Finite Group Databasebeta

← #393531

Subgroups — #393531

Order 512.

1657 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35157C211yesmaximal, characteristic
2#33996C211yesmaximal, characteristic, complemented
3#20979C212yesmaximal
4C2×#1822C211yesmaximal, characteristic, complemented
5#10949C214yesmaximal, complemented
6#10888C214yesmaximal, complemented
7#10716C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C22×#117C2212yes—
10C22×#117C2211yescharacteristic
11C2×#421C2214yes—
12C2×#407C2212yes—
13#2853—22no—
14#2622—22no—
15#2500C414yescomplemented
16#2499C414yescomplemented
17#2485—24no—
18#2317C2212yes—
19#2311C2212yes—
20#2280C2211yescharacteristic
21#2274—24no—
22#2274C2214yes—
23#2273—22no—
24C4×#117—24no—
25C4×#117—22no—
26#1897—24no—
27#1896—24no—
28#1822—24no—
29C2×#343C2214yes—
30#1733C2214yes—
31C2×H(3,ℤ/4)C2218yes—
32C22×#41C2311yescharacteristic
33C24×C4C2311yescharacteristic
34C22×M16C2314yes—
35#525—22no—
36#525—22no—
37#525—24no—
38C2×#123—21no—
39C2×C22⋊Q8C2×C412yes—
40C2×#121C2×C412yes—
41C2×#120—21no—
42C2×C4×D4—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—24no—
48C2×#117—216no—
49C2×#117C2×C414yes—
50C2×#117D414yes—
51C2×#117D414yes—
52C2×#117C2311yescharacteristic
53C2×#117C2312yes—
54C2×#117C2314yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D412yes—
59C22×C42D414yes—
60C22×C42D412yes—
61#421—216no—
62C2×#104—28no—
63#407—24no—
64#407—22no—
65#407—22no—
66C2×C22⋊C8—24no—
67#397—28no—
68#394—28no—
69#391C2×C414yescomplemented
70#390C2×C414yescomplemented
71#389—24no—
72#388—28no—
73#387—28no—
74#386—24no—
75#385—24no—
76#384C2×C414yescomplemented
77#382C2×C414yescomplemented
78#381—21no—
79#380—22no—
80#380—24no—
81#379—21no—
82C4×C4⋊C4—28no—
83C4×C4⋊C4—216no—
84C4×C22⋊C4—216no—
85C4×C22⋊C4—28no—
86C4×C22⋊C4C2×C414yes—
87#376—24no—
88C43—24no—
89#343—28no—
90#339—28no—
91H(3,ℤ/4)—216no—
92C25C2×D411yescharacteristic
93C2×#41—22no—
94C2×#41—22no—
95C2×#41—28no—
96C22×Q8C22×C411yescharacteristic
97C22×D4—21no—
98C22×D4—22no—
99C22×D4C22×C411yescharacteristic
100C23×C4—28no—
101C23×C4—24no—
102C23×C4—21no—
103C23×C4C22×C412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D414yes—
109C23×C4#4114yes—
110C23×C4C2411yescharacteristic
111C2×M16—24no—
112C2×M16—24no—
113C2×M16—28no—
114C2×M16—28no—
115C22×C8—24no—
116#123—41no—
117#123—41no—
118#123—42no—
119C22⋊Q8—24no—
120C22⋊Q8—24no—
121#121—24no—
122#121—24no—
123#120—42no—
124#120—42no—
125C4×D4—44no—
126C4×D4—44no—
127#117—216no—
128#117—216no—
129#117—28no—
130#117—28no—
131#117—28no—
132#117—28no—
133#117—28no—
134#117—28no—
135#117—216no—
136#117—22no—
137#117—22no—
138#117#41116yes—
139C2×C4⋊C4—216no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C2×D414yes—
150C2×C22⋊C4—216no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4C22⋊C418yes—
162C2×C42—24no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—24no—
166C2×C42—216no—
167C2×C42—21no—
168C2×C42—216no—
169C2×C42C22⋊C418yes—
170C2×C42C22×C414yes—
171C2×C42C2×D418yes—
172#104—416no—
173C22⋊C8—44no—
174C22⋊C8—44no—
175#95—216no—
176#95—24no—
177#95—28no—
178#95—28no—
179C24—24no—
180C24—21no—
181C24—82no—
182C24C2×C22⋊C412yes—
183C24#12012yes—
184C24#12312yes—
185C24C22×D411yescharacteristic
186#41—44no—
187#41—44no—
188#41—48no—
189C2×Q8—22no—
190C2×Q8—24no—
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—84no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—216no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—216no—
209C22×C4—24no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—84no—
216C22×C4—88no—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C418yes—
219C22×C4C4×D414yes—
220C22×C4C4×D414yes—
221C22×C4#12011yescharacteristic
222C22×C4#12011yescharacteristic
223C22×C4#12114yes—
224C22×C4#12114yes—
225C22×C4#12312yes—
226C22×C4C23×C411yescharacteristic
227C22×C4C22×D412yes—
228C22×C4C22×D411yescharacteristic
229C22×C4C2×#4112yes—
230C22×C4C2×#4118yes—
231M16—44no—
232M16—44no—
233M16—48no—
234C2×C8—44no—
235C2×C8—48no—
236C4⋊C4—216no—
237C4⋊C4—28no—
238C4⋊C4—28no—
239C4⋊C4—216no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—88no—
245C22⋊C4—216no—
246C22⋊C4—216no—
247C22⋊C4—216no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—88no—
255C22⋊C4—84no—
256C42—216no—
257C42—28no—
258C42—28no—
259C42—28no—
260C42—28no—
261C42—42no—
262C42—816no—
263C42#123116yes—
264C23—22no—
265C23—22no—
266C23—24no—
267C23—24no—
268C23—44no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—44no—
273C23—44no—
274C23—41no—
275C23—42no—
276C23—88no—
277C23#38614yes—
278C23C22×C22⋊C411yescharacteristic
279C23C2×#12011yescharacteristic
280C23C2×#12311yescharacteristic
281C23#52514yes—
282Q8—44no—
283D4—44no—
284D4—88no—
285D4—82no—
286D4—82no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—28no—
295C2×C4—216no—
296C2×C4—28no—
297C2×C4—416no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—84no—
308C2×C4—816no—
309C2×C4—84no—
310C2×C4#38612yes—
311C2×C4#38612yes—
312C2×C4C2×C4×D412yes—
313C2×C4C4×#4114yes—
314C2×C4C2×#12112yes—
315C2×C4C2×#12314yes—
316C2×C4#52514yes—
317C2×C4#52918yes—
318C2×C4#53214yes—
319C8—84no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—44no—
327C22—44no—
328C22—44no—
329C22—84no—
330C22—84no—
331C22—84no—
332C22—82no—
333C22—81no—
334C22C2×#38611yescharacteristic
335C22#229712yes—
336C22#284912yes—
337C22#285312yes—
338C4—22no—
339C4—22no—
340C4—416no—
341C4—48no—
342C4—84no—
343C4—82no—
344C4—81no—
345C4C4×#12314yes—
346C4#285414yes—
347C2—22no—
348C2—22no—
349C2—22no—
350C2—44no—
351C2—161no—
352C2#1470611yescharacteristic
353C2#2101212yes—
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33996C2semidirect
4. C2×#1822C2semidirect
5. #10949C2semidirect
6. #10888C2semidirect
7. #10716C2semidirect
15. #2500C4semidirect
16. #2499C4semidirect
69. #391C2×C4semidirect
70. #390C2×C4semidirect
76. #384C2×C4semidirect
77. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#351572
#339962
#209792
C2×#18222
#109492
#108882
#107162