Finite Group Databasebeta

← #4130877

Subgroups — #4130877

Order 512.

2404 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17271C211yesmaximal, characteristic, complemented
2#16014C211yesmaximal, characteristic, complemented
3#15694C214yesmaximal, complemented
4#15606C214yesmaximal, complemented
5#15381C214yesmaximal, complemented
6#15261C212yesmaximal
7#14201C214yesmaximal, complemented
8#14154C211yesmaximal, characteristic
9#14145C212yesmaximal, complemented
10#14095C214yesmaximal, complemented
11#14069C214yesmaximal, complemented
12C22×#127C2211yescharacteristic
13C22×#123C2212yes—
14C2×#397C2214yes—
15C2×#394C2214yes—
16C2×#390C2214yes—
17C2×#390C2214yes—
18C2×#388C2214yes—
19C2×#388C2214yes—
20C2×#387C2214yes—
21C2×#386C2212yes—
22C2×#386C2212yes—
23C2×C4×C4⋊C4C2214yes—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C22⋊C4C2214yes—
26#1444C2214yes—
27#1443C2214yes—
28#1442C2214yes—
29#1441C2214yes—
30#1440C2214yes—
31#1440C2214yes—
32#1439C2214yes—
33#1439C2214yes—
34#1439C2214yes—
35#1439C2214yes—
36#1439C2214yes—
37#1438C2214yes—
38#1438C2212yes—
39#1438C2211yescharacteristic
40#1438C2212yes—
41#1438C2211yescharacteristic
42#1438C2212yes—
43#1438C2212yes—
44#1438C2214yes—
45#1437C2214yes—
46#1437C2214yes—
47#1437C2214yes—
48#1437C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1434C2214yes—
55#1434C2214yes—
56C22⋊#95C2214yes—
57C23×D4C2312yes—
58C23×D4C2312yes—
59C23×D4C2311yescharacteristic
60C23×D4C2311yescharacteristic
61C24×C4C2312yes—
62C2×#127—24no—
63C2×#127—44no—
64C2×#123—216no—
65C2×#123—44no—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C42C2311yescharacteristic
84C22×C42C2314yes—
85C22×C42C2314yes—
86C22×C42C2314yes—
87#397—48no—
88#394—216no—
89#390—48no—
90#390—48no—
91#388—48no—
92#388—48no—
93#387—216no—
94#386—28no—
95#386—44no—
96C4×C4⋊C4—48no—
97C4×C4⋊C4—44no—
98C4×C22⋊C4—48no—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2312yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2312yes—
118C25C2411yescharacteristic
119C25C2411yescharacteristic
120C25C2412yes—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—28no—
124C22×D4—28no—
125C22×D4—22no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4#4114yes—
137C23×C4—24no—
138C23×C4—28no—
139C23×C4—42no—
140C23×C4—44no—
141C23×C4—42no—
142C23×C4C2414yes—
143C23×C4C2414yes—
144C23×C4C2414yes—
145C23×C4C2414yes—
146C23×C4C2414yes—
147C23×C4C2411yescharacteristic
148C23×C4C2414yes—
149C23×C4C2412yes—
150#127—416no—
151#123—432no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4#41116yes—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—42no—
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—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—28no—
203C2×C42—28no—
204C2×C42—28no—
205C2×C42—44no—
206C2×C42—44no—
207C2×C42—44no—
208C2×C42—44no—
209C2×C42—44no—
210C2×C42—44no—
211#95—28no—
212#95—28no—
213#95—28no—
214#95—28no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—42no—
225#95—42no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230C24—24no—
231C24—24no—
232C24—24no—
233C24—24no—
234C24—22no—
235C24—22no—
236C24—24no—
237C24—24no—
238C24—24no—
239C24—24no—
240C24—21no—
241C24—21no—
242C24—24no—
243C24—24no—
244C24—22no—
245C24—22no—
246C24—24no—
247C24—24no—
248C24C2×#4114yes—
249C24C2511yescharacteristic
250C2×D4—44no—
251C2×D4—48no—
252C2×D4—48no—
253C2×D4—48no—
254C2×D4—416no—
255C2×D4—416no—
256C2×D4—44no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—88no—
260C2×D4—88no—
261C2×D4—84no—
262C2×D4—84no—
263C2×D4—84no—
264C2×D4—88no—
265C2×D4—88no—
266C22×C4—22no—
267C22×C4—22no—
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—22no—
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—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—48no—
311C22×C4—416no—
312C22×C4—88no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4C2×#4112yes—
316C22×C4C2×#4118yes—
317C22×C4C2×#4118yes—
318C22×C4C2×#4118yes—
319C22×C4H(5,𝔽2)18yes—
320C22×C4H(5,𝔽2)14yes—
321C4⋊C4—232no—
322C4⋊C4—416no—
323C4⋊C4—416no—
324C4⋊C4—88no—
325C22⋊C4—416no—
326C22⋊C4—48no—
327C22⋊C4—416no—
328C22⋊C4—416no—
329C22⋊C4—416no—
330C22⋊C4—416no—
331C22⋊C4—88no—
332C22⋊C4—88no—
333C22⋊C4—88no—
334C22⋊C4—84no—
335C22⋊C4—88no—
336C22⋊C4—88no—
337C22⋊C4—84no—
338C42—28no—
339C42—416no—
340C42—88no—
341C42—88no—
342C23—28no—
343C23—28no—
344C23—24no—
345C23—24no—
346C23—24no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—42no—
384C23—42no—
385C23C22×#4114yes—
386C23C22×#4111yescharacteristic
387C23C2×H(5,𝔽2)12yes—
388C23C2×H(5,𝔽2)14yes—
389C23#58514yes—
390D4—816no—
391D4—816no—
392D4—88no—
393D4—88no—
394C2×C4—28no—
395C2×C4—24no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—28no—
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—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—816no—
444C2×C4#55118yes—
445C22—44no—
446C22—44no—
447C22—48no—
448C22—48no—
449C22—44no—
450C22—84no—
451C22—84no—
452C22—84no—
453C22—84no—
454C22—82no—
455C22—82no—
456C22—84no—
457C22—84no—
458C22—84no—
459C22—84no—
460C22—84no—
461C22—84no—
462C22—81no—
463C22—81no—
464C22C2×#55112yes—
465C22#348014yes—
466C22#348114yes—
467C22#349011yescharacteristic
468C22#349112yes—
469C22#350614yes—
470C22#351714yes—
471C22#351914yes—
472C22#353414yes—
473C22#354014yes—
474C22#354312yes—
475C4—28no—
476C4—48no—
477C4—48no—
478C4—48no—
479C4—42no—
480C4—84no—
481C4—84no—
482C4—84no—
483C2—84no—
484C2—161no—
485C2—161no—
486C2#4670414yes—
487C2#4699811yescharacteristic
488C2#5972414yes—
489C2#5975414yes—
490C2#5994012yes—
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. #17271C2semidirect
2. #16014C2semidirect
3. #15694C2semidirect
4. #15606C2semidirect
5. #15381C2semidirect
7. #14201C2semidirect
9. #14145C2semidirect
10. #14095C2semidirect
11. #14069C2semidirect

Maximal subgroups

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

subgroupindex
#172712
#160142
#156942
#156062
#153812
#152612
#142012
#141542
#141452
#140952
#140692