Finite Group Databasebeta

← #393464

Subgroups — #393464

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33996C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3#14128C212yesmaximal
4#13958C211yesmaximal, characteristic
5#10918C218yesmaximal
6#10726C212yesmaximal
7C22×#117C2212yes—
8C22×#117C2211yescharacteristic
9C2×#385C2214yes—
10C2×#381C2211yescharacteristic
11C2×#381C2212yes—
12#2274—22no—
13#2274—22no—
14#2274—21no—
15#2274—21no—
16#2273C2214yes—
17C4×#117—24no—
18C4×#117—22no—
19C22×#95C2211yescharacteristic
20#1820C418yescomplemented
21C2×#342C2218yes—
22C2×#342C2214yes—
23#1445—21no—
24#1445—21no—
25#1444—22no—
26#1442—22no—
27#1436—24no—
28#1434—22no—
29C22⋊#95C2214yes—
30C22⋊#95C2214yes—
31C24×C4C2312yes—
32C24×C4C2311yescharacteristic
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—24no—
40C2×#117—216no—
41C2×#117C2×C414yes—
42C2×#117Q814yes—
43C22×C4⋊C4—24no—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C2314yes—
50C22×C22⋊C4C2312yes—
51C22×C22⋊C4C2314yes—
52C22×C42D414yes—
53C22×C42D412yes—
54#385—216no—
55#381—24no—
56#381—24no—
57#381C2×C418yes—
58C4×C4⋊C4—24no—
59C4×C4⋊C4—24no—
60C4×C4⋊C4—24no—
61C4×C4⋊C4—24no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—24no—
64C4×C4⋊C4—22no—
65C4×C22⋊C4—24no—
66C4×C22⋊C4—24no—
67C4×C22⋊C4—28no—
68C4×C22⋊C4—28no—
69#376—24no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—24no—
73C2×#95—24no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—24no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—24no—
82C2×#95—24no—
83C43—24no—
84#342—216no—
85#342C2×C4116yes—
86C25C2411yescharacteristic
87C23×C4—28no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—28no—
94C23×C4—44no—
95C23×C4—82no—
96C23×C4C22×C412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D414yes—
104C23×C4C2×D414yes—
105C23×C4C2×Q812yes—
106C23×C4#4114yes—
107#117—216no—
108#117—216no—
109#117—216no—
110#117—216no—
111#117—28no—
112#117—28no—
113#117—28no—
114#117—28no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—48no—
127C2×C4⋊C4—84no—
128C2×C4⋊C4C4⋊C414yescomplemented
129C2×C4⋊C4C4⋊C414yescomplemented
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4C22×C418yes—
141C2×C22⋊C4C22×C414yes—
142C2×C22⋊C4C2×Q814yes—
143C2×C22⋊C4#41116yes—
144C2×C42—24no—
145C2×C42—22no—
146C2×C42—24no—
147C2×C42—22no—
148C2×C42—28no—
149C2×C42—28no—
150C2×C42—24no—
151C2×C42—28no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42C4⋊C418yescomplemented
155#95—24no—
156#95—28no—
157#95—24no—
158#95—28no—
159#95—28no—
160#95—48no—
161#95—48no—
162#95—41no—
163#95—41no—
164#95—42no—
165#95—42no—
166#95—42no—
167#95—48no—
168#95—84no—
169#95—84no—
170C24—42no—
171C24—41no—
172C24—41no—
173C24C23×C411yescharacteristic
174C24C22×D411yescharacteristic
175C24C22×D412yes—
176C24C22×Q811yescharacteristic
177C24C2×#4112yes—
178C24C2×#4118yes—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—216no—
187C22×C4—28no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—816no—
229C22×C4C2×C4⋊C412yes—
230C22×C4C2×C4⋊C412yes—
231C22×C4C2×C4⋊C414yes—
232C22×C4C2×C4⋊C414yes—
233C22×C4C4×D414yes—
234C22×C4C4×D414yes—
235C22×C4#12011yescharacteristic
236C22×C4#12012yes—
237C22×C4#12011yescharacteristic
238C22×C4C22⋊Q814yes—
239C22×C4C22⋊Q814yes—
240C22×C4#12312yes—
241C22×C4#12312yes—
242C22×C4#12318yes—
243C22×C4#12318yes—
244C4⋊C4—28no—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—816no—
251C22⋊C4—28no—
252C22⋊C4—28no—
253C22⋊C4—216no—
254C22⋊C4—216no—
255C22⋊C4—216no—
256C22⋊C4—48no—
257C22⋊C4—48no—
258C42—216no—
259C42—216no—
260C42—28no—
261C42—28no—
262C42—88no—
263C42—84no—
264C42—84no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—48no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—48no—
280C23C22×C4⋊C411yescharacteristic
281C23C2×C4×D412yes—
282C23C4×#4114yes—
283C23C2×#12011yescharacteristic
284C23C2×C22⋊Q812yes—
285C23C2×#12314yes—
286C23C2×#12311yescharacteristic
287C23#52514yes—
288C23#52514yes—
289C23#52918yes—
290C23#53314yes—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—28no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—82no—
343C2×C4#38512yes—
344C2×C4#38512yes—
345C2×C4#38514yes—
346C22—21no—
347C22—22no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—22no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—24no—
360C22—44no—
361C22—48no—
362C22—44no—
363C22C2×#38511yescharacteristic
364C22#229612yes—
365C22C4×#12314yes—
366C22#284912yes—
367C22#285312yes—
368C22#285514yes—
369C22#302512yes—
370C22#302711yescharacteristic
371C22#302811yescharacteristic
372C4—22no—
373C4—24no—
374C4—22no—
375C4—44no—
376C4—44no—
377C4—48no—
378C4—48no—
379C4—88no—
380C4—84no—
381C4—162no—
382C2—24no—
383C2—22no—
384C2—24no—
385C2—22no—
386C2#1470511yescharacteristic
387C2#2088811yescharacteristic
388C2#2089011yescharacteristic
389C2#3302112yes—
390C2#3302312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1820C4semidirect
128. C2×C4⋊C4C4⋊C4semidirect
129. C2×C4⋊C4C4⋊C4semidirect
154. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#339962
C2×#18202
#141282
#139582
#109182
#107262