Finite Group Databasebeta

← #161200

Subgroups — #161200

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2#20682C212yesmaximal
3#7646C214yesmaximal, complemented
4#7632C214yesmaximal, complemented
5#7593C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C411yescharacteristic
8C23×C42C2211yescharacteristic
9C2×#407—22no—
10#2296—22no—
11#2296—22no—
12#2296C412yes—
13#2296C412yes—
14#2296C2212yes—
15#2296C2212yes—
16C2×#389C412yes—
17C2×#389C2212yes—
18C2×#385—22no—
19C2×C4×C4⋊C4—22no—
20#1817—22no—
21#1817—22no—
22#1817C2214yes—
23#1817C2214yes—
24#1728—22no—
25#1728C2214yes—
26#1271C22116yescomplemented
27C24×C4C2×C411yescharacteristic
28C24×C4D412yes—
29C22×M16—24no—
30C2×#117—21no—
31C2×#117—21no—
32C2×#117—21no—
33C2×#117—21no—
34C2×#117—28no—
35C2×#117—44no—
36C2×#117C2×C412yes—
37C2×#117C2×C412yes—
38C2×#117C2×C412yes—
39C2×#117C2×C412yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C412yes—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C42—21no—
47C22×C42—21no—
48C22×C42—44no—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42D412yes—
53C22×C42D412yes—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#407—48no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8—24no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389C2×C414yes—
66#389C2×C414yes—
67#389C2×C414yes—
68#389C2314yes—
69#385—24no—
70#385—24no—
71#385—24no—
72#385—24no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4—24no—
77C2×#95—22no—
78C2×#95—22no—
79#336—24no—
80#336—24no—
81#336—24no—
82#336C2318yes—
83C25C22⋊C411yescharacteristic
84C23×C4—24no—
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—28no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—44no—
91C23×C4—44no—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C2×M16—48no—
105C2×M16—48no—
106C2×M16—48no—
107C22×C8—28no—
108#117—28no—
109#117—28no—
110#117—28no—
111#117—28no—
112#117—44no—
113#117—44no—
114#117—44no—
115#117—44no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4M1618yescomplemented
127C2×C4⋊C4C22×C414yes—
128C2×C4⋊C4C22×C414yes—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—44no—
135C2×C42—28no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42—44no—
144C2×C42—44no—
145C2×C42—44no—
146C2×C42—42no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42—44no—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C412yes—
152C2×C42C22⋊C412yes—
153C2×C42C22×C411yescharacteristic
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2411yescharacteristic
160C22⋊C8—48no—
161C22⋊C8—48no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166C24—22no—
167C24—22no—
168C24—42no—
169C24—42no—
170C24#9914yes—
171C24C2×C22⋊C411yescharacteristic
172C24C2×C22⋊C411yescharacteristic
173C24C2×C22⋊C411yescharacteristic
174C22×C4—232no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—28no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—432no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4#10114yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×M1614yes—
226C22×C4C23×C411yescharacteristic
227C22×C4C22×D412yes—
228M16—816no—
229C2×C8—48no—
230C2×C8—48no—
231C2×C8—48no—
232C4⋊C4—24no—
233C4⋊C4—28no—
234C4⋊C4—44no—
235C4⋊C4—48no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4C2×M1618yes—
241C22⋊C4—28no—
242C22⋊C4—28no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C42—28no—
246C42—28no—
247C42—42no—
248C42—42no—
249C42—42no—
250C42—41no—
251C42—41no—
252C42—41no—
253C42—42no—
254C42—42no—
255C42—42no—
256C42—41no—
257C42—44no—
258C42—44no—
259C23—22no—
260C23—22no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—28no—
270C23—48no—
271C23—44no—
272C23—48no—
273C23—44no—
274C23#40712yes—
275C23C2×#9912yes—
276C23C2×#9912yes—
277C23C2×#9912yes—
278C23C2×#10112yes—
279C23C22×C22⋊C411yescharacteristic
280C2×C4—216no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—416no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—88no—
318C2×C4#40712yes—
319C2×C4C2×#10112yes—
320C2×C4C2×#10112yes—
321C2×C4#42114yes—
322C2×C4C22×M1612yes—
323C2×C4#57514yes—
324C8—88no—
325C22—22no—
326C22—24no—
327C22—22no—
328C22—24no—
329C22—24no—
330C22—24no—
331C22—22no—
332C22—22no—
333C22—416no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22#146314yes—
341C22C2×#40711yescharacteristic
342C22C22×#9911yescharacteristic
343C22C22×#10111yescharacteristic
344C22#289612yes—
345C22C2×#42112yes—
346C4—24no—
347C4—24no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—48no—
353C4—42no—
354C4—42no—
355C4—84no—
356C2—24no—
357C2—42no—
358C2—42no—
359C2C2×#146311yescharacteristic
360C2#1745012yes—
361C2#1758712yes—
362C2#1766411yescharacteristic
363C2#1773911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7646C2semidirect
4. #7632C2semidirect
5. #7593C2semidirect
26. #1271C22semidirect
126. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#206822
#76462
#76322
#75932