Finite Group Databasebeta

← #409336

Subgroups — #409336

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14348C211yesmaximal, characteristic, complemented
2#13785C211yesmaximal, characteristic
3#11323C211yesmaximal, characteristic, complemented
4#11315C212yesmaximal, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11121C212yesmaximal
7#11051C211yesmaximal, characteristic
8#11049C211yesmaximal, characteristic
9#10908C212yesmaximal, complemented
10#10796C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×#106C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×#376C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31#1442—22no—
32#1442—22no—
33#1437—22no—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C22⋊#95—22no—
39C23×D4C2311yescharacteristic
40C24×C4—21no—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4—42no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C22⋊C4—21no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#106—24no—
61C2×#106—42no—
62C2×#106—42no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—42no—
67C2×#102—42no—
68C2×#102—42no—
69#390—22no—
70#390—22no—
71#390—22no—
72#390Q814yes—
73#384—42no—
74#382—42no—
75C4×C4⋊C4—42no—
76#376—24no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95D412yes—
86C2×#95D412yes—
87#340—24no—
88#340—24no—
89#340—42no—
90#340—42no—
91#340—42no—
92#336—24no—
93#336—22no—
94#336—22no—
95#336—24no—
96C25—21no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—21no—
100C22×D4—21no—
101C22×D4—21no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4C2×Q812yes—
105C22×D4#4112yes—
106C22×D4#4112yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—24no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—82no—
116C23×C4—82no—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8#4114yes—
136C4×D4—48no—
137C4×D4—48no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
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—44no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C2×Q812yes—
163C2×C4⋊C4#4112yes—
164C2×C4⋊C4#4112yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—82no—
171C2×C22⋊C4—82no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—82no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—42no—
184C2×C42—82no—
185C2×C42C2×Q812yes—
186C2×C42#4112yes—
187C2×C42#4114yes—
188#106—48no—
189#102—48no—
190#102—48no—
191#95—24no—
192#95—24no—
193#95—42no—
194#95—44no—
195#95—82no—
196#95—82no—
197#95—82no—
198#95—82no—
199#95—82no—
200#95—82no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—41no—
205C24—41no—
206C24—41no—
207C24—41no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24C22×D411yescharacteristic
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4#12514yes—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—48no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—88no—
281C22×C4—88no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4C22⋊Q814yes—
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4C22×Q811yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4H(5,𝔽2)12yes—
296C22×C4H(5,𝔽2)12yes—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—88no—
326C22⋊C4—48no—
327C22⋊C4—88no—
328C22⋊C4—88no—
329C42—24no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—42no—
334C42—42no—
335C42—84no—
336C23—24no—
337C23—42no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23—42no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—82no—
350C23—82no—
351C23—82no—
352C23—82no—
353C23—81no—
354C23—81no—
355C23—82no—
356C23—81no—
357C23—81no—
358C23—82no—
359C23C2×C22⋊Q811yescharacteristic
360C23C2×#12311yescharacteristic
361C23#52511yescharacteristic
362C23#52511yescharacteristic
363C23#53211yescharacteristic
364C23#53511yescharacteristic
365C23#53511yescharacteristic
366C23#57612yes—
367C23#57712yes—
368C23#57712yes—
369C23#57712yes—
370D4—28no—
371D4—44no—
372D4—44no—
373D4—44no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—816no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—82no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4C2×#12511yescharacteristic
429C2×C4#52912yes—
430C2×C4#52911yescharacteristic
431C2×C4#52912yes—
432C2×C4#52911yescharacteristic
433C2×C4#53311yescharacteristic
434C2×C4#53812yes—
435C2×C4#53812yes—
436C2×C4#55112yes—
437C2×C4#55112yes—
438C2×C4#55112yes—
439C2×C4#55312yes—
440C2×C4#55312yes—
441C2×C4#55412yes—
442C2×C4#55412yes—
443C2×C4#55512yes—
444C8—44no—
445C8—82no—
446C22—48no—
447C22—44no—
448C22—82no—
449C22—82no—
450C22—84no—
451C22—82no—
452C22—82no—
453C22—84no—
454C22—82no—
455C22—82no—
456C22—84no—
457C22#239511yescharacteristic
458C22#248211yescharacteristic
459C22#248711yescharacteristic
460C22#256611yescharacteristic
461C22#261011yescharacteristic
462C22#263111yescharacteristic
463C22#285111yescharacteristic
464C22#308112yes—
465C22#308412yes—
466C22#309612yes—
467C22#309912yes—
468C22#311812yes—
469C22#311912yes—
470C22#312312yes—
471C22#312512yes—
472C22#317312yes—
473C22#317912yes—
474C22#322011yescharacteristic
475C22#322111yescharacteristic
476C22#322112yes—
477C22#322211yescharacteristic
478C22#322211yescharacteristic
479C22#322412yes—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4—162no—
488C4—162no—
489C4#250512yes—
490C4#251812yes—
491C4#257412yes—
492C4#257612yes—
493C2—84no—
494C2#1501711yescharacteristic
495C2#2377112yes—
496C2#2759411yescharacteristic
497C2#2759711yescharacteristic
498C2#2768812yes—
499C2#2919911yescharacteristic
500C2#2920411yescharacteristic
501C2#2957012yes—
502C2#3004911yescharacteristic
503C2#3005311yescharacteristic
504C2#3307612yes—
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. #11323C2semidirect
4. #11315C2semidirect
5. #11268C2semidirect
9. #10908C2semidirect
10. #10796C2semidirect

Maximal subgroups

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

subgroupindex
#143482
#137852
#113232
#113152
#112682
#111212
#110512
#110492
#109082
#107962
#107882