Finite Group Databasebeta

← #190088

Subgroups — #190088

Order 512.

1044 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#34052C211yesmaximal, characteristic
2#17357C212yesmaximal
3#8743C214yesmaximal, complemented
4#8742C214yesmaximal, complemented
5#7953C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C411yescharacteristic
8C22×#117C2211yescharacteristic
9#2920C2214yes—
10#2317—22no—
11#2317—22no—
12#2316—22no—
13#2315—22no—
14#2296C412yes—
15#2296C412yes—
16#2296C2212yes—
17#2296C2212yes—
18#2285C412yes—
19#2285C2212yes—
20#1464C2214yes—
21#1464C2214yes—
22#1463—22no—
23#1462—22no—
24#1458—22no—
25#1458—22no—
26#1285C22116yescomplemented
27C24×C4C2×C411yescharacteristic
28C2×#117—21no—
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—21no—
32C2×#117—24no—
33C2×#117—24no—
34C2×#117—44no—
35C2×#117—44no—
36C2×#117C2×C412yes—
37C2×#117C2×C412yes—
38C2×#117C2×C412yes—
39C2×#117C2×C412yes—
40C2×#117C2×C411yescharacteristic
41C2×#117C2×C411yescharacteristic
42C2×#117D412yes—
43C2×#117D412yes—
44C2×#117C2311yescharacteristic
45C2×#117C2311yescharacteristic
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D412yes—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4D412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C412yes—
54C22×C42C2×C412yes—
55C22×C42C2311yescharacteristic
56#432—24no—
57#432C2318yes—
58#431—24no—
59#431—24no—
60#423—24no—
61#407—24no—
62C4×M16—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—24no—
67#385—24no—
68#385—24no—
69#385—24no—
70#385—24no—
71#384—22no—
72#384—22no—
73#382—22no—
74#382—22no—
75#380—22no—
76#380—22no—
77#380—22no—
78#380—22no—
79C4×C4⋊C4C2×C414yes—
80C4×C4⋊C4C2×C414yes—
81C4×C4⋊C4C2×C414yes—
82C4×C4⋊C4C2314yes—
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×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D412yes—
102C2×M16—44no—
103#117—24no—
104#117—24no—
105#117—24no—
106#117—24no—
107#117—24no—
108#117—24no—
109#117—24no—
110#117—24no—
111#117—44no—
112#117—44no—
113#117—44no—
114#117—44no—
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—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4C22⋊C412yes—
144C2×C22⋊C4C22⋊C412yes—
145C2×C22⋊C4C2×D412yes—
146C2×C22⋊C4C2×D412yes—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—44no—
152C2×C42—44no—
153C2×C42C22⋊C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C414yes—
157C2×C42C22×C414yes—
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D412yes—
160C2×C42#4114yes—
161C2×C42#4114yes—
162C2×C42C2411yescharacteristic
163C4⋊C8—28no—
164C4⋊C8—44no—
165C22⋊C8—28no—
166C22⋊C8—44no—
167#97—28no—
168C4×C8—44no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173C24—21no—
174C24—21no—
175C24—22no—
176C24—22no—
177C24—82no—
178C24C2×C22⋊C411yescharacteristic
179C24C2×C22⋊C411yescharacteristic
180C24C2×C22⋊C411yescharacteristic
181C22×C4—216no—
182C22×C4—216no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—82no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—82no—
209C22×C4#10014yes—
210C22×C4#10114yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4#11712yes—
218C22×C4#11712yes—
219C22×C4#12314yes—
220C22×C4#12314yes—
221C22×C4C23×C411yescharacteristic
222C22×C4C22×D412yes—
223M16—88no—
224C2×C8—48no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—22no—
249C42—44no—
250C42—44no—
251C42—44no—
252C42—44no—
253C42#11714yes—
254C42#12314yes—
255C42#12314yes—
256C42C2×#4114yes—
257C23—24no—
258C23—24no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—82no—
269C23—82no—
270C23—84no—
271C23—84no—
272C23#38111yescharacteristic
273C23#38111yescharacteristic
274C23C2×#10012yes—
275C23C2×#10112yes—
276C23C22×C22⋊C411yescharacteristic
277C2×C4—28no—
278C2×C4—28no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—88no—
308C2×C4—88no—
309C2×C4#38114yes—
310C2×C4#38112yes—
311C2×C4C2×#10012yes—
312C2×C4C2×#10012yes—
313C2×C4C2×#10112yes—
314C2×C4C2×#10112yes—
315C2×C4C2×#11712yes—
316C2×C4C2×#12314yes—
317C8—88no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—84no—
334C22—84no—
335C22C2×#38111yescharacteristic
336C22C22×#10011yescharacteristic
337C22C22×#10111yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—24no—
342C4—24no—
343C4—42no—
344C4—42no—
345C4—84no—
346C4—84no—
347C4#182114yes—
348C2—21no—
349C2—21no—
350C2—41no—
351C2—41no—
352C2—42no—
353C2—42no—
354C2C2×#182111yescharacteristic
355C2#2070812yes—
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ 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. #7953C2semidirect
26. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#340522
#173572
#87432
#87422
#79532