Finite Group Databasebeta

← #413450

Subgroups — #413450

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2074C211yesmaximal, characteristic, complemented
2C2×#2073C211yesmaximal, characteristic
3C2×#1983C212yesmaximal
4#14273C211yesmaximal, characteristic
5#13779C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#11050C211yesmaximal, characteristic, complemented
9#11049C211yesmaximal, characteristic
10#10849C212yesmaximal, complemented
11#10833C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×#376C2211yescharacteristic
27#2074—24no—
28#2073—24no—
29#1983—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1439—22no—
38#1439—22no—
39#1436—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C23×Q8C2311yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#124—22no—
47C2×#124—24no—
48C2×#124—24no—
49C2×#124D414yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C42—22no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—24no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102—42no—
79#391—22no—
80#391—22no—
81#391—42no—
82#390—22no—
83#390—22no—
84#390—42no—
85#384—24no—
86#384—24no—
87#382—22no—
88#382D414yes—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—24no—
91#376—24no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100C2×#95D412yes—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#336—22no—
106#336—22no—
107#336—42no—
108C25—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8C2×D412yes—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4C2×D412yes—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
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—
153#124—216no—
154#124—48no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4#4114yes—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—44no—
198C2×C42—82no—
199C2×C42—82no—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42#4112yes—
203C2×C42#4114yes—
204C2×C42#4114yes—
205C2×C42#4112yes—
206C4⋊C8—48no—
207#103—48no—
208#103—48no—
209#102—48no—
210#102—48no—
211#95—28no—
212#95—24no—
213#95—24no—
214#95—42no—
215#95—42no—
216#95—44no—
217#95—44no—
218#95—82no—
219#95—82no—
220C24—22no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24C22×D411yescharacteristic
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—42no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
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—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4#12114yes—
316C22×C4#12114yes—
317C22×C4#12314yes—
318C22×C4#12414yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4112yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4112yes—
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)12yes—
328C22×C42−1+411yescharacteristic
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—88no—
356C4⋊C4—88no—
357C22⋊C4—416no—
358C22⋊C4—88no—
359C22⋊C4—84no—
360C22⋊C4—84no—
361C42—24no—
362C42—24no—
363C42—42no—
364C42—42no—
365C42—88no—
366C42#12318yes—
367C42#12618yes—
368C23—42no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—82no—
374C23—82no—
375C23—81no—
376C23—81no—
377C23—82no—
378C23—82no—
379C23—81no—
380C23—82no—
381C23—82no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—81no—
391C23C2×#12111yescharacteristic
392C23C2×#12111yescharacteristic
393C23C2×#12311yescharacteristic
394C23#52511yescharacteristic
395C23C2×#12411yescharacteristic
396C23#53511yescharacteristic
397C23#53711yescharacteristic
398C23#57612yes—
399C23#57712yes—
400C23#57712yes—
401C23#57812yes—
402Q8—44no—
403Q8—44no—
404Q8—82no—
405Q8—82no—
406D4—44no—
407D4—44no—
408D4—82no—
409D4—82no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4#39718yes—
465C2×C4C2×#12312yes—
466C2×C4C2×#12612yes—
467C2×C4#52912yes—
468C2×C4#53212yes—
469C2×C4#54012yes—
470C2×C4#54212yes—
471C2×C4D4211yescharacteristic
472C2×C4#54612yes—
473C2×C4#54711yescharacteristic
474C2×C4#54711yescharacteristic
475C2×C4#55012yes—
476C2×C4#55312yes—
477C2×C4#55511yescharacteristic
478C2×C4#55612yes—
479C2×C4#55912yes—
480C2×C4#55912yes—
481C8—82no—
482C8—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—84no—
487C22—84no—
488C22—161no—
489C22—161no—
490C22—162no—
491C22—162no—
492C22—161no—
493C22—161no—
494C22C2×#39711yescharacteristic
495C22#253511yescharacteristic
496C22#253711yescharacteristic
497C22#254811yescharacteristic
498C22#259211yescharacteristic
499C22#262411yescharacteristic
500C22#285411yescharacteristic
501C22#304512yes—
502C22#304512yes—
503C22#304612yes—
504C22#304612yes—
505C22#309812yes—
506C22#309912yes—
507C22#311812yes—
508C22#312112yes—
509C22#313612yes—
510C22#313712yes—
511C22#322011yescharacteristic
512C22#322111yescharacteristic
513C22#322211yescharacteristic
514C22#322311yescharacteristic
515C22#326212yes—
516C22#326312yes—
517C4—28no—
518C4—44no—
519C4—84no—
520C4—82no—
521C4—84no—
522C4—81no—
523C4—81no—
524C4—161no—
525C4—161no—
526C4#244512yes—
527C4#244712yes—
528C4#251212yes—
529C4#251312yes—
530C2—162no—
531C2#1514711yescharacteristic
532C2#2363412yes—
533C2#2817111yescharacteristic
534C2#2817311yescharacteristic
535C2#2826212yes—
536C2#2876011yescharacteristic
537C2#2876111yescharacteristic
538C2#2936212yes—
539C2#2993111yescharacteristic
540C2#2993211yescharacteristic
541C2#3311512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2074C2semidirect
5. #13779C2semidirect
6. #11190C2semidirect
8. #11050C2semidirect
10. #10849C2semidirect
11. #10833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20742
C2×#20732
C2×#19832
#142732
#137792
#111902
#111892
#110502
#110492
#108492
#108332