Finite Group Databasebeta

← #415247

Subgroups — #415247

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#21322C212yesmaximal
3C2×#1986C212yesmaximal
4#13910C211yesmaximal, characteristic
5#11296C214yesmaximal
6#11210C214yesmaximal
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×#389—22no—
18C2×#384C2212yes—
19C2×#384C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1992—24no—
23#1990—22no—
24#1990—22no—
25#1986—28no—
26C2×#343C2212yes—
27C2×#343C2212yes—
28#1736C2218yes—
29C2×#339C2218yes—
30#1444—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—24no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C42D412yes—
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56C4×M16—24no—
57C4×M16—24no—
58C2×#97—22no—
59C2×C4×C8—22no—
60#398—28no—
61#395—28no—
62#391—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—24no—
69#389—48no—
70#384—22no—
71#384—28no—
72#384—22no—
73#384—22no—
74#380—24no—
75#380—24no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4D414yes—
80C2×#95—21no—
81C2×#95—24no—
82C2×#95D412yes—
83#343—24no—
84#343Q818yescomplemented
85#340—28no—
86#340—28no—
87#339—216no—
88C22×Q8—22no—
89C22×Q8—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2411yescharacteristic
97C2×M16—24no—
98C2×M16—24no—
99C2×M16—24no—
100C2×M16—24no—
101C2×M16—24no—
102C2×M16C2×Q818yes—
103C22×C8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—84no—
109#125—28no—
110#125—28no—
111C22⋊Q8—44no—
112C22⋊Q8—44no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C4×Q8—44no—
116C4×Q8—44no—
117C4×Q8—44no—
118C4×Q8—44no—
119#117—24no—
120#117—24no—
121#117—24no—
122#117—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—48no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—84no—
149C2×C4⋊C4C2×D414yes—
150C2×C4⋊C4C2×D414yes—
151C2×C4⋊C4C2×Q814yes—
152C2×C4⋊C4#4114yes—
153C2×C22⋊C4—24no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—22no—
157C2×C42—24no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—44no—
161C2×C42C2×D414yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42#4114yes—
166C2×C42#4112yes—
167#97—44no—
168C4×C8—44no—
169#95—28no—
170#95—28no—
171#95—24no—
172#95—24no—
173#95—48no—
174C24C22×D411yescharacteristic
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—82no—
180C2×Q8—82no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4#12012yes—
208C22×C4#12011yescharacteristic
209C22×C4#12011yescharacteristic
210C22×C4#12112yes—
211C22×C4#12112yes—
212C22×C4#12412yes—
213C22×C4#12414yes—
214C22×C4#12412yes—
215C22×C4C4⋊Q814yes—
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D412yes—
218C22×C4C22×Q812yes—
219C22×C4C2×#4112yes—
220C22×C4C2×#4111yescharacteristic
221C22×C4C2×#4112yes—
222C22×C42−1+412yes—
223C22×C42−1+412yes—
224M16—28no—
225M16—28no—
226M16—44no—
227M16—44no—
228C2×C8—28no—
229C2×C8—44no—
230C2×C8—44no—
231C4⋊C4—216no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—48no—
243C4⋊C4—82no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C4⋊C4—84no—
249C4⋊C4—816no—
250C4⋊C4—82no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C42—28no—
254C42—44no—
255C42—44no—
256C42—44no—
257C42—44no—
258C42—82no—
259C42#12114yes—
260C42#12114yes—
261C42#12314yes—
262C42#12314yes—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23C2×#12011yescharacteristic
268C23C2×#12111yescharacteristic
269C23C2×#12411yescharacteristic
270C23C2×#12411yescharacteristic
271C23C2×C4⋊Q811yescharacteristic
272C23#53611yescharacteristic
273C23#53611yescharacteristic
274Q8—88no—
275Q8—88no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
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—84no—
308C2×C4—84no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4#39412yes—
313C2×C4#39414yes—
314C2×C4#39412yes—
315C2×C4#45312yes—
316C2×C4#45612yes—
317C2×C4C2×#12112yes—
318C2×C4C2×#12312yes—
319C2×C4#53314yes—
320C2×C4#54114yes—
321C2×C4#54712yes—
322C2×C4#54814yes—
323C2×C4D4×Q812yes—
324C2×C4D4×Q814yes—
325C2×C4#55612yes—
326C2×C4#55712yes—
327C8—48no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22C2×#39411yescharacteristic
341C22#243411yescharacteristic
342C22#243611yescharacteristic
343C22#254611yescharacteristic
344C22#254711yescharacteristic
345C22#264211yescharacteristic
346C22#286211yescharacteristic
347C22C2×#45311yescharacteristic
348C22C2×#45611yescharacteristic
349C22#302111yescharacteristic
350C22#302211yescharacteristic
351C4—22no—
352C4—24no—
353C4—48no—
354C4—44no—
355C4—44no—
356C4—84no—
357C4—84no—
358C4—162no—
359C4#203914yes—
360C4#242914yes—
361C4#244614yes—
362C2—21no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2#1518811yescharacteristic
367C2C2×#203911yescharacteristic
368C2#2350011yescharacteristic
369C2#2582411yescharacteristic
370C2#2582511yescharacteristic
371C2#2591811yescharacteristic
372C2#2591911yescharacteristic
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
84. #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
#139102
#112962
#112102
C4×#3432