Finite Group Databasebeta

← #480353

Subgroups — #480353

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34211C211yesmaximal, characteristic
2#17369C212yesmaximal
3#11535C214yesmaximal, complemented
4#11531C214yesmaximal, complemented
5#11447C212yesmaximal, complemented
6#11434C212yesmaximal, complemented
7C22×#124C411yescharacteristic
8C2×#519C411yescharacteristic
9C2×#518C2211yescharacteristic
10#2975—22no—
11#2367—22no—
12#2367—22no—
13#2342—24no—
14#2326C412yes—
15#2326C2212yes—
16#2310C2211yescharacteristic
17#2309C411yescharacteristic
18C4×#124C412yes—
19C4×#124C2212yes—
20#2285C2211yescharacteristic
21C4×#117C411yescharacteristic
22#1465C2218yes—
23#1462—24no—
24#1460—22no—
25#1458C2214yes—
26#1325C2214yescomplemented
27#1324C2214yescomplemented
28#1319C2218yescomplemented
29C23×Q8C2×C411yescharacteristic
30C23×D4C2×C411yescharacteristic
31C2×#124—24no—
32C2×#124—22no—
33C2×#124—24no—
34C2×#124C2×C412yes—
35C2×#124C2×C412yes—
36#519—44no—
37#518—44no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8C2×C412yes—
41C2×C4×Q8C2×C412yes—
42C2×C4×D4C2×C412yes—
43C2×C4×D4D412yes—
44C2×C4×D4D412yes—
45C2×C4×D4C2312yes—
46C2×#117—22no—
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117C2×C411yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4D414yes—
53C22×C42C2×C411yescharacteristic
54#436—48no—
55#431—24no—
56#407—28no—
57#391—22no—
58#391C2×C412yes—
59#391C2312yes—
60#390—22no—
61#390C2×C412yes—
62#390C2×C412yes—
63#388—24no—
64#388—24no—
65#388—24no—
66#388—24no—
67#385—24no—
68#385—24no—
69#384C2×C411yescharacteristic
70#384C2311yescharacteristic
71#382C2×C411yescharacteristic
72#382C2311yescharacteristic
73#381—22no—
74#380—22no—
75#379—22no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—22no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—22no—
82#376C2×C412yes—
83C43C2×C412yes—
84#324C2318yes—
85#323—24no—
86#323—28no—
87C25C22⋊C412yes—
88C22×Q8—21no—
89C22×Q8—22no—
90C22×Q8—42no—
91C22×Q8—42no—
92C22×Q8M1614yescomplemented
93C22×Q8C22×C412yes—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—42no—
97C22×D4—42no—
98C22×D4C22⋊C412yes—
99C22×D4C22×C412yes—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C22⋊C414yes—
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D414yes—
105C2×M16—48no—
106#124—216no—
107#124—24no—
108#124—24no—
109#124—24no—
110#124—24no—
111C4×Q8—44no—
112C4×Q8—44no—
113C4×Q8—44no—
114C4×Q8—44no—
115C4×D4—28no—
116C4×D4—44no—
117C4×D4—44no—
118C4×D4—44no—
119#117—24no—
120#117—42no—
121#117—48no—
122#117—42no—
123#117—42no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22×C411yescharacteristic
132C2×C4⋊C4C22×C411yescharacteristic
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C4⋊C4C2411yescharacteristic
135C2×C22⋊C4—216no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—48no—
143C2×C22⋊C4C22⋊C412yes—
144C2×C22⋊C4C22⋊C414yes—
145C2×C22⋊C4C2×D412yes—
146C2×C22⋊C4C2×D414yes—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—21no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42M1614yescomplemented
155C2×C42C22×C412yes—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C412yes—
159C2×C42C2×D412yes—
160C4⋊C8—44no—
161C22⋊C8—28no—
162C22⋊C8—48no—
163C4×C8—44no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95—24no—
168#95—22no—
169#95—24no—
170C24—24no—
171C24—42no—
172C24—42no—
173C24—82no—
174C24—82no—
175C24C2×C22⋊C414yes—
176C24C2×C22⋊C411yescharacteristic
177C2×Q8—22no—
178C2×Q8—24no—
179C2×Q8—48no—
180C2×Q8—48no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8C2×M1614yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—48no—
194C2×D4—48no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—28no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—84no—
212C22×C4—84no—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×M1612yes—
218C22×C4C2×M1612yes—
219C22×C4C23×C411yescharacteristic
220C22×C4C22×D412yes—
221M16—168no—
222C2×C8—48no—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4#9914yes—
234C22⋊C4—216no—
235C22⋊C4—216no—
236C22⋊C4—24no—
237C22⋊C4—28no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—84no—
243C22⋊C4—84no—
244C42—21no—
245C42—21no—
246C42—24no—
247C42—24no—
248C42—24no—
249C42—42no—
250C42—44no—
251C42—44no—
252C42—42no—
253C42—82no—
254C42—82no—
255C42C2×M1612yes—
256C42C2×M1612yes—
257C23—21no—
258C23—21no—
259C23—24no—
260C23—24no—
261C23—24no—
262C23—41no—
263C23—41no—
264C23—44no—
265C23—44no—
266C23—48no—
267C23—88no—
268C23—162no—
269C23—162no—
270C23—164no—
271C23#40712yes—
272C23C22×C22⋊C411yescharacteristic
273Q8—44no—
274Q8—44no—
275Q8—48no—
276D4—44no—
277D4—44no—
278D4—48no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—48no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—88no—
308C2×C4—84no—
309C2×C4—164no—
310C2×C4#40712yes—
311C2×C4C2×#9914yes—
312C2×C4C2×#9912yes—
313C2×C4#41312yes—
314C2×C4C22×M1611yescharacteristic
315C2×C4C22×M1611yescharacteristic
316C8—88no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—48no—
323C22—41no—
324C22—41no—
325C22—44no—
326C22—44no—
327C22—44no—
328C22—84no—
329C22—168no—
330C22C2×#40711yescharacteristic
331C22C22×#9911yescharacteristic
332C22C2×#41311yescharacteristic
333C4—24no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—44no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4#146514yes—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—84no—
350C2C2×#146511yescharacteristic
351C2#3336111yescharacteristic
352C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11535C2semidirect
4. #11531C2semidirect
5. #11447C2semidirect
6. #11434C2semidirect
26. #1325C22semidirect
27. #1324C22semidirect
28. #1319C22semidirect
92. C22×Q8M16semidirect
154. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
#342112
#173692
#115352
#115312
#114472
#114342