Finite Group Databasebeta

← #5228725

Subgroups — #5228725

Order 512.

1835 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17198C211yesmaximal, characteristic, complemented
2#16957C214yesmaximal, complemented
3#16548C214yesmaximal
4#16528C214yesmaximal, complemented
5#15963C214yesmaximal, complemented
6#15855C214yesmaximal, complemented
7#15668C212yesmaximal
8#15616C211yesmaximal, characteristic, complemented
9#15433C214yesmaximal, complemented
10#14658C211yesmaximal, characteristic
11#14537C212yesmaximal, complemented
12C22×#123C2211yescharacteristic
13C2×#399C2214yes—
14C2×#397C2214yes—
15C2×#396C2214yes—
16C2×#393C2212yes—
17C2×#393C2212yes—
18C2×#391C2212yes—
19C2×#391C2214yes—
20C2×#389C2214yes—
21C2×#388C2214yes—
22C2×#387C2214yes—
23C2×#380C2212yes—
24#1446C2214yes—
25#1445C2212yes—
26#1445C2214yes—
27#1444C2214yes—
28#1444C2214yes—
29#1443C2214yes—
30#1442C2214yes—
31#1442C2214yes—
32#1442C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1441C2214yes—
37#1441C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1440C2211yescharacteristic
41#1440C2214yes—
42#1440C2214yes—
43#1440C2211yescharacteristic
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1438C2214yes—
48#1438C2212yes—
49#1437C2212yes—
50#1437C2212yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56C22⋊#95C2214yes—
57C23×Q8C2312yes—
58C23×D4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C2×#123—24no—
61C2×#123—44no—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2311yescharacteristic
78C22×C42C2314yes—
79C22×C42C2312yes—
80#399—48no—
81#397—48no—
82#396—48no—
83#393—28no—
84#393—44no—
85#391—44no—
86#391—48no—
87#389—48no—
88#388—48no—
89#387—48no—
90#380—44no—
91C2×#95C2314yes—
92C2×#95C2314yes—
93C2×#95C2312yes—
94C2×#95C2314yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2312yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C25C2411yescharacteristic
116C25C2411yescharacteristic
117C22×Q8—24no—
118C22×Q8—28no—
119C22×Q8—42no—
120C22×Q8—44no—
121C22×Q8—42no—
122C22×D4—22no—
123C22×D4—42no—
124C22×D4—44no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—44no—
128C23×C4C2414yes—
129C23×C4C2414yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2411yescharacteristic
133C23×C4C2412yes—
134C23×C4C2414yes—
135C23×C4C2412yes—
136C23×C4C2414yes—
137#123—416no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
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×C22⋊C4—28no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
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#4114yes—
183C2×C42—44no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42—42no—
187C2×C42—44no—
188#95—28no—
189#95—24no—
190#95—28no—
191#95—28no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—28no—
196#95—28no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—42no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212C24—24no—
213C24—22no—
214C24—24no—
215C24—24no—
216C24—21no—
217C24—24no—
218C24—22no—
219C24—24no—
220C24—24no—
221C24C2×#4112yes—
222C24C2511yescharacteristic
223C2×Q8—416no—
224C2×Q8—48no—
225C2×Q8—84no—
226C2×Q8—88no—
227C2×Q8—84no—
228C2×D4—44no—
229C2×D4—84no—
230C2×D4—88no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—28no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
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—88no—
284C22×C4—84no—
285C22×C4C2×#4114yes—
286C22×C4H(5,𝔽2)14yes—
287C22×C4H(5,𝔽2)18yes—
288C22×C42−1+418yes—
289C4⋊C4—416no—
290C4⋊C4—416no—
291C4⋊C4—48no—
292C4⋊C4—88no—
293C4⋊C4—88no—
294C4⋊C4—88no—
295C4⋊C4—88no—
296C22⋊C4—44no—
297C22⋊C4—48no—
298C22⋊C4—416no—
299C22⋊C4—88no—
300C22⋊C4—88no—
301C22⋊C4—88no—
302C22⋊C4—88no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C42—84no—
306C42—88no—
307C23—22no—
308C23—24no—
309C23—28no—
310C23—44no—
311C23—44no—
312C23—42no—
313C23—42no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—44no—
318C23—41no—
319C23—42no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—42no—
324C23—42no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23C22×#4111yescharacteristic
330C23C2×H(5,𝔽2)12yes—
331C23C2×H(5,𝔽2)14yes—
332C23C2×2−1+414yes—
333C23#58514yes—
334Q8—816no—
335D4—88no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—88no—
392C22—42no—
393C22—48no—
394C22—44no—
395C22—84no—
396C22—84no—
397C22—81no—
398C22—84no—
399C22—84no—
400C22—84no—
401C22—82no—
402C22#349011yescharacteristic
403C22#351012yes—
404C22#352014yes—
405C22#353414yes—
406C22#353614yes—
407C22#353812yes—
408C22#354912yes—
409C22#355414yes—
410C22#355414yes—
411C22#355514yes—
412C22#356014yes—
413C4—44no—
414C4—48no—
415C4—44no—
416C4—48no—
417C4—81no—
418C4—84no—
419C4—84no—
420C4—84no—
421C4—84no—
422C2—82no—
423C2—161no—
424C2#4645711yescharacteristic
425C2#4966012yes—
426C2#5136014yes—
427C2#5366814yes—
428C2#5448614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17198C2semidirect
2. #16957C2semidirect
4. #16528C2semidirect
5. #15963C2semidirect
6. #15855C2semidirect
8. #15616C2semidirect
9. #15433C2semidirect
11. #14537C2semidirect

Maximal subgroups

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

subgroupindex
#171982
#169572
#165482
#165282
#159632
#158552
#156682
#156162
#154332
#146582
#145372