Finite Group Databasebeta

← #393521

Subgroups — #393521

Order 512.

2009 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3C2×#1899C212yesmaximal
4#14684C211yesmaximal, characteristic, complemented
5#10960C214yesmaximal
6#10912C214yesmaximal, complemented
7#10736C212yesmaximal, complemented
8C22×#123C2211yescharacteristic
9C22×#117C2212yes—
10C23×C42C2211yescharacteristic
11C22×#99C2214yes—
12#2853—24no—
13#2853—21no—
14#2853—21no—
15#2853—22no—
16C2×#388C414yescomplemented
17C2×#387C414yescomplemented
18C2×#386C2212yes—
19C2×#385C2212yes—
20C2×#381C2212yes—
21C2×#381C2211yescharacteristic
22C4×#117—28no—
23C2×C4×C4⋊C4—22no—
24C2×C4×C22⋊C4C2214yes—
25C2×#376—21no—
26C2×C43—21no—
27#1899—28no—
28C2×#342C2214yes—
29C2×#342C2218yes—
30#1434—24no—
31C22⋊#95C2214yes—
32C23×D4C2311yescharacteristic
33C24×C4C2311yescharacteristic
34C24×C4C2312yes—
35C2×#123—21no—
36C2×#123—21no—
37C2×#123—22no—
38C2×#123—24no—
39C2×#123—24no—
40C2×#117—24no—
41C2×#117—216no—
42C2×#117D418yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C414yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2×C414yes—
48C22×C22⋊C4C2314yes—
49C22×C22⋊C4C2311yescharacteristic
50C22×C22⋊C4C2312yes—
51C22×C22⋊C4C2314yes—
52C22×C42—24no—
53C22×C42—22no—
54C22×C42—24no—
55C22×C42—24no—
56C22×C42C2×C414yes—
57C22×C42D414yes—
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D412yes—
61C2×#99—28no—
62C2×#99—216no—
63#388—24no—
64#388—24no—
65#388—24no—
66#388—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—24no—
71#386—22no—
72#386—24no—
73#386—22no—
74#385—22no—
75#385—24no—
76#385—22no—
77#381—21no—
78#381—21no—
79#381—22no—
80#381—24no—
81#381—24no—
82C4×C4⋊C4—216no—
83C4×C22⋊C4—216no—
84#376—28no—
85C2×#95—24no—
86C2×#95—28no—
87C2×#95—24no—
88C2×#95C2×C414yes—
89C2×#95C2×C414yes—
90C2×#95C2×C414yes—
91C43—28no—
92#342—28no—
93#342—216no—
94C25C22×C411yescharacteristic
95C25C2411yescharacteristic
96C22×D4—21no—
97C22×D4—21no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—28no—
101C23×C4—28no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—48no—
107C23×C4C22×C411yescharacteristic
108C23×C4C22×C414yes—
109C23×C4C22×C412yes—
110C23×C4C22×C412yes—
111C23×C4C22×C414yes—
112C23×C4C2×D412yes—
113C23×C4C2×D414yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D414yes—
118C23×C4C2×D412yes—
119C23×C4#4114yes—
120#123—42no—
121#123—42no—
122#123—48no—
123#123—42no—
124#123—42no—
125#117—232no—
126#117—232no—
127C2×C4⋊C4—216no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—48no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—48no—
154C2×C22⋊C4C2×D418yes—
155C2×C22⋊C4#41116yes—
156C2×C42—216no—
157C2×C42—216no—
158C2×C42—28no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—24no—
162C2×C42—216no—
163C2×C42—22no—
164C2×C42—24no—
165C2×C42—22no—
166C2×C42—216no—
167C2×C42—416no—
168C2×C42C22⋊C414yes—
169C2×C42C22⋊C414yes—
170C2×C42C22⋊C414yes—
171C2×C42C22⋊C414yes—
172#99—416no—
173#95—216no—
174#95—24no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—416no—
181#95—48no—
182C24—21no—
183C24—22no—
184C24—24no—
185C24—44no—
186C24—44no—
187C24C23×C411yescharacteristic
188C24C22×D411yescharacteristic
189C24C22×D411yescharacteristic
190C24C22×D412yes—
191C24C2×#4112yes—
192C24C2×#4118yes—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—41no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—48no—
200C2×D4—44no—
201C22×C4—28no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—28no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—216no—
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—22no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—22no—
216C22×C4—216no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—28no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—416no—
230C22×C4—416no—
231C22×C4—416no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—416no—
236C22×C4—44no—
237C22×C4—48no—
238C22×C4C2×C22⋊C414yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C414yes—
242C22×C4C4×D418yes—
243C22×C4#12012yes—
244C22×C4#12012yes—
245C22×C4#12118yes—
246C22×C4#12312yes—
247C22×C4#12312yes—
248C4⋊C4—232no—
249C4⋊C4—48no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—48no—
253C22⋊C4—28no—
254C22⋊C4—216no—
255C22⋊C4—48no—
256C22⋊C4—41no—
257C22⋊C4—42no—
258C22⋊C4—41no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—48no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—48no—
267C22⋊C4#123116yes—
268C42—232no—
269C42—28no—
270C42—28no—
271C42—28no—
272C42—28no—
273C42—432no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23—44no—
278C23—44no—
279C23—416no—
280C23—44no—
281C23—41no—
282C23—42no—
283C23—44no—
284C23—88no—
285C23—84no—
286C23C22×C22⋊C411yescharacteristic
287C23C2×C4×D412yes—
288C23C4×#4114yes—
289C23C2×#12011yescharacteristic
290C23C2×#12112yes—
291C23C2×#12311yescharacteristic
292C23C2×#12314yes—
293C23#52514yes—
294C23#52514yes—
295C23#52918yes—
296C23#53214yes—
297D4—82no—
298D4—82no—
299D4—84no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—216no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—216no—
312C2×C4—216no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—416no—
317C2×C4—416no—
318C2×C4—416no—
319C2×C4—432no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—88no—
331C2×C4—84no—
332C2×C4—816no—
333C2×C4—88no—
334C2×C4—84no—
335C2×C4#38614yes—
336C2×C4#38614yes—
337C2×C4#41018yes—
338C22—28no—
339C22—28no—
340C22—28no—
341C22—22no—
342C22—22no—
343C22—24no—
344C22—24no—
345C22—44no—
346C22—44no—
347C22—48no—
348C22—81no—
349C22—84no—
350C22—82no—
351C22—164no—
352C22C2×#38611yescharacteristic
353C22#229712yes—
354C22C4×#12314yes—
355C22#284912yes—
356C22#285312yes—
357C22#285414yes—
358C22C2×#41014yes—
359C4—24no—
360C4—24no—
361C4—28no—
362C4—416no—
363C4—44no—
364C4—81no—
365C4—84no—
366C4—82no—
367C4—164no—
368C2—22no—
369C2—24no—
370C2—24no—
371C2—22no—
372C2—161no—
373C2#1470611yescharacteristic
374C2#2098312yes—
375C2#2142114yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#117C2semidirect
4. #14684C2semidirect
6. #10912C2semidirect
7. #10736C2semidirect
16. C2×#388C4semidirect
17. C2×#387C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C2×C4×#1172
C2×#18992
#146842
#109602
#109122
#107362