Finite Group Databasebeta

← #359942

Subgroups — #359942

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#20705C212yesmaximal
4#20688C211yesmaximal, characteristic
5#11212C214yesmaximal, complemented
6#10893C214yesmaximal, complemented
7#10720C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C22×#117C2211yescharacteristic
11#2924—28no—
12#2921—22no—
13#2920—22no—
14#2905C2212yes—
15C2×#407C2212yes—
16#2873C2212yes—
17#2317C412yescomplemented
18#2317C412yescomplemented
19#2317C414yescomplemented
20#2316—22no—
21#2315—22no—
22#2296—22no—
23#2296—22no—
24#2285C2212yes—
25#2285C2212yes—
26#2280C2212yes—
27#1826—28no—
28#1820—22no—
29#1819—22no—
30#1740C2212yes—
31#1740C2212yes—
32#1733C2218yes—
33C2×#339C2218yes—
34C24×C4C2311yescharacteristic
35C22×M16C2314yes—
36C2×#117—24no—
37C2×#117—21no—
38C2×#117—21no—
39C2×#117—24no—
40C2×#117—21no—
41C2×#117—21no—
42C2×#117—28no—
43C2×#117—24no—
44C2×#117—24no—
45C2×#117C2×C412yes—
46C2×#117C2×C412yes—
47C2×#117C2×C412yes—
48C2×#117C2×C411yescharacteristic
49C2×#117C2×C411yescharacteristic
50C2×#117D412yes—
51C2×#117D412yes—
52C2×#117C2314yes—
53C2×#117C2312yes—
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4D412yes—
62C22×C42—21no—
63C22×C42D412yes—
64C22×C42C2312yes—
65#433—216no—
66#432—28no—
67#431—28no—
68#424—28no—
69#423—24no—
70#407—22no—
71#407—22no—
72#407D418yes—
73C2×C22⋊C8—24no—
74#405—28no—
75C4×M16—24no—
76#389—28no—
77#389—24no—
78#389—24no—
79#385—28no—
80#385—24no—
81#385—24no—
82#384C2×C414yes—
83#384C2×C414yes—
84#382C2×C414yes—
85#382C2×C414yes—
86#381—22no—
87#380—24no—
88#380—24no—
89#380—24no—
90#379—22no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C22⋊C4—24no—
94#342—24no—
95#342—24no—
96#339—216no—
97#339—216no—
98C25C2×D411yescharacteristic
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C22⋊C414yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D414yes—
110C23×C4#4114yes—
111C23×C4C2411yescharacteristic
112C2×M16—24no—
113C2×M16—28no—
114C2×M16—28no—
115C2×M16C2×D418yes—
116C22×C8—24no—
117#117—28no—
118#117—28no—
119#117—28no—
120#117—24no—
121#117—28no—
122#117—24no—
123#117—22no—
124#117—22no—
125#117—22no—
126#117—22no—
127#117—24no—
128#117—24no—
129#117—24no—
130#117—24no—
131#117—28no—
132#117—24no—
133#117—24no—
134#117—28no—
135#117—28no—
136#117—42no—
137#117—44no—
138#117—42no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4C22⋊C414yes—
151C2×C4⋊C4C22×C414yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4C22⋊C414yes—
165C2×C22⋊C4C2×D412yes—
166C2×C22⋊C4C2×D412yes—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42—24no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42C22⋊C414yes—
177C2×C42C22×C412yes—
178C2×C42C22×C412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42#4118yes—
182C4⋊C8—216no—
183C22⋊C8—28no—
184C22⋊C8—28no—
185#97—28no—
186C4×C8—28no—
187#95—28no—
188#95—28no—
189#95—44no—
190C24—22no—
191C24—22no—
192C24—82no—
193C24C2×C22⋊C411yescharacteristic
194C24C2×C22⋊C411yescharacteristic
195C24#12114yes—
196C24C22×D411yescharacteristic
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—216no—
208C22×C4—216no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—28no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C414yes—
223C22×C4C2×C22⋊C414yes—
224C22×C4C4×D414yes—
225C22×C4C4×D414yes—
226C22×C4#12012yes—
227C22×C4#12012yes—
228C22×C4#12114yes—
229C22×C4#12312yes—
230C22×C4#12312yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D412yes—
233C22×C4C22×D411yescharacteristic
234C22×C4C2×#4114yes—
235C22×C4C2×#4112yes—
236C22×C42−1+414yes—
237M16—216no—
238M16—44no—
239M16—44no—
240C2×C8—28no—
241C2×C8—48no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—44no—
250C4⋊C4—48no—
251C4⋊C4—88no—
252C4⋊C4—88no—
253C22⋊C4—28no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—28no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—88no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—28no—
271C42—28no—
272C42—42no—
273C42—42no—
274C42—44no—
275C42—48no—
276C42#11714yes—
277C42#11714yes—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—28no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—44no—
287C23—44no—
288C23—84no—
289C23—84no—
290C23#38612yes—
291C23#38612yes—
292C23C22×C22⋊C411yescharacteristic
293C23C2×#12112yes—
294C23#52512yes—
295C23#53612yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—28no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—48no—
317C2×C4—44no—
318C2×C4—84no—
319C2×C4—88no—
320C2×C4—88no—
321C2×C4—88no—
322C2×C4—84no—
323C2×C4#38612yes—
324C2×C4#38612yes—
325C2×C4C2×#11712yes—
326C2×C4C2×C4×D412yes—
327C2×C4#51912yes—
328C2×C4C2×#12011yescharacteristic
329C2×C4C2×#12311yescharacteristic
330C2×C4#52512yes—
331C2×C4#53612yes—
332C2×C4#54718yes—
333C2×C4#55218yes—
334C8—48no—
335C22—22no—
336C22—24no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—24no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—88no—
349C22#228011yescharacteristic
350C22C2×#38611yescharacteristic
351C22#231011yescharacteristic
352C22#241314yes—
353C4—22no—
354C4—22no—
355C4—44no—
356C4—44no—
357C4—44no—
358C4—44no—
359C4—84no—
360C4—164no—
361C4#234414yes—
362C4#240612yes—
363C4#248412yes—
364C2—21no—
365C2—24no—
366C2—21no—
367C2—42no—
368C2—42no—
369C2#1387011yescharacteristic
370C2#2094712yes—
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11212C2semidirect
6. #10893C2semidirect
17. #2317C4semidirect
18. #2317C4semidirect
19. #2317C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#340522
#207052
#206882
#112122
#108932
#107202