Finite Group Databasebeta

← #359939

Subgroups — #359939

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34179C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#20700C212yesmaximal
4C2×#1822C211yesmaximal, characteristic
5#11210C214yesmaximal, complemented
6#10892C214yesmaximal, complemented
7#10720C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#407C2212yes—
12#2347—24no—
13#2346—24no—
14#2317—24no—
15#2317C418yescomplemented
16#2316C2212yes—
17#2312—22no—
18C2×#389—22no—
19C2×#385—22no—
20C2×#384C2212yes—
21#2283—22no—
22C2×#382C2212yes—
23#2280C2212yes—
24#2274C2212yes—
25#1824—24no—
26#1823—24no—
27#1822—24no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30#1736C2218yes—
31C2×#339C2218yes—
32C24×C4—21no—
33C24×C4C2311yescharacteristic
34C22×M16C2314yes—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—24no—
39C2×#117—28no—
40C2×#117C2×C414yes—
41C2×#117C2×C414yes—
42C2×#117D414yes—
43C2×#117C2314yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—21no—
56C22×C22⋊C4D412yes—
57C22×C42D412yes—
58C22×C42C2312yes—
59#407—24no—
60#407—24no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63#389—28no—
64#389—216no—
65#388—216no—
66#385—216no—
67#384—24no—
68#384C2×C418yes—
69#383—24no—
70#382—24no—
71#382—24no—
72#382C2×C418yes—
73#381—22no—
74#380—24no—
75#380D414yes—
76#380D414yes—
77#379—22no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—28no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—22no—
82C4×C22⋊C4—22no—
83#376—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—22no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—84no—
98C23×C4C22×C412yes—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4#4114yes—
106C23×C4C2411yescharacteristic
107C2×M16—24no—
108C2×M16—28no—
109C2×M16—24no—
110C2×M16—28no—
111C22×C8—24no—
112#117—216no—
113#117—28no—
114#117—28no—
115#117—28no—
116#117—28no—
117#117—216no—
118#117—28no—
119#117—28no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—216no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—88no—
140C2×C4⋊C4C22×C414yes—
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C22×C412yes—
143C2×C4⋊C4C2×D418yes—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4C22⋊C418yes—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—22no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—28no—
161C2×C42C22⋊C418yes—
162C2×C42C22×C414yes—
163C2×C42#4118yes—
164C22⋊C8—44no—
165C22⋊C8—44no—
166#95—28no—
167#95—28no—
168#95—216no—
169#95—24no—
170#95—24no—
171C24—22no—
172C24—22no—
173C24—42no—
174C24—42no—
175C24C2×C22⋊C412yes—
176C24#12012yes—
177C24#12312yes—
178C24C22×D411yescharacteristic
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—28no—
186C22×C4—28no—
187C22×C4—216no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—28no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—28no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—48no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—48no—
204C22×C4—88no—
205C22×C4—88no—
206C22×C4—88no—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C414yes—
209C22×C4C2×C22⋊C414yes—
210C22×C4C4×D414yes—
211C22×C4C4×D414yes—
212C22×C4#12011yescharacteristic
213C22×C4#12011yescharacteristic
214C22×C4#12114yes—
215C22×C4#12114yes—
216C22×C4#12312yes—
217C22×C4C23×C411yescharacteristic
218C22×C4C22×D412yes—
219C22×C4C22×D411yescharacteristic
220C22×C4C2×#4114yes—
221C22×C4C2×#4112yes—
222C22×C42−1+414yes—
223M16—44no—
224M16—44no—
225M16—48no—
226C2×C8—44no—
227C2×C8—48no—
228C4⋊C4—216no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—28no—
234C4⋊C4—216no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—88no—
239C4⋊C4—816no—
240C22⋊C4—216no—
241C22⋊C4—216no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—84no—
245C22⋊C4—84no—
246C42—24no—
247C42—28no—
248C42—28no—
249C42—28no—
250C42—48no—
251C42—88no—
252C42#11718yes—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—24no—
257C23—24no—
258C23—24no—
259C23—24no—
260C23—22no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—48no—
266C23—48no—
267C23#38614yes—
268C23C22×C22⋊C411yescharacteristic
269C23C2×#12011yescharacteristic
270C23C2×#12311yescharacteristic
271C23#52512yes—
272C23#53612yes—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—416no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—48no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—84no—
300C2×C4—816no—
301C2×C4—84no—
302C2×C4#38612yes—
303C2×C4#38612yes—
304C2×C4C2×#11712yes—
305C2×C4C2×C4×D412yes—
306C2×C4#51912yes—
307C2×C4C2×#12112yes—
308C2×C4#52512yes—
309C2×C4#53612yes—
310C2×C4#54718yes—
311C2×C4#55218yes—
312C8—84no—
313C22—24no—
314C22—24no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—24no—
321C22—24no—
322C22—44no—
323C22—44no—
324C22—48no—
325C22—48no—
326C22#228011yescharacteristic
327C22C2×#38611yescharacteristic
328C22#231011yescharacteristic
329C22#240612yes—
330C22#248412yes—
331C22#302112yes—
332C22#302212yes—
333C4—22no—
334C4—22no—
335C4—48no—
336C4—44no—
337C4—44no—
338C4—88no—
339C4—84no—
340C4#234414yes—
341C4#241314yes—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—44no—
346C2#1387011yescharacteristic
347C2#2095511yescharacteristic
348C2#2095611yescharacteristic
349C2#2408014yes—
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. #11210C2semidirect
6. #10892C2semidirect
15. #2317C4semidirect

Maximal subgroups

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

subgroupindex
#341792
C2×#23172
#207002
C2×#18222
#112102
#108922
#107202