Finite Group Databasebeta

← #190073

Subgroups — #190073

Order 512.

1236 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#17357C212yesmaximal
3#8743C214yesmaximal, complemented
4#8742C214yesmaximal, complemented
5#7950C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C2211yescharacteristic
8C23×C42C411yescharacteristic
9#2920C2214yes—
10#2296—22no—
11#2296—22no—
12#2296C412yes—
13#2296C412yes—
14#2296C2212yes—
15#2296C2212yes—
16C2×#389—22no—
17C2×#385—22no—
18C2×C4×C4⋊C4C412yes—
19C2×C4×C4⋊C4C2212yes—
20#1464C2214yes—
21#1464C2214yes—
22#1463—22no—
23#1462—22no—
24#1458—22no—
25#1458—22no—
26#1285C22116yescomplemented
27C24×C4—21no—
28C24×C4C2×C411yescharacteristic
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—24no—
32C2×#117—24no—
33C2×#117—44no—
34C2×#117C2×C412yes—
35C2×#117C2×C412yes—
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×#117D412yes—
39C2×#117D412yes—
40C2×#117C2311yescharacteristic
41C2×#117C2311yescharacteristic
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4D412yes—
46C22×C22⋊C4—21no—
47C22×C22⋊C4D412yes—
48C22×C42—21no—
49C22×C42—21no—
50C22×C42—21no—
51C22×C42—44no—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C412yes—
54C22×C42C2×C412yes—
55C22×C42C2×C412yes—
56C22×C42C2×C412yes—
57C22×C42C2311yescharacteristic
58#432—24no—
59#432C2318yes—
60#431—24no—
61#431—24no—
62#423—24no—
63#407—24no—
64C4×M16—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—24no—
69#385—24no—
70#385—24no—
71#385—24no—
72#385—24no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4C2×C414yes—
78C4×C4⋊C4C2×C414yes—
79C4×C4⋊C4C2×C414yes—
80C4×C4⋊C4C2314yes—
81C2×#95—22no—
82C2×#95—22no—
83#324—24no—
84#324—24no—
85#323—24no—
86#323—24no—
87C25C22⋊C411yescharacteristic
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—48no—
97C23×C4—44no—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D412yes—
106C2×M16—44no—
107#117—24no—
108#117—24no—
109#117—24no—
110#117—24no—
111#117—24no—
112#117—24no—
113#117—24no—
114#117—24no—
115#117—44no—
116#117—44no—
117#117—44no—
118#117—44no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—48no—
128C2×C4⋊C4—48no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4C22⋊C412yes—
140C2×C22⋊C4C22⋊C412yes—
141C2×C22⋊C4C2×D412yes—
142C2×C22⋊C4C2×D412yes—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—44no—
150C2×C42—44no—
151C2×C42—44no—
152C2×C42—44no—
153C2×C42—44no—
154C2×C42—44no—
155C2×C42—44no—
156C2×C42C22⋊C412yes—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C414yes—
160C2×C42C22×C414yes—
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D412yes—
163C2×C42#4114yes—
164C2×C42#4114yes—
165C2×C42C2411yescharacteristic
166C4⋊C8—28no—
167C4⋊C8—44no—
168C22⋊C8—28no—
169C22⋊C8—44no—
170#97—28no—
171C4×C8—44no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—24no—
176C24—22no—
177C24—22no—
178C24—22no—
179C24—44no—
180C24C2×C22⋊C411yescharacteristic
181C24C2×C22⋊C411yescharacteristic
182C24C2×C22⋊C411yescharacteristic
183C22×C4—216no—
184C22×C4—216no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—28no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—44no—
208C22×C4—48no—
209C22×C4—48no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—432no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4#10014yes—
220C22×C4#10114yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4#11712yes—
228C22×C4#11712yes—
229C22×C4#12314yes—
230C22×C4#12314yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D412yes—
233M16—88no—
234C2×C8—48no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—48no—
240C4⋊C4—48no—
241C4⋊C4—48no—
242C4⋊C4—48no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—44no—
256C42—44no—
257C42—44no—
258C42—44no—
259C42—44no—
260C42—44no—
261C42#11714yes—
262C42#12314yes—
263C42#12314yes—
264C42C2×#4114yes—
265C23—24no—
266C23—21no—
267C23—21no—
268C23—28no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—42no—
274C23—42no—
275C23—48no—
276C23—48no—
277C23—44no—
278C23—44no—
279C23#38112yes—
280C23C2×#10012yes—
281C23C2×#10112yes—
282C23C22×C22⋊C411yescharacteristic
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—28no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—416no—
307C2×C4—44no—
308C2×C4—416no—
309C2×C4—416no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—88no—
318C2×C4#38114yes—
319C2×C4#38111yescharacteristic
320C2×C4#38111yescharacteristic
321C2×C4C2×#10012yes—
322C2×C4C2×#10012yes—
323C2×C4C2×#10112yes—
324C2×C4C2×#10112yes—
325C2×C4C2×#11712yes—
326C2×C4C2×#12314yes—
327C8—88no—
328C22—24no—
329C22—24no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—416no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22#182714yes—
343C22C2×#38111yescharacteristic
344C22C22×#10011yescharacteristic
345C22C22×#10111yescharacteristic
346C4—22no—
347C4—22no—
348C4—22no—
349C4—24no—
350C4—24no—
351C4—48no—
352C4—42no—
353C4—42no—
354C4—84no—
355C4#182112yes—
356C4#182112yes—
357C2—21no—
358C2—24no—
359C2—21no—
360C2—42no—
361C2—42no—
362C2C2×#182111yescharacteristic
363C2C2×#182712yes—
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#381
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8743C2semidirect
4. #8742C2semidirect
5. #7950C2semidirect
26. #1285C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#173572
#87432
#87422
#79502