Finite Group Databasebeta

← #329595

Subgroups — #329595

Order 512.

864 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 154 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#32905C212yesmaximal
2#29876C211yesmaximal, characteristic, complemented
3#21786C211yesmaximal, characteristic
4#10621C212yesmaximal, complemented
5#10476C214yesmaximal, complemented
6#10264C212yesmaximal, complemented
7#9130C212yesmaximal, complemented
8#8599C211yesmaximal, characteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#500C2211yescharacteristic, complemented
11C2×#461C2214yes—
12C2×#450—22no—
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2857C2212yes—
16#2086—22no—
17#2075—22no—
18#1990C2212yescomplemented
19#1954—22no—
20#1954C2212yes—
21#1954C2212yes—
22#1877—24no—
23#1872—22no—
24#1834C2211yescharacteristic, complemented
25#1802C2214yes—
26#1797C2212yes—
27#1779—22no—
28C2×#335C2212yescomplemented
29C2×#332C2214yescomplemented
30#1296C2214yescomplemented
31#1277C2212yescomplemented
32C23×D4D411yescharacteristic
33C24×C4—21no—
34C22×SD16—24no—
35C2×C4⋊Q8C2312yes—
36C2×C22⋊Q8—22no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—42no—
40C2×C4×D4—42no—
41C2×C4×D4D412yes—
42C2×C4×D4C2312yes—
43C22×C4⋊C4D411yescharacteristic
44C22×C22⋊C4—21no—
45C22×C42C2311yescharacteristic
46#500—42no—
47#461—44no—
48#461—44no—
49#450—48no—
50#431—42no—
51#431—42no—
52C2×#107—22no—
53C2×C4⋊C8D412yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2314yes—
56C2×#103—24no—
57C2×#102—24no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8D412yes—
63C2×C4×C8—21no—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#391—22no—
67#389—22no—
68#389—22no—
69#384C2312yes—
70#380—22no—
71#380—22no—
72#340—22no—
73#340—24no—
74#340—24no—
75#336—22no—
76#335—42no—
77#332—44no—
78C25—21no—
79C22×Q8—22no—
80C22×D4—21no—
81C22×D4—22no—
82C22×D4—42no—
83C22×D4—42no—
84C22×D4D814yes—
85C22×D4C2×D412yes—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—84no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4#4111yescharacteristic
95C2×SD16—48no—
96C2×SD16—48no—
97C2×SD16—48no—
98C22×C8—24no—
99C22×C8—22no—
100C22×C8C2×D412yes—
101C22×C8C2×D412yes—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C22⋊Q8—44no—
105C22⋊Q8—44no—
106C4×D4—24no—
107C4×D4—24no—
108C4×D4—44no—
109C4×D4—42no—
110C4×D4—42no—
111C4×D4—44no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4C2×D412yes—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—42no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42C2411yescharacteristic
131#107—44no—
132C4⋊C8—24no—
133C4⋊C8—24no—
134C4⋊C8—24no—
135C4⋊C8—42no—
136#103—48no—
137#102—48no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140C22⋊C8—44no—
141C22⋊C8—84no—
142C22⋊C8—84no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—42no—
146#95—42no—
147#95—42no—
148C24—22no—
149C24—24no—
150C24—42no—
151C24—42no—
152C24—82no—
153C24—82no—
154C24#12111yescharacteristic
155C2×Q8—44no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—44no—
159C2×D4—24no—
160C2×D4—22no—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—48no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—48no—
167C2×D4C2×D814yes—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—84no—
185C22×C4—82no—
186C22×C4—88no—
187C22×C4—88no—
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4C2×D812yes—
192C22×C4#13512yes—
193C22×C4C22×D411yescharacteristic
194C22×C4C22×D411yescharacteristic
195C22×C4C2×#4111yescharacteristic
196SD16—816no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—44no—
202C2×C8—42no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—44no—
206C4⋊C4—24no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C22⋊C4—84no—
219C42—21no—
220C42—22no—
221C42—21no—
222C42—22no—
223C42H(5,𝔽2)12yes—
224C42H(5,𝔽2)12yes—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23—24no—
230C23—48no—
231C23—42no—
232C23—44no—
233C23—42no—
234C23—44no—
235C23—88no—
236C23—84no—
237C23—84no—
238C23—88no—
239C23#46612yes—
240C23C2×#12111yescharacteristic
241Q8—84no—
242Q8—84no—
243D4—28no—
244D4—48no—
245D4—44no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—82no—
260C2×C4—88no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#46612yes—
272C2×C4#53412yes—
273C2×C4#53411yescharacteristic
274C2×C4#53511yescharacteristic
275C2×C4#53812yes—
276C2×C4C22×D811yescharacteristic
277C2×C4C2×#13511yescharacteristic
278C2×C4#57512yes—
279C2×C4#57714yes—
280C8—42no—
281C8—42no—
282C8—84no—
283C8—82no—
284C22—22no—
285C22—24no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—44no—
290C22—48no—
291C22—816no—
292C22—82no—
293C22—82no—
294C22—84no—
295C22#261711yescharacteristic
296C22#304212yes—
297C22C2×#46611yescharacteristic
298C22#305411yescharacteristic
299C22#306912yes—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—162no—
306C4—162no—
307C4#318612yes—
308C4#319112yes—
309C4#320012yes—
310C4#322212yes—
311C2—42no—
312C2—84no—
313C2#1939212yes—
314C2#1943112yes—
315C2#2972311yescharacteristic
316C2#2973011yescharacteristic
317C2#2975211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×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
2. #29876C2semidirect
4. #10621C2semidirect
5. #10476C2semidirect
6. #10264C2semidirect
7. #9130C2semidirect
8. #8599C2semidirect
10. C2×#500C22semidirect
18. #1990C22semidirect
24. #1834C22semidirect
28. C2×#335C22semidirect
29. C2×#332C22semidirect
30. #1296C22semidirect
31. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#329052
#298762
#217862
#106212
#104762
#102642
#91302
#85992