#253811yescharacteristic
535C22#254311yescharacteristic
536C22#261911yescharacteristic
537C22#269212yes—
538C22#271412yes—
539C22C2×#46611yescharacteristic
540C22C2×#46911yescharacteristic
541C22#306011yescharacteristic
542C22#306111yescharacteristic
543C22#320512yes—
544C22#320712yes—
545C22#322412yes—
546C22#322512yes—
547C22#329412yes—
548C22#330112yes—
549C22#330812yes—
550C22#331812yes—
551C22Q8×D811yescharacteristic
552C22#338011yescharacteristic
553C22#338412yes—
554C22#339111yescharacteristic
555C22#339311yescharacteristic
556C22#339712yes—
557C4—24no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—81no—
562C4—81no—
563C4—84no—
564C4—162no—
565C4—162no—
566C4#256611yescharacteristic
567C4#262611yescharacteristic
568C4#266211yescharacteristic
569C4#278012yes—
570C4#280012yes—
571C4#281311yescharacteristic
572C2—164no—
573C2#1682211yescharacteristic
574C2#2826311yescharacteristic
575C2#2827311yescharacteristic
576C2#2855111yescharacteristic
577C2#2856111yescharacteristic
578C2#2980611yescharacteristic
579C2#2981311yescharacteristic
580C2#3151812yes—
581C2#3152312yes—
582C2#3216212yes—
583C2#3216712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
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×#1916C2semidirect
3. #15134C2semidirect
4. #11328C2semidirect
5. #11256C2semidirect
6. #11076C2semidirect
7. #11066C2semidirect
8. #11049C2semidirect
11. #10841C2semidirect
191. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#19702
C2×#19162
#151342
#113282
#112562
#110762
#110662
#110492
#110222
#108412