Finite Group Databasebeta

← #190184

Subgroups — #190184

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2#17357C212yesmaximal
3#8757C214yesmaximal, complemented
4#8756C214yesmaximal
5#8109C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C411yescharacteristic
8C22×#117C2211yescharacteristic
9#2920C2214yes—
10#2317—22no—
11#2317—22no—
12#2316C412yes—
13#2316C2212yes—
14#2315C412yes—
15#2315C2212yes—
16#2296—22no—
17#2296—22no—
18#2285C412yes—
19#2285C2212yes—
20#1464—22no—
21#1464—22no—
22#1463C2214yes—
23#1462C2214yes—
24#1458—22no—
25#1458—22no—
26#1305C22116yes—
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×#117D412yes—
41C2×#117D412yes—
42C2×#117D412yes—
43C2×#117D412yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4C2×C412yes—
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2311yescharacteristic
52C22×C42—21no—
53C22×C42—21no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2311yescharacteristic
56#432—24no—
57#432—24no—
58#431—24no—
59#431—24no—
60#423—24no—
61#407C2318yes—
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#380C2×C414yes—
76#380C2×C414yes—
77#380C2×C414yes—
78#380C2314yes—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—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×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2411yescharacteristic
104C2×M16—28no—
105#117—24no—
106#117—24no—
107#117—24no—
108#117—24no—
109#117—24no—
110#117—24no—
111#117—24no—
112#117—24no—
113#117—44no—
114#117—44no—
115#117—44no—
116#117—44no—
117#117—44no—
118#117—44no—
119#117—44no—
120#117—44no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4C22⋊C412yes—
134C2×C4⋊C4C22⋊C412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
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⋊C4C22×C414yes—
146C2×C22⋊C4C22×C414yes—
147C2×C22⋊C4C2×D412yes—
148C2×C22⋊C4C2×D412yes—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—44no—
156C2×C42—44no—
157C2×C42C22⋊C412yes—
158C2×C42C22⋊C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42#4114yes—
162C2×C42#4114yes—
163C4⋊C8—44no—
164C4⋊C8—44no—
165C22⋊C8—28no—
166C22⋊C8—28no—
167#97—44no—
168C4×C8—44no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173C24—22no—
174C24—22no—
175C24—22no—
176C24—81no—
177C24—81no—
178C24C2×C22⋊C411yescharacteristic
179C24C2×C22⋊C411yescharacteristic
180C24C23×C411yescharacteristic
181C22×C4—216no—
182C22×C4—216no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
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—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4#10014yes—
215C22×C4#10114yes—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4#11712yes—
224C22×C4#11712yes—
225C22×C4#11714yes—
226C22×C4C22×D412yes—
227C22×C4C2×#4114yes—
228M16—88no—
229C2×C8—48no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—84no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C4⋊C4—84no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—44no—
255C42—44no—
256C42—44no—
257C42—44no—
258C42#12314yes—
259C42#12314yes—
260C42#12314yes—
261C42#12314yes—
262C23—28no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—82no—
276C23—82no—
277C23—84no—
278C23—84no—
279C23C2×#10012yes—
280C23C2×#10112yes—
281C23C22×C22⋊C411yescharacteristic
282C23C2×#11712yes—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—28no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—28no—
295C2×C4—28no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—88no—
315C2×C4—88no—
316C2×C4#38111yescharacteristic
317C2×C4#38111yescharacteristic
318C2×C4#38114yes—
319C2×C4#38112yes—
320C2×C4C2×#10012yes—
321C2×C4C2×#10012yes—
322C2×C4C2×#10112yes—
323C2×C4C2×#10112yes—
324C2×C4C2×#12314yes—
325C8—88no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—88no—
344C22C2×#38111yescharacteristic
345C22C22×#10011yescharacteristic
346C22C22×#10111yescharacteristic
347C4—21no—
348C4—21no—
349C4—22no—
350C4—22no—
351C4—24no—
352C4—24no—
353C4—42no—
354C4—42no—
355C4—84no—
356C4—84no—
357C4#182114yes—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—42no—
362C2—42no—
363C2C2×#182111yescharacteristic
364C2#2071011yescharacteristic
365C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8757C2semidirect
5. #8109C2semidirect

Maximal subgroups

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

subgroupindex
#340522
#173572
#87572
#87562
#81092