Finite Group Databasebeta

← #5236477

Subgroups — #5236477

Order 512.

2108 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C211yesmaximal, characteristic, complemented
2#16589C214yesmaximal, complemented
3#16451C214yesmaximal, complemented
4#16359C214yesmaximal, complemented
5#16014C211yesmaximal, characteristic, complemented
6#15918C214yesmaximal, complemented
7#15606C214yesmaximal, complemented
8#15387C214yesmaximal, complemented
9#15261C212yesmaximal
10#14661C211yesmaximal, characteristic
11#14410C212yesmaximal, complemented
12C22×#127C2211yescharacteristic
13C22×#123C2212yes—
14C2×#398C2214yes—
15C2×#397C2214yes—
16C2×#394C2214yes—
17C2×#388C2214yes—
18C2×#388C2214yes—
19C2×#386C2212yes—
20C2×#386C2212yes—
21C2×#383C2212yes—
22C2×#380C2214yes—
23#1445C2214yes—
24#1443C2214yes—
25#1443C2214yes—
26#1442C2214yes—
27#1442C2214yes—
28#1442C2214yes—
29#1442C2214yes—
30#1441C2214yes—
31#1440C2214yes—
32#1440C2214yes—
33#1439C2214yes—
34#1439C2214yes—
35#1439C2214yes—
36#1439C2214yes—
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yes—
40#1438C2214yes—
41#1438C2212yes—
42#1438C2211yescharacteristic
43#1438C2211yescharacteristic
44#1438C2212yes—
45#1438C2214yes—
46#1438C2212yes—
47#1438C2214yes—
48#1438C2214yes—
49#1438C2212yes—
50#1437C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1434C2214yes—
56C22⋊#95C2214yes—
57C23×D4C2312yes—
58C23×D4C2312yes—
59C23×D4C2311yescharacteristic
60C23×D4C2311yescharacteristic
61C24×C4C2312yes—
62C2×#127—24no—
63C2×#127—44no—
64C2×#123—48no—
65C2×#123—44no—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C42C2311yescharacteristic
84C22×C42C2314yes—
85#398—48no—
86#397—48no—
87#394—48no—
88#388—48no—
89#388—48no—
90#386—28no—
91#386—44no—
92#383—44no—
93#380—48no—
94C2×#95C2314yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2312yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2312yes—
114C2×#95C2314yes—
115C25C2411yescharacteristic
116C25C2411yescharacteristic
117C25C2412yes—
118C22×D4—24no—
119C22×D4—24no—
120C22×D4—22no—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—44no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4—42no—
127C22×D4—42no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—44no—
132C22×D4—44no—
133C22×D4#4114yes—
134C23×C4—24no—
135C23×C4—28no—
136C23×C4—42no—
137C23×C4—44no—
138C23×C4—42no—
139C23×C4C2414yes—
140C23×C4C2414yes—
141C23×C4C2414yes—
142C23×C4C2414yes—
143C23×C4C2414yes—
144C23×C4C2411yescharacteristic
145C23×C4C2414yes—
146C23×C4C2412yes—
147#127—416no—
148#123—432no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
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—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—44no—
200C2×C42—44no—
201C2×C42—44no—
202#95—28no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—44no—
217#95—42no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223C24—24no—
224C24—24no—
225C24—22no—
226C24—22no—
227C24—24no—
228C24—24no—
229C24—24no—
230C24—24no—
231C24—21no—
232C24—21no—
233C24—24no—
234C24—24no—
235C24—22no—
236C24—22no—
237C24—24no—
238C24—24no—
239C24—24no—
240C24—24no—
241C24C2×#4114yes—
242C24C2511yescharacteristic
243C2×D4—44no—
244C2×D4—48no—
245C2×D4—48no—
246C2×D4—48no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—48no—
250C2×D4—84no—
251C2×D4—84no—
252C2×D4—84no—
253C2×D4—88no—
254C2×D4—88no—
255C2×D4—88no—
256C2×D4—88no—
257C2×D4—88no—
258C2×D4—88no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—48no—
307C22×C4—416no—
308C22×C4—88no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4C2×#4112yes—
312C22×C4H(5,𝔽2)14yes—
313C22×C4H(5,𝔽2)18yes—
314C4⋊C4—416no—
315C4⋊C4—88no—
316C4⋊C4—88no—
317C4⋊C4—88no—
318C22⋊C4—416no—
319C22⋊C4—416no—
320C22⋊C4—416no—
321C22⋊C4—48no—
322C22⋊C4—88no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C22⋊C4—88no—
326C22⋊C4—88no—
327C22⋊C4—88no—
328C22⋊C4—88no—
329C22⋊C4—88no—
330C22⋊C4—88no—
331C42—28no—
332C42—88no—
333C23—24no—
334C23—24no—
335C23—24no—
336C23—28no—
337C23—28no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—44no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23C22×#4111yescharacteristic
377C23C2×H(5,𝔽2)12yes—
378C23C2×H(5,𝔽2)14yes—
379C23#58514yes—
380C23#58514yes—
381D4—88no—
382D4—816no—
383D4—816no—
384D4—88no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—28no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—816no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—48no—
440C22—48no—
441C22—84no—
442C22—84no—
443C22—84no—
444C22—84no—
445C22—82no—
446C22—82no—
447C22—84no—
448C22—84no—
449C22—81no—
450C22—81no—
451C22—84no—
452C22—84no—
453C22—84no—
454C22—84no—
455C22#349011yescharacteristic
456C22#349112yes—
457C22#351314yes—
458C22#353212yes—
459C22#353312yes—
460C22#353914yes—
461C22#354614yes—
462C22#355114yes—
463C22#355114yes—
464C22#355814yes—
465C22#356614yes—
466C4—48no—
467C4—44no—
468C4—42no—
469C4—84no—
470C4—84no—
471C4—84no—
472C4—84no—
473C4—84no—
474C2—84no—
475C2—161no—
476C2—161no—
477C2#4719511yescharacteristic
478C2#4986414yes—
479C2#5199312yes—
480C2#5496214yes—
481C2#5533114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17273C2semidirect
2. #16589C2semidirect
3. #16451C2semidirect
4. #16359C2semidirect
5. #16014C2semidirect
6. #15918C2semidirect
7. #15606C2semidirect
8. #15387C2semidirect
11. #14410C2semidirect

Maximal subgroups

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

subgroupindex
#172732
#165892
#164512
#163592
#160142
#159182
#156062
#153872
#152612
#146612
#144102