Finite Group Databasebeta

← #5236485

Subgroups — #5236485

Order 512.

1902 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#17271C211yesmaximal, characteristic
2#17029C214yesmaximal, complemented
3#16524C214yesmaximal, complemented
4#16339C214yesmaximal, complemented
5#15949C211yesmaximal, characteristic
6#15791C214yesmaximal, complemented
7#15718C214yesmaximal, complemented
8#15440C214yesmaximal, complemented
9#15262C212yesmaximal
10#14662C211yesmaximal, characteristic
11#14506C212yesmaximal
12C22×#124C2211yescharacteristic
13C22×#123C2212yes—
14C2×#400C2214yes—
15C2×#400C2214yes—
16C2×#395C2214yes—
17C2×#388C2214yes—
18C2×#385C2214yes—
19C2×#385C2212yes—
20C2×#385C2212yes—
21C2×#382C2212yes—
22C2×#382C2214yes—
23#1446C2212yes—
24#1445C2214yes—
25#1444C2214yes—
26#1444C2214yes—
27#1444C2214yes—
28#1444C2214yes—
29#1443C2214yes—
30#1442C2214yes—
31#1442C2214yes—
32#1442C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1441C2214yes—
36#1441C2214yes—
37#1440C2214yes—
38#1440C2212yes—
39#1440C2212yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1438C2211yescharacteristic
48#1438C2211yescharacteristic
49#1438C2212yes—
50#1437C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53C4⋊#95C2214yes—
54#1434C2214yes—
55#1434C2214yes—
56C22⋊#95C2214yes—
57C23×Q8C2311yescharacteristic
58C23×D4C2311yescharacteristic
59C24×C4C2312yes—
60C2×#124—24no—
61C2×#124—44no—
62C2×#123—28no—
63C2×#123—48no—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2312yes—
80C22×C42C2311yescharacteristic
81C22×C42C2314yes—
82#400—48no—
83#400—48no—
84#395—48no—
85#388—48no—
86#385—48no—
87#385—44no—
88#385—44no—
89#382—48no—
90#382—44no—
91C2×#95C2314yes—
92C2×#95C2314yes—
93C2×#95C2314yes—
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×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2312yes—
112C2×#95C2312yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C25C2412yes—
116C22×Q8—22no—
117C22×Q8—42no—
118C22×Q8—44no—
119C22×D4—44no—
120C22×D4—42no—
121C22×D4#4114yes—
122C23×C4—24no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4C2411yescharacteristic
128C23×C4C2411yescharacteristic
129C23×C4C2414yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2414yes—
133C23×C4C2411yescharacteristic
134C23×C4C2412yes—
135C23×C4C2414yes—
136C23×C4C2414yes—
137#124—416no—
138#123—432no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
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×C4⋊C4—44no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C42—22no—
185C2×C42—24no—
186C2×C42—28no—
187C2×C42—28no—
188C2×C42—44no—
189#95—28no—
190#95—28no—
191#95—28no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—28no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—42no—
204#95—42no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213C24—24no—
214C24—24no—
215C24—22no—
216C24—22no—
217C24—24no—
218C24—24no—
219C24—24no—
220C24—24no—
221C24C2×#4114yes—
222C24C2511yescharacteristic
223C2×Q8—44no—
224C2×Q8—84no—
225C2×Q8—88no—
226C2×D4—48no—
227C2×D4—44no—
228C2×D4—88no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
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—22no—
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—48no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4—88no—
285C22×C4—88no—
286C22×C4C2×#4112yes—
287C22×C4H(5,𝔽2)14yes—
288C22×C4H(5,𝔽2)18yes—
289C22×C42−1+418yes—
290C4⋊C4—416no—
291C4⋊C4—88no—
292C4⋊C4—88no—
293C4⋊C4—88no—
294C4⋊C4—88no—
295C4⋊C4—88no—
296C4⋊C4—88no—
297C22⋊C4—216no—
298C22⋊C4—48no—
299C22⋊C4—416no—
300C22⋊C4—48no—
301C22⋊C4—48no—
302C22⋊C4—416no—
303C22⋊C4—48no—
304C22⋊C4—88no—
305C22⋊C4—88no—
306C42—28no—
307C42—416no—
308C23—24no—
309C23—24no—
310C23—24no—
311C23—28no—
312C23—28no—
313C23—44no—
314C23—44no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—42no—
319C23—44no—
320C23—42no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23C22×#4111yescharacteristic
330C23C2×H(5,𝔽2)14yes—
331C23C2×H(5,𝔽2)12yes—
332C23C2×2−1+414yes—
333C23#58514yes—
334Q8—88no—
335D4—88no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—28no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
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—44no—
392C2×C4—816no—
393C22—48no—
394C22—48no—
395C22—44no—
396C22—44no—
397C22—44no—
398C22—84no—
399C22—84no—
400C22—84no—
401C22—84no—
402C22#349012yes—
403C22#349011yescharacteristic
404C22#351514yes—
405C22#353514yes—
406C22#353812yes—
407C22#354014yes—
408C22#354012yes—
409C22#354914yes—
410C22#355214yes—
411C22#355914yes—
412C22#355914yes—
413C4—48no—
414C4—44no—
415C4—42no—
416C4—48no—
417C4—84no—
418C4—84no—
419C4—84no—
420C4—84no—
421C4—81no—
422C4—81no—
423C2—84no—
424C2#4756311yescharacteristic
425C2#4855414yes—
426C2#5164612yes—
427C2#5538114yes—
428C2#5731014yes—
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
2. #17029C2semidirect
3. #16524C2semidirect
4. #16339C2semidirect
6. #15791C2semidirect
7. #15718C2semidirect
8. #15440C2semidirect

Maximal subgroups

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

subgroupindex
#172712
#170292
#165242
#163392
#159492
#157912
#157182
#154402
#152622
#146622
#145062