Finite Group Databasebeta

← #474774

Subgroups — #474774

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14277C211yesmaximal, characteristic, complemented
2#14273C211yesmaximal, characteristic
3#11267C212yesmaximal, complemented
4#11067C212yesmaximal, complemented
5#11066C212yesmaximal
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×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#376C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1444—22no—
35#1442—22no—
36#1439—22no—
37#1439—22no—
38C4⋊#95C2212yes—
39C23×Q8C2311yescharacteristic
40C23×D4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#124—22no—
44C2×#124—42no—
45C2×#124—42no—
46C2×#124Q814yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
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#389—42no—
80#389—42no—
81#384—42no—
82#384—42no—
83#382—24no—
84#376—42no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D412yes—
93C2×#95D412yes—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#336—24no—
99#336—22no—
100#336—22no—
101#336—42no—
102C25—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8C2×Q812yes—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4C2×Q812yes—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2411yescharacteristic
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—
145C22×C8—22no—
146C22×C8—22no—
147#124—48no—
148#124—48no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
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—22no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4#4114yes—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42C2×Q812yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200#103—48no—
201#103—48no—
202#102—48no—
203#102—48no—
204#95—28no—
205#95—42no—
206#95—42no—
207#95—44no—
208#95—42no—
209#95—42no—
210#95—44no—
211#95—82no—
212#95—82no—
213C24—22no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24C22×D411yescharacteristic
225C2×Q8—24no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—24no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C22×C4—22no—
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—24no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
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×C4C22⋊Q814yes—
311C22×C4#12414yes—
312C22×C4C22×Q811yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4H(5,𝔽2)12yes—
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C4H(5,𝔽2)12yes—
321C22×C4H(5,𝔽2)12yes—
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—
334C2×C8—42no—
335C2×C8—42no—
336C4⋊C4—24no—
337C4⋊C4—24no—
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—42no—
347C4⋊C4—42no—
348C4⋊C4—88no—
349C4⋊C4—88no—
350C4⋊C4—88no—
351C22⋊C4—416no—
352C22⋊C4—88no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C42—24no—
356C42—24no—
357C42—24no—
358C42—24no—
359C42—42no—
360C42—42no—
361C23—42no—
362C23—44no—
363C23—44no—
364C23—42no—
365C23—42no—
366C23—81no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—81no—
372C23—81no—
373C23—82no—
374C23—82no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—81no—
384C23C2×C22⋊Q811yescharacteristic
385C23#52511yescharacteristic
386C23C2×#12411yescharacteristic
387C23#53411yescharacteristic
388C23#53511yescharacteristic
389C23#53511yescharacteristic
390C23#53511yescharacteristic
391C23#57612yes—
392C23#57712yes—
393C23#57712yes—
394C23#57812yes—
395Q8—44no—
396Q8—44no—
397Q8—82no—
398Q8—82no—
399D4—44no—
400D4—44no—
401D4—82no—
402D4—82no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—42no—
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—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
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—41no—
432C2×C4—41no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4#53812yes—
459C2×C4#53812yes—
460C2×C4#53812yes—
461C2×C4#54012yes—
462C2×C4#54012yes—
463C2×C4#54312yes—
464C2×C4#55312yes—
465C2×C4#55312yes—
466C2×C4#55411yescharacteristic
467C2×C4#55411yescharacteristic
468C2×C4#55412yes—
469C2×C4#55511yescharacteristic
470C2×C4#55511yescharacteristic
471C2×C4#55912yes—
472C2×C4#56012yes—
473C2×C4#56012yes—
474C8—82no—
475C8—82no—
476C22—84no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—84no—
481C22—161no—
482C22—161no—
483C22—162no—
484C22—162no—
485C22—161no—
486C22—161no—
487C22#256311yescharacteristic
488C22#258011yescharacteristic
489C22#259111yescharacteristic
490C22#262611yescharacteristic
491C22#263111yescharacteristic
492C22#264111yescharacteristic
493C22#281511yescharacteristic
494C22#308312yes—
495C22#308412yes—
496C22#312412yes—
497C22#312512yes—
498C22#313612yes—
499C22#313712yes—
500C22#319812yes—
501C22#320112yes—
502C22#322011yescharacteristic
503C22#322111yescharacteristic
504C22#322211yescharacteristic
505C22#322311yescharacteristic
506C22#322412yes—
507C22#322412yes—
508C22#322512yes—
509C22#322512yes—
510C4—44no—
511C4—44no—
512C4—84no—
513C4—82no—
514C4—81no—
515C4—81no—
516C4—161no—
517C4—161no—
518C4—162no—
519C4#278012yes—
520C4#278012yes—
521C4#280312yes—
522C4#280312yes—
523C2—162no—
524C2#1700711yescharacteristic
525C2#2915112yes—
526C2#2926811yescharacteristic
527C2#2926911yescharacteristic
528C2#2934512yes—
529C2#2996112yes—
530C2#3005512yes—
531C2#3022211yescharacteristic
532C2#3022311yescharacteristic
533C2#3276511yescharacteristic
534C2#3276711yescharacteristic
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. #14277C2semidirect
3. #11267C2semidirect
4. #11067C2semidirect
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
#142772
#142732
#112672
#110672
#110662
#110522
#110512
#110502
#110492
#110092
#107962