Finite Group Databasebeta

← #382770

Subgroups — #382770

Order 512.

1283 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34057C211yesmaximal, characteristic
2C2×#1799C212yesmaximal
3#11351C214yesmaximal, complemented
4#11169C212yesmaximal
5#11166C212yesmaximal
6#10854C214yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C22×#99C2214yes—
9#2318—24no—
10#2315C2212yes—
11#2313C412yescomplemented
12#2312C412yescomplemented
13#2308C2211yescharacteristic
14#2308C2211yescharacteristic
15#2290—24no—
16#2283C412yescomplemented
17#2280C412yescomplemented
18C2×#380C2214yes—
19C2×C4×C22⋊C4C2212yes—
20#1799—24no—
21#1799—24no—
22C2×#342C2214yes—
23C2×#342C2218yes—
24C2×#342C2218yes—
25C23×D4C2311yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2311yescharacteristic
28C2×C4×D4—24no—
29C2×C4×D4—24no—
30C2×C4×D4D414yes—
31C2×C4×D4D414yes—
32C2×#117—22no—
33C22×C4⋊C4—24no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2318yes—
37C22×C22⋊C4C2312yes—
38C22×C22⋊C4C2312yes—
39C22×C22⋊C4C2311yescharacteristic
40C22×C42—22no—
41C22×C42C2×C411yescharacteristic
42C2×#99—28no—
43C2×#99—48no—
44#388—24no—
45#388—24no—
46#387—24no—
47#387—24no—
48#386—24no—
49#386—24no—
50#385—24no—
51#385—24no—
52#383—22no—
53#382—22no—
54#381—22no—
55#381C2×C412yes—
56#381C2×C412yes—
57#380—22no—
58#380—24no—
59#380—24no—
60#380—24no—
61#380—24no—
62#380C2×C414yes—
63#380C2×C414yes—
64#379C2×C412yes—
65#379C2×C412yes—
66C4×C4⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4C2×C414yes—
71C4×C22⋊C4C2×C414yes—
72#376—22no—
73C2×#95—22no—
74C2×#95—24no—
75#342—28no—
76#342—48no—
77#342—48no—
78C25C22×C411yescharacteristic
79C25C2411yescharacteristic
80C22×D4—24no—
81C22×D4—24no—
82C22×D4—24no—
83C22×D4C2×D414yes—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—28no—
89C23×C4—82no—
90C23×C4—82no—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4#4114yes—
101C23×C4#4114yes—
102C4×D4—28no—
103C4×D4—28no—
104C4×D4—28no—
105C4×D4—28no—
106#117—84no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—88no—
113C2×C4⋊C4C22⋊C414yes—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—28no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4C22⋊C414yes—
133C2×C22⋊C4C22×C414yes—
134C2×C22⋊C4C22×C414yes—
135C2×C22⋊C4C2×D414yes—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—84no—
141C2×C42C22⋊C412yes—
142C2×C42C22⋊C412yes—
143#99—416no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—22no—
148#95—22no—
149#95—24no—
150#95—22no—
151#95—22no—
152#95—44no—
153#95—44no—
154C24—22no—
155C24—24no—
156C24—81no—
157C24—81no—
158C24—81no—
159C24—81no—
160C24C2×C22⋊C412yes—
161C24C23×C411yescharacteristic
162C24C22×D412yes—
163C24C2×#4114yes—
164C24H(5,𝔽2)14yes—
165C242−1+414yes—
166C2×D4—28no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—28no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—44no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—44no—
184C22×C4—48no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—84no—
194C22×C4—81no—
195C22×C4—82no—
196C22×C4—84no—
197C22×C4—84no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—88no—
206C22×C4—88no—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C412yes—
213C22×C4#11712yes—
214C22×C4#11712yes—
215C22×C4#11714yes—
216C22×C4#12314yes—
217C22×C4#12314yes—
218C22×C4#12314yes—
219C22×C4#12314yes—
220C4⋊C4—28no—
221C4⋊C4—88no—
222C4⋊C4—84no—
223C4⋊C4—84no—
224C4⋊C4—88no—
225C22⋊C4—28no—
226C22⋊C4—28no—
227C22⋊C4—28no—
228C22⋊C4—28no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—44no—
232C22⋊C4—48no—
233C22⋊C4—48no—
234C22⋊C4—48no—
235C22⋊C4—48no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C42—22no—
241C42—24no—
242C42—22no—
243C42—48no—
244C42—84no—
245C23—22no—
246C23—24no—
247C23—42no—
248C23—44no—
249C23—41no—
250C23—42no—
251C23—44no—
252C23—41no—
253C23—42no—
254C23—42no—
255C23—44no—
256C23—82no—
257C23—82no—
258C23—84no—
259C23—84no—
260C23—88no—
261C23C22×C22⋊C411yescharacteristic
262C23C2×#11712yes—
263C23#52014yes—
264C23C2×#12314yes—
265C23#53512yes—
266C23#53612yes—
267C23#53714yes—
268C23#54014yes—
269C23#54114yes—
270C23#54214yes—
271C23#54214yes—
272D4—48no—
273D4—48no—
274C2×C4—22no—
275C2×C4—28no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4—88no—
305C2×C4—88no—
306C2×C4—88no—
307C2×C4—88no—
308C2×C4—168no—
309C2×C4—168no—
310C2×C4#38111yescharacteristic
311C2×C4#38112yes—
312C2×C4#38114yes—
313C2×C4#38111yescharacteristic
314C22—21no—
315C22—24no—
316C22—22no—
317C22—21no—
318C22—24no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—44no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—84no—
332C22—88no—
333C22C2×#38111yescharacteristic
334C22#231112yes—
335C22#231414yes—
336C22#262414yes—
337C22#262514yes—
338C4—21no—
339C4—22no—
340C4—24no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—84no—
345C4—84no—
346C4—168no—
347C4#182812yes—
348C2—22no—
349C2—24no—
350C2—24no—
351C2—24no—
352C2—82no—
353C2#1434311yescharacteristic
354C2C2×#182811yescharacteristic
355C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11351C2semidirect
6. #10854C2semidirect
11. #2313C4semidirect
12. #2312C4semidirect
16. #2283C4semidirect
17. #2280C4semidirect

Maximal subgroups

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

subgroupindex
#340572
C2×#17992
#113512
#111692
#111662
#108542