Finite Group Databasebeta

← #190025

Subgroups — #190025

Order 512.

1381 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#518C211yesmaximal, characteristic
2C2×#1465C212yesmaximal
3#8735C214yesmaximal, complemented
4#8734C214yesmaximal, complemented
5#7929C214yesmaximal, complemented
6C2×#518C411yescharacteristic
7C2×#518C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C2×#407C2214yes—
10#2275—24no—
11#2274C2212yes—
12#2273C412yes—
13#2273C2212yes—
14C42×D4—24no—
15C4×#117C412yes—
16C2×C4×C22⋊C4C412yes—
17C2×C4×C22⋊C4C2212yes—
18#1465—24no—
19#1465—24no—
20C2×#324C2214yes—
21C2×#323C2214yes—
22#1282C22116yes—
23C23×D4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×M16—24no—
27#518—28no—
28#518—28no—
29C2×C4×D4—24no—
30C2×C4×D4—24no—
31C2×C4×D4—24no—
32C2×C4×D4—24no—
33C2×C4×D4D414yes—
34C2×C4×D4D414yes—
35C2×#117C2×C412yes—
36C2×#117C2×C412yes—
37C22×C4⋊C4C2311yescharacteristic
38C22×C22⋊C4C2×C412yes—
39C22×C22⋊C4C2×C412yes—
40C22×C22⋊C4C2×C412yes—
41C22×C22⋊C4C2×C411yescharacteristic
42C22×C22⋊C4C2×C412yes—
43C22×C22⋊C4C2311yescharacteristic
44C22×C42—22no—
45C22×C42—22no—
46C22×C42C2×C411yescharacteristic
47#407—24no—
48#407—24no—
49#407—24no—
50#407—24no—
51C2×C22⋊C8—24no—
52C2×C22⋊C8C2318yes—
53C4×C4⋊C4—24no—
54C4×C4⋊C4—24no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—22no—
57C4×C22⋊C4—22no—
58C4×C22⋊C4—22no—
59C4×C22⋊C4—22no—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—24no—
62C4×C22⋊C4—24no—
63C4×C22⋊C4—24no—
64C4×C22⋊C4—24no—
65C4×C22⋊C4C2×C414yes—
66C4×C22⋊C4C2×C414yes—
67C4×C22⋊C4C2×C414yes—
68C4×C22⋊C4C2314yes—
69#376—22no—
70C2×#95—22no—
71C2×#95—22no—
72C43—22no—
73#324—28no—
74#323—28no—
75C25C22×C411yescharacteristic
76C25C22×C411yescharacteristic
77C22×D4—24no—
78C22×D4—24no—
79C22×D4—24no—
80C22×D4C22⋊C414yes—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—82no—
88C23×C4—82no—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4#4114yes—
98C23×C4#4114yes—
99C23×C4C2411yescharacteristic
100C2×M16—44no—
101C2×M16—44no—
102C2×M16—88no—
103C22×C8—28no—
104C4×D4—232no—
105C4×D4—232no—
106C4×D4—28no—
107C4×D4—28no—
108C4×D4—28no—
109C4×D4—28no—
110#117—24no—
111#117—28no—
112#117M1618yescomplemented
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4C2×D414yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4C22⋊C414yes—
128C2×C22⋊C4M1618yescomplemented
129C2×C22⋊C4C22×C414yes—
130C2×C22⋊C4C22×C414yes—
131C2×C22⋊C4C2×D414yes—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—24no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—84no—
141C2×C42—84no—
142C2×C42C22⋊C412yes—
143C2×C42C22⋊C412yes—
144C22⋊C8—28no—
145C22⋊C8—28no—
146C22⋊C8—44no—
147C22⋊C8—44no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152C24—22no—
153C24—22no—
154C24—22no—
155C24—22no—
156C24—81no—
157C24—81no—
158C24—81no—
159C24—81no—
160C24C2×C22⋊C412yes—
161C24C2×C22⋊C412yes—
162C24#11714yes—
163C24C2×M1614yes—
164C24C23×C411yescharacteristic
165C2×D4—232no—
166C2×D4—28no—
167C2×D4—28no—
168C2×D4—28no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—44no—
183C22×C4—48no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—84no—
187C22×C4—82no—
188C22×C4—84no—
189C22×C4—81no—
190C22×C4—84no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C22⋊C412yes—
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4#11712yes—
208C22×C4#11712yes—
209C22×C4#12314yes—
210C22×C4#12314yes—
211C22×C4#12314yes—
212C22×C4#12314yes—
213C22×C4C2×M1614yes—
214C22×C4C22×D412yes—
215C22×C4C2×#4114yes—
216M16—88no—
217M16—88no—
218C2×C8—48no—
219C2×C8—88no—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—28no—
223C22⋊C4—28no—
224C22⋊C4—28no—
225C22⋊C4—28no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C22⋊C4—28no—
229C22⋊C4—28no—
230C22⋊C4—28no—
231C22⋊C4C2×M1618yes—
232C22⋊C4C2×M1618yes—
233C42—28no—
234C42—22no—
235C42—24no—
236C42—22no—
237C42—28no—
238C42—48no—
239C42—84no—
240C42—84no—
241C42—84no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—28no—
247C23—28no—
248C23—42no—
249C23—41no—
250C23—42no—
251C23—41no—
252C23—44no—
253C23—44no—
254C23—42no—
255C23—42no—
256C23—82no—
257C23—82no—
258C23—84no—
259C23—84no—
260C23—84no—
261C23—84no—
262C23#38114yes—
263C23#40714yes—
264C23C22×C22⋊C411yescharacteristic
265C23C2×#11712yes—
266C23C22×M1612yes—
267C23C22×M1612yes—
268D4—232no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—28no—
274C2×C4—24no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—88no—
298C2×C4—88no—
299C2×C4—88no—
300C2×C4—88no—
301C2×C4#38111yescharacteristic
302C2×C4#38112yes—
303C2×C4#38111yescharacteristic
304C2×C4#40714yes—
305C2×C4C2×#12314yes—
306C8—88no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—22no—
311C22—24no—
312C22—24no—
313C22—24no—
314C22—24no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—88no—
324C22—88no—
325C22C2×#38111yescharacteristic
326C22C2×#40711yescharacteristic
327C22C2×#40711yescharacteristic
328C22#292014yes—
329C4—24no—
330C4—24no—
331C4—21no—
332C4—22no—
333C4—48no—
334C4—48no—
335C4—42no—
336C4—42no—
337C4#182812yes—
338C2—21no—
339C2—21no—
340C2—24no—
341C2—44no—
342C2—44no—
343C2#2068211yescharacteristic
344C2C2×#182811yescharacteristic
345C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8735C2semidirect
4. #8734C2semidirect
5. #7929C2semidirect
112. #117M16semidirect
128. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5182
C2×#14652
#87352
#87342
#79292