Finite Group Databasebeta

← #415248

Subgroups — #415248

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#21322C212yesmaximal
3C2×#1986C212yesmaximal
4#13911C211yesmaximal, characteristic
5#11296C214yesmaximal
6#11211C214yesmaximal
7C4×#343C211yesmaximal, characteristic
8C2×#533C2212yes—
9C2×C4×M16C2212yes—
10#2857—22no—
11#2659—22no—
12#2584—24no—
13#2567—22no—
14#2546—22no—
15#2316C2212yes—
16#2316C2212yes—
17C2×#384C2211yescharacteristic
18C2×#382C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1992—24no—
22#1990—22no—
23#1990—22no—
24#1986—28no—
25C2×#343C2212yes—
26C2×#343C2212yes—
27#1736C2218yes—
28C2×#339C2218yes—
29#1442—22no—
30C4⋊#95—22no—
31C4⋊#95C2212yes—
32C22×M16C2314yes—
33#533—24no—
34#533—24no—
35#533—24no—
36#533—24no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8D412yes—
40C2×#125D414yes—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×#117C2314yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C42D412yes—
53C22×C42C2312yes—
54C22×C42C2311yescharacteristic
55C4×M16—24no—
56C4×M16—24no—
57C2×#97—22no—
58C2×C4×C8—22no—
59#398—28no—
60#395—28no—
61#391—24no—
62#389—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#384—22no—
69#384—22no—
70#384—22no—
71#382—28no—
72#380—24no—
73#380—24no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4D414yes—
78C2×#95—21no—
79C2×#95—24no—
80C2×#95—24no—
81C2×#95D412yes—
82#343—24no—
83#343Q818yescomplemented
84#340—28no—
85#340—28no—
86#339—216no—
87C22×Q8—22no—
88C22×Q8—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C2×M16—24no—
97C2×M16—24no—
98C2×M16—24no—
99C2×M16—24no—
100C2×M16—24no—
101C2×M16C2×Q818yes—
102C22×C8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C4⋊Q8—24no—
107C4⋊Q8—84no—
108#125—28no—
109#125—28no—
110C22⋊Q8—44no—
111C22⋊Q8—44no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114C4×Q8—44no—
115C4×Q8—44no—
116C4×Q8—44no—
117C4×Q8—44no—
118#117—24no—
119#117—24no—
120#117—24no—
121#117—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—82no—
145C2×C4⋊C4C2×D414yes—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×Q814yes—
148C2×C4⋊C4#4114yes—
149C2×C22⋊C4—24no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—44no—
157C2×C42C2×D414yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42#4114yes—
162C2×C42#4112yes—
163#97—44no—
164C4×C8—44no—
165#95—28no—
166#95—28no—
167#95—24no—
168#95—24no—
169#95—48no—
170#95—84no—
171C24C22×D411yescharacteristic
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—82no—
177C2×Q8—82no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4#12012yes—
205C22×C4#12011yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12112yes—
208C22×C4#12112yes—
209C22×C4#12412yes—
210C22×C4#12414yes—
211C22×C4#12412yes—
212C22×C4C4⋊Q814yes—
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D412yes—
215C22×C4C22×Q812yes—
216C22×C4C2×#4112yes—
217C22×C4C2×#4111yescharacteristic
218C22×C4C2×#4112yes—
219C22×C42−1+412yes—
220C22×C42−1+412yes—
221M16—28no—
222M16—28no—
223M16—44no—
224M16—44no—
225C2×C8—28no—
226C2×C8—44no—
227C2×C8—44no—
228C4⋊C4—28no—
229C4⋊C4—28no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—48no—
235C4⋊C4—48no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—48no—
240C4⋊C4—82no—
241C4⋊C4—84no—
242C4⋊C4—84no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C4⋊C4—82no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—28no—
250C42—44no—
251C42—44no—
252C42—44no—
253C42—44no—
254C42—82no—
255C42#12114yes—
256C42#12114yes—
257C42#12314yes—
258C42#12314yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23C2×#12011yescharacteristic
264C23C2×#12111yescharacteristic
265C23C2×#12411yescharacteristic
266C23C2×#12411yescharacteristic
267C23C2×C4⋊Q811yescharacteristic
268C23#53611yescharacteristic
269C23#53611yescharacteristic
270Q8—88no—
271Q8—88no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—84no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4#39412yes—
309C2×C4#39414yes—
310C2×C4#39412yes—
311C2×C4#45412yes—
312C2×C4#45512yes—
313C2×C4C2×#12112yes—
314C2×C4C2×#12312yes—
315C2×C4#53314yes—
316C2×C4#54114yes—
317C2×C4#54712yes—
318C2×C4#54814yes—
319C2×C4D4×Q812yes—
320C2×C4D4×Q814yes—
321C2×C4#55612yes—
322C2×C4#55712yes—
323C8—48no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22C2×#39411yescharacteristic
337C22#243411yescharacteristic
338C22#243611yescharacteristic
339C22#254611yescharacteristic
340C22#254711yescharacteristic
341C22#264211yescharacteristic
342C22#286211yescharacteristic
343C22C2×#45511yescharacteristic
344C22C2×#45411yescharacteristic
345C22#302111yescharacteristic
346C22#302211yescharacteristic
347C4—22no—
348C4—24no—
349C4—44no—
350C4—44no—
351C4—84no—
352C4—84no—
353C4—84no—
354C4—162no—
355C4#204014yes—
356C4#242914yes—
357C4#244614yes—
358C2—21no—
359C2—21no—
360C2—21no—
361C2—21no—
362C2#1518811yescharacteristic
363C2C2×#204011yescharacteristic
364C2#2350011yescharacteristic
365C2#2582311yescharacteristic
366C2#2582611yescharacteristic
367C2#2591711yescharacteristic
368C2#2592011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
83. #343Q8semidirect

Maximal subgroups

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

subgroupindex
#351732
#213222
C2×#19862
#139112
#112962
#112112
C4×#3432