Finite Group Databasebeta

← #215121

Subgroups — #215121

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2#21902C212yesmaximal, complemented
3#9651C214yesmaximal, complemented
4#9650C214yesmaximal, complemented
5#8301C212yesmaximal
6#8299C212yesmaximal, complemented
7C22×#123C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10#2322—28no—
11C2×#389C2212yes—
12C2×#388C2212yes—
13C2×#387C412yes—
14C2×#386C411yescharacteristic
15C2×#385C412yes—
16C2×#385C411yescharacteristic
17C2×#381C411yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1923—28no—
20#1797C2214yes—
21#1779C2214yes—
22C4×C22⋊C8C2214yes—
23C2×#324C2218yes—
24C2×#324C2214yes—
25C2×#323C2214yes—
26C23×D4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4C2311yescharacteristic
31C2×#123—24no—
32C2×#123—24no—
33C2×#123C818yescomplemented
34C2×C4×D4—24no—
35C2×C4×D4—44no—
36C22×C4⋊C4—24no—
37C22×C4⋊C4—44no—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2311yescharacteristic
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4C2311yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42C2312yes—
51C2×C4⋊C8—24no—
52C2×C22⋊C8—24no—
53C2×C22⋊C8—24no—
54C2×C22⋊C8C2318yes—
55C2×C4×C8—24no—
56#389—24no—
57#389—24no—
58#388—24no—
59#388—24no—
60#387—28no—
61#386—24no—
62#385—24no—
63#385—28no—
64#381—24no—
65C4×C22⋊C4D418yes—
66C2×#95C2×C412yes—
67C2×#95C2×C412yes—
68C2×#95C2×C412yes—
69C2×#95C2×C412yes—
70C2×#95C2×C412yes—
71#336—24no—
72#336—24no—
73#324—48no—
74#324—44no—
75#323—44no—
76C25C22×C411yescharacteristic
77C25C22×C411yescharacteristic
78C22×D4—22no—
79C22×D4—22no—
80C22×D4—42no—
81C22×D4—42no—
82C22×D4C2×C814yes—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—24no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—82no—
93C23×C4—82no—
94C23×C4—82no—
95C23×C4C2×C814yes—
96C23×C4C22×C412yes—
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C412yes—
101C23×C4C22×C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C412yes—
104C23×C4C2×D414yes—
105C23×C4C2411yescharacteristic
106C22×C8—28no—
107#123—232no—
108C4×D4—416no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—416no—
120C2×C4⋊C4—82no—
121C2×C4⋊C4C2×C818yes—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—82no—
131C2×C22⋊C4C22⋊C414yes—
132C2×C22⋊C4C22⋊C414yes—
133C2×C22⋊C4C2×C818yes—
134C2×C22⋊C4C2×C814yes—
135C2×C22⋊C4M1614yes—
136C2×C42—22no—
137C2×C42—42no—
138C2×C42—82no—
139C2×C42C2×D414yes—
140C2×C42C2×D414yes—
141C4⋊C8—84no—
142C22⋊C8—28no—
143C22⋊C8—28no—
144C22⋊C8—84no—
145C22⋊C8—84no—
146C4×C8—84no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95C22⋊C418yes—
152C24—22no—
153C24—22no—
154C24—21no—
155C24—21no—
156C24—22no—
157C24—22no—
158C24—42no—
159C24—42no—
160C24—42no—
161C24—42no—
162C24C2×C22⋊C412yes—
163C24C22×C812yes—
164C24C22×C812yes—
165C24C2×M1612yes—
166C24C23×C411yescharacteristic
167C2×D4—28no—
168C2×D4—48no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—48no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—28no—
182C22×C4—24no—
183C22×C4—28no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—48no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—84no—
209C22×C4—84no—
210C22×C4—88no—
211C22×C4—88no—
212C22×C4—88no—
213C22×C4C2×C22⋊C414yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C414yes—
216C22×C4C22×C814yes—
217C22×C4C22×C812yes—
218C22×C4C22×C814yes—
219C22×C4C2×M1614yes—
220C22×C4C22×D412yes—
221C22×C4H(5,𝔽2)14yes—
222C2×C8—48no—
223C2×C8—48no—
224C2×C8—48no—
225C4⋊C4—216no—
226C4⋊C4—48no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—84no—
231C22⋊C4—216no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C22⋊C4—84no—
236C22⋊C4C22⋊C8116yes—
237C42—44no—
238C42—44no—
239C42—44no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—24no—
245C23—24no—
246C23—24no—
247C23—24no—
248C23—22no—
249C23—24no—
250C23—44no—
251C23—44no—
252C23—44no—
253C23—44no—
254C23—48no—
255C23—44no—
256C23—44no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—82no—
263C23—82no—
264C23—84no—
265C23C2×C22⋊C814yes—
266C23C22×C22⋊C411yescharacteristic
267C23#51812yes—
268C23C23×C811yescharacteristic
269C23C22×M1611yescharacteristic
270C23#56812yes—
271D4—416no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—28no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—416no—
289C2×C4—44no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—88no—
304C2×C4—84no—
305C2×C4—88no—
306C2×C4—88no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4C2×C22⋊C818yes—
311C2×C4#53514yes—
312C8—88no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—24no—
319C22—28no—
320C22—48no—
321C22—48no—
322C22—42no—
323C22—42no—
324C22—44no—
325C22—42no—
326C22—44no—
327C22—84no—
328C22—84no—
329C22—81no—
330C22—81no—
331C22—82no—
332C22#230911yescharacteristic
333C22C22×C22⋊C811yescharacteristic
334C22#287911yescharacteristic
335C22#288312yes—
336C22#288912yes—
337C22#297212yes—
338C22#297512yes—
339C4—24no—
340C4—28no—
341C4—44no—
342C4—42no—
343C4—42no—
344C4—44no—
345C4—81no—
346C4—82no—
347C4—81no—
348C2—22no—
349C2—22no—
350C2—44no—
351C2—82no—
352C2#1735112yes—
353C2#1740612yes—
354C2#2178611yescharacteristic
355C2#2179511yescharacteristic
356C2#2180211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21902C2semidirect
3. #9651C2semidirect
4. #9650C2semidirect
6. #8299C2semidirect
33. C2×#123C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
#219022
#96512
#96502
#83012
#82992