Finite Group Databasebeta

← #381014

Subgroups — #381014

Order 512.

1356 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2#14347C211yesmaximal, characteristic
3#13796C211yesmaximal, characteristic
4#11381C214yesmaximal
5#11353C214yesmaximal
6#10854C214yesmaximal
7C22×C22⋊Q8C2212yes—
8#2659—24no—
9#2659—24no—
10C2×#398C414yescomplemented
11C2×#395C414yescomplemented
12C2×#380C2212yes—
13C2×#380C2212yes—
14C2×#380C2212yes—
15C2×C4×C22⋊C4C2212yes—
16C22×#95C2211yescharacteristic
17C2×#342C22116yes—
18#1442—24no—
19#1434—24no—
20C22⋊#95C2212yes—
21C22⋊#95C2212yes—
22C22⋊#95C2212yes—
23C22⋊#95C2212yes—
24C23×Q8C2×C412yes—
25C24×C4C2311yescharacteristic
26C24×C4C2311yescharacteristic
27C24×C4C2311yescharacteristic
28C2×C22⋊Q8—24no—
29C2×C22⋊Q8—24no—
30C2×C22⋊Q8—48no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4C2×C414yes—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C412yes—
38C22×C22⋊C4C2314yes—
39C22×C22⋊C4C2314yes—
40C22×C22⋊C4C2314yes—
41C22×C42—22no—
42#398—216no—
43#395—216no—
44#380—24no—
45#380—24no—
46#380—24no—
47#380—24no—
48#380—28no—
49C4×C22⋊C4—24no—
50C4×C22⋊C4Q818yescomplemented
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95C2×C414yes—
62C2×#95C2×C414yes—
63C2×#95C2×C414yes—
64C2×#95D414yes—
65C2×#95D414yes—
66#342—416no—
67C25C2411yescharacteristic
68C22×Q8—24no—
69C22×Q8—44no—
70C22×Q8—82no—
71C22×Q8—82no—
72C23×C4—24no—
73C23×C4—24no—
74C23×C4—24no—
75C23×C4—24no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4—81no—
80C23×C4—81no—
81C23×C4—81no—
82C23×C4C22×C414yes—
83C23×C4C22×C414yes—
84C23×C4C22×C412yes—
85C23×C4C22×C412yes—
86C23×C4C22×C412yes—
87C23×C4C2×D412yes—
88C23×C4C2×D412yes—
89C23×C4C2×D412yes—
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×Q814yes—
94C23×C4#4114yes—
95C22⋊Q8—416no—
96C22⋊Q8—416no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—48no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—82no—
107C2×C4⋊C4—82no—
108C2×C4⋊C4—82no—
109C2×C4⋊C4—82no—
110C2×C4⋊C4—82no—
111C2×C4⋊C4—82no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—28no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—216no—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—28no—
121C2×C22⋊C4C2×Q818yes—
122C2×C42—28no—
123C2×C42—82no—
124C2×C42—82no—
125#95—28no—
126#95—28no—
127#95—28no—
128#95—44no—
129#95—82no—
130#95—82no—
131#95—82no—
132#95—82no—
133#95—82no—
134#95—82no—
135#95—82no—
136#95—82no—
137#95C22⋊C4116yes—
138C24—41no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24C23×C411yescharacteristic
143C24C22×D411yescharacteristic
144C24C22×D411yescharacteristic
145C24C22×Q812yes—
146C24C2×#4112yes—
147C24H(5,𝔽2)14yes—
148C242−1+414yes—
149C2×Q8—48no—
150C2×Q8—84no—
151C2×Q8—84no—
152C2×Q8—816no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—28no—
162C22×C4—28no—
163C22×C4—28no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—84no—
200C22×C4—84no—
201C22×C4—88no—
202C22×C4—88no—
203C22×C4C2×C22⋊C414yes—
204C22×C4C2×C22⋊C414yes—
205C22×C4C2×C22⋊C414yes—
206C22×C4C4×Q818yes—
207C22×C4C22⋊Q814yes—
208C22×C4C22⋊Q814yes—
209C22×C4#12414yes—
210C22×C4C4⋊Q814yes—
211C4⋊C4—48no—
212C4⋊C4—416no—
213C4⋊C4—84no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C4⋊C4—88no—
218C22⋊C4—216no—
219C22⋊C4—48no—
220C22⋊C4—48no—
221C22⋊C4—48no—
222C22⋊C4—416no—
223C42—88no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—42no—
237C23—42no—
238C23—44no—
239C23—44no—
240C23—42no—
241C23—42no—
242C23—44no—
243C23—44no—
244C23C22×C22⋊C411yescharacteristic
245C23C2×C4×Q812yes—
246C23#51812yes—
247C23#51912yes—
248C23C2×C22⋊Q812yes—
249C23C2×#12411yescharacteristic
250C23C2×C4⋊Q811yescharacteristic
251C23#53512yes—
252C23#53512yes—
253C23#53612yes—
254C23#53612yes—
255C23#54218yes—
256C23#54418yes—
257Q8—816no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—48no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—84no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—84no—
290C2×C4—88no—
291C2×C4—88no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4#39118yes—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—22no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—21no—
320C22—48no—
321C22—48no—
322C22C2×#39111yescharacteristic
323C22#230911yescharacteristic
324C22#231011yescharacteristic
325C22#233114yes—
326C22#262814yes—
327C22#264012yes—
328C22#266012yes—
329C22#288311yescharacteristic
330C22#288311yescharacteristic
331C22#288311yescharacteristic
332C22#288411yescharacteristic
333C4—28no—
334C4—44no—
335C4—44no—
336C4—88no—
337C4—84no—
338C4—84no—
339C4—164no—
340C2—24no—
341C2—22no—
342C2—24no—
343C2—22no—
344C2#1428211yescharacteristic
345C2#2133311yescharacteristic
346C2#2133411yescharacteristic
347C2#2179511yescharacteristic
348C2#2179611yescharacteristic
349C2#2182711yescharacteristic
350C2#2182811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×#398C4semidirect
11. C2×#395C4semidirect
50. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
#143472
#137962
#113812
#113532
#108542