Finite Group Databasebeta

← #470617

Subgroups — #470617

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14348C211yesmaximal, characteristic, complemented
2#14273C211yesmaximal, characteristic
3#11267C212yesmaximal, complemented
4#11190C212yesmaximal, complemented
5#11189C212yesmaximal
6#11052C211yesmaximal, characteristic, complemented
7#11051C211yesmaximal, characteristic
8#11050C211yesmaximal, characteristic, complemented
9#11049C211yesmaximal, characteristic
10#11009C212yesmaximal
11#10796C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#376C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1442—22no—
34#1442—22no—
35#1439—22no—
36#1439—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C23×Q8C2311yescharacteristic
40C23×D4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#124—22no—
44C2×#124—22no—
45C2×#124—24no—
46C2×#124—42no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#103—24no—
62C2×#103—42no—
63C2×#103—42no—
64C2×#103—42no—
65C2×#103—42no—
66C2×#103—42no—
67C2×#102—24no—
68C2×#102—42no—
69C2×#102—42no—
70C2×#102—42no—
71C2×#102—42no—
72C2×#102—42no—
73#391—22no—
74#391—22no—
75#391—42no—
76#390—22no—
77#390—22no—
78#390—42no—
79#384—42no—
80#384—42no—
81#382—42no—
82#376—24no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#336—24no—
97#336—22no—
98#336—22no—
99#336—42no—
100C25—21no—
101C22×Q8—21no—
102C22×Q8—22no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8#4112yes—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4#4112yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145#124—48no—
146#124—48no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
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—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42#4114yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198#103—48no—
199#103—48no—
200#102—48no—
201#102—48no—
202#95—24no—
203#95—24no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—82no—
208#95—82no—
209#95—82no—
210#95—82no—
211C24—22no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24C22×D411yescharacteristic
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
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—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
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#12312yes—
308C22×C4#12312yes—
309C22×C4#12414yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4H(5,𝔽2)12yes—
316C22×C4H(5,𝔽2)12yes—
317C22×C4H(5,𝔽2)12yes—
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C42−1+411yescharacteristic
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—88no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—88no—
352C22⋊C4—88no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—24no—
357C42—42no—
358C42—42no—
359C23—42no—
360C23—44no—
361C23—44no—
362C23—42no—
363C23—42no—
364C23—81no—
365C23—82no—
366C23—82no—
367C23—81no—
368C23—82no—
369C23—82no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—81no—
382C23C2×#12311yescharacteristic
383C23#52511yescharacteristic
384C23C2×#12411yescharacteristic
385C23#53511yescharacteristic
386C23#53511yescharacteristic
387C23#53511yescharacteristic
388C23#53711yescharacteristic
389C23#57612yes—
390C23#57712yes—
391C23#57712yes—
392C23#57812yes—
393Q8—44no—
394Q8—44no—
395Q8—82no—
396Q8—82no—
397D4—44no—
398D4—44no—
399D4—82no—
400D4—82no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4#53812yes—
457C2×C4#53812yes—
458C2×C4#54012yes—
459C2×C4#54012yes—
460C2×C4#54012yes—
461C2×C4#54212yes—
462C2×C4#55112yes—
463C2×C4#55112yes—
464C2×C4#55211yescharacteristic
465C2×C4#55211yescharacteristic
466C2×C4#55311yescharacteristic
467C2×C4#55311yescharacteristic
468C2×C4#55312yes—
469C2×C4#55312yes—
470C2×C4H(3,𝔽4)12yes—
471C2×C4#56212yes—
472C8—82no—
473C8—82no—
474C22—82no—
475C22—84no—
476C22—82no—
477C22—82no—
478C22—84no—
479C22—161no—
480C22—161no—
481C22—161no—
482C22—161no—
483C22—162no—
484C22—162no—
485C22#248211yescharacteristic
486C22#250411yescharacteristic
487C22#256211yescharacteristic
488C22#262411yescharacteristic
489C22#263111yescharacteristic
490C22#264111yescharacteristic
491C22#281911yescharacteristic
492C22#309812yes—
493C22#309912yes—
494C22#312412yes—
495C22#312512yes—
496C22#313612yes—
497C22#313712yes—
498C22#322011yescharacteristic
499C22#322111yescharacteristic
500C22#322211yescharacteristic
501C22#322311yescharacteristic
502C22#322412yes—
503C22#322412yes—
504C22#322512yes—
505C22#322512yes—
506C22#325712yes—
507C22#325812yes—
508C4—44no—
509C4—42no—
510C4—42no—
511C4—82no—
512C4—82no—
513C4—161no—
514C4—161no—
515C4—162no—
516C4—162no—
517C4#275312yes—
518C4#275312yes—
519C4#276112yes—
520C4#276112yes—
521C2—162no—
522C2#1669111yescharacteristic
523C2#2759511yescharacteristic
524C2#2759711yescharacteristic
525C2#2798412yes—
526C2#2913712yes—
527C2#2993512yes—
528C2#3005512yes—
529C2#3022211yescharacteristic
530C2#3022311yescharacteristic
531C2#3281211yescharacteristic
532C2#3281311yescharacteristic
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. #14348C2semidirect
3. #11267C2semidirect
4. #11190C2semidirect
6. #11052C2semidirect
8. #11050C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
#143482
#142732
#112672
#111902
#111892
#110522
#110512
#110502
#110492
#110092
#107962