Finite Group Databasebeta

← #194546

Subgroups — #194546

Order 512.

1528 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1528 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3152C211yesmaximal, characteristic
2#34368C211yesmaximal, characteristic
3#21324C212yesmaximal, complemented
4C2×#1525C211yesmaximal, characteristic
5#9003C218yesmaximal, complemented
6#8028C212yesmaximal, complemented
7C22×#137—22no—
8C2×#532C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3152—24no—
11#3152C418yescomplemented
12C2×#497C2214yes—
13C2×#494—21no—
14C2×#492—21no—
15C2×C4×M16C2212yes—
16C2×C4×M16C2211yescharacteristic
17#2379C2214yes—
18#2369—21no—
19#2366—21no—
20#2354—22no—
21D4×C4⋊C4—22no—
22#2324—24no—
23#2311—22no—
24#2285C2212yes—
25#1993C2218yes—
26#1990—22no—
27#1990—22no—
28#1736—24no—
29#1525—24no—
30C2×#329C2214yes—
31#1295C2218yes—
32C22×#41D412yes—
33C23×Q8—21no—
34C2×#137—216no—
35C2×#137—48no—
36C2×#137—48no—
37C22×Q16—24no—
38C22×SD16—24no—
39C22×M16—21no—
40C22×M16—22no—
41#532—24no—
42#532C2×C414yes—
43#532D414yes—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8C2×C414yescomplemented
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8C2312yes—
51C2×#127D411yescharacteristic
52C2×#124C2312yes—
53C2×#121—22no—
54C2×C4×D4—24no—
55C2×C4×D4—22no—
56C2×#117—22no—
57C2×#117D414yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C42C2311yescharacteristic
61#497—28no—
62#497C2×C4116yescomplemented
63#494—28no—
64#492—28no—
65C4×M16—22no—
66C4×M16—24no—
67C4×M16—24no—
68C4×M16C2×C414yescomplemented
69C2×#97C2314yes—
70C2×#97C2312yes—
71C2×C4×C8—21no—
72C2×C4×C8—22no—
73#391—24no—
74#390—24no—
75#390—24no—
76#389—24no—
77#388—28no—
78#386—24no—
79#386—24no—
80#385—28no—
81#384D412yes—
82#384D412yes—
83#382C2314yes—
84#380—24no—
85#380—22no—
86#380—22no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90#340—28no—
91#340—28no—
92#329—28no—
93C2×#41—24no—
94C2×#41—44no—
95C2×#41—44no—
96C2×#41C22⋊C418yes—
97C22×Q8—24no—
98C22×Q8—22no—
99C22×Q8—44no—
100C22×Q8—44no—
101C22×Q8—41no—
102C22×Q8—42no—
103C22×Q8C2×D412yes—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4C2×D412yes—
107C23×C4—22no—
108C23×C4—24no—
109C23×C4—41no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112#137—464no—
113C2×Q16—216no—
114C2×Q16—48no—
115C2×Q16—48no—
116C2×SD16—216no—
117C2×SD16—48no—
118C2×SD16—48no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—28no—
123C2×M16—44no—
124C2×M16—44no—
125C22×C8—22no—
126C22×C8—24no—
127C4⋊Q8—24no—
128C4⋊Q8—28no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—24no—
132C4⋊Q8—44no—
133C4⋊Q8—42no—
134C4⋊Q8C22⋊C414yes—
135C4⋊Q8C22×C414yes—
136#127—22no—
137#127C22⋊C414yes—
138#124—24no—
139#124C22×C414yes—
140#124C2×D414yes—
141#121—28no—
142#121—28no—
143C4×D4—28no—
144C4×D4—28no—
145C4×D4—88no—
146#117—48no—
147#117—84no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—84no—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D414yes—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—24no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42C22×C412yes—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42#4114yes—
172C2×C42C2411yescharacteristic
173#97—22no—
174#97—24no—
175#97—24no—
176#97C22×C414yes—
177C4×C8—24no—
178C4×C8—44no—
179#95—28no—
180#95—24no—
181#95—24no—
182C24—22no—
183C24#12011yescharacteristic
184#41—416no—
185#41—88no—
186C2×Q8—22no—
187C2×Q8—24no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—44no—
191C2×Q8—44no—
192C2×Q8—416no—
193C2×Q8—42no—
194C2×Q8—44no—
195C2×Q8—88no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8C2×C22⋊C414yes—
199C2×D4—28no—
200C2×D4—22no—
201C2×D4—28no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—82no—
207C2×D4—82no—
208C2×D4C2×C22⋊C414yes—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—28no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—84no—
222C22×C4—88no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C2×C22⋊C412yes—
226C22×C4C4×D414yes—
227C22×C4#12011yescharacteristic
228C22×C4#12012yes—
229C22×C4#12114yes—
230C22×C4#12312yes—
231C22×C4C22×D411yescharacteristic
232C22×C4C22×D412yes—
233Q16—432no—
234SD16—432no—
235M16—216no—
236M16—416no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—28no—
241C2×C8—44no—
242C2×C8—44no—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—48no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—84no—
254C22⋊C4—28no—
255C22⋊C4—28no—
256C22⋊C4—84no—
257C22⋊C4—84no—
258C22⋊C4—88no—
259C42—48no—
260C42—84no—
261C42C2×C22⋊C412yes—
262C42C4×D414yes—
263C42#12114yes—
264C42#12312yes—
265C42C23×C411yescharacteristic
266C42C22×D411yescharacteristic
267C42C2×#4112yes—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—42no—
274C23—42no—
275C23—44no—
276C23—44no—
277C23#38612yes—
278C23C2×#12011yescharacteristic
279Q8—44no—
280Q8—48no—
281Q8—82no—
282Q8—84no—
283Q8—88no—
284D4—44no—
285D4—416no—
286D4—82no—
287D4—88no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—416no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—88no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—168no—
311C2×C4#38614yes—
312C2×C4#38612yes—
313C2×C4#45312yes—
314C2×C4#45312yes—
315C2×C4#45514yes—
316C2×C4#45614yes—
317C2×C4C22×C22⋊C411yescharacteristic
318C2×C4C2×C4×D412yes—
319C2×C4C2×#12112yes—
320C2×C4C2×#12311yescharacteristic
321C8—28no—
322C8—48no—
323C22—24no—
324C22—22no—
325C22—41no—
326C22—42no—
327C22—41no—
328C22—48no—
329C22—44no—
330C22—82no—
331C22—84no—
332C22—82no—
333C22C2×#38611yescharacteristic
334C22C2×#45311yescharacteristic
335C22C2×#45311yescharacteristic
336C22C2×#45512yes—
337C22C2×#45612yes—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—44no—
343C4—84no—
344C4—84no—
345C4—84no—
346C4#188218yes—
347C4#188714yes—
348C2—42no—
349C2—84no—
350C2#1907411yescharacteristic
351C2#1907612yes—
352C2#1907811yescharacteristic
353C2C2×#188212yes—
354C2C2×#188711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21324C2semidirect
5. #9003C2semidirect
6. #8028C2semidirect
11. #3152C4semidirect
48. C2×C4⋊Q8C2×C4semidirect
62. #497C2×C4semidirect
68. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31522
#343682
#213242
C2×#15252
#90032
#80282