Finite Group Databasebeta

← #325992

Subgroups — #325992

Order 512.

712 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 162 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
1#27813C212yesmaximal, complemented
2#25385C212yesmaximal
3#10574C211yesmaximal, characteristic, complemented
4#10432C211yesmaximal, characteristic
5#10068C212yesmaximal, complemented
6#9819C211yesmaximal, characteristic, complemented
7#9562C212yesmaximal, complemented
8#9217C212yesmaximal, complemented
9C8×#102C212yesmaximal, complemented
10C2×#498C2211yescharacteristic
11C2×#465—22no—
12C2×#461C2212yes—
13C2×C8×D4C2212yes—
14#2551C2211yescharacteristic
15#2542C2212yes—
16#2537C2211yescharacteristic
17#2099—22no—
18#2035—22no—
19#2023—22no—
20#2000—22no—
21#1990—22no—
22#1990C2212yes—
23#1988C2212yes—
24#1956C2212yes—
25#1948C2212yes—
26#1934—22no—
27#1892—22no—
28#1844C2211yescharacteristic
29#1802C2212yes—
30C8×C4⋊C4C2212yes—
31C4×#102C2212yes—
32C2×#332C2212yes—
33C2×D4⋊C8C2212yes—
34C2×C82C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1277C2212yes—
39C22×SD16—22no—
40C23×C8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2312yes—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×#121—22no—
46C2×C4×D4C2312yes—
47#498—22no—
48#498—41no—
49#465—48no—
50#461—24no—
51#461—42no—
52C8×D4—24no—
53C8×D4—42no—
54C2×#106—22no—
55C2×#106—41no—
56C2×#106—41no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#102—22no—
63C2×#102D412yes—
64C2×#102D412yes—
65C2×C22⋊C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8C2312yes—
68C2×C4×C8C2311yescharacteristic
69#399—42no—
70#395—41no—
71#394—41no—
72#390—22no—
73#389—22no—
74#384—22no—
75#384C2311yescharacteristic
76#384C2311yescharacteristic
77#380—22no—
78C4×C4⋊C4C2312yes—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340D412yes—
86#340D412yes—
87#336—22no—
88#332—24no—
89D4⋊C8—24no—
90C82—41no—
91C22×Q8—22no—
92C22×Q8—41no—
93C22×D4—41no—
94C22×D4C2×D412yes—
95C23×C4—22no—
96C2×SD16—44no—
97C2×SD16—44no—
98C2×SD16—82no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—44no—
103C22×C8—44no—
104C22×C8—82no—
105C22×C8C2×D412yes—
106C22×C8C2×D412yes—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—44no—
112C4⋊Q8—82no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C22⋊Q8—82no—
116C22⋊Q8—82no—
117#121—44no—
118#121—82no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C42—22no—
140C2×C42C2411yescharacteristic
141#106—44no—
142#106—44no—
143#106—82no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8#4114yes—
148#103—44no—
149#103—44no—
150#103—82no—
151#102—24no—
152#102—82no—
153#102SD1618yescomplemented
154C22⋊C8—44no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—21no—
158C4×C8—82no—
159C4×C8#4112yes—
160#95—22no—
161#95—41no—
162#95—41no—
163C24—22no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—84no—
171C2×Q8—161no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—84no—
178C2×D4—161no—
179C2×D4C2×SD1614yes—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4C22×D412yes—
193C22×C4H(5,𝔽2)11yescharacteristic
194SD16—88no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—88no—
210C2×C8C2×SD1614yes—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—84no—
224C4⋊C4—84no—
225C4⋊C4—84no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4C2×SD1614yes—
229C4⋊C4#13514yes—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—84no—
233C22⋊C4—84no—
234C22⋊C4—84no—
235C42—42no—
236C42—42no—
237C42C2×#4111yescharacteristic
238C42C2×#4112yes—
239C42H(5,𝔽2)11yescharacteristic
240C23—24no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23D4211yescharacteristic
248Q8—82no—
249Q8—82no—
250Q8—82no—
251Q8—82no—
252Q8—164no—
253D4—24no—
254D4—24no—
255D4—42no—
256D4—42no—
257D4—164no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4#53212yes—
281C2×C4#53811yescharacteristic
282C2×C4#54612yes—
283C2×C4#55511yescharacteristic
284C2×C4C22×SD1612yes—
285C2×C4C2×#13512yes—
286C2×C4#57612yes—
287C2×C4#57712yes—
288C8—24no—
289C8—24no—
290C8—44no—
291C8—42no—
292C8—82no—
293C22—44no—
294C22—44no—
295C22—44no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22#255511yescharacteristic
301C22#328012yes—
302C22D4×SD1612yes—
303C22#329111yescharacteristic
304C22#329311yescharacteristic
305C4—21no—
306C4—21no—
307C4—44no—
308C4—44no—
309C4—82no—
310C4—161no—
311C4—161no—
312C4—162no—
313C4#316712yes—
314C4#317912yes—
315C4#329512yes—
316C4#330412yes—
317C2—84no—
318C2SD16211yescharacteristic
319C2#1997912yes—
320C2#1998711yescharacteristic
321C2#2890512yes—
322C2#2893911yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radical#2537C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #27813C2semidirect
3. #10574C2semidirect
5. #10068C2semidirect
6. #9819C2semidirect
7. #9562C2semidirect
8. #9217C2semidirect
9. C8×#102C2semidirect
153. #102SD16semidirect

Maximal subgroups

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

subgroupindex
#278132
#253852
#105742
#104322
#100682
#98192
#95622
#92172
C8×#1022