Finite Group Databasebeta

← #474836

Subgroups — #474836

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2042C212yesmaximal
2#14276C211yesmaximal, characteristic, complemented
3#14273C211yesmaximal, characteristic
4#11207C212yesmaximal, complemented
5#11190C212yesmaximal, complemented
6#11189C212yesmaximal
7#11052C211yesmaximal, characteristic, complemented
8#11051C211yesmaximal, characteristic
9#11050C211yesmaximal, characteristic, complemented
10#11049C211yesmaximal, characteristic
11#10796C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#2042—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1441—22no—
36#1441—22no—
37#1439—22no—
38#1439—22no—
39C4⋊#95C2212yes—
40C23×Q8C2311yescharacteristic
41C23×D4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#124—22no—
45C2×#124—42no—
46C2×#124—42no—
47C2×#124D414yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—42no—
66C2×#103—42no—
67C2×#103—42no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—42no—
72C2×#102—42no—
73C2×#102—42no—
74#391—22no—
75#391—22no—
76#391—42no—
77#390—22no—
78#390—22no—
79#390—42no—
80#389—24no—
81#389—24no—
82#384—24no—
83#384—42no—
84#382—42no—
85#376—42no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95D412yes—
94C2×#95D412yes—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#336—24no—
100#336—22no—
101#336—22no—
102#336—42no—
103C25—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8C2×D412yes—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4C2×D412yes—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2411yescharacteristic
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—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148#124—48no—
149#124—48no—
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—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
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—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4#4114yes—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C2×D412yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201#103—48no—
202#103—48no—
203#102—48no—
204#102—48no—
205#95—42no—
206#95—42no—
207#95—44no—
208#95—42no—
209#95—42no—
210#95—44no—
211#95—44no—
212#95—82no—
213#95—82no—
214C24—22no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24C22×D411yescharacteristic
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
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—24no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
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#12114yes—
312C22×C4#12414yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4H(5,𝔽2)12yes—
319C22×C4H(5,𝔽2)12yes—
320C22×C4H(5,𝔽2)12yes—
321C22×C42−1+411yescharacteristic
322C22×C42−1+411yescharacteristic
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—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—88no—
350C4⋊C4—88no—
351C4⋊C4—88no—
352C22⋊C4—416no—
353C22⋊C4—88no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C42—24no—
357C42—24no—
358C42—24no—
359C42—24no—
360C42—42no—
361C42—42no—
362C23—42no—
363C23—44no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—81no—
368C23—81no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—81no—
374C23—81no—
375C23—82no—
376C23—82no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23C2×#12111yescharacteristic
386C23#52511yescharacteristic
387C23C2×#12411yescharacteristic
388C23#53411yescharacteristic
389C23#53511yescharacteristic
390C23#53511yescharacteristic
391C23#53611yescharacteristic
392C23#57612yes—
393C23#57712yes—
394C23#57712yes—
395C23#57812yes—
396Q8—44no—
397Q8—44no—
398Q8—82no—
399Q8—82no—
400D4—44no—
401D4—44no—
402D4—82no—
403D4—82no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4#53412yes—
460C2×C4#53812yes—
461C2×C4#54012yes—
462C2×C4#54012yes—
463C2×C4#54212yes—
464C2×C4#54212yes—
465C2×C4#54612yes—
466C2×C4#54711yescharacteristic
467C2×C4#54711yescharacteristic
468C2×C4#55312yes—
469C2×C4#55312yes—
470C2×C4#55312yes—
471C2×C4#55611yescharacteristic
472C2×C4#55611yescharacteristic
473C2×C4#56212yes—
474C2×C4#56212yes—
475C8—82no—
476C8—82no—
477C22—82no—
478C22—82no—
479C22—84no—
480C22—84no—
481C22—82no—
482C22—161no—
483C22—161no—
484C22—161no—
485C22—161no—
486C22—162no—
487C22—162no—
488C22#253511yescharacteristic
489C22#254711yescharacteristic
490C22#259111yescharacteristic
491C22#261711yescharacteristic
492C22#263211yescharacteristic
493C22#264111yescharacteristic
494C22#281611yescharacteristic
495C22#304512yes—
496C22#304612yes—
497C22#311812yes—
498C22#312112yes—
499C22#313612yes—
500C22#313712yes—
501C22#320312yes—
502C22#320412yes—
503C22#322011yescharacteristic
504C22#322111yescharacteristic
505C22#322211yescharacteristic
506C22#322311yescharacteristic
507C22#322412yes—
508C22#322512yes—
509C22#324312yes—
510C22#324412yes—
511C4—44no—
512C4—44no—
513C4—84no—
514C4—81no—
515C4—81no—
516C4—82no—
517C4—161no—
518C4—161no—
519C4—162no—
520C4#270112yes—
521C4#270112yes—
522C4#280012yes—
523C4#280012yes—
524C2—162no—
525C2#1701111yescharacteristic
526C2#2817111yescharacteristic
527C2#2817311yescharacteristic
528C2#2873012yes—
529C2#2934212yes—
530C2#2972912yes—
531C2#3006812yes—
532C2#3022211yescharacteristic
533C2#3022311yescharacteristic
534C2#3278311yescharacteristic
535C2#3278511yescharacteristic
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
2. #14276C2semidirect
4. #11207C2semidirect
5. #11190C2semidirect
7. #11052C2semidirect
9. #11050C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20422
#142762
#142732
#112072
#111902
#111892
#110522
#110512
#110502
#110492
#107962