Finite Group Databasebeta

← #474766

Subgroups — #474766

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14708C211yesmaximal, characteristic
2#14707C211yesmaximal, characteristic, complemented
3#11268C212yesmaximal, complemented
4#11067C212yesmaximal, complemented
5#11066C212yesmaximal
6#11052C211yesmaximal, characteristic, complemented
7#11052C211yesmaximal, characteristic, complemented
8#11051C211yesmaximal, characteristic
9#11051C211yesmaximal, characteristic
10#11009C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2212yes—
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C43C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1437—22no—
35#1437—22no—
36#1436—22no—
37#1436—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⋊C4C2312yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—22no—
56C22×C42D412yes—
57C22×C42D412yes—
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#382—24no—
82#382—42no—
83#382—42no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C43—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—24no—
101#336—22no—
102#336—22no—
103#336—42no—
104C25—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8C2×Q812yes—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4C2×Q812yes—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2411yescharacteristic
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—
148C22×C8—22no—
149#124—48no—
150#124—48no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
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—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4#4114yes—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42C2×Q812yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202#103—48no—
203#103—48no—
204#102—48no—
205#102—48no—
206#95—24no—
207#95—24no—
208#95—44no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—42no—
217C24—22no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24C22×D411yescharacteristic
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
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—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
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—42no—
302C22×C4—41no—
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—42no—
312C22×C4C22⋊Q814yes—
313C22×C4#12412yes—
314C22×C4#12412yes—
315C22×C4#13514yes—
316C22×C4C22×Q811yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4H(5,𝔽2)12yes—
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
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—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—88no—
353C22⋊C4—416no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C22⋊C4—88no—
357C42—24no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—48no—
362C42—48no—
363C42—42no—
364C42—42no—
365C23—42no—
366C23—44no—
367C23—44no—
368C23—42no—
369C23—42no—
370C23—82no—
371C23—81no—
372C23—82no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—81no—
388C23C2×C22⋊Q811yescharacteristic
389C23#52511yescharacteristic
390C23C2×#12411yescharacteristic
391C23#53411yescharacteristic
392C23#53511yescharacteristic
393C23#53511yescharacteristic
394C23#53511yescharacteristic
395C23C2×#13512yes—
396C23#57512yes—
397C23#57512yes—
398C23#57512yes—
399Q8—44no—
400Q8—44no—
401Q8—82no—
402Q8—82no—
403D4—44no—
404D4—44no—
405D4—82no—
406D4—82no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
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—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#53812yes—
463C2×C4#53812yes—
464C2×C4#53812yes—
465C2×C4#54012yes—
466C2×C4#54012yes—
467C2×C4#54312yes—
468C2×C4#55312yes—
469C2×C4#55312yes—
470C2×C4#55411yescharacteristic
471C2×C4#55411yescharacteristic
472C2×C4#55412yes—
473C2×C4#55511yescharacteristic
474C2×C4#55511yescharacteristic
475C2×C4#55912yes—
476C2×C4#56012yes—
477C2×C4#56012yes—
478C8—82no—
479C8—82no—
480C22—84no—
481C22—84no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—161no—
486C22—161no—
487C22—162no—
488C22—161no—
489C22—161no—
490C22—162no—
491C22#256311yescharacteristic
492C22#258011yescharacteristic
493C22#259111yescharacteristic
494C22#262611yescharacteristic
495C22#263111yescharacteristic
496C22#264111yescharacteristic
497C22#281511yescharacteristic
498C22#307612yes—
499C22#308012yes—
500C22#310212yes—
501C22#311112yes—
502C22#313111yescharacteristic
503C22#313511yescharacteristic
504C22#314011yescharacteristic
505C22#314011yescharacteristic
506C22#319112yes—
507C22#319712yes—
508C22#321112yes—
509C22#321112yes—
510C22#321212yes—
511C22#321912yes—
512C22#321912yes—
513C22#321912yes—
514C4—44no—
515C4—42no—
516C4—42no—
517C4—44no—
518C4—84no—
519C4—82no—
520C4—161no—
521C4—161no—
522C4—162no—
523C4#278012yes—
524C4#278012yes—
525C4#280312yes—
526C4#280312yes—
527C2—162no—
528C2#1700711yescharacteristic
529C2#2914211yescharacteristic
530C2#2914611yescharacteristic
531C2#2926112yes—
532C2#2933311yescharacteristic
533C2#2933911yescharacteristic
534C2#2995812yes—
535C2#3004412yes—
536C2#3022111yescharacteristic
537C2#3022511yescharacteristic
538C2#3276312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14707C2semidirect
3. #11268C2semidirect
4. #11067C2semidirect
6. #11052C2semidirect
7. #11052C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
#147082
#147072
#112682
#110672
#110662
#110522
#110512
#110092
#107952