Finite Group Databasebeta

← #359936

Subgroups — #359936

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic
3#20700C212yesmaximal
4C2×#1821C211yesmaximal, characteristic
5#11211C214yesmaximal, complemented
6#10892C214yesmaximal, complemented
7#10719C212yesmaximal
8C23×C4⋊C4C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#407C2212yes—
12#2345—24no—
13#2344—24no—
14#2316C2212yes—
15#2313—22no—
16C2×#382—22no—
17C2×#382—24no—
18C2×#382C418yescomplemented
19C2×#382C2212yes—
20C2×#382C2212yes—
21#2280—22no—
22#2280C2212yes—
23#2274C2212yes—
24C22×#95—22no—
25#1824—24no—
26#1823—24no—
27#1821—24no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30#1736C2218yes—
31C2×#339C2218yes—
32C24×C4—21no—
33C24×C4—21no—
34C24×C4D412yes—
35C24×C4C2311yescharacteristic
36C22×M16C2314yes—
37C2×#117—24no—
38C2×#117C2314yes—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2×C414yes—
48C22×C4⋊C4C2×C414yes—
49C22×C4⋊C4D412yes—
50C22×C4⋊C4D414yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42—21no—
55C22×C42—22no—
56C22×C42—28no—
57C22×C42C2×C414yes—
58C22×C42C2312yes—
59#407—24no—
60#407—24no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63#389—28no—
64#387—216no—
65#382—216no—
66#382—24no—
67#382—24no—
68#382C2×C418yes—
69#382C2×C418yes—
70#381—24no—
71#381—22no—
72#380—24no—
73#380—24no—
74#380D414yes—
75#380D414yes—
76#379—22no—
77C4×C4⋊C4—28no—
78C4×C22⋊C4—24no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82C2×#95—216no—
83C2×#95—24no—
84C2×#95—24no—
85#343—24no—
86#343—24no—
87#340—216no—
88#339—216no—
89C25C2×D411yescharacteristic
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—28no—
93C23×C4—24no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—44no—
101C23×C4—44no—
102C23×C4—48no—
103C23×C4—48no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—42no—
107C23×C4C22⋊C418yes—
108C23×C4C22×C412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4#4114yes—
116C23×C4C2411yescharacteristic
117C2×M16—24no—
118C2×M16—28no—
119C2×M16—24no—
120C2×M16—28no—
121C22×C8—24no—
122#117—28no—
123#117—28no—
124#117—88no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C22⋊C418yes—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C2×D418yes—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D412yes—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C42—28no—
157C2×C42—216no—
158C2×C42—24no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—216no—
162C2×C42—24no—
163C2×C42—28no—
164C2×C42C22×C414yes—
165C2×C42#4118yes—
166C22⋊C8—44no—
167C22⋊C8—44no—
168#95—28no—
169#95—28no—
170#95—216no—
171#95—216no—
172#95—24no—
173#95—24no—
174C24—22no—
175C24—22no—
176C24—24no—
177C24—24no—
178C24C2×C22⋊C412yes—
179C24#12011yescharacteristic
180C24#12011yescharacteristic
181C24#12312yes—
182C24C22×D411yescharacteristic
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—28no—
190C22×C4—28no—
191C22×C4—216no—
192C22×C4—216no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—28no—
197C22×C4—28no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—28no—
201C22×C4—24no—
202C22×C4—28no—
203C22×C4—432no—
204C22×C4—48no—
205C22×C4—48no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—48no—
209C22×C4—416no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—416no—
213C22×C4—48no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—88no—
219C22×C4—84no—
220C22×C4—88no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C414yes—
225C22×C4C2×C22⋊C414yes—
226C22×C4C4×D414yes—
227C22×C4C4×D414yes—
228C22×C4#12012yes—
229C22×C4#12114yes—
230C22×C4#12114yes—
231C22×C4#12312yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4114yes—
236C22×C4C2×#4112yes—
237C22×C42−1+414yes—
238M16—44no—
239M16—44no—
240M16—48no—
241C2×C8—44no—
242C2×C8—48no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—216no—
248C4⋊C4—28no—
249C4⋊C4—216no—
250C4⋊C4—44no—
251C4⋊C4—48no—
252C4⋊C4—88no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—48no—
256C22⋊C4—48no—
257C42—216no—
258C42—24no—
259C42—28no—
260C42—48no—
261C42—88no—
262C42#11718yescomplemented
263C23—216no—
264C23—216no—
265C23—22no—
266C23—24no—
267C23—24no—
268C23—24no—
269C23—24no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23#38612yes—
279C23#38612yes—
280C23#45414yes—
281C23#45514yes—
282C23C22×C22⋊C411yescharacteristic
283C23C2×#12011yescharacteristic
284C23C2×#12311yescharacteristic
285C23#52512yes—
286C23#53612yes—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—28no—
295C2×C4—28no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—432no—
299C2×C4—416no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—416no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—44no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—88no—
315C2×C4—84no—
316C2×C4—88no—
317C2×C4—88no—
318C2×C4—84no—
319C2×C4#38614yes—
320C2×C4C2×#11712yes—
321C2×C4C2×C4×D412yes—
322C2×C4#51912yes—
323C2×C4C2×#12112yes—
324C2×C4#52512yes—
325C2×C4#53612yes—
326C2×C4#54718yes—
327C2×C4#55218yes—
328C8—84no—
329C22—28no—
330C22—28no—
331C22—216no—
332C22—24no—
333C22—24no—
334C22—28no—
335C22—28no—
336C22—22no—
337C22—22no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22#188514yes—
345C22#188814yes—
346C22#228011yescharacteristic
347C22C2×#38611yescharacteristic
348C22#231011yescharacteristic
349C22#240612yes—
350C22#248412yes—
351C22C2×#45512yes—
352C22C2×#45412yes—
353C4—24no—
354C4—48no—
355C4—44no—
356C4—44no—
357C4—88no—
358C4—84no—
359C4#234414yes—
360C4#241314yes—
361C2—21no—
362C2—21no—
363C2—24no—
364C2—24no—
365C2—22no—
366C2#1387011yescharacteristic
367C2C2×#188511yescharacteristic
368C2C2×#188811yescharacteristic
369C2#2407714yes—
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. #11211C2semidirect
6. #10892C2semidirect
18. C2×#382C4semidirect
262. C42#117semidirect

Maximal subgroups

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

subgroupindex
#341782
C22×#3822
#207002
C2×#18212
#112112
#108922
#107192