Finite Group Databasebeta

← #393498

Subgroups — #393498

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3#20911C212yesmaximal
4C2×#1821C211yesmaximal, characteristic
5#10957C214yesmaximal, complemented
6#10891C214yesmaximal, complemented
7#10733C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C2×#419C2214yes—
12C2×#407C2212yes—
13#2502—24no—
14#2502C418yescomplemented
15C2×#398—22no—
16C2×#391C2212yes—
17C2×#385—22no—
18C2×#384C2212yes—
19#2274—24no—
20#2274—22no—
21#2274C2214yes—
22C2×C4×C4⋊C4—24no—
23C2×C4×C22⋊C4—22no—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1873—24no—
26#1872—24no—
27#1821—24no—
28C2×#343C2214yes—
29#1736C2214yes—
30C2×#339C2218yes—
31C23×Q8C2311yescharacteristic
32C24×C4D412yes—
33C24×C4C2311yescharacteristic
34C22×M16C2314yes—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8C2×C414yes—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—216no—
43C2×#117D414yes—
44C2×#117C2314yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4D414yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D414yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C42—22no—
60C22×C42—24no—
61C22×C42C2312yes—
62#419—216no—
63C2×#103—28no—
64#407—24no—
65#407—24no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68#398—216no—
69#391—24no—
70#391C2×C418yescomplemented
71#385—216no—
72#384—24no—
73#384C2×C418yescomplemented
74C4×C4⋊C4—216no—
75C4×C4⋊C4—216no—
76C4×C22⋊C4—28no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4—28no—
79C4×C22⋊C4C2×C414yes—
80C2×#95—24no—
81C2×#95—21no—
82C2×#95—24no—
83#343—28no—
84#340—28no—
85#339—216no—
86C25C2×D411yescharacteristic
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—28no—
91C22×Q8C22×C412yes—
92C23×C4—28no—
93C23×C4—24no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—84no—
101C23×C4C22⋊C418yes—
102C23×C4C22×C412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D414yes—
108C23×C4#4114yes—
109C23×C4C2411yescharacteristic
110C2×M16—24no—
111C2×M16—28no—
112C2×M16—24no—
113C2×M16—28no—
114C22×C8—24no—
115C22⋊Q8—28no—
116C22⋊Q8—28no—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119#117—216no—
120#117—216no—
121#117—28no—
122#117—216no—
123#117#41116yes—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—216no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—216no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—88no—
142C2×C4⋊C4C22⋊C418yes—
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C2×D414yes—
146C2×C4⋊C4C2×D418yes—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—42no—
156C2×C42—216no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—28no—
160C2×C42—22no—
161C2×C42—216no—
162C2×C42—88no—
163C2×C42C22×C414yes—
164#103—416no—
165C22⋊C8—44no—
166C22⋊C8—44no—
167#95—216no—
168#95—24no—
169#95—216no—
170C24—24no—
171C24—24no—
172C24—42no—
173C24C2×C22⋊C412yes—
174C24#12011yescharacteristic
175C24#12011yescharacteristic
176C24#12312yes—
177C24C22×D411yescharacteristic
178C2×Q8—24no—
179C2×Q8—28no—
180C2×Q8—42no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×Q8—48no—
185C22×C4—24no—
186C22×C4—28no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—216no—
192C22×C4—28no—
193C22×C4—216no—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—28no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—28no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—48no—
204C22×C4—48no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—44no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—44no—
215C22×C4—88no—
216C22×C4—84no—
217C22×C4—88no—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C418yes—
220C22×C4C4×D414yes—
221C22×C4C4×D414yes—
222C22×C4#12012yes—
223C22×C4#12114yes—
224C22×C4#12114yes—
225C22×C4#12312yes—
226C22×C4C23×C411yescharacteristic
227C22×C4C22×D412yes—
228C22×C4C22×D411yescharacteristic
229C22×C4C2×#4112yes—
230C22×C4C2×#4118yes—
231M16—48no—
232M16—48no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—44no—
236C4⋊C4—216no—
237C4⋊C4—28no—
238C4⋊C4—28no—
239C4⋊C4—28no—
240C4⋊C4—216no—
241C4⋊C4—24no—
242C4⋊C4—42no—
243C4⋊C4—48no—
244C4⋊C4—816no—
245C4⋊C4#123116yes—
246C22⋊C4—216no—
247C22⋊C4—216no—
248C22⋊C4—216no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C42—216no—
252C42—216no—
253C42—28no—
254C42—44no—
255C42—816no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—24no—
263C23—24no—
264C23—22no—
265C23—22no—
266C23—24no—
267C23—42no—
268C23—42no—
269C23—416no—
270C23#38612yes—
271C23#38612yes—
272C23#45614yes—
273C23C22×C22⋊C411yescharacteristic
274C23C2×#12011yescharacteristic
275C23C2×#12311yescharacteristic
276C23#52514yes—
277Q8—48no—
278Q8—84no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—216no—
289C2×C4—28no—
290C2×C4—44no—
291C2×C4—416no—
292C2×C4—432no—
293C2×C4—416no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—84no—
304C2×C4—88no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4#38614yes—
310C2×C4C2×C4×D412yes—
311C2×C4C4×#4114yes—
312C2×C4C2×#12112yes—
313C2×C4C2×#12314yes—
314C2×C4#52514yes—
315C2×C4#52918yes—
316C2×C4#53214yes—
317C8—84no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—21no—
324C22—28no—
325C22—28no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—48no—
330C22—48no—
331C22—44no—
332C22#188614yes—
333C22C2×#38611yescharacteristic
334C22#229712yes—
335C22#284912yes—
336C22#285312yes—
337C22C2×#45612yes—
338C22#301912yes—
339C4—24no—
340C4—48no—
341C4—48no—
342C4—416no—
343C4—82no—
344C4—82no—
345C4C4×#12314yes—
346C4#285414yes—
347C2—22no—
348C2—22no—
349C2—24no—
350C2—42no—
351C2#1470611yescharacteristic
352C2C2×#188611yescharacteristic
353C2#2095311yescharacteristic
354C2#3300014yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10957C2semidirect
6. #10891C2semidirect
14. #2502C4semidirect
70. #391C2×C4semidirect
73. #384C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#22742
#209112
C2×#18212
#109572
#108912
#107332